3145035047 Electromagnetic Waves llnoK» Prom Iii: 1.1. 'I'l.l.ll'IKlNi: I.aiioratorii'.s ). By IllVlNO B. CrANDALL, Late Member of the Technical Staff, Bell Telephone Laboratories. CONTEMPORARY PHYSICS. By Karl K. Darrow, Member of the Technical Staff, Bell Telephone Laboratories. Second Edition. SPEECH AND HEARING. By Harvey Fletcher, Acoustical Research Director, Bell Telephone Laboratories. With an Introduction by H. D. Arnold, Director of Research, Bell Telephone Laboratories. PROBABILITY AND ITS ENGINEERING USES. By Thornton C. Fry, Member of the Technical Staff, Bell Telephone Laboratories. ELEMENTARY DIFFERENTIAL EQUATIONS. By Thornton C. Fry. Second Edition. • TRANSMISSIONS CIRCUITS FOR TELEPHONIC COMMUNICATION. METHODS OF ANALYSIS AND DESIGN. By K. S. Johnson, Member of the Technical Staff, Bell Telephone Laboratories. A FUGUE IN CYCLES AND BELS. By John Mills, Director of Publication, Bell Telephone Laboratories. TRANSMISSION NETWORKS AND WAVE FILTERS. By T. E. Shea, Special Products Engineer, Bell Telephone Laboratories. ECONOMIC CONTROL OF QUALITY OF MANUFACTURED PRODUCT. By W. A. Shewhart, Member of Technical Staff, Bell Telephone Laboratories. THE APPLICATION OF ELECTROMECHANICAL IMPEDANCE ELEMENTS IN TRANSDUCERS AND WAVE FILTERS. By Warren P. Mason, Member of the Technical Staff, Bell Telephone Laboratories, Inc. RHOMBIC ANTENNA DESIGN. By A. E. Harper, Bell Telephone Laboratories. POISSON'S EXPONENTIAL BINOMIAL LIMIT. By E. C. Molina, Switching Theory Engineer, Bell Telephone Laboratories. ELECTROMAGNETIC WAVES. By S. A. Sciielkunoff, Member of the Technical Staff, Bell Telephone laboratories. FUNDAMENTAL THEORY OF SERVO-MECHANISMS. By LeRoy A. MacColl, Member if the Technical Staff, Bell Telephone Laboratories, Inc. QUARTZ CRYSTALS FOR ELECTRICAL CIRCUITS. By Raymond A. Heisino, Radio RtUarch Engineer, Bell Telephone Laboratories, Inc. I'll IILINllttl) by D, Van Nosthand company, Inc. ELECTROM AGNETIC WAVES By S. A. SCHELKUNOFF Member of the Technical Staff Bell Telephone Laboratories, Ihc, FOURTH PRINTING cm NEW YORK D. VAN NOSTRAND COMPANY, Inc. 250 Fourth Avenue PREFACE COPYRIGHT, 1943 D. VAN NOSTRAND COMPANY, Inc. Alilj RIGHTS RESERVED This book, or any parts thereof, may not be reproduced in any form without written permission from the author and publisher. First Published, April 1943 Reprinted July 1943, May 1944, October 1945 In the- summer of 1942 it was my pleasure to give a course on Electro-iii i 'in.-tic Waves at Brown University in connection with its Program of .....11 Instruction and Research in Mechanics. There I not only enjoyed I hr opportunity to test this book in manuscript but, through generous .......gemer.ts made by the University, I was enabled to put it in final shape l"i |it11>lii ation. To the Officers of Brown University, and particularly to \i i .. I). Richardson, Dean of the Graduate School, I am grateful for their lilt rest in the book and for the facilities which they put at my disposal. is ft whole, the book is an outgrowth of my research and consulting in 11vitics in Bell Telephone Laboratories. Its first draft was prepared in connection with courses of lectures in the Laboratories' " Out-of-Hour " I.......am. Courses were given in 1933-34 and 1934-35, for which the lectures mimeographed under the title "Electromagnetic Theory and Its Applications." A third course was given in 1941-42, when the notes were ii \ i id under the present title " Electromagnetic Waves." 11' this book proves to be a " practical theory " of electromagnetic waves it will be largely due to my close association with experimentalists in the Hi II 1 .aboratories. Some credit for its final issuance is due to Dr. H. T. Friis who for years urged me to publish my notes. To Dr. M. J. Kelly and I )i. Thornton C. Fry I am grateful for arranging a leave of absence for my Work at Brown University. I am particularly indebted to Miss Marion C. Gray for her invaluable USlStancc throughout the entire preparation of this book. S. A. S. N..w York, N; Y. January, 1943 Produced by TECHNICAL COMPOSITION CO. ÜOäTON, MASS. PRINTED IN THE UNITED STATES OF AMERICA V TO THE READER Since 1929 the opportunities for practical applications of electromagnetic theory have increased so spectacularly that a new approach has become ilniuit a necessity. The old practice of working out each boundary value pinlilcm as if it were a new problem is being abandoned as repetitious and 1111' i < momical because it fails to coordinate the various results. In the interest Bi unity, simplicity, compactness and physical interpretation, the con-. 1'iitids of one-dimensional wave theory are being extended to waves in i luxe dimensions and field theory is no longer considered as something apart I Him circuit and transmission line theories. All physical fields are three dimensional; but in some circumstances i iiIht two or all three dimensions arc unimportant; then they may be " integrated out" and thus "concealed"; in the first case the problem In lungs to "transmission line theory" and in the second to "network theory." This suppression of some or all physical dimensions is analogous Id (he method of " ignoration of coordinates " in mechanics; and it may or nuy not involve approximations. It is a mistake to say that the circuit and 11in- theories are approximate while only the field theory is exact. In fact in many important cases a three-dimensional problem is rigorously re-duclble to a set of one-dimensional problems. Once the one-dimensional problem has been solved in sufficiently general terms, the results can be used repeatedly in the solution of more general problems. This point of view leads to a better understanding of wave phenomena; il: saves time and labor; and it benefits the mathematician by suggesting to him more direct methods of attacking new problems. Once these ideas mi- more generally disseminated, large sections of electromagnetic theory can be explained in terms intelligible to persons with elementary engineering education. The classical physicist, being concerned largely with isolated transmission systems, has emphasized only one wave concept, that of the velocity of propagation or more generally of the propagation constant. But the communication engineer who is interested in " chains " of such systems from the very start is forced to adopt a more general attitude and introduce the second important wave concept, that of the impedance. The physicist II mcentrates his attention on one particular wave: a wave of force, or a w:ive of velocity or a wave of displacement. His original differential equations may be of the first order and may involve both force and velocity; but vu viii TO rill''. KKADI-.lt TO THE HEADER ix by tradition he eliminates one of these variables, obtains a second order differentia] equation in the other and calls it the " wave equation." Thus he loses sight of the interdependence of force and velocity waves and he does not stress the difference which may exist between waves in different media even though the velocity of wave propagation is the same. The engineer, on the other hand, thinks in terms of the original " pair of wave equations " and keeps constantly in mind this interdependence between force and velocity waves. In this book I have injected the communication engineer's attitude into an orderly development of " field theory." If the modern theory of electromagnetism were to be presented in four ideal volumes, then the first volume would treat the subject broadly rather than thoroughly, with emphasis on more elementary topics. The second volume would be devoted to electromagnetic waves in passive media free from space charge; in this volume electric generators would appear merely as given data, either as electric intensities tangential to the boundaries of the " generator regions " or as given currents inside these regions. Another volume, on " electromechanical transducers," would deal with interaction between mechanical and electrical forces and the final volume on " space charge waves " would be devoted to phenomena in vacuum tubes. The present book is confined to the material which would properly belong to the second of these volumes. It is intended as a textbook and for reference. In it a practicing engineer will find basic theoretical information on radiation, wave propagation, wave guides and resonators. Those engaged in theoretical research will find a stock of equations which may serve as a starting point for further investigations. Chapters 1 and 3, dealing with vector analysis and special functions, such as Bessel functions and Legendre functions, are intended for ready reference. These chapters are brief because it is only necessary for thi reader to be familiar with the language of vectors and, in most cases, only elementary properties of the special functions are needed. Chapter 2 deals with applications of complex variables to the theory of oscillations and waves and Chapter 4 reviews the fundamental conceptions and equations. Elements of circuit theory are presented in Chapter 5; there the three-dimensional character of electromagnetic fields is suppressed and the discussion is conducted in terms of resistance, inductance and capacitance. Chapter 6 is concerned with some general aspects of waves in free space, on wires, and in wave guides. Its last few sections cover electrostatics and magnetostatics to the extent needed in wave theory. The one-dimensional wave theory is presented in great detail in Chapter 7. The following chapter treats the simplest types of waves in free space and in wave guides. Chapter 10 contains a more general, systematic treatment of such waves. Chapter 9 is devoted to radiation from known current distributions and to the directive properties of antennas, antenna arrays and electric horns. Chapter 11 presents a recent antenna theory and, finally, Chapter 12 deals with certain impedance discontinuities in wave guides. There is enough material for an intensive six-hour course; the particular order adopted is best suited to students of communication engineering and microwave transmission. In the case of radio engineers, the first four sections of Chapter 8 may be followed by Chapter 9; and in the case of students of physics or applied mathematics these four sections may be followed directly by Chapter 10. For a shorter course the instructor will find it easy to select the material best suited to the needs of his students. THE AUTHOR CONTENTS UHAPTEft 1. VECTORS AND COORDINATE SYSTEMS. PAGE 1 Vectors...................................................... } Functions of position.......................................... 3 Divergence. Line integral, circulation, curl.................... Coordinate systems.............................. Differential expressions for gradient, divergence, curl. Differential invariants and Green's theorems........ Miscellaneous equations.......................... 6 7 8 10 12 13 II. MATHEMATICS OF OSCILLATIONS AND WAVES................. 14 2.1 Complex variables............................................. 14 2.2 Exponential functions......................................... l| 2.3 Exponential and harmonic oscillations........................... 19 2.4 Waves....................................................... 22 2.5 Nepers, bels, decibels.......................................... 25 2.6 Stationary waves.............................................. 26 2.7 Impedance concept............................................ 26 2.8 Average power and complex power.............................. 3| 2.9 Step and impulse functions..................................... 31 2.10 Natural and forced waves...................................... 38 III. BESSEL AND LEGENDRE FUNCTIONS............................ 44 3.1 Reduction of partial differential equations to ordinary differential equations.................................................. 44 3.2 Boundary conditions.......................................... 46 3.3 Bessel functions............................................... 47 3.4 Modified Bessel functions...................................... 50 3.5 Bessel functions of order n + j and related functions ............. 51 3.6 Spherical harmonics and Lcgcndre functions...................... 53 3.7 Miscellaneous formulae........................................ 55 IV. FUNDAMENTAL ELECTROMAGNETIC EQUATIONS.............. 60 4.1 Fundamental equations in the MKS system of units............... 60 4.2 Impressed forces.............................................. 2§ 4.3 Currents across a closed surface................................. '2 4.4 Differential equations of electromagnetic induction and boundary con- ditions..................................................... 73 4.5 Conditions in the vicinity of a current sheet...................... 74 4.6 Conditions in the vicinity of linear current filaments............... 75 4.7 Moving surface discontinuities.................................. /_■> 4.8 Energy theorems.............................................. 77 4.9 Secondary electromagnetic constants............................ g! 4.10 Waves in dielectrics and conductors............................. 86 4.11 Polarization.................................................. 90 4.12 Special forms of Maxwell's equations in source-free regions......... 94 xi XII CONTENTS ciiArn it PAtii: V. JMI'KDORS, TRANSDUCERS, NETWORKS........................ 97 5.1 Impcdors and networks........................................ 97 5.2 Transducers.................................................. 104 5.3 Iterated structures............................................ 108 5.4 Chains of symmetric T-networks............................... 110 5.5 Chains of symmetric 11-networks.................... ........... Ill 5.6 Continuous transmission lines.................................. 112 5.7 Filters......................................................... 112 5.8 Forced oscillations in a simple series circuit....................... 115 5.9 Natural oscillations in a simple series circuit...................... IIS 5.10 Forced oscillations in a simple parallel circuit..................... 119 5.11 F,xpansion of the input impedance function....................... 121 VI. ABOUT WAVES IN GENERAL..................................... 126 6.0 Introduction................................................. 126 6.1 The field produced by a given distribution of currents in an infinite homogeneous medium........................................ 126 6.2 The field of an electric current clement.......................... 129 6.3 Radiation from an electric current element....................... 133 6.4 The mutual impedance between two current elements and the mutual radiated power.............................................. 134 6.5 Impressed currents varying arbitrarily with rime.................. 138 6.6 Potential distribution on perfectly conducting straight wires........ 140 6.7 Current and charge distribution on infinitely thin perfectly conducting wires........■............................................... 142 6.8 Radiation from a wire energized at the center.................... 144 6.9 The mutual impedance between two current loops; the impedance of a loop...................................................... 144 6.10 Radiation from a small plane loop carrying uniform current........ 147 6.11 Transmission lines and wave guides............................. 148 6.12 Reflection.................................................... 156 6.13 The induction theorem........................................ 158 6.14 The equivalence theorem....................................... 158 6.15 Stationary fields.............................................. 159 6.16 Conditions in the vicinities of simple and double layers of charge. . . . 160 6.17 Equivalence of an electric current loop and a magnetic double layer. 162 6.18 Induction and equivalence theorems for stationary fields........... 164 6.19 Potential and capacitance coefficients of a system of conductors..... 165 6.20 Representation of a system ol conductors by an equivalent network of capacitors................................................ 166 6.21 Energy theorems for stationary fields............................ 168 6.22 The method of images......................................... 169 6.23 Two-dimensional stationary fields............................... 173 6.24 The inductance of a system of parallel currents................... 176 6.25 Functions of complex variables and stationary fields............... 179 VII. TRANSMISSION THEORY.......................................... 188 7.0 Introduction................................................. 188 7.1 Impressed forces and currents.....................;............ 189 7.2 Point sources....................................'............. 189 7.3 The energy theorem........................................... 191 7.4 Fundamental sets of wave functions for uniform lines.............. 192 7.5 Characteristic constants of uniform transmission lines............. 195 7.6 The input impedance.......................................... 197 7.7 Transmission lines as transducers............................... 201 7.8 Waves produced by point sources............................... 201 7.9 Waves produced by arbitrary distributions of sources.............. 204 7.10 Nonuniform transmission lines.................................. 205 7.11 Calculation of nonuniform wave function.! by successive approxima- tions...................................................... 207 CONTENTS Mil 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 paci; Slightly nonuniform transmission lines........................... 209 Reflection in uniform lines..................................... 210 Kcllcction coefficients as functions of the impedance ratio.......... 212 Induction anil equivalence theorems for transmission lines.........1 217 Conditions for maximum delivery of power to an impedance........ 218 Transformation and matching of impedances . . . .................. 219 Tapered transmission lines and impedance matching............... 222 Transmission across a section of a uniform line................... 223 Reflection in nonuniform lines.................................. 226 Formation of wave functions with the aid of reflection coefficients. . . 227 Natural oscillations in uniform transmission lines................. 229 Conditions for impedance matching and natural oscillations in terms of the reflection coefficient................................... 232 Expansions in partial fractions.................................. 232 Multiple transmission lines..................................... 235 Iterative structures............................................ 236 Resonance in slightly nonuniform transmission lines............... 237 VIII. WAVES, WAVE GUIDES AND RESONATORS —1.................. 242 8.0 Introduction................................................ 8.1 Uniform plane waves......................................... 8.2 Elliptic-ally polarized plane waves.............................. 8.3 Wave impedances at a point................................... 8.4 Reflection of uniform plane waves at oblique incidence............ 8.5 Uniform cylindrical waves..................................... 8.6 Cylindrical cavity resonators.............•..................... 8.7 Solenoids and wedge transmission lines......................... 8.8 Wave propagation along coaxial cylinders....................... 8.9 Transverse electromagnetic plane waves......................... 8.10 Transverse electromagnetic waves on parallel wires............... 8.11 Transverse electromagnetic spherical waves..................... 8.12 Transverse electromagnetic waves on coaxial cones............... 8.13 Transverse electromagnetic waves on a cylindrical wire........... 8.14 Waves on inclined wires...................................... 8.15 Circular magnetic waves in ide a hollow metal sphere............. 8.16 Circular electric waves inside a hollow sphere.................... 8.17 Two-dimensional fields........................................ 8.18 Shielding theory............................................. 8.19 Theory of laminated shields................................... 8.20 A diffraction problem......................................... 8.21 Dominant waves in wave guides of rectangular cross-section (TEi, mode)........................•........................ 8.22 Dominant waves in circular wave guides (TEi.i-mode) ........... 8.23 The effect of curvature on wave propagation.................... 242 242 248 249 251 260 267 273 275 281 283 285 286 290 292 294 298 299 303, 312 315 316 322 324 L RADIATION AND DIFFRACTION.................................. 331 9.0 Introduction...............................................'.. 331 9.1 The distant field.................,............................. 331 9.2 A general radiation formula.......................'............. 333 9.3 On calculation of radiation vectors.............................. 334 9.4 Directivity................................................... 335 9.5 Directive properties of an electric current clement................. 336 9.6 Directive properties of a small electric current loop................ 338 9.7 Directive properties of a vertical antenna........................ 339 9.8 The effect of the radius of the wire on the radiated power.......... 341 9.9 Linear arrays with uniform amplitude distribution................ 342 9.10 The gain o(* end-fire arrays of current elements.................... 345 9.11 The gain of broadside arrays of current elements.................. 348 9.12 Radiation from progressive waves on a wire............... 348 XIV U>N TKNTS 9.13 Arrays with nonuniform amplitude distribution................... 7.1-1 The solid angle of the major radiation IuIjl-, the form factor, and the gain...................;................................... 9.15 Broadside arrays of highly directive elements..................... 9.16 Ground effect................................................. 9.17 Rectangular arrays............................................ 9.18 Radiation from plane electric and magnetic current sheets.......... 9.19 Transmission through a rectangular aperture in an absorbing screen. 9.20 Transmission through a circular aperture and reflection from a circu- lar plate................................................... 9.21 Transmission through a rectangular aperture: oblique incidence..... 9.22 Radiation from an open end of a rectangular wave guide........... 9.23 Electric horns................................................ 9.24 Fresnd diffraction............................................. 9.25 The field of sinusoidally distributed currents...................... 9.26 The mutual power radiated by two parallel wires................. 9.27 Power radiated by a straight antenna energized at the center....... 9.28 Power radiated by a pair of parallel wires....................■ ■ - ■ 349 350 352 353 353 354 355 356 358 359 360 365 369 372 373 373 X. WAVES, WAVE GUIDES, AND RESONATORS — 2.................. 375 30.1 Transverse magnetic plane waves................,.............. 375 10.2 Transverse electric plane waves........, , . ...................... 3SO 10.3 Genera! expressions for electromagnetic fields in terms of two scalar wave functions............................................. 382 10.4 Natural waves in cylindrical wave guides........................ 383 10.5 Natural waves in rectangular wave guides........................ 387 10.6 Natural waves in circular wave guides........................... 389 10.7 Natural waves between coaxial cylinders........................- 390 10.8 Wave guides of miscellaneous cross-sections...................... 392 10.9 Slightly noncircular wave guides................................ 397 10.10 Transverse magnetic spherical waves............................ 399 10.11 Transverse electric spherical waves.............................. 403 10.12 Wave guides of variable cross-section............................ 405 10.13 Cylindrical waves............................................. 406 10.14 Circulating waves............................................. 409 10.15 Relations between plane, cylindrical, and spherical waves.......... 410 10.16 Waves on an infinitely long wire...............................- 417 10.17 Waves on coaxial conductors................................... 418 10.18 Waves on parallel wires........................................ 421 10.19 Forced waves in metal tubes................................... 423 10.20 Waves in dielectric wires....................................... 425 10.21 Waves over a dielectric plate................................... 428 10.22 Waves over a plane earth...................................... 431 10.23 Wave propagation between concentric spheres.................... 435 10.24 Natural oscillations in cylindrical cavity resonators................ 437 XI. ANTENNA THEORY................................................ 441 11.1 Biconical antenna...........................................■■ 411 11.2 General considerations concerning the input impedance and admit- tance of a conical antenna.................................... 449 11.3 Current distribution in the antenna and the terminal impedance. . . . 450 11.4 Calculation of the inverse of the terminal impedance.............. 452 11.5 The input impedance and admittance of a conical antenna......... 454 11.6 The input impedance of antennas of arbitrary shape and end effects. 459 11.7 Current distribution in antennas................................ 466 11.8 Inclined wires and wires energized un symmetric ally............... 469 11.9 Spherical antennas............................................ 471 11.10 The reciprocity theorem....................................... 476 11.11 Receiving antennas............................................ 478 CONTKNTS (MAPTI " pAriK Ml. 1111': IMITOANCT CONCT.iT...................................... 480 12.1 In retrospect................................................. 480 12.2 Wave propagation between two impedance sheets................. 484 I:\J ()n impedance and reflection of waves at certain irregularities in wave _ guides..................................................... 490 12,4 TJlO impedance seen by a transverse wire in a rectangular wave guide 494 PROBLEMS................................................................ 497 QUESTIONS AND EXERCISES............................................ 510 IIIIII.IOGUAITIICAL NOTE.................................................513 SYMBOLS USED IN TEXT................................................ 515 INDEX.................................................................... si7 CHAPTER I In the manufacture of this book, the publishers have observed the recommendations of the War Production Board and any variation from previous printings of tho same book is the result of this effort to conserve paper and other critical materials as an aid to the war effort. Vectors and Coordinate Systems I.I. Vectors Vector is a generic name for such quantities as velocities, forces, electric Intensities) etc. A vector can be represented graphically by a directed q Ci Fiq. 1.1. Equal vectors. segment PQ (Fig. 1.1) whose length is proportional to the magnitude of the vector. Two parallel vectors PQ and P'Q' having the same magnitude and direction are considered equal. d c a b Flo. 1.2. Addition of two vectors. The method of adding vectors is what distinguishes them from other quantities. This method consists in obtaining the diagonal of the parallelogram constructed on two vectors as adjacent sides (Fig. 1.2); thus* AB + AD = AC. *We shall use no special marks to designate vectors if the meaning is clear from the context; otherwise we shall use a bar over the letters. 1 ELECTROMAGNETIC WAVES 9 Any number nl wi„r:: may In- at Kin I liy using the end of one vector as the origin of the next; the vector drawn from the origin of the first to the end of the last is the sum (Fig. 1.3). Since by definition AB + BA = 0, or BA = -AB, Fig. 1.3. Addition of several vectors. Fic. 1.4. Subtraction of vectors. subtraction of vectors is essentially the same as addition; thus (Fig. 1.4) PQ - PR = PQ + RP = RQ. Hence the difference of two vectors drawn from the same origin is the vector connecting the end of the second to the end of the first. The scalar product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them; thus A ■ B = (A,B) = ab cos The scalar product of two unit vectors is the cosine of the angle between them. Two vectors are perpendicular if their scalar product is zero. Scalar multiplication obeys commutative and distributive laws A- B = BA, (A + B) ■ C = A- C + B- C. The component of a vector in the direction defined by a given unit vector is the scalar product of these two vectors; that is, the projection of the given vector on the unit vector. The direction components of a vector drawn from point P{xi,y\fi\) to Q(x2,y2,z2), taken in the positive directions of the coordinate axes, are the differences x2 — #i, y2 — z2 — Z\. If / is the length of the vector and a, /8, y are the angles the vector makes with the coordinate axes, then PQx = x2 — xi - I cos a, PQv = J2 - }'i = /cos /3, PQz = z2 — zx = / cos 7. The scalar product of any two vectors may be expressed as the sum of the VKCTOUS AND mnUMNATk SYSTEMS Fig. l.S. The vector product. I.....Iiitls ill their direction components A' ■ A" = A'XA'J + A'vA'y' + AW,'- I Icikt the cosine of the angle i£ between the vectors is the sum of the products of their direction cosines cos Tp = cos a'x cos a'J + cos av cos a'J + cos a£ cos a'J. Tlic vector product A X B or [A,B] of A and />' i. a vector perpendicular to both, pointing in i lie direction in which a right-handed I ii w would advance if turned from vector / lo vector B through the smaller angle i Til'.. 1.5); the magnitude of the vector prod-tn ( is the product of the magnitudes of A and // and of the sine of the angle between them, 11i.i i is, the area of the parallelogram con-Htt'iicted on A and B as adjacent sides. For vector products we have AXB = -BXA, (A + B)XC = AXC+BXC. The components of a vector product are expressed in terms of the direction BOmponents of the constituent vectors as follows {A1 X A")x — AyA'J — A'zAy , {A' X A")v = A'ZA" - AWJ, {A' X A"\ = A'xA'y ~ A'yA". I .' Functions oj Position A function of position or a point function is a function f(x,y,z) depending only on the position of points. Loci of equal values of a point function lire called level surfaces or contour surfaces; in the two dimensional case We have level lines or contour lines. Some level surfaces bear special names luch as equipotential or isothermal or isobaric surfaces. Figure 1.6 illustrates how a two-dimensional point function may be represented jritphically by drawing contour lines. The solid lines are the contour lines lov ;/ = log pi/p2, where p\ and o2 are the distances from two fixed points; ......I t he dotted lines are the contour lines for the angle # made by BP with I'/t as shown in Fig. 1.6(a). The rate of change of a point function depends not only on the position "I ;i point but also on the particular direction of travel. If AV is the i Kange in the value of a point function V{x$) as we pass from A(x,y) to B{x E Ax,y + Ay) an£l if A* is the distance AB (Fig. 1.7), the ratio AV/As ELECTRi MAGNETIC WAVES C'llAI'. I VECTORS AND moRDINATK SYSTEMS is I In- aoeragt rate />/ change of F{x,y) in the direction AB. The limit of this ratio as li approaches A while remaining on the same straight line is the directional derivative of V{x,y) in the direction AB. This derivative is denoted by dF/ds. Partial derivatives dV/dx and dV/dy are simply the directional derivatives taken along coordinate axes. p--t--- \p. A =-rt \ J =-n ^ =+rr b 0 /////////////A ---Ci ^ > o (a) Fig. 1.6. Two families of contour lines. The maximum rate of change is along the normal to the level line through A (Fig. 1.8). The gradient of V is defined as a vector along this normal , „ dF _ grad jr = — «, where » is the unit vector orthogonal to the level line. For an infinitesi- ttnil mi vilincur triangle, we have Am = (Aj) cos uV, H lit I I lirrcforc W dF — = —- cos yf>. ds dn I. m i ihr directional derivative of a point junction is the component of its tt,idii ut in that particular direction. (2-1) ay B(x+c.x,y+c.y) Alx,yi Fiq. 1.7. Illustrating directional increments. Fio. 1.8. Illustrating the gradient. these equations are of course equally true for three-dimensional point functions. If a, 0, 7 are the angles made with the coordinate axes by the normal to the level surface at A, we have by (1) dF dF dF dF a dF dF dx dn By dn dz dn (2-2) Thus the partial derivatives are the direction components of the gradient, .iikI we have _ ELECTRi (MAGNETIC WAVES Anothei expression for the normal derivative can be obtained if the equations in the .set (2) are multiplied by cos a, cos 0, cos 7 respectively and then added BV BV bV BV — = — cos a + — cos p + — cos y. Bn Bx By Bz This relation can be written down directly if we consider that the gradient is the sum of the projections of its components upon itself. A complex point function is a function whose real and imaginary parts are point functions V{x,y,z) = Pifc&jfy + iF2(x,y,z). We cannot speak of level surfaces of complex point functions since there is one family of level surfaces for the real part, another for the imaginary part, a third for the absolute value, etc. Loci of equal phase * = tan iTi—k Vy (x,y,z) of a complex point function are called eqtiiphase surfaces; they are used in the classification into plane, cylindrical, spherical, etc., waves. The gradient of a complex point function is denned as the complex vector whose components are the partial derivatives of the function. A vector point function is a vector whose direction components are ordinary point functions. 1.3. Divergence The flux of a vector F(x,y,z) through a surface S is defined as the surface integral JJFndS, where Fn is the component normal to the surface of integration. The outward flux of F through a simply connected closed surface S divided by the volume v enclosed by S is called the average divergence of F. The limit of the average divergence as S contracts to a point is the divergence of F at that point; thus div F = \\ dS , as S -» 0. Dividing the total volume v surrounded by the surface S into elementary cells, we observe that the total flux of F across the surface is the sum of the fluxes through the boundaries of the elementary cells, the fluxes through VECTORS AND COORDINATE SYSTEMS ,|.....mmmi partitions between the cells contributing nothing to the whole. HImm 1 lit- Mux through the boundary of a typical cell is div Fdv, we have I In turface divergence is defined similarly; thus (3-1) div' F m lim ds , as s —> 0, ,,. s is the boundary of the elementary area S. The linear divergence is ......|y 1 lie ordinary derivative. I 'I Line integral, Circulation, Curl I he line integral of a vector F along a path AB (Fig. 1.9) is defined as ||u integral / Fs ds of the tangential component of the vector. IfFisa UB) r. ' 1 I..... this integral represents the work done by F on a particle moving 1 la) Fio. 1.9. Hlustrating the line integral. along AB. If the curve is closed, the line integral is called the circulation. Tlir circulation per unit area of an infinitely small loop so oriented that •the circulation is maximum is denoted by curl F; it is a vector perpendicular to the plane of the loop. The positive directions of this vector ,iii.| circulation are related as shown in Fig. 1.10. Consider a surface 5 bounded by a simple closed curve. Dividing S mi0 elements, we observe that the circulation of F along the boundary of $ |S the sum of the circulations round the boundaries of the elements, since i'.i.r.ctuomaonftjc wavks Chap. I lie contributions due to the boundaries common to adjacent elements cancel out. Since the circulation round the boundary of each element is curl,t FdS we have fp,ds = j fcurln FdS. (4-1) 1.5. Coordinate Systems In practical applications the most frequently used coordinates are rectangular, cylindrical, and spherical; in these systems a typical point P is denoted by (x,y,z), (P, Fig. 1.11. Cartesian, cylindrical, and spherical systems of coordinates. coordinates is explained in Fig. 1.11; x, y, z are the distances from three mutually perpendicular planes; p is the distance from the z-axis; r is the VECTORS AND COORDINATE SYSTEMS 9 |i i.inir he nil the origin; the " polar an^lc " I) is the angle between the i j. I in, i .mi I I In- v.-axis; I lie " longitude '' tp is the angle between the xz-plane hi. I I he plane determined by the a axis and the point P. In a general system of coordinates, a point P(u,v,w) is specified as a point of intersection of three surfaces /i (*,jy,z) = «> />(**?>*) = v, M*&&) = w-The lines of intersection of these coordinate surfaces are coordinate lines; ilnr. rt-lines are intersections of v- and w-surfaces. V-LINE Fig. 1.12. An elementary coordinate cell. If the coordinates are orthogonal, the differential distances along coordinate lines are proportional to the differentials of the coordinates (Fig. 1.12); thus dsu = ei du, dsv = KI>lf\ \ If. systems 11 \\y definition, (he component curl,, /''of curl /-'in the ^-direction is the Circulation ol Fper unit area iti the a-surface passing through P (Fig. 1.13). II we picture F as a mechanical force, curl,, V is the work done by F per ...... area in the » surface. Consider an elementary area about P bounded by v- and W-lines. The work clone by F along the w-line through P is /••„, ,/.,„„ its rate of change in the ^direction DV(FW dsw), and the total work in i he counter-clockwise direction along the w-paths of the loop bounding I the elementary area is DV(FW dsw) dv. Similarly, the work done along the I remaining two sides of the loop is DW(FV dsv) dw in the clockwise direction. When the total work round the loop Dv(Fv,dsw) dv - Dw(Fvdsv)dw is divided by dSu, the area enclosed by the loop, we have curl„ F. The remaining components are obtained by the cyclic permutation of a, v, w. Substituting the corresponding expressions for the differential elements end differential areas in the various coordinate systems, we obtain Fig. 1.13. Illustrating the derivation of the curl of a vector. ■ F = -^ curl, F = - — , curl„ F = curlr F — curie F = dF dy dF_x dz 9F. dx 1 or i, oi x i m curl, F = —1 — curl,F = ŠFy dz dF, dx dF* dy curl 1 dF, P dtp dFn dz 1 dz m dp dp r sin 0 1 r sin 6 [Do(s\n OF?) - DV(F0){, [DJFr) - sin 6 Dr{rFv)}, electromagnetic waves curly fm flPrirFt) - D0{Fr)U cur],, F = — [Dv(e3Fw) - D„,(e2Fv)], curl„ F = — {Dw{eiFu) - Du{e5Fw)\, curlw F = — [Du(e2F,) - DMFu)]- Chap, i eie2 1.7. Differential Invariants and Green's Theorems The Laplacian or the second differential invariant is defined as the divergence of the gradient of a point function; symbolically AF = div grad V. In the above considered coordinate systems, we obtain 0 = B\V + D\V + DlV, 1 . _ ... . 1 AF AV AV [DP(P D„V) + ~ (D%F) + P{Dini p p = jrr-Q [^n 9 Dr(r2 DrV) + D9(sin 6 DeF) + ~ D%F\, r sin 6 • sin 0 = — \D. Duv) + Dv f& DA + Dw jfe* The Laplacian of a vector F is the vector whose cartesian components are the Laplacians of the cartesian components of F. The first differential invariant is defined as the scalar product of the gradient of two point functions; symbolically MV,?) = (grad V, grad V) = DXUDXV + D»UDyV + DJJ DZV. Green's theorems are, then, expressed by the following equations: and fffA(U,n dv = JfuS£dS - JJjUAVdv, (7-1) Jj'f(UAF-FAV)dv n ff{u^-Fd£)dS, (7-2) where the surface integration is extended over the boundary of the volume and the normal derivatives are taken along the outward normals. The vectors and coordinate systems 13 .....,,| theorem la the consequence of the first: if Uand Tare interchanged , (I) and the result is subtracted from the original equation, (2) will l,,||,,u Equation (1) is pro veil by integrating the left hand side by parts. IK. Miscellaneous Equations div curl F = 0, curl grad F = 0, (8-1) curl curl F = grad div F - AF, (8-2) div FF = V div F + F • grad F, (8-3) curl FF = F curl F — FX grad F, (8-4) div [F X G] = G ■ curl F — F ■ curl G, (8-5) mathematics of oscillations and WAVES is CHAPTER II Mathematics of Oscillations and Waves 2.1. Complex Variables A complex number z = x iy is a combination of real numbers x and_)> and an " imaginary " unit * subject to the following condition: ■2 i I — —1. The quantities x andjy are called the real and imaginary parts of z; thus we write x — re(z), _y = im(z). Complex numbers are represented graphically by points in a plane (Fig. 2.1) or by vectors drawn from the origin to these points. Complex Fig. 2.1. Representation of a complex variable by points in a plane. numbers obey the same arithmetical laws as real numbers. In the complex plane the addition and subtraction of complex numbers correspond to the addition and subtraction of vectors (Figs. 2.2 and 2.3). In polar coordinates we have (Fig. 2.1) z — p(cos

ma<;nktk: waves Ghai'. 2 MA'l'l II1'MATH 'S OL OSCILLATIONS AND WAVES l<> In transmission theory wo shall use I lie following theorem: // ■/. describes a circle in the /.-plane {including straight lines as special cases), then w also describes a circle in the w-plaue. This theorem is proved by substituting for z in (1) and showing that the resulting equation in w is of the same form. 2.2. Exponential Functions Consider two variables z and to, real or complex. The ratios Az/z and Aw/w are called the relative increments. Just as the limit of the ratio Aw/As of two absolute increments represents the rate of change of w with respect to 2, the limit of the ratio Aw/wAz of the relative increment in w to the absolute increment in z represents the relative rate of change of re> with respect to z. The former rate of change is the derivative of w with respect to z and the latter the relative derivative or the logarithmic derivative. An exponential function is a function whose relative derivative is constant dw (2-1) 1 dio , - — = k, w dz or -■- =a kw. dz In particular the function whose relative derivative is unity and which becomes unity when the independent variable vanishes is designated as follows :* thus by definition is — exp z — — exp z = exp z, exp 0= 1. dz In terms of this function the general solution of (1) may be expressed as follows: w = A exp kz — Aekz. Since all the derivatives of exp z are equal to the function itself, we obtain from Taylor's series z2 z3 zn e* - exp z = 1 + z + - + - + ...+- + .... In particular we have e - expT= 1 + 1 +'~+ |j + ■ ' = 2.71828 Either, from (1) or by multiplying the power series the following addition theorem may be obtained: exp (2! + z2) = (exp zi)(exp z2). * The second notation anticipates some properties of exponential functions. 11 i-, evident from I he dcliiiilioii thai: nil derivatives of an exponential liiiuinm are proportional to lite function itself. Hence a general linear ilillcrcntial equation with constant coefficients d"w . rf"_1TO + «o = aekl (2-2) (2-3) I* w 14 W W + an~l ~dl^ + will possess a particular solution of the following form: w — behr. H.....nstanl b is obtained by substituting in (2); thus A nolution of this form exists for any value of & which is not a zero of Z{k). i in I he other hand if z(L) = o, i In ii (2) possesses the following solution w = bm^ when a = 0. The most general solution of (2) when a 0 is then (2-7) (2-5) (2-6) w - Z(k) 2,3. Exponential and Harmonic Oscillations Let the position of a point P(z) in the complex plane be an exponential linn lion of time; thus z = Aept, (3-1) Where A and^i are complex con-Bunts I A = • then p = £ + iw; (3-2) z = aeeteitu't+'pr,). (3-3) I I ii r; the vector OP, drawn from i In origin 0 to the point z, re-polves about 0 with a constant lingular velocity w and its length j ji ics exponentially with time Fig. 2.7. Uniform rotation and harmonic oscillations. to • i.KiTiumaonktic wavks CiiAf. 2 a o •a o 0 1 M A 1111■.M A I I( S OF OSCILLATIONS AND VVA VIiS 1\ If f ■■ 0, the length of the vector OP remains constant and the point P ItinvcN along the circle of radius a with a uniform speed (Fig. 2.7). When 0, ilie .....veinent is counterclockwise. The projection Q of the point /' mi I lie real axis is .said to oscillate harmonically about 0. The distance il (' from the center of the oscillation is a sinusoidal function of time (Fig. he quantity .v = re(z) = a cos («/ -f tp0). If) = (Sit + tpQ | i idled the phase of the oscillation and a is the amplitude of the oscillation. flPJir constant is the initial phase. Fig. 2.9. Constant relative increments in the complex plane. Two phases which differ by an integral multiple of 2ir are regarded as the ..... I.e. .m.se the points P and Q occupy the same positions. The interval riiel w. en two successive co-phase instants is called the period of revolution ■III /' iiiul the period of oscillation of Q; this period is T = (3-1) The number of revolutions of P or oscillations of Q is the frequency f in flyclcs per second; thus / = J, > w = 2ir/. (3-5) he constant o> is the angular velocity or ihefrequency in radians per second. 11 i 9^ 0, we have from (3) dp ~ p£ »fi dtp — w dt, 22 KLECTftl (MAGNETIC WAVES Chal. > and, therefore, dp = - p dip. [■fente the angle ij/ between the radius and the trajectory (Fig. 2.9) is obtained from « f tan ip — - , or cot ^ = — . The trajectories of point P(z) are, thus, equiangular or logarithmic spirals; Fig. 2.10 shows several such spirals for different values of f/cu. In this case MATHEMATICS OF OSCILLATIONS AND WAV ICS M Fig. 2.10. Logarithmic spirals illustrating exponential functions of time; z = = e{V*)v+i)¥>. The so-called "£>" associated with oscillations denned as the magnitude of cj/2,:. may be the distance OQ = x :? a sinusoidal function with an exponentially varying amplitude x = re(z) = aeil cos (a>/ -f- ^0). (3-6) The constant £ may be called the growth constant. The growth constant per cycle \jf is the logarithmic increment (or decrement). Since there is one-to-one correspondence between points P(z) moving in accordance with equation (]) and their projections on the real axis, expo- .....nal OSClllfttiona defined by equation (6) may be represented symboli- cully liy complex exponentials of the form (I). The complex constant p r, > allal the oscillation constant. The constant A gives simultaneous infor-malion about the initial amplitude and phase. 2.4. Waves \ wave function is a function of coordinates and time. function F(x,t) = mr*m% VI here //, F, p are complex constants, is a wave function. A = aei9\ p = £ + iu>, r = « + IP; i In- real part of V, given by tc(F) = ae-"x+H cos (w/ - (3* + 450), For instance the (4-1) Let (4-2) (4-3) 1 also a wave function. At any point x this function is a sinusoidal func-111'ii11 of time, with exponentially varying amplitude; at any instant /, rii(//) is a sinusoidal function of the coordinate x, also with exponentially varying amplitude. Physical phenomena expressed by wave functions .in called waves. As there is one-to-one correspondence between exponential functions of, the form (1) and sinusoidal functions of the form (3), we hiay use the former to represent the latter. The constant T is called the propagation constant; its real part a is the attenuation constant and its pi aginary part /3 the phase constant. Tin: quantity * = oil - fix + di - P dx = 0; v = dx Tt 03 0) p = -■ V (4-6) * Except, of course, when w = 0, ■24 electromagnetic WAVES Chap. 2 From (5), (6), and (3-5) wc obtain /X - v. (4-7) Consider now a general three-dimensional harmonic wave function V = A^y^e'^^™1 in which A and <& are two real functions. The surfaces of equal phase (at the same instant), given by ®(xi}'>z) = constant, are called equiphase swfaces* The waves represented by V are called plane, cylindrical, spherical, etc., if the equiphase surfaces are plane, cylindrical, spherical, etc. For any pair of infinitely close points in an equiphase surface, we have ■— dx + ■— dry H--= 0. d,v dy dz Replacing the differentials dx, dy, dz'm this equation by (X — x), (Y ~ y), (Z — z) where (x,y^) is a point in the equiphase surface and (X,Y,Z) is a typical point in space, we obtain the equation of the tangent plane 3* a$ t># — (X - x) + — (Y - y) + - (Z - z) = 0. ox dy dz The family of curves everywhere normal to the equiphase surfaces is given by the simultaneous differential equations (4-8) dx dy dz ai> ~ a* = a* dx dy dz These curves are called wave normals. Equation (8) implies that any wave normal is tangential to grad y>. Consequently, wave nonnals are curves along which the phase changes most rapidly. At two points infinitely close in spaee-dme the phases are the same if , a* _ a* a* aat — -— dx---ay--— dz — 0. dx dy dz If dy = dz = 0, the instantaneous rate with which the phase changes along an jf-line is d$/dx. The " phase constants " along coordinate lines, which * For purposes of this definition phases differing by an integral multiple of tt are regarded as equal. MATHEMATICS UK OSCILLATIONS AND WAVES i......nllv arts nnt constants at all, arc 25 8* dy ; a* 52 ' Tin' |ili:i:ic constant along a wave normal is |3C>W) = I Srad * j- In i liter t|imansions the phase constant may be regarded as a vector whose dm , tiiiii components are di'/dx, d$/dy, d$/3zs The phase velocity is usually defined by v{x,y,z) (3 grad * | I'hi'i is the instantaneous velocity along a wave normal. We may also |. il, ui the phase velocities along the coordinate axes 01 ai' dx ai' by v. = ft ai' as ■If In evident, however, that vs, vy, vz are not the cartesian components of the pIlHMC velocity along the wave normal. On the other hand, the reciprocals nl these phase velocities, being proportional to the phase constants, behave ii'i \ rrt:or components should. Thus, we define phase slowness S S — - grad $ . 03, 2.5. Nepers, Bels, Decibels The logarithmic measure has come into use because in certain measurements the logarithm of a ratio of two quantities is more significant than the I'd I in itself. When the ratio of two quantities of the same kind is expressed In nepers, the number of nepers is computed from knmi this we have N = log -j- nepers. (5-1) Ay = A2eN, A2 *> Axe .-AT Originally the logarithmic unit was introduced for the evaluation of flower ratios and present laboratory units are the bel and the decibel. The number of bels and decibels (abbreviated db), expressing a power ratio tl'i/lVg, is computed from X! = login „,- bels = 10 log,0 — decibels. W 2 rr 2 26 ELECTROMAGNET!C WAVES' Chap. 2 MATHEMATICS OF OSCILLATIONS AND WAVIvS 27 More recently I lie use "I logarithmic units has linn extended to " intensity ratios," that is, to voltage and current: ratios; then the number of bcls ami decibels has been defined as E E N = 1 logi0 — bels = 20 log]0 — decibels. -C.2 E2 It is unfortunate that the size of the " bel " or " decibel " is not uniform but depends upon the nature of the measured quantity. We shall keep the neper as a fixed unit defined by (1) regardless of the nature of A. Thus in translating from nepers into decibels, at least when dealing with electrical quantities, we must multiply by different conversion factors. For power ratios we have 1 neper = 10 log10 e ~ 4.343 db; and for voltage ratios, current ratios, field intensity ratios, we have 1 neper = 20 logm e e=! 8.686 db. In accordance with the above definitions the attenuation constant is measured in nepers per unit length or in decibels per unit length. The phase constant is measured in radians per unit length. Similarly the time growth constant is measured in nepers or decibels per unit time and the frequency oi in radians per unit time. 2.6. Stationary Waves The waves discussed in section 4 are called progressive waves. The sum (or the difference) of two unattenuated progressive waves, of equal amplitude, moving in opposite directions is called a stationary wave because different points oscillate always either in phase or 180° out of phase. For instance, < a X 0 / \ Ft ir to its time integral Vi f Rli, V* = L dlj. dt dVi the above coefficients of proportionality are called respectively the resistance, the inductance, the capacitance. In diagrams resistors, inductors, ELECTROMAGNETIC WAVES Crap, 2 :mil capacitors are represented as shown in Eig. 2.14. For exponential and harmonic voltages and currents we have pC V = 72/, V = iuZJy V = R + i X G + IB Fic. 2.13. Diagrammatic representation of impedances and admittances. '.. h 1, Ft<;. 2.14. Diagrammatic representation of resistances, inductances, and capacitances. The impedance is a function of the frequency or more generally a function of the oscillation constant. To any exponential voltage Vept there corresponds a finite response in the impedor (Fig. 2.11), given by Vevi (7-2) Je*' = Zip)' provided p does not satisfy the following equation Zip) = 0. Similarly, since . Vevt m Z(p)Jept, (7-3) (7-A) a definite exponential voltage exists across the terminals except when p is an infinity of the impedance function or a zero of the admittance function Zip) = », Y{p) = 0. (7-5) MATHEMATICS < >E OSCILLATIONS AND WAVES 29 When /j in a zero of the impedance function, then a finite exponential muh m may exist when /' 0, that is, when the terminals of the impedor Hi In irt circuited; the /.en >s ol '/,{[>) are the natural oscillation constants of flu* impedor, with the terminals short-circuited. Similarly, if p is a pole ill' /(/')> a finite voltage may exist if f tin terminals of the impedor are open; 0 * VV\AA/ I Ihm- poles determine the natural os-tillution constants of the impedor, with the terminals open. The corresponding frequencies are the natural frequencies of the impedor. As we ■ ,11 see, natural oscillations may be . . in il by an impulsive voltage. citor, and an inductor connected series. °-), (7-8) then by (7) Z(-m) = R(o>) - iX(o>). On the other hand, replacing w by -co in (8), we have Z(-m) = ic(-w) + zX(-to). Comparing the last two equations, we find that the resistance function is an even function of the frequency and the reactance function is an odd function; thus R(-w) = £(«), Xt-u) = -Xi \\ \ VI IS .11 'I'he impedanci- of a Unite combination of resistors, inductors, ami capaci 1111 is a rational fraction; the difference between the degrees of the numerator a in I t he t km mi mat m- is ei i her unity or zero or negative unity. The ...... and poles of a nondissipative impedor are simple; they lie on the .....Iginary axis and they separate each other. The zeros and poles of a passive dissipative impedor are always in the left half of the oscillation .....slant: plane; they are not necessarily simple but usually so. .Mi. Average Power and Complex Power The work performed by an applied voltage driving an electric current (through an impedor is & = / Vili dt = VaIa j cos (úst + I'* Y = -= — VV* II* 2* ' (8-1) (8-2) (8-3) (8-4) II* 2** 2.9. Step and Impulse Functions* Three functions are particularly important in wave theory: the sinusoidal function, W more generally the exponential function; the step function (Fig. 2.16) and the impulse function (Fig. 2.17). In the last case it is frequently assumed that the , Mint t of the step is infinitely small and the step itself is infinitely large, while the ttrettgth of the impulse, represented by the area under the step is finite. The independent variable is usually either time or distance. A step function is called a unit * Most of the contents of this and following sections are needed only in Chapter 10 and thus may be omitted on the first reading. v. ia.i';n'uoMA(;Ni"i-k: waves- I MAI'. I step ill lie sudden rise itt from zero to unity. An impulse function is a unit impulse (or 11 unit source) if its strength is unir.y. These functions are important in their own right; besides, by superposing either a finite or an infinite number of them one can obtain any function that may be met in practice. It is easy to see that this is so with impulse and step functions (Fig. 2.18); t OR x Fig. 2.16. A step function. it requires some analysis to show that a function can be expanded either in a " Fourier series " or a " Fourier integral," representing addition of sinusoidal functions. In order to obtain the response of a linear system to an almost arbitrary force we need only find its response to any one of the three above mentioned standard functions t OR x t or s Fie. 2.17. An impulse function. and integrate the result. Naturally, the principal requirement is that the integrals be convergent. For example, to find the electric current I(t) through the terminals of a given jmpedor due to an electromotive force V{i) applied across the terminals, we write 1(f) = r rOmJ)^, (9-D where Fit,?) is the current at time/in response to a unit impulse of electromotive force at time t. In the present case we know a priori some properties of F{tJ). This function is identically zero for t < t since the electromotive force is not retroactive; for mathematics oh1 OSCILLATIONS AND waves 33 I > 1 its value depends only on (/ - ?). Lot P(tfl) - Fit); then F{t,t) = Fit - 1) .HIil v 1 no f V(t)F{( - /) dl = / F{t)F(t - f) it. (9-2) In potential theory, the function corresponding to F(ttt) is called Green's function; '.vi- may apply tills tunic to all responses to unit impulses. )i'lO. 2.18. Representation of arbitrary functions by superposition of impulse functions . and step functions. If A{t) is the current in the network in response to a unit voltage step at t = t\, then = IKjIi - ti) + f V'(t)A(t - t) at.' (9-3) I lere we have assumed that prior to.-i = ti the voltage is zero and that subsequently the voltage is a continuous function of time so that it changes in infinitely small steps, P"f$) dl, where W is the derivative. If there are finite discontinuities in V{t), these must be taken care of in the same manner as the sudden change from zero to V(ti). For instance, if the electromotive force ceases to act at t = tit then I{t) - F{ti)AU - h) + P F'(t)A(t ~1)dt~ Vih)Ait - h). (9-4) The upper limit of the integral can be h just as well as f since A it) vanishes for negative values of I. More generally, we can replace (3) and (4) by /DC -q Ait - i) dF(i), (9-5) ELECTRH (MAGNETIC WAVES ClMI'. 1 MATHEMATICS < )!•' OSCILLATIONS AND WAVES 35 provided we interpret 1 liin inIcp.ful in I he Slicltjcs sense instead of the Kicmanii scii.se. This interpretation consists In taking the EUemann integral and adding to it |C(r-|- 0) — F(t — 0)]/J(t — t) at each point t = r where V{t) is discontinuous. Jn this sense the dilferential dV is permitted to be finite as well as infinitesimal. In electric circuit theory A(t) is called the indhiat admittance of the network. hi " 2m J(c) Fie.. 2.19. The contour (C) involved in the representation of functions by contour integrals which can be interpreted as " sums of sinusoidal functions of infinitely small amplitudes." We shall now express the unit step and the unit impulse functions as contour integrals. Consider the following integral ■ dp (9-6) (O P in the complex ^-plane (Fig. 2.19). The contour (C) is along the imaginary axis indented to the right at the origin, li t < t, we can add to this contour a semicircle (Ci) of infinite radius without changing the value of the integral. On this semicircle the real part of p is positive and e^^/p vanishes exponentially when —w/2 < ph(/>) < tt/2; hence the integral over (Ci) is zero except, perhaps, over the portions corresponding to the phase angles infinitely close to tt/2 or —ir/2. A closer study of the integral in these regions would show that their contributions vanish as the radius of the semicircle becomes infinite. The integrand is single-valued and has no poles within (C 4- Ci); hence the integral (6) is zero for t < i. if / > /, then we can add to (C) an infinite semicircle (C2) in the left half of the plane, without changing the value of (6). Within this contour there is a simple pole at p = 0; since the residue is unity, the value of the integral is unity. Thus the integral (6) represents a unit step at / = /. lly superposing two step functions, of magnitude I i and — !/r, the first beginning nt / — t/2 and the second at / = r/2, we find that the unit impulse function, centered at t " 0, is pT e*ldp, as r^O. , (9-7) In this expression it is not permissible to let t = 0 since the resulting integral is divergent in the usual sense. At times, however, the integrals derived from (7) converge for r = 0 and the substitution is permissible. I f the unit impulse is spatial rather than temporal, we write (7) in the form : dy, as s —0. (9-8) If in (7) and (8) we replace / by / — t and x by x impulse from the origin to a typical point pT 1 2tt/ i, we shift the center of the 1 2jti (9-9) In the preceding equations p is pure imaginary on (C) except in the immediate vicinity of p = 0. Thus the unit step and the unit impulse have been represented by superimposing sinusoids of infinitely small amplitudes with frequencies ranging from —» to +°°. The above contour integrals can be turned into more conventional forms of "Fourier Integrals" depending only on positive frequencies; but t be present form is, on the whole, more useful. The values of the integrals will not be changed if (C) is deformed into any other contour provided this deformation takes place in the finite part of the plane and no poles are crossed in the process. We shall now represent an arbitrary function/(/) which is equal to zero for t < 0, .in the form m I S{p)e^'dp, S(P) 2ttí J o f(t)e-pt dt. (9-10) The function S(p) is called the complex spectrum or simply the spectrum* of/(/). The * In mathematical books S(p) is also called the Laplace transform of/(/). 36 ELECTROMAGNETIC WAVES Chap. 2 ; unction/(/) can bs regarded u the lim'il of the sum of impulse; funciiiona of strength /(}) A/, as A/ approaches /flu; thus * /(/) A/ /(0 = hm D sinh /> At 1 = 0 2iri p At ■ e^'-^dp, as A?->0, p A? sinh-— = lim -i-. f

A/ 2 i f e^dp C mr^di, and we have (10). It should be noted that if/(/) is given by (10), then in - ® = f dp. The spectrum of this function is If now the electromotive force V{f) impressed on an impedor is expressed as an exponential contour integral, the current through the terminals may be obtained by superposition of the responses to each elementary exponential electromotive force. Thus V{t) = f S(p)e*dp, then /(/) = f (9-11) In particular the responses to a unit voltage step beginning at t = 0 and to a unit impulse centered at / = 0, are respectively 1 C evl 1 m = / -^-Jp, bw-—. sinh— 1 2wi 1 PiZ(p) (C) ? e*dp. (9-12) For circuits consisting of physical elements, the second integral converges as r —>0 and consequently so) = — r -^-jp. 2 th J(C) Z(p) (9-13) However, in some applications it is necessary to deal with impulses of finite width rather than with idealized impulses of zero width. MATHEMATICS OK OSCILLATIONS- AND WAVES 37 Substituting for/(/) in (10) the functions A (f) and H(i) from (12), we have 1 /"° 1 ("* wri B(,,<~"'*- X A 4(tyi dt, p - plane Since the response /(/) is zero prior to the application of the electromotive force, 'In contour (C) should be to the right of the poles of the integrand. If some poles are on the imaginary axis, the contour is indented as shown in Fig. 2.20. Consider now a special example. Take a resistor and an inductor in series (Fig. 2.21); the impedance function is Z{p) = R + pL. Hence the current B{l) flowing in response to an infinitely short unit voltage impulse is 1 C ept 2mJ[C)pL + R ~ - When / < 0, this integral is automatically zero because of our choice of (C). When / > 0, we may ) ) ) IC) ) WMAAA Fio. 2.21. A resistor and an inductor in series. Km. 2.20. Infinitely small indentations in the contour (C) when the poles of the integrand (the close (c) w;th an ;nfinite sermcjrc[e jn the left half of infinities of the admittance rune- . . . . , tion) pass from the left half of the Plalle- 1 here 13 °nl>' 0ne Pole * = ~R'L w,th" Che plane to the imaginary axis, m this contour and therefore This condition exists in non-dissipative (purely reactive) networks. BQ) = - f-WlX. (9-14) If the zeros of Z(p) are simple, we can obtain the current Bit) in response to a unit voltage impulse in a simple form. In the vicinity of a typical zero we have %{p) = (P — Pm)Z'{pm); hence the residue of the admittance is Z'{pm) and (13) hecomes Z'{pm) The summation is extended over all the zeros of Z(p). Similarly, for an impulse of finite duration, we obtain from (12) (9-15) B(t) - % „ . , pmT 2 sinh — ep»* PmtZ'U)m) (9-16) 18 KU'XTI<<>MA(;NKTIC WAV KS Chai>. J MAT1IKMATICS Oh' OSU 1.1 ATIONH AND WAVES 39 for / > r/2, tlnil is, iti the interval after the impulse has censed. During the operation of the- impulse, we have — r/2 < / < r/2 anil {(.') in (12) cannot be closed with infinite semicircles since their con tri hu duns become infinite instead of infinitesimal. Substituting sinh^r/2 = - e-^l2) in (12), we obtain pP.H-(t/2)1 Jf Mi-ítU)] "1 -* Within our interval / + r/2 is positive and (C) in the first integral can be closed on the left; at the same time I — r/2 is negative and (C) in the second integral can be closed on the right. The second integral vanishes and we have 1 1 íí>Jh-W»I (9-17) assuming that Z(0) ^ 0. The first term arises from the pole at the origin. Evidently (17) represents the response to a voltage step of magnitude 1/r at I = —r/2. This is not surprising, since during the operation of the impulse the circuit does not know that the voltage will cease to operate. The significance of the first term in (17) will be understood if we apply this equation to the circuit shown in Fig, 2,21. White the impulse is operating, we have B{t) J_ Rt I lir' (9-18) The response (14) to an impulse of infinite magnitude but of zero duration starts with a finite value. For a physical impulse of finite duration, the response (18) is zero at the instant the voltage begins to operate and builds up to the value § I -Sr ti. Rr at the instant l — r/2, when the voltage is off. Subsequently the response decreases exponentially. If the interval r is so short that Rr/L is much less than unity, B{r/2) becomes approximately 1/L; this agrees with (14). 2.1.0. Natural and Forced Waves Generally speaking, several wave functions are associated with a physical wave. When a wave is traveling along a string under tension, a typical point is not only displaced from the neutral position but it is also moving with some velocity and it is acted upon by some force. Thus we have a wave of displacement, a wave of velocity, and a wave of force. In electromagnetic waves we are confronted with interdependent waves of electric and magnetic intensities. The ratios of certain space-time wave functions play just as important a part in wave theory as the ratios of wave functions depending only on time play in the theory of oscillations. These generalized ratios are called wave impedances and are differentiated among themselves by qualifying adjectives and phrases. For the present we shall confine ourselves to the simplest type of wave motion: waves in ;i i ran.....issioti line. Tin- equal n.......;overning these waves are (').v ~-(&*<■+Lfj-Eti* A § + (10- 1) where: Vi and It are respectively the instantaneous transverse voltage across the line and the longitudinal electric current in it; Ei is the voltage per unit length, impressed along the line in " series " with it; R, L, G, C are constants representing the series It Ei Fio. 2.22. A diagram explaining the convention regarding the positive directions of the transverse electromotive force V and the longitudinal electric current / in the lower wire (that is, the transverse magnetomotive force around the wire) of a transmission line consisting of two parallel wires. resistance, the series inductance, the shunt conductance, the shunt capacitance — all per unit length of the line. A pair of parallel wires (Fig. 2.22) is a concrete example of a transmission line; the arrows in the figure explain the convention with regard to the positive directions of the variables Vit Ti, Ei. Let Ei, li, and Vi be harmonic functions; then Ei{x,t) = ÉeíjŠ?""*). Vi = re(^tot), h = re(f«?tof). (10-2) As we have seen these complex exponentials will also satisfy equations (1). Substituting them in these equations and canceling the time factor ewts we obtain a set of ordinary differential equations dx -(R + iosL)t + E(x), -jr-H -(G + icoC)P. ax (10-3) The expressions Z = R + Y = G + Í<úC, are known as the series impedance and the shunt admittance per unit length. Let us now suppose that the applied force E is distributed exponentially along the line. To solve the equations we assume that the response P, I is also exponential and we write tentatively 1 E{x) = E?% P(x) = Vev\ l(x) = (10-4) 11» electk< magnetic waves ClIAl'. mathematics OF oscillations AND waves ■II Substituting ill (3) and solving, we obtain yV+ZT-E, YF+yl (10-5) v■ = yt-ZY ' " ZF-72J Thus we find that a response of type (4) is possible except when the propagation constant y satisfies the following equation ' 72 - ZY = 0. (10-6) This response is given by (5) and the corresponding voltage and current waves are called forced waves, by analogy with forced oscillations. When 7 is a root of (6), that is, when 7 = ± there is no finite response of type (4), unless the impressed force E vanishes. Exponential waves may exist in the transmission line without the operation of an applied force along the entire line. These waves arc called natural waves and the corresponding propagation constants are called natural propagation constants. Elm Fig. 2.23. A spatial impulse function representing a highly localized impressed electromotive force. Just as natural oscillations can be produced by a temporal impulse of force, natural waves can be produced by a spatial impulse of force (Fig. 2.23). The magnitude of this impulse is the applied or impressed voltage Xs/3 i(x)dx, (10-7) a/2 and it is represented by the area of the impulse. By (9-8) we have 7J- *** dy, where (C) is a contour in the 7-planc, shown in Fig. 2,24. Consequently from (4) and (5) wc obtain ys 1» f sinh¥ yy* I ' 1« J\oy*-ZY tirs J(o sinh yW- 72) er*dy. (10-8) Fig. 2.24. The contour (C) in the propagation constant plane. These integrals are convergent as s -»0; thus for a point generator we have yi n ye"< ZY dy, I(x) _ 2l C ltd J(C \dy. (10-9) ho ZY-y* Each integrand in (9) has two poles (Fig. 2.24) corresponding to the natural propagation constants; in the present case the origin isjiot a pole and thus contributes nothing to the value of the integral. Let V = V' ZY be that value of the square root which is located in the first quadrant of the 7-plane or on its boundaries* If x > 0, (C) can be closed with an infinite semicircle in the left halFof the 7-plane. Evaluating (9) we have V{x) = lf*e-T*, I(x) - r^; x>0. (10-10) If a: < 0, (C) can be closed in the right half of the 7-plane; then we obtain from (9) V{x) = /(*) = e^; x < 0. (10-11) : Since R, L, G, C are positive, VZY is either in the first quadrant or in the third. 42 MU I Hi (MAGNETIC WAVES C11a i Thus If l 11.n■mi mic voltage l^e*"' is inserted at some point x = 0 of an infinitely long transmission line (Fig. 2.25) in scries with it, two waves are originated, one traveling to the right and the other to the left. The current through the generator is v1 vf*o) = i vl Fig. 2.25. The conditions existing in a transmission line extending to infinity in both directions when an electric generator is inserted in series with the line. continuous but the transverse voltage suffers a sudden jump F1 in passing across the generator. The input impedance seen by the generator is 1(0) 2T Y 4 It will be remembered that in dealing with oscillations in electric circuits there was a question regarding the disposition of the contour (C). For dissipative circuits, tire correct result was obtained automatically by choosing (C) along the imaginary axis; and for nondissipative circuits, if regarded as limits of the dissipative circuits, it was natural to indent (C) as in Fig. 2.20. Nevertheless the only valid reason for making (C) pass on the right of the poles (natural oscillation constants) is to satisfy the physical condition, not included in the differential equation, to the effect that there should be no response to a force before it begins to operate. A similar situation exists in the present case; this time, however, we expect the waves to travel in both directions from the point source and hence we want (C) to separate the poles (the natural propagation constants). But these poles might be separated as shown in Fig. 2.26, in which case we should obtain PM-ifV, Hx) = -—e^; *>0; YVi Vix) = -^-r» I(x) = -— e-r*; * < 0; (10-12) instead of (10) and (11). We can object to this result on two counts. In the first place, for dissipative lines the real part of T is positive and (12) states that the voltage and the current increase exponentially with the distance from the generator. This is contrary to our experience and would imply that infinite power could be dissipated in the line when finite power is supplied by the generator. In the second place, equations M A I'll] m A riCS OF < >sc MXATIONS A NI > WAVES ll,') imply that power im not supplied by i he generator to the line but that the line .....mUiti-:; pnwer to the generator. Thus, by (12), the input impedances of dissipa- m\ r mid nondissipative lines are respectively ■II 7 _ * __2r -z._ _n f. '~I(0)~ Y' * "\C y-p lane c > IC! Flo. 2.26. A " forbidden " form of the Fio. 2.27. Indentations in the contour of contour of integration. integration when the natural propagation constants move to the imaginary axis as the dissipation in the transmission line approaches zero. hence the current flows in opposition to the impressed force. Therefore, we come to I he conclusion that (C) must separate the natural propagation constants as shown in big. 2.24. For nondissipative lines this contour assumes the form shown in Fig. 2.27. We have dismissed at the start the possibility that the wave could be started on one side of the generator and not on the other; for, when an alternating voltage is impressed at some point of the line, there is nothing to indicate on which side the wave should be if it is to be on one side only. In any case this possibility is contrary to experience. The method just explained is particularly useful in more advanced chapters on wave theory. In the present case there is a simpler way of looking at this type of problem (see Chapter 7). CHAPTER III bessel and lf.oendre functions 3.1. Reduction of Partial Diffei'cntial Equations to Ordinary Differential Equations Numerous problems of electromagnetic theory depend on solving the following equation subject to certain boundary conditions. The usual method consists in seeking solutions of the form V = X(x)Y(y)Z(z)> (1-2) and forming the desired solutions by either adding or integrating functions of this type. Substituting from (2) in (1) and dividing by XYZ, we obtain X dx2 Y dy2 Z dz2 ~ ' On the left-hand side we have three terms, each depending on one variable only; the sum of these terms is a constant. The only way we can satisfy this equation is to set each term equal to a constant; thus d2X iv d2Y ■-- = 0~xXt -- : dx2 dy2 The constants ax, p)Z{z\ substituting in (5), and dividing by R$Z, we have i d ( dR\ , l d2® , imz pR dp V dp 44 Z dz (1-5) (1-6) (1-7) bessel \nd legendre functions 45 Since all the terms except the third are independent of , the third term must also be independent of s; thus 1 d2Z fffi or Z dz2 ' d. ,fZ Substituting in (7) and multiplying by p2, we have dR\ + _1 d2®

. d2® dtp- d ( dR\ , {o-2 - dtp2 — ?»2tf, sin 8- dO + (£2sin2 0 - m2)Q = 0. (1-13) (1-14) This is the Associated Legendre Equation. 4<5 electromagnetic waves ClIAl'. .1 BKSSBL AND I I .c.l ,NI>luihlnns ii r, real and livi|iiciil iy an integer. The eonslant k. may nisi) he complex hut often it: is real and of the form k" = h(h -|- 1), where it is an integer. When m is an integer, 'I' is a periodic function with period 2ir. When n is an integer, some of the solutions of (14) are finite for all values of 0; for all other values of n the solutions of (14) become infinite either for 6 = 0 or for d = it. There is an equation related to (12), which is important in subsequent work. Rewriting (12) in the form dr1 dR dr = (*» + °% are arbitrary. Various supplementary physical conditions will restrict these constants in one way or another. For instance, let us suppose that physical conditions require that V should vanish for * = 0 and x = a, that it should be independent of y, and that it should vanish at z = «>. As a function of x, V will vanish as required if ^(0) and X(a) vanish. Writing the general solution for X in the form X{x) = A cosh o~xx + B sinh cxx, we find that X(0) vanishes only if A = 0; X(a) vanishes if sinh ffxa = 0, ffxa = tntr, ax =- a where n is an integer. The constant B remains arbitrary. (2-1) (2-2) I ivill be independent oi t il V Is a constant] then +2™' J~v{-%) = m?0 m\(-V + m)\2-'+2™t (3~2) where the generalized factorial is defined in terms of the Gamma function p\ = Y{p+\). (3-3) The point z = 0 is seen to be a branch point. The most general solution of (1) is a linear combination of these two functions. If v is a positive integer », then 7_„(z) = (-)»/*(=), (3-4) and we are left with only one solution, regular at z = 0. A second solution is defined as follows. For any nonintegral order v the function /,(z) cos vir — J-,{z) Ny{z) (3-5) i:; klectr< MAGNETIC waves" Chap. .1 is a solution ill (In- Itrs.i I equation, lis limit us i' iippnincites an integer n N„(■:,-) - lim N„(z) as v -* n (3-6) continues to he a solution nf the Bessel equation. A series expansion for this function is i («- m - vm>rm 2 mm---£--r^t~— + - opb z + c - log 2/»(2 i' z™— 2m ir^02n+2mm\(m + »)! [*»(»») + ¥>(« + «)L = - (log z + C - log 2)/«« + - £ 22m(w,)2 *<*), where the auxiliary function 0 in the expression for Np. For large values of z we have asymptotic expressions J,{t) - - f - l), A'.fol - - f - 0. (3-17) fY^fz These asymptotic expansions are valid only within certain phase limits: in the expres-imiis for /„ and N„, the phase of 2 must lie in the interval ( — 7T,tt), in the expression for H(l' the phase is in the interval ( —7r,2x),and in the expression for H1^ the phase is in the interval ( — 2x,x). These restrictions are needed because Bessel functions lire multiple-valued functions. Complete asymptotic expansions of the various Bessel functions are 2 \ 1/2 irz 2 \ 1/2 cxp f (z (2/z)- 00 cxp ii-z + hir + lir) L 1 2V/T r i 1 ^ v l~ J cos (z — a^lT — i7t) 2^ — Sill (2 — \VTT — £x) £ 1/1=0 m=o (2z; (-)»^,2m + 1) 2\1'2 K{z) ~ ( — sin (a — \vir — Jx) Jl m=0 + cos (z — \v-ti — jx) X! (22)2m+l (-)m(^,2w) (2z)Sm (-)'"(i',2m + 1)" (3-18) (2z) SO electromagni'.rit; WAVJiS Qu». 3 Tin; auxiliary fun, i'mn (iy«) is defined by , . + w - j)I (4yg- la)(4^ - m ■ • ■ [4«» - (2w - l)sl mil,-m-Dl" • <3"19) When v = » -f then jjg + 7»)! (3-20) The expansions (IS) are divergent; but if a finite number of terms is retained, then as z increases the sum of these terms will represent the corresponding function with increasing accuracy. 3.4. Modified Beuel Functions The modified Bessel equation is tdhv dz* dw -I- % ~~r - {zt + vi)w = 0 aZ (4-1) and its solutions are called modified Bessel functions. For nomntegral values of v a set of two linearly independent solutions is -h-3m =0m!(»' + ffl)!2'+i!»' * "V~J fflir,0m!(-f.-|-m}!2~'+:!m* Another important solution is a linear combination of these two functions Kr(z) If f is a positive integer k, then 2 sin vir V-,(?) ~ m% (4-3) I-aiz) = $j§| (4-4) and the equations (2) represent only one solution of the modified Bessel equation. A second solution is obtained from (3) by allowing v to approach n and passing to the limit Kn(z) = lim Kr{z) as v—>nw (4-5) 2 /a/-n _ a/A \ 9« ö» / ' cos kit From this definition the following series are obtained = £ - mlgn-j'-+ (-)tt+1dog *+c - log mm + (-)» Z '02n+im+lm\(.n + m)\ [(m + m)\ (4-6) -(log,+ C-log2)/0(a)+Si22m(M!)2 = - (log a + C - log 2), provided re(v) > 0 in the expression for J^,. 11 Z is large, we have asymptotically K^~{^) ni+^mt*—^zr—+ provided —3x/2 .' II It TIM »\1.\t .Ml ■■|'lf WAVES' Chap, .1 t/l A',,,!/.(«). (S-l) The £ and / functions arc solutions of the following differentia] equation, related to (1-16), dho r «(» + 1)1 ■5?-L1 + —?—J1 The / and iV functions satisfy X2 »(»+!) From (4-10) and (1) we have The analytic expressions for the functions defined in (1) are (5-2) (5-3) (5-4) fc„(z) = E (« + m)\ o *»!(» - m)\(2z)m' L m=om\(n - m)\{2z)m i u\ ( ,nr ■ - ™ Yr" /, z = ( cos — sin z — sin — cos z 1 >. V 2 2 y mfo (« + w)! m=o'»!(« — «0 l(2z) '-)*(» + 2»)! T + I cos — cos z + sin — sin z I >. - \ 2 2 / £o P* + 1)!(» - / »* . Mr . V^" (-)m(» «»(:) = —| cos — cos z 4- sin — sin z I >. - \ 2 ^2 / „fr0 2m\{n - S 2ro!(» - 2OT)!(2z)2m (S-S) -)m(» + 2»? + 1)! 2m - 1)!(2z)2"<+1 + 2m)\ Kir . . »t \ + cos — sin z — sin •— cos z >_ 2 2 / ^0 (2m + 1)!(h - 2« - l)!(2z)2»+1 2w)!(2z)2"' (-)"(» + 2?» + 1)! In particular we have Aft) = r*, &KE EUNtTlONS 53 In. Spherical Harmonics and Legendre Functions Solutions of (1-11) are called spherical harmonics. We have seen that it has I llutioni of the form T(0,) + 2 £ 7n(±<7p) cos »«9 11=1 (7-4) «5 50 e<-V»*>*-J0$p) + 2 £ Jin(ßp) cos 2n/ -s/2 O -s/2 u/2 ■m/2 (7-7) I (7-9) 1 /"''2 /,s''2 X íV lf-3í 0 Jot J to t r* i - cos/ 7o / = C + log x — Ci Si os = Gx = logx + C-^ + ^---?- + ■■■, C= 0.577- Si x Ci x * cos* " (-)"(»! sin x " (-)"(2» + 1)1 2^ ..(a — 2_/ * «=o x- X it=0 sin* ™ (-)"(»! cos* " (-)"(2» + 1)! x n=0 * x „=0 *2B+l £*'( ±w) = Ci x ± / Si x Si (—#) = — Si x 1 — COS 4 Cxl- cos t 0 / Jo t Jo /„(*) «& = 1 r Mkx) i 1 -ax = —, n = 1,2,3, • • • Jo v » Chap, 3 (7-13) (7-141 (7-15) (7-16) (7-17) (7-18) (7-19) (7-20) (7-21) (7-22) (7-23) (7-24) (7-25) (7-26) (7-27) (7-28) Hi ssi I. AND legendre functions * 1 • 3 • 5 • • • (2 n- 3)Jn-i(x) '0 * M i I /•'./■ v) J, #m dx ~ — /n+l(*) _ (m + » + l)/n+2(y) (w + w-f l)(w4-» + 3)/^W ■<# = C(x) + iS(x) / cos — dt = C(x), \ sm — dt= S(x) J o 2 t/ o * £7(0) - .5(0) = 0, C(oo) = S(«) = 0.5 CM = E b2So(2«)!(4» + 1) (-)-t tI_2ri+lv,4n+3 nf0 22«+H2» + 1)1(47/ + 3) irA-" its;'' C(x) ~0.5 4- P(.v) cos — - Q(x) sin — S{x) ~ 0.5 + P(«) sin — + 0(.v) cos — J_ f (-)"+'! -3-5 ••■ (4*4-1) ^ (™2)2»+* 1 - (_)»+!! ■ 3 ■ s • ■ • (4w - 1) (vr*2)2" 57 (7-29)* (7-30) (7-31) (7-32) (7-33) (7-34) (7-35) (7-36) (7-37)t Z J*+Wi>(iv**), S(x) = E hn-\-(m)(^x") (7-38) Ti =0 n =0 *»cos/3(r + 2) dz = Ci/Sfrz + za) -Gpin + zi) r = VP2 + z"2, n = Vp* 4- z2, r2 = Vp2 4- z\ rz" sin gQr + z) Jz, r dz = Si P(ra + 22) - Si 0(n + 21) -2) Jf^cos S(r ■ ■-—--<&=Ci/3(r1-2i)-Ci/3(r!!-Z2) P Mng(r~-i!? ^z = s; /s(n - Zi) - si jfa - 22) i/a, r * For n = 1 the numerical coefficient is unity. | The first term of Q(x) is —l/irx. (7-39) (7-40) (7-41) (7-42) 58 ELECTROMAGNET!* WAVES i tiAv. :i /Via (cos 0) = 1\ (cos 0) + 2A |VB (cos 0) log cos^ + 5«J Ph+A (- cos 9) = (-)"/J„(cos 0) + 2A |"(-)n/J„ (cos 0) log sin | + Jn'J = E —TIT-vi I , + —;-7 +----TT Jsin h „ = i a!a;!(» — a)! \» 1 1 + a » + a — 1 « + l) _ Sill ?7,V 7T — „v E - = —— , 0 < x < 2x »=i n 2 -§ log 2(1 — cos .v) = — log (7-45)'* (7-46)* (7-47) cos nx x- jr.v .v n-l » 6"i + 4' 0^-<2- 1 — COS RJf 7TW * » = 1 » , - - , 0 '2£ * <2ir 2 4 6 ~T+12' °--V<27r 2 sin - J, 0 < * < 2ir (7-48) (7-49) (7-50) (7-51) E —5 ■= — # log l/A3 J_M5 9\2/ +450\2/ »=1 ™J 288 + ■ ■ •, 0 < x < 2w (7-52) (7-53) CO X « F{6,® # J" = ™S 9, /I = COS 9, -Pr cos 0 2x (« + #;)'■■ 2« + l (» - m)\ f2v C1 2w (n + m)\ 1 (/») sin m*»* - - - E (2» + A03>-)/'. (cos o) ,tfr«« £ (2„ + l)i»)n(flr)P9 (cos 9) 59 (7-57) (7-58) rxp - iVr2 + a2 - 2ar cos 0 1 ™ * ,, ,— = -pr- S (2»+ i)^n(r«)/n(rr)Pn (cos e), r < a Vr + a2 - 2a;-cos 9 rar„=0 _ (7-59) cxp [-<)3\'/H + a2-2arcos 9] _ Vr2 + a3 - 2ar cos 9 J OO E (2» + l)[/„(j3a) - ifi.(fi*)]jMr)P» (cos 9), r < a (7-60) CHAPTER IV Fundamental Electromagnetic Equations 4.1. Fundamental Equations in the MKS System of Units It is assumed that the reader is familiar with fundamental electromagnetic concepts. An excellent description of a set of experiments underlying these concepts and the laws of electromagnetic induction may be found in the first four chapters of " Physical Principles of Electricity and Magnetism" by R. W. Pohl.* We shall use exclusively the meter-kilogram-second-coulomb system of units, commonly known as the Giorgi or the MKS system. In this system electromagnetic equations are particularly simple and correspond closely to physical ideas and measurements, com.-mon in engineering laboratories. Table I gives a list of quantities, symbols, names of units, and some dimensional equivalents of these units. The definitions of common electromagnetic terms may be summarized as follows. The basic idea of electric charge has to be gained from experience. When two bodies are electrified certain forces between them are attributed to the " electric charges " in them and the force E per unit " positive " charge is called the electric intensity. Electricity appears to be atomic in structure and the smallest particle of negative charge is called the electron. In some substances, called conductors, there are many " free " electrons, easily detachable from atoms; in such substances the electric current density /, defined as the time rate of flow of electric charge per unit area normal to the lines of flow, is proportional to the electric intensity (Ohm's law); thus , J = lE, where the coefficient of proportionality is the conductivity of the substance. By definition, the direction of the electric current coincides with the direction of moving positive charge and is opposite to the direction of moving negative charge. The electromotive force V (or the " voltage ") acting along a path joining points P and Q is defined as the line integral of the electric intensity V : Blackie & Son Limited (1930). E s ds. (1-1) 60 FUNDAMENTAL ELECTROMAGNETIC EQUATIONS <\i Tahi.k I Name of Quantity Sym-■ bol Name of Unit Dimensional Equivalent 1 .CMgtll — meter — Mass — kilogram — Time t second 1— Energy — joule volt-coulomb, newton- meter Power — watt joule per second 1 'i iree — newton joule per meter Electric charge, electric dis- ? coulomb ampere-second placement Displacement density D coulomb per meter2 Electric current I ampere coulomb per second Current density J ampere per meter2 Electromotive force V volt joule per coulomb Electric intensity E volt per meter newton per coulomb Impedance (electric) Z,K oh in volt per ampere Admittance (electric) Y,M mho ampere per volt 1 nductance L henry ohm-second Permeability henry per meter Capacitance C farad Dielectric constant e farad per meter Conductivity S mho per meter Magnetomotive force U ampere Magnetic intensity H ampere per meter Magnetic flux é weber volt-second Magnetic charge m weber volt-second Magnetic flux density B weber per meter Magnetic current K1 volt Magnetic current density M1 volt per meter2 1 It will be clear from the context whether K stands for magnetic current or impedance. 2 It will be clear from the context whether M designates magnetic current density or admittance. Thus the electromotive force represents the work done by £ on a unit positive charge moving along PQ and the work done on a charge q is Vq. The electromotive force is not a true force in the usual mechanical sense.* Consider now a homogeneous conducting rod of length / and let its cross- *The electromotive force may be regarded as the generalized force and the electric charge as the generalized coordinate in Lagrange's sense. 62 kif.c'i K< )IV1.\< ;ni . i K waves OtAI1. 4 gei iimi lir .v; I hen I In- in ri ii 11 / in the mil i:; S'J, llu- electromotive force \r, //',', arid therefore I=GF, G = ^f; V=RI, i? = ^. The coefficients of proportionality R and G are called respectively the resistance and the conductance of the rod. The work done by V per second is Fq/tox Vl = GV2 watts; this work appears as heat. The electric power dissipated in heat per unit volume is evidently JE — gE2. The magnitudes of the volt and the ampere have been chosen to suit the most common experience. Thus the voltages supplied to the homes for lighting and cooking purposes are between 110 and 120 volts; the current through a 100 watt electric bulb is about 0.9 ampere and the resistance of the bulb is about 122 ohms. For the mechanical units, similarly, the weight of 1 kilogram is about 9.8 newtons; the potential energy of a kilogram-mass 1 meter above ground is 9.8 joules; and the kinetic energy of 1 kilogram moving with a velocity equal to 1 meter per second is half a joule. A perfect dielectric is a medium possessing no detachable electric particles; in such a medium g = 0. Vacuum is the only physical example of a perfect dielectric; in the first approximation, however, many other media may be treated as perfect dielectrics. Consider now the field produced by electric charges placed in a perfect dielectric. If a neutral conductor is introduced in the field, the free electrons will move under the influence of E until inside the conductor the electric intensity of the separated or ' induced " charges becomes equal to — E. At the surface of the conductor the total tangential component of the electric intensity must be zero or the electrons would still be moving; on the other hand, the normal component will be different from zero. In order to explore this " electrostatic induction " quantitatively we may take a conductor formed by two separable thin flat discs of small area and insert it at some point in the field. The discs are then separated, removed from the field, and the charges on them measured. It will be found that these charges are equal and opposite and that their magnitudes are proportional to the product of the electric intensity, the area of the discs, and the cosine of the angle formed by the normal to the discs with some fixed direction. The maximum charge per unit area is called the displacement density D at the point in question. Generally the components of D are linear functions of the components of E and the directions of D and E are different; but in isotropic media D and E are in the same direction and D = eE. FUNDAMENTAL ELECTR< MAGNETIC EQ1IATIONS 63 The coefficient: of proportionality < is called the dielectric constant. In this book we are'concerned only with isotropic media. For example, consider two parallel conducting plates with equal and opposite charges. If the distance between the plates is small compared with the dimensions of the plates, the electric field between the plates is nearly uniform except near the edges and electric lines (lines tangential + ++ + + + ++ +>++ + + -M4 Fig. 4.1. Electric lines of force between the plates of a capacitor. In E) look like those in Fig. 4.1. Thus experience indicates that between two infinite uniformly charged plates the field would be uniform and the charges on the two conducting plates introduced in the field would separate as shown in Fig. 4.2. The displacement density is found to be equal to the electric charge per unit area on the positively charged plate. + + -i- + + + + + + + +■»+++ + ++ ++ + + + + + Pig. 4.2. Electrostatic induction in the field between FlQ. 4.3. Concentric spheres, two equally and oppositely charged planes. In his original experiments on electrostatic induction Faraday employed two metal spheres (Fig. 4.3). He placed a fixed charge q on one sphere and then enclosed it within the other. After connecting the outer sphere to ground, he measured the charge remaining on the sphere. He found that this charge is equal and opposite to the charge on the inner sphere regardless of the dimensions of the spheres and the medium between them. In the case of concentric spheres electric lines of force are radial and there- ii i ( i k< >m.\i;ni-:i u WAVES Chap. I Imr the infill di'/>/,iccntciil through liny sphere concentric with the sphere containing y is dS (1-2) The voltage is found to be proportional to q; hence D is proportional to E. In the region between concentric spheres we have then Dr = 4irr Er = (1-3) where r is the distance from the center of the inner sphere. The voltage between the spheres is J a Aire \a b) The ratio q/V is called the capacitance; hence the capacitance of two concentric spheres is itreab b — a When the outer sphere is removed to infinity, then C = 4wea. Let the outer sphere be infinite and the inner vamshingly small. Then any other " point charge " q will not affect the field of q and the force exerted by one charge on the other is t F - 4xer2 This is Coulomb's law. In free space e is approximately equal to (l/36ir)10~9 farads per meter (8.854 X 10-12); thus small charges produce large electric intensities. By direct integration it may be shown that for a point charge and therefore for any distribution of charge the displacement through a closed surface is equal to the enclosed charge so that (2) is true regardless of the shape of (S). An electric current exerts a force on each end of a compass needle and thus is surrounded by a magiieticfield. In the case of two coaxial cylindrical sheets or two plane sheets, carrying equal and oppositely directed steady currents (Figs. 4.4^ and 4.4£), the field is confined to the region between the cylinders or the planes. In the first case magnetic lines (that is, lines tangential to the forces on the ends of the compass needle) are circles coaxial with the cylinders and in the second case they are straight lines parallel to the planes and perpendicular to the direction of current flow. The FUNDAMENTAL ELECTROMAGNETIC EQUATIONS 65 hi rows in Fig. 4.4 indicate the direction of the force on the north-seeking hi I he " positive " end of the needle; -|-/ is supposed to be Rowing toward i he reader. Between two plane current sheets the magnetic intensity H miiloi'iu; it depends on the linear current density in the sheets but not mii i he distance between them. This linear current density is taken as the +i Via. 4.4. The magnetic intensity between coaxial cylinders Eind parallel planes. measure of H; hence the unit of H is the ampere per meter. Similarly I he magnetic field is uniform inside a closely wound cylindrical coil; the magnetic intensity is independent of the shape and si/.e of the cross-section of the cylinder, is parallel to the generators of the cylinder, and is equal to the circulating current per unit length of the coil. In this case Jrl is also equal to the number of " ampere-turns," that is, to the product of the current in the wire and the number of turns per unit length of the coil. FlG- 4-s- A wire W It is an experimental fact, discovered by Faraday, that a voltage exists across the terminals of" a loop (Fig. 4.5) in a varying magnetic field. For a small loop this voltage is proportional to the cosine of the angle between H and the axis of the loop; the maximum voltage is proportional to the product of the area S of the loop and the time derivative of H; the coefficient of proportionality n depends on the medium and is called the permeability. The time integral of the voltage is called the magnetic flux or the magnetic displacement $ through the loop; thus for a small locp $ = SB, B = pH, (1-4) where B is called the magnetic flux density. The magnetic flux through any surface is defined as the surface integral of the normal component of B Bn dS. (1-5) ELECTROMAGNETIC WAVES The following time derivutives ot' at' Chapi j (I 61 are called respectively the magnetic current and the magnetic current density, The first law of electromagnetic induction (Faraday's law) may then In expressed as follows V = -K, or f E,ds = - ff MH dS, (1-7) where the line integral is taken round the closed curve forming the edge of a surface ( (1-15) j E. ds - // (gEn + . charge in a magnetic field is assumed hi be Urn. The dimensions are col rect but a numerical factor might have been included. The torque on n magnetic doublet (two charges m, — »/ separated by distance /) of mmw.nl ml placed normally to the lines of force is then ilml. In Chapter G we shall find that with the above definitions in mind the field of a magnetic doublet is the same as that of an elementary electric current loop provided ml — p.SI, where / is the current and S is the area of the loop. The torque on the loop whose axis is normal to the magnetic lines would seem to be tiHSI-= BSI; this happens to be the case and we have a machinery for replacing in calculations circulating electric currents by equivalent magnetic doublets. Coulomb's law for the force between magnetic charges as above defined is evidently m^niil^-nyr1. Maxwell's equations in the form in which we have expressed them possess considerable symmetry; E and //correspond to each other, the first being measured in volts per meter and the second in amperes per meter; D and li correspond to each other, the first being measured in ampere-seconds per square meter and the second in volt-seconds per square meter; electric and magnetic currents correspond to each other, the first being measured in amperes and the second in volts. In literature one finds arguments tc the effect that " physically " E and B (and D and H) are similar and that B is more " basic " than H. All such arguments seem sterile since electric and magnetic quantities are physically different; whatever similarity there is comes from the equations. 4.2. Impressed Forces In equations (1-15) and (1—16) electric generators are represented by the current densities / and M whose values are supposed to be given. Of course, in order to obtain given values of the generator currents we must have properly distributed impressed inte?isities which sustain these currents against the forces of the electromagnetic field. These impressed intensities are not included in E and H in equations (1-15) and (1-16); they are equal and opposite to the field intensities against which the charges in the generator have to be moved. For example, in an electrolytic cell there exist contact forces tending to separate positive and negative charges. In Fig. 4.9 the lightly shaded region represents an electrolyte and the heavily shaded regions are greatly Fig. 4.9. A diagram of an electrolytic cell. FUNDAMENTAL ELECTROMAGNETIC EQUATIONS 71 lungnifiecl regions of contact between the electrolyte and I he metal plates . /.....1 /), The contael forces nmvc I he negative charge to the metal plates until the electromotive forces of the separated charges just balance the oniaet electromotive forces. Contact lours are different for different metals and when two plates A and D are submerged there may exist a net impressed electromotive force between A and D (Fig. 4.10) which is equal l+ + ++'-f+-*--*-t--l--«- + + + -t--l--l-+ + ++ -»--4--H- + I :.;.4.KI. The electromotive force between the metal plates of the cell due to the electric charges separated within it is the same for any path either inside or outside the cell. Inside the cell diere are also contact forces acting on the charges. and opposite to the voltage Vaqd produced by the separated charges. The voltage Fj.pd = — Vaqd and the total electromotive force of the field round the closed path is zero, which is consistent with (1-15) since there is no magnetic current through the circuit. If A and D are connected by a conducting wire, the electric intensity of the field of separated charges will move the electrons in the wire and a current will be produced. —- As another example let us take a j fixed pair of wires AB and CD and a p/ sliding wire MN of length / equal to the separation between AB and CD _\_ (Fig. 4.11). Let there be a uniform A V___- magnetic field H perpendicular to pIGi 4 ii_ a conducting wire sliding alons the plane of the paper and directed a pair of parallel wires in a magnetic field, toward the reader. Let the velocity of MTV be it. Consider a circuit MNPM of which MN is the only moving part. In time dt the magnetic flux through this circuit changes by Blv dt and the electromotive force in the circuit due to the motion of MN is Vi = -Blv. This force is independent of the fixed part of the circuit and hence resides in the moving wire MN; for this reason it is called the motional electromotive force. The relative directions of H, v and the motional E are shown in Fig. 4.12. More generally the force on a charge moving with velocity v i:LECTR( (MAGNETIC WAVES in a magnetic field is F = go X B. In subsequent work we shall assume as given either impressed or genei ator currents or impressed electromotive forces, according to circum stances. In a pair of parallel wires (Fig. 4.13) for instance, we'may star) with a given current P flowing from B to A, determine the charge ami Fig. 4.12. The relative directions of the motional electric intensity, the magnetic intensity, and the velocity. Fic. 4.13. A diagram illustrating the simplified conception of an electric generator in wave theory. current distribution in the wires, the field due to these charges and currents and hence the electromotive force ^of this field acting from A to B. The impressed electromotive force needed to sustain P against V\s Vi = — V. Or we may start with a given V% and determine the corresponding I*. 4.3. Currents across a Closed Surface The total electric and magnetic currents across a closed surface vanish. This theorem follows immediately from (1-15) when these equations are assumed to hold independently of the choice of (S). We simply draw a closed curve on a closed surface (S), and apply (1-15) to each part of the surface. Thus we have n»Bjtds=-nM»ds=-K> (3-1) where I and K are the impressed currents flowing out of the volume bounded by (S). In perfect dielectrics g = 0. Substituting in (1) and integrating with respect to t, we have J J eEndS = J j DndS= -f Idt=q, (3-2) FUNDAMENTAL ELECTROMAGNETIC EQUATIONS 73 f j pi/,, dS = f j' Bn dS - - J Kdt - m, (3-2) where q and m are the electric and magnetic charges inside (S) at time t. 1.1. Differentia! Equations of Electromagnetic Induction and Boundary Conditions Applying the integral equations (1-15) to an infinitely small loop and using the definition of the curl of a vector, we obtain curlE= -y.^--M, curltf = gE + *^+J. (4-1) Similarly, equations (1-16) for harmonic fields become curl E = -iapH - M, curl H = (g + icoe)E + J. (4-2) At a boundary (S) between two media the above equations are not necessarily satisfied because E and H may be discontinuous. A connection between the fields on opposite sides of (S) is obtained from the integral Fig. 4.14. A cross-section of a boundary between two media and a rectangle having two sides parallel to this boundary and the other two sides vanishingly small. equations. Thus, assuming that all variables and constants in these equations are finite, and applying the equations to a typical rectangle with two sides, one in each medium, close to and parallel to (S), such as the rectangle A'B'B"A" in Fig. 4.14, we have £,' = #', H't=H['. (4-3) Hence the tangential components of E and H are continuous at the interface of two media. Since the circulation of the tangential component of H per unit area is the normal component of /, the latter is continuous across S. The normal component of M is also continuous and we have /i = 7*, Ml - Mil. (4-4) For harmonic fields in source-free regions, these equations become (/ + i^')E'n = Or" + ft*")**. M = (4-5) /1 ELECTRl MAGNETIC WAVES Chap, -l FUNDAMENTAL ELECTROMAGNETIC EQl 1ATH )NS 75 For main lirlil:; ill pel In r i licit* 11 n :,, [lie- i cilltlilh ills air c ■*■-■* If > ///,: = n»HU. (4-6) In perfect conductors {g<= oo) the electric intensity is zero for finite currents and the condition at the boundary is Et = 0, or Hn = 0. (4-7) The conception of perfect conductors is valuable chiefly because it helps to simplify mathematical calculations and to provide approximations to solutions of problems involving good conductors. In the future we shall assume all perfect conductors to be infinitely thin sheets. For reasons that will become evident later we may describe perfectly conducting sheets as sheets of zero impedance. A sheet of infinite impedance is defined by the boundary conditions complementary to (7)j that is by Ht = 0, or = 0. (4-8) Such a sheet can be pictured as having an infinite permeability and it is useful as an auxiliary concept for simplifying certain problems. 4.5. Conditions in the Vicinity of a Current Sheet Another auxiliary concept is that of a current sheet, defined as an infinitely thin sheet carrying finite current per unit length normal to the lines of flow. Let us suppose that Fig. 4.14 shows a cross-section of an electric current sheet whose linear current density f is normal to the plane of the figure and is directed to the reader. Applying (1—15) to the rectangle A' B'B" A' obt am (5-1) The positive directions of the current density, the tangential component of H and the normal to the sheet are assumed to form a right-handed triad. Similarly for a magnetic current sheet of density M, we have E[ - E[' = -M, H't = Hi' (5-2) The discontinuities in the tangential components of the field intensities imply discontinuities in the normal components of the field current densities. Imagining a pill box with its broad faces infinitely close and parallel to the electric current sheet on its opposite sides and then calculating the current into the pill box and out of it, we have Ji! - J'n= -div' /, M'n = Mil. (5-3) Similarly for the magnetic current sheet, we obtain M'J -M'n= -div' M, Ji = Ji'. (5-4) ■Id. Conditions in tln} Vicinity of [,'tiicttr Current Vilumvuts These conditions arc obtained directly from (1.-7) and (1-9). Thus in the immediate vicinity of an infinitely thin electric current filament /, and magnetic current filament K, we have *V=-^, (6-1) Irrp Fic. 4.15. A cross-section of the wave-front, that is, the boundary separating a field from field free space. assuming that the filaments coincide with the z-axis. '1.7. Moving Surface Discontinuities We shall now consider the case in which the time derivatives of E and H .in infinite as, for example, at a wavefront defined as the boundary between a finite moving field and a field-free space. Without loss of generality we may ignore the impressed currents. I'iii- reasons of simplification our discussion is restricted to homogeneous perfect dielectrics. Since there is no surface charge on the wavefront (S) (Fig. 4.15), the normal components of the electric and magnetic displacement densities are continuous; in homogeneous media this means that the normal components of the electric and magnetic intensities are also continuous. Since the field is identically zero on one side of the wavefront, the normal components vanish and E and H are tangential to the wavefront. Let us assume that the positive directions of E, H and the velocity v of the field (normal to S) form a right-handed triplet and consider a rec-i angle A'B'B"A" (Fig. 4.15) in which A'B' is normal to H. The magnetic displacement through the rectangle increases at a rate fiHvl, where / is the length of A'B'. This must be equal to the electromotive force El around the rectangle, where E is the electric intensity along A'B'; in view of our convention regarding the positive directions of E, H, v, we have E = jivH. Similarly, if we choose A'B' in the direction of H, we obtain H = evE. These equations connect H and the component of E normal to it. If there were a residual component of E, then, starting with this component and proceeding as above, we should have to acknowledge the existence of H normal to it which would be inconsistent with the original assumption that we have started with the total H. Multiplying and dividing the above equations we have 1 jltV =1, v = ± ± T)H, 7] = 76 emu 11«jmac;ni.'iit waves Vhav, -i ITINHAMENTAE ELECTROMAGNETIC EQUATIONS 77 The velocity of the wnvefront unci the ratio E/Hon it arc thus fixed by the properties of the medium; this velocity will be called the characteristic velocity and the ratio j> = E/H the intrinsic impedance. In free space m have approximately* v{] = 3 X 10s m/sec, rj0 = 120*-377 ohms; in pore water rv = \ X 3G8 m/sec, ?? ^ 42 ohms. Since £ is positive when the product of // and tJ is positive, the field is moving in the direction in which a right-h anded screw would advance when turned from E toH through 90°. Depending upon the relative directions of E and H, the field is either moving into the field-free space or receding from it. Imagine for instance a uniform " field slice " (Fig. 4.16) in which E and 11 have constant values between two parallel planes. As we have shown, (0,0) Fig. 4.16. The cross-sec don of a " field slice. " Fig. 4.17. An electric current sheet generating a field slice. such a slice cannot remain stationary but must move with the characteristic velocity. That such a moving field could conceivably be generated may be seen as follows. Imagine an infinite plane sheet containing uniformly distributed equal and opposite charges and let a constant impressed intensity E? set these charges in motion at the instant t = 0, At this moment magnetic intensities must appear on each side of the current sheet (Fig. 4.17) and H+ = -H~ = where f is the current density in the sheet. For the electric intensities on the two sides we have = E~ = —E\ Considering the relative directions of E and H, we find that on both sides the field will be propagated away from the sheet. Between the two wavefronts the field remains uniform until Ei ceases to operate, at the instant / = T, let us say. Thereafter we shall have two field slices of thickness / = vT moving in opposite directions. The work per unit area performed by E{ in sustaining p during the interval (0,7*) is E\J{T and the energy contributed to the field is carried away by the field slices. Similar but spherical field shells expanding outwards are created whenever an electric particle is accelerated or decelerated. * The subscript zero is used to indicate specifically that the constants refer to free space. 4.8. Energy Theorems Starting with the fundamental equations of electromagnetic induction (4-1), let us take the scalar product of the first equation and H and subtract from it the scalar product of the second equation and E m _ be H-curl E-E- curl H = —M-H — E - J — gE2 — p,H■ dt *E- dt Integrating over a volume (r) bounded by a closed surface (S), using equa-(ions (1.8-5) and then (1.3-1), and rearranging the terms we obtain -III E-Ii- fflMHJ* - SSL sE°* +s III »* *+1 //£>hH'dr*$L{EXH)-M (8-1) As usual the positive normal n to ($} 'is directed outwards. Integrating (1) with respect to / in the interval (- «V) and assuming that originally the space was field-free, we obtain " f & f f f / + M-H)dr = f dt f f f gE2dr +j J J &E2 + ^H2)dr + f dt J (EXH)ndS. (8-2) The left side in (1) is the rate at which work is done by the impressed forces against the forces of the field in sustaining the impressed currents and the left side in (2) is the total work performed by the impressed forces up to the instant t. In accordance with the principle of conservation of energy we say that this work appears as electromagnetic energy and we explain the various terms as follows. The first term on the right of (1) is the rate at which electric energy is converted into heat and the first term in (2) is the total energy so converted.* The second term in (1) may be interpreted as the time rate of change of the electric energy within (S) and the third term as the time rate of change of magnetic energy; the corresponding terms in (2) represent the electric and magnetic energies within {S) at the instant t. The last term in (1), being a surface integral, is interpreted as the rime rate of energy flow across (S); similarly the last * Since gE is the conduction current density gE dt is the electric charge moving in response to E; consequently gE2 dt is the work done by the field and must appear as some other form of energy. This form is heat and gE2 is the power conversion pe' unit volume. 78 KI.M TUI MAGNETIC WAVES CiiAť. 4 term in (2) is interpreted as the total How of energy across (.V) up ftt the instant /. In this interpretation we assume that the electromagnetic energies arc distributed throughout the field just as the energy dissipated in heat is known to be distributed. In conformity with (2) the volume densities of electric and magnetic energies are assumed to be Sc = hE2, % = l„/72. (8-3) It is consistent with (1) and (2) to interpret the Poynting vector P = E X // (8-4) as a vector representing the time rate of energy flow per unit area. Certainly the surface integral of this vector over a closed surface represents Fig. 4.18 The field of a magnetic doublet and an electric charge at the center of the doublet. (0,0) Fig. 4.19. The cross-section of a moving field slice and a " pill-box" with the flat faces parallel to the wave-front at vanishingly small distances from it. the difference between the energy contributed to the field inside (S) and the energy accounted for within (S). On the other hand it is also true that the value of this integral remains the same if the curl of an arbitrary function is added to P. Furthermore, in the case of a magnet and an electric charge at its center (Fig. 4.18), P does not vanish; and yet in this case we are averse to assuming an actual flow of energy even though such an assumption is permissible. In another instance, however, the interpretation of P as power flow per unit area is attractive. Consider a uniform field slice, moving in a perfect dielectric, and a pill box with its broad surfaces parallel to and on opposite sides of the wavefront (Fig. 4,19). In this case / = M = 0, g = 0, and (1) becomes EHS = &E2 + %pH*)vS, (8-5) M IN i >am i'.ntai, EI.ECTRl >ma< JNETIC EOJ lath >ns where A' is the area of each of the lun.nl surfaces of the pill bqX. The v ret or /' is in the direct inn of the advancing wave and it seem/as if the in m \ associated with the wave were actually traveling with velocity v, whit h is reasonable since I he wave itself is advancing with this velocity. II in the volume bounded by (o1) there are no sources of energy, the energy dissipated in heat should enter the volume ac/bss (S). Consider lor instance a direct current I in a cylindrical wire of/radius a. If E and H .ue the components tangential to the wire, then the energy flowing into a section of length / in time / is (2iraH)(lE)t = Vít, (8-6) where V is the voltage along the surface.of the wire. Since // is the charge which has passed through the wire in jíme /, Vit is indeed the work done by i In forces of the field. We shall now derive another energy theorem, particularly suitable to harmonic fields. Multiplying scalarly the first equation of the set (4-2) 1. v //*, the conjugate of thysecond by E, and subtracting, we obtain //* • curl E - E ■ cmyíl* = -M ■ U* - E ■ J* - gE ■ E* + ioěEE* - wpH< H*. Integrating this over a volume (t) bounded by a closed surface (S), using (1.8-5) and (1.3-1), rearranging the terms and dividing by two, we have -I f f f {E.J* + M'H*)dr = % f f f gEE*dr \m J J ■ H* dr - \iu Jff iB ■ £* dr r\ f f {EXll*)ndS. (8-7) The real part of the expression on the left of this equation is the average power spent by the impressed forces in sustaining the field. Some of this power is transformed into heat and the precise amount is given by the first term on the right; the rest flows out of the volume across (S) and the amount is represented by the real part of the last term. The second and third terms on the right are equal to the product of 2ío and the difference between the average magnetic and electric power stored inside (S). The last term in (7) is called the complex power flow across (S) and is designated by Ý + *=$ jj {ExH*)«dS. (8-8) KU ELECTROMAGNETIC WAVES' I'llAI', 4 PI "NDAMENTAL ELECTROMAGNETIC EQUATIONS HI The vector P «■ %E x H* t9 the complex Poynting uector\ its retil purl) It] the average power (low per unit area. If (<$') is a perfect conductor, the tangential component of /',' vanishes hence there is no flow of energy across (A') and /' is parallel to the sin I'm < A closed perfectly conducting sheet separates space into two electronuiu, netically independent regions. Similar complete separation is afforded by n closed surface of infinite impedance (Ht = 0). In the physical world metals are in some respects good approximations to perfect conductor! but there are no good approximations to infinite impedance sheets exeepl at zero frequency (or nearly zero) when substances with extremely high permeability act as high impedance media. Only the tangential components of E and H contribute to SI'. If u and u are orthogonal coordinates on (S) and if u, v, n form a right-handed triplet of directions, then * = (EJf* - EvHt) dS. Introducing the ratios Zuv Eu y H,±' we obtain If now then Zuv ~' \ J J{ZvuHuHl + ZmHvll%) dS. hf fzn(HJI* +. HVH*) dS. (8-9) (8-10)| (8-11 (8-12) (8-131 In this case Zn is called the impedance normal to the surface (S). Consider now a conducting surface of thickness t. The linear current density / is equal to //, where J is the surface current density. If g is the conductivity, then / = gE and consequently / = gtE. If / approache zero and g increases so that the product G = gl remains constant, we have / = G£, E = Rf, (8-14) where G and R are called respectively the surface conductance and the surface resistance of the sheet. More generally we define the surface admittance Ys and the surface impedance Z, by equations similar to the above' / = YSE, E = Z J, where the constants of proportionality are complex. (8-15) Imagine now two infinitely close sheets, one with infinite surface imped- .....■ and the other with finite impedance Z„. Since the component of H tangential to an infinite impedance sheet is zero, the component of H tan-in r11i.il to thai side of the finite impedance sheet which is not adjacent io i lie infinite impedance sheet is equal to the linear current density /; I hus Hu — J Vs Hv — J u (8-16) mid the impedance normal to the combination of the two sheets* is equal i11 i lie surface impedance. Equation (13) then becomes * = */ fz'^»/i+ Aft)dS- (8-17) <\.>) Secondary Electromagnetic Constants The conductivity g, the dielectric constant e, and the permeability n .ni i he primary electromagnetic constants of the medium in the sense that iIm y appear directly in the formulation of the electromagnetic equations. In equations (4-2) the terms on the left are three dimensional derivatives ' responding to ordinary derivatives in one-dimensional problems. The transmission line equations (2.10-3) represent a special case of Maxwell's . i |n.itions and the terminology of the former may be extended to the latter. Thus we may call mil the distributed series impedance of the medium and (i; I tat) the distributed shunt admittance. The constants n, g, e are m ,|actively the distributed series inductance, shunt conductance, shunt i ap'icitance. In wave theory the important constants are not the primary iiinstants. Thus in transmission line theory two secondary constants an- introduced: the propagation constant V and the characteristic impedance K, the first being defined as the square root of the product of the series Impedance and the shunt admittance and the second as the square root of {heir ratio. Likewise, in three dimensional theory the important constants lire the intrinsic propagation constant e is negligible compared with g (except at optical frequencies) and both a and 17 are on the bisector. In general a and rj are complex quantities cr = a + iß, (9-2) The quantities a and are respectively the attenuation constant and the phase constant of the medium; £R. and ®C are the intrinsic resistance and reactance. Bearing in mind the wave terminology introduced in section 2.4 we have the following expressions for perfect dielectrics a = iß, ß — wV 1 2tt v = ->= , X = — : Vpt ß 2tt X n V (9-3) 1_ rjv The phase velocity V as defined by these equations is called the characteristic velocity of the medium. For some electromagnetic waves the phase velocity is equal to this characteristic velocity; but in general the characteristic velocity is only one factor in determining the actual phase velocity of a wave. Similarly the wavelength as defined above is called the characteristic wavelength; it is one of the factors determining the actual wavelength. For free space we have the following numerical values of various constants 376.7 es* 377 120t ohms, v0 = = 2.998 X 108 3 X 10s meters/second, Vo = 2.654 X 1(T3 I I Po 1 20tt * At optical frequencies e may be negative mhc (9-4) ••UNI)AMENTAL ELECTR<>MAd'NI.1'Ic i•; (9-12) -1/4 H M ]$] #§ | = I - - tan-1 Q, a = Vig (l - II + iQ* + ^ - Jgf - + ■■■), hi IN DAMKNTAI. ITiXTKOMAUNETIC EQUATIONS 85 fi - v£g (i + iff + i(?a - tVC?3 - i S g'G* + *W + wV^M -f- 1 4 + •)• (1 (9-12) (cont'd.) 8S3 + K_ I ft, + _3_____> The last two series are appropriate for Q > 1 and the preceding pair for Q < 1. While there exist rapidly converging series expansions in the neighborhood of 0 = 1 it is more practicable in this range to compute directly from the first four equations. The first terms of the last two series are first approximations for quasi -diclectrics (Q 1) and the first two terms in the preceding pair are first approximations for quasi-conductors (Q 1). The frequency and free-\pace wavelength for which Q — 1 are determined from (10) or still more conveniently from 1.8 X 10llV — ■■ *-y- » r (9-13) The following table illustrates the extent to which media may differ from each other electromagnetically, Mica, ^l.lX 10"14, 5.7 < er< 7, 2.8 X 10~52fie (10-3) in nondiaaipaclve media we have vi 4- rj 4- n = -ß2 = iL* X* ' (10-4) Thus the laws of induction impose only one condition on the three propagation contains. Two of these constants are controlled by the distribution of sources producing the field. If this distribution is uniform in planes parallel to the .vy-plane, for example, we should expect the field to be similarly uniform and Tx = r„ = 0; then ilu propagation consrani in the ss-direction is equal to the intrinsic propagation eon- ■ I.ml. Consider a typical plane through the origin x cos A + y cos B -\- z cos C — 0, (10-5) where cos A, cos B and cos C are the direction cosines of a normal to the plane. The distance s from this plane along the normal passing through the origin may be expressed as s = .v cos A + y cos B + z cos C. I lence if the field is uniform in planes parallel to (5), we have y _ «— trs g—a(x coh A 4- y cob B + z com C) The propagation constants along the coordinate axes are Tx = a cos A, Yy = cr cos B, T% = cr cos C, and for uniform plane waves in nondissipative media, we have j3x = 8 cos A, j8j = j8 cos B, ft = jS cos C; \x — X sec A, Xj, = X sec B} \e = X sec C; vx = v sec A, Dy — v sec B, v3 = v sec C; (10-6) (10-7) (10-8) 1 1 1 1 x2 + x2 + xi x2' 1 1 1 1 m k »I v- Thus in the case of uniform plane waves the phase velocities in various directions are never smaller than the characteristic phase velocity and the phase constants Bx, /3„, 8Z never greater than B. If the propagation constants Tx, Tv, Tt are all imaginary, then ffi + Pi + 0| = /32. (10-9) None of the phase constants is greater than the characteristic phase constant and we can identify planes in which the field is uniform as those normal to the straight line whose direction cosines are . ßx j, ßv r ft cos A = — , cos B = — , cos 6 = — . ß ß ß (10-10) But if the phase constant in some direction is greater than 0, then there must be a real propagation constant in some perpendicular direction. Thus let j8s > fi; then HK I.I IX I K( >MAC;NI. I IC WAVES Chap, -i RIND A M KNTAI. I.I .T.C "I K< )M A( i NI'.' I' K' EQUATIONS XV r* + Tl = ft - ft" > 0. I'c.1- instance if (i, fiV 1 and therefore t; v/\fl, and if Tv = 0, then r,"j}» 2ir/X, l\X = 2ir. The attenuation in the .v-dircction per characteristic wavelength is about 6.28 nepers or 54.6 decibels; the field intensity is reduced to 0.00187 of its value if x is increased by X. In the general case of complex propagation constants we have y = e-<."xx+"yv-i-'*ii!) e-HPzx+iiyy+0*) From (4) we obtain (for nondissipative media) 05 + j| + M -al-al- «$ = /3s, agx + ay&y + a& = 0. (10-11) The second equation shows that the equiamplitude planes axx + ctj,j + a„z = constant are perpendicular to the equiphase planes fix* + fiy}' + PzZ = constant. Thus in nondissipative media equiamplitude planes either coincide with or are normal to equiphase planes. In the former case the waves are uniform on equiphase planes in the sense that E and H each have constant values at all points of a given equiphase plane at a given instant; in the other case the amplitude varies exponentially, the fastest variation being in the direction given by the direction components In dissipative media the second equation of the set (11) becomes and equiamplitude planes are no longer perpendicular to equiphase planes. The foregoing general conclusions concerning waves of exponential type (2) have a broader significance than appears at first sight. The constant Tx represents die relative rate of change of V in the ^-direction and we have i V dx2' 1 bV V dx r2 The second equation is also satisfied by — Vx and in it V may be a sum of two exponential terms, one proportional to e~ Tx" and the other to er**. If the wave function is not exponential we may still define rx by the second of the above equations; Tv and I\. may be defined similarly. If these quantities vary slowly from point to point, the solutions of the wave equation will be approximately exponential, and the above properties of exponential waves will be applicable in sufficiently small regions. Some broad conclusions may be drawn with regard to waves at an interface between two homogeneous media whose intrinsic propagation constants are of different orders of magnitude, as is the case for conductors and dielectrics. Consider a plane interface (the Jcy-plane) between air (substantially free space) above the plane and some conductor below the plane (Fig. 4.20). For an exponential wave of type (2) the propagation constants Tx> Ty in directions parallel to the boundary must be the same in both media in order that the boundary conditions may be satisfied at all points of the air-conductor interface. This is evidently so if V represents a component of cidier /'.' or // parallel to the boundary; in this instance /' is continuous across the boundary and the continuity cannot be satisfied at all points unless l\, l\ are the mine on both sides of the boundary. The same argument applies to the normal com- conductor * zn Fig. 4.20. A plane boundary between two semi-infinite media, ponents of the current densities (g + i(x>t)E2 and ?w^/7j. Thus we have r;+ F|+ %= Subtracting the first from the second, we have Tl = cr2 + $ + IS We have seen that the propagation constant for a conductor is always much larger than that for free space. Tip is the propagation constant in free space in the direction normal to the interface; it may be comparable to p0 if the direction of the wave is nearly normal to the conductor, or much smaller than j3o if the wave direction is nearly parallel to the conductor. Hence in the conductor the propagation constant normal to the interface is substantially equal to the intrinsic propagation constant at all engineering frequencies. Since the current density normal to the interface is continuous, we have hence the normal component of E in the conductor is negligibly small compared with the normal component of E in the air. Even at moderately high frequencies the attenuation constant in the conductor is large and the field becomes quite small at rather small distances from the interface. For frequencies of 103, 106, and 109 cycles per second the attenuation constant for copper is respectively 0.478,15.1,478 nepers per millimeter; or 4.15, 131,4150 decibels per millimeter. Each 20 decibels represents a 10 to 1 intensity ratio; thus at a million cycles the field intensity one millimeter from the surface of the conductor is less than one millionth of the intensity at the surface. Except at low frequencies the fields are confined largely to thin skins of conductors. The current density at the surface of the conductor is gEt and elsewhere it isgEte~", where z is the normal distance from the surface. Then the total current per unit length normal to the lines of flow is 0 gEic ■■ďz = - Et m -£t. (10-12) On the other hand, Ht = j and therefore the impedance normal to the interface is equal to the intrinsic impedance of the conductor. The conductor may be replaced I I IX I ly the conductor. The resistance normal to a sufficiently thick metal plate may be expressed as 11 *A = -; (10-13) a = ^ = hence this resistance is equal to the d-c resistance of a plate of thickness /, defined by the reciprocal of the attenuation constant This thickness is called the " skin depth "5 but the term should not be interpreted as meaning that the rest of the conductor could be removed without changing its a-c resistance. The attenuation through the skin depth is only 1 neper and the field is reduced to only 0.368 of its value on the surface. If the entire current were compelled to flow in the " skin " of thickness t, given by the above equation, the a-c resistance would be 8.6 per cent higher than the actual resistance. The field is reduced to one tenth of its value when the distance from the surface is 2.3 times the skin depth. It will follow from the equations of section 8.1 that the surface impedance of a conducting plate whose thickness / is small compared with the radius of curvature is jj coth at. 4.11. Polarization In electromagnetic wave theory the differences between various media are expressed by three primary constants g, e, p. Inasmuch as material media are regions of free space in which are imbedded various material and electric particles, one can expect that in the final analysis there is only one medium, this being free space, with g = 0, e = eo, M = Mo- An electromagnetic field acts on the electric particles of the medium; their spatial distribution and velocities are changed; and they act as secondary sources of the field. The macroscopic effect of these secondary sources is described by g, e — eo, /i — gfg, In wave theory we are not interested in physical explanations of the electromagnetic differences between various media, the constants g, e, jji are supposed to be known, and we are not concerned whether their values have been obtained experimentally or somehow computed; but the principle of replacing one medium by another with compensating secondary sources is sometimes useful and can profitably be examined. Before passing to generalities let us consider a few examples. Take a pair of concentric conducting spheres (Fig. 4.21). If the electric charge on the inner sphere is q, that on the outer (after being grounded) is —q, regardless of the dielectric between the spheres. By (1-3) the electric intensity is while in free space it would be Er The difference is 4xeor2 -gfj — eo) 4xeoer2 (11-D (11-2) (11-3) hi IN DAMENTAl. El .KCTR< )MA< ;NETIC EQUATIONS ')\ Fig. 4.21. Polarization of dielectrics. I In difference in intensities could be produced in free space by an electric charge ■ — (e«/e)| on the inner sphere and q[i — («u/e)] on the outer. If we postulate (Iii-mc- charges on (he surfaces of tin- dielectric adjacent In the spheres, we can account for the actual electric intensity on the assumption that the dielectric constant between the spheres is £o instead of e. In order to explain the postulated charges we may assume a reservoir of equal and opposite quantities of electricity in l lie dielectric so distributed as to render it neutral in ill, ib> hit nt' an electric force. Alter the inner .sphere has received an electric charge +q and the outer — q, an electric field (2) is established. Under the influence ■ if this field the electrified particles in the dielectric are displaced, negative particles toward the inner sphere and positive toward the outer. The total effect is to give rise to surface charges on the boundaries of the dielectric. These charges produce a field acting against the field (2), thus reducing die intensity to the value given by (1). The displacement density between the spheres is DT = *Er = e»ET + (e - ea)Er. (11-4) It differs from the displacement density that would have been produced by the same intensity in free space. The difference is called the polarization of the dielectric P = (e -€„)£. (11-5) As another example let us take a pair of conducting planes with a stratified dielectric between them (Fig. 4.22). If on the lower plane we have charge q and on the upper plane —q then D = q = tiEi m e2£2. (11-6) We can now say that the dielectric constant is ei everywhere between the conducting planes but that in the shaded region (Fig. 4.22) the medium has become polarized relative to the surrounding medium. The relative polarization P is defined by P = (ft - «i)£. (H-7) In the polarized region we now have D = ^E+P. (11-8) On the boundaries between the polarized and the unpolarized regions E and ei£ are discontinuous and since this discontinuity is no longer ascribed to a difference in dielectric constants, it must be explained by surface charges. At the upper boundary the discontinuity in eiE is ti(Ei — Ei); by (6) this is equal to (e2 — ei)£s and therefore to the polarization P. Thus the density of the surface charge on the upper boundary is P; similarly on the lower boundary the density is — P If e2 < ei, P is negative and the surface charges on the boundaries are of the same sign as those on the conductors nearest to them. ku'.itkoma<;nktk: wavks Ciiai'. 4 [f inatead of, stratified dielectric between the parallel plates we have ■ stratified conductor through winch an electric current of density J « flowing, wc Sh,.....Z exactly the results ok above with conductivities gl and Si in place of diel,,,, „ constants «a and e2 and with / in place of q. + ++ + + 4 + 4 + 4 + + ++ ♦+ + +■4 4++ + ++ + 44 + ++ ++ +4 + Fig. 4.22. The medium may be regarded as electrically homogeneous if we assume the existence of compensating charges. More generally let us consider a medium which is homogeneous except for an " island " (Fig. 4.23) and suppose that the island is source free. Widiin the island , , , we have curl e = -mii'% curl H = (g" + iae")E, (11-9) and at the boundary e', = e';, Hi = h"u ///: = m"<, (&' + i™')e'a = Qr" + »W')£'„'. (11-10) Since equations (9) can be written in the form , curl e = —io>p!h — m, curl//= (g' + ,W)£+ /, (11-11) Fig. 4.23. The cross-section of an " island " in an orhcr-wisehomogeneousmcdium. where / = is" ~ s')E + MjP - <■')£, M = itafo" - fi*)H, (11-12) it is theoretically possible to assume that the electromagnetic constants of the island are the same as those of the surrounding medium and that the field external to the island has induced in the latter the electric and magnetic currents given by (12). The island is said to be polarized relative to the external medium and /, M are called polarization currents. The latter act as impressed currents in addition to tiiose producing the field- I'l INI )AIY11'.NTAl, "l'K< (MM iN FLIC KQIIATN e)PEx; (12-6) dip dp dp dip dll. . . . . _ 3H, , . . d , „ . dEp „ dip dp dp dip (12-7) II I he field is circularly symmetric, that is, if it is independent of the ip-coordinate, linn ilie general set of six equations again breaks into two independent sets. In < vlindiieal coordinates we have = ip,pHz, —-2 - -r^ = + ^t)£v; (12-8) oz op dz op *~ = -(g+io}e)E„ X (p// ) = Or + ioi€)pEz, -r!—--zTTs" iofiHv. dz dp dp dz (12-9) In spherical coordinates we have d d —(sin 8 Ev) = — iwprsm 8 Hr, —-(rE9) = iccprHg, dd dr I" (rHt) - —'■ = Or + M^E*; or — (sin 8 Hv) = Or + >o°*)r sin & E" r (r^f) = _ (ir + io}e)rEg, dd dr d . „ , 3£r . „ - (^a) - — = -tLcprHy. dr d8 (12-10) (12-11) 1 f the field is independent of two coordinates, * and y let us say, then equations (1) heeome -(g+ive)Ex; (12-12) dE, ■ rr dfI» dEv = icop.Hx dll dz ff.«0 (g + iue)£y; (12-13) (12-14) dz £. = 0, We shall also have occasion to use the following sets, substantially the same as (4) and (5). If the field is independent of the x-coordinate, then -- = -tosp.ily, - = iwpHz, —---— = (g+iax)tx; (12-13) dz dy dy dz 96 dz &- ELECTROMAGNETIC WAVES . I. Impedors and Networks An impedor is any combination of conductors and dielectrics, with two accessible terminals (Fig. 2.11). It may be as simple in structure as a laboratory resistor or as complicated as an antenna. In the latter case i he impedor includes the wires of the antenna proper and the surrounding medium, including the earth, the Heaviside layer, etc. The impedor is linear if in the steady state the harmonic electromotive force between the terminals is proportional to the current V= ZI, (1-1) where the coefficient Z, called the impedance of the impedor, is a function of I he frequency and in general of the oscillation constant. Strictly speaking, we should specify the path between the terminals A and B, along which we compute or measure the electromotive force. For any two paths the difference of the electromotive forces is equal to the magnetic current through a surface bounded by these two paths. If H is the average magnetic intensity over the surface whose area is S, then the difference between the two voltages is* 2¥)ir2SHa. (1-2) In the immediate vicinities of the wires the magnetic intensity is equal to the current divided by the length / of their circumference; hence Hm, is certainly less than I/I, in fact considerably less since the integral of H along the radius will vary as log /. But even with / in the denominator of (2), the voltage difference is small if the distance between the terminals is a small fraction of the wavelength. Thus at low frequencies it becomes unnecessary to specify the path, except in precision calculations. At high frequencies we shall assume that the path is a straight line connecting the terminals, unless otherwise specified. In the unrestricted frequency range the impedance is a complicated * Assuming that the paths are in free space. 97 ELECTUOMAííNFTIC WAVES Ciiai'. 5 function of the firequencyj but its expansion in the vicinity of w = 0 gen- erally is* I Fig, 5.1. A wire loop as an electric circui t. IccC (1-3) If C = co and R 7^ 0, then for sufficiently low frequencies the impedor is a resistor; if C = °o and R = 0, then the impedor is an inductor; and if R = L = 0 but C 7^ oo, it is a capacitor. In practice R and L may be very small but they never vanish. Consider for example a conducting loop (Fig. 5.1). From Faraday's law (4.1-7) we have / E,ds+ i Esds^-iu®, j{acb) j(ha) (1-4) the first integral being taken along the conducting wire and the second along the straight line joining the terminals. The impressed electromotive force V, needed to transfer the charge from B to A against the field produced by the charge and the current in the wire must be equal and opposite to the second integral V *J(ba) Eads. d-5) Substituting in (4), rearranging the terms, and dividing by the current flowing out of A, we have {ace) Esds J (1-6) The first term, representing the ratio of the electromotive force along the surface of the wire to the input current, is called the internal impedance or the surface impedance of the wire; the second term is the external impedance. At a = 0 for a homogeneous wire of length / and of uniform cross-section S, we have E,ds = -z = RI. * Always, for actual physical structures; but for idealized physical structures the origin may sometimes be a branch point. IMPED' >us, TRANSDUCERS, ME'I'VVi >RKS 99 I he impedance of I lie loop is jusl its resistance. The magnetic flux '1> is proportional to 7. If L is the coefficient of proportionality atw = 0, then hi low frequencies the external impedance is approximately proportional to i he frequency. In the next chapter we shall obtain the next higher term in i he expansion for the impedance of the loop. From (2) we find that except at rather high frequencies the impedance I ij a loop of practical dimensions is small. This impedance may be in-i n asetl by winding the wire into a coil (Fig. 5.2). Thus inside a long coil ihr magnetic intensity is approximately equal to the number of ampere- -o o 6 B A Fig. 5.2. A solenoid as an inductor. Fig. 5.3. An electric circuit containing a capacitor. turns per unit length. The magnetic flux through the coil is then pSnl, where n is the number of turns per unit length. The electromotive force round each turn is —ioifiSnl, and per unit length it is — /'tow^w2/. The impressed electromotive force needed to drive the current through the coil against this electromotive force of " self-induction " is equal and opposite. Hence by increasing the number of turns, the impedance of the coil may be raised. Consider now another structure consisting of a conducting wire and a pair of closely spaced conducting plates (Fig. 5.3). Applying the first induction law to the circuit ACDEFBA, we have f E3ds+ fEsds+ f sds — — iod$>, (1-7) where the second integral is taken along ACD and EFB. The first term is equal and opposite to the impressed electromotive force V. The time derivative of the charge q on the lower plate is the current Ic flowing into the capacitor and iwq = Ic, g-'= . (1-8) When co = 0, the charge is proportional to the voltage across the capacitor and therefore A iuC' f v (DE) Es ds — (1-9) 100 ELECTROMAGNETIC WAV KS except for tin- iu KS mi Consider a number of impedors connected in series (Fig. 5.6). Assuming ih.il their impedances are not too small, we may neglect the impedance of I he connecting wires and write VUC + Vdk + tm + ?BA = 0, (1-12) where the separate terms are the electromotive forces of the field which act between the various terminals in the order indicated. The last term is — V, where V is the impressed electromotive force between H and A. The above equation expresses the first Kirchhoff law. From (12) we then have V - (Zi + Za 4- Za)7, Z = Zí + Z2 + Z3, (1-13) where Z is the impedance of the entire circuit. The internal impedances of the connecting wires and the external impedance of the connecting loop < mtkl he added to (13). In the above equations we have assumed that the current through each impedor is equal to the input current of the entire H o ao_ -«-6l Fig. 5.6. A series connection of impedors. Fig. 5.7. A parallel connection of impedors. circuit and thus disregarded the displacement currents between the connecting wires. Since the dielectric constant is small, it may be anticipated that these currents are negligible even at comparatively high frequencies. In order to obtain more precise information about their magnitudes and effect on the input impedance of the circuit we shall have to consider wave propagation on wires. Consider now a number of impedors in parallel (Fig. 5.7). At a branch point the total current flowing in or out is zero; this is the second Kirchhoff law and it follows from Ampere's law of induction if we neglect the displacement current flowing from the branch point. Thus i = 4 +1% + u (1-14) Applying Faraday's law to various circuits in Fig. 5.7 and to a circuit in which the parallel combination is replaced by an " equivalent " impedor Z, 102 wc liave Consequently I I .I'C I K( iMACiNI-yriC WAVF.S V - Zj/j = z2i2 - z3/3 = z/. t'llAI 1 1 1 1 Z, Zj z2 z3 (1-15) Let us now consider a more general network (Fig. 5.8). We could apply KirchhofF's laws to different circuits or meshes of the network and write a number of equations connecting the impressed voltages with the currents through various impedors or branches of the network. A simpler set of Fig, 5.8. A network of impedors. equations is obtained, however, in terms of mesh currents as shown in Fig. 5.8. Mesh currents satisfy automatically the conditions at a branch point. Applying the circuital law to the chosen fundamental set of meshes, and substituting the mesh currents for the branch currents, we obtain the following typical set of equations for an M-mesh electric network Zn/j + W2 + Z13/3 + ■ ■ ■ "h Z\.llIri = $% Z21/1 + Z22/2 + Z23/3 ■ + Z2ři/,j = Z31/1 + Z32/2 + Z33Í3 ' + Z3nIri — r*, (1-16) Zni/i + zn2i2 + zn%h + ■ "F ZnnIn — V r 71* where the F's are the total applied voltages in the corresponding meshes. The coefficient Zmm is called the impedance of the mth mesh and the coefficients Zm!c are the mutual impedances between meshes m and k. If the electric current in the £th mesh is 1 ampere and the currents in the remaining meshes are equal to zero, then the voltage in the rath mesh is Zmk volts. In Fig. 5.8 we have: Z12 = -ZBF, Z,3 = 0, Z2S = -ZCF, etc. If the matrix of the coefficients in (16) is nonsingular, we can solve the IMPEDORS, TRANSDUCERS, NI ÍTW( )RKS 103 1 ip .11 inns and obtain /, = y,xVy + yviv2 + yun + ■■■ + YtnrH) h = YnVi + y22V2 + Y»r* + ■■■ + ymk Afcm A (1-19) where A is the determinant of the coefficients in (17) and Afcm is the co-factor of the element Ykm in A. The impedance seen by the generator in the ?Hth mesh is the ratio Vm/Im when all the V's, except Vm, are equal to zero; hence this impedance is 1 D Y /) I* mm J-y?ttm The impedance and admittance matrices are symmetric ^mk = ^kmj ^mh = ^/cm- (1-20) (1-21) That is, the electromotive force in the mth mesh due to a unit current in the kth mesh is the same as the electromotive force in the kth mesh due to a unit current in the mth mesh, and also the current in the mth mesh due to a unit electromotive force in the kth mesh is the same as the current in the kth mesh due to a unit electromotive force in the mth mesh. This is the Reciprocity Theorem. Since it is possible to choose the fundamental meshes in such a way that any two given branches belong to two different meshes and to no others, the reciprocity theorem implies that an interchange of the positions of a generator and an ammeter does not change the ammeter reading. KM ELE< TROMAGNETIC WAVES Chai To prove the theorem we shall first establish the following lemma: let F'u F2,- • • V'n be the electromotive forces in the various meshes of the electric network and l\, /», ■ • • /',' the corresponding currents, and let ll;■■»■ Jfa:j• ' In be the currents in response to another set of electromotive forces V'{, V'i, ■ ■ ■ V'l; then E Ki'l = E VllL (1-22) On the left side of this equation we replace a typical voltage Vra by the sum of the branch voltages in that mesh and group the terms having common branch voltages. If any particular branch PQ is common to several meshes, the voltage FpQ multiplied by the respective mesh currents will occur in several terms, the sum of which will be FpQIpQ; therefore E V'j-'L — E VpqIpq — E Zpnlpolj «=0 (PQ) U-Q) PQ-lPQlpQ, where the last two summations are taken over all the branches. The last expression is symmetric in the primed and double primed 7s; hence (22) is true and our lemma is proved. Since /], I2, ■ • ■ In and 11,11,' ■ • are two independent sets of quantities, we may set - i, if a = m: I" = 0, if a m; Substituting in (22) we obtain ß = 1, = 0, M = v" and Z km if ß = k; if ß^k. Zmk' Similarly choosing K = 1, if a = m; V'j = 1, if ß = k; = 0, if a ^ m; = 0, if ß ^ k; wc obtain 7« — JJ» and Ymic = Ykm. Thus the reciprocity theorem has been proved. 5.2. Transducers A four-terminal transducer or simply a transducer is any combination of conductors and dielectrics with two pairs of accessible terminals (Fig. 5.9). The pairs of terminals may be those of a transformer,, or of a telephone transmission line between two cities, or of two antennas. In the last case the transducer includes the space between the antennas, the ground, etc. [MPEDORS, TRANSDUCERS, SETW< IRKS II the transducers are linear we have d priori equations V\ m Ziil i + Z12/2» (2-1) f% - Zai/i + Z2272; in if uly the currents are linear func mis of the voltages /, = YUVX + Y12F2, (2-2) h = ^1^1 + Y22V2. 105 Be Fie. 5.9. A diagram for a four terminal transducer. I I . . "■. .111.1 )"s an..- functions of the electrical properties of the transducer .....I of the frequency but not of the F's and I's. The coefficient Zn is (iidled 1 he impedance seen from the first pair of terminals and Z22 the im-Iwilimce seen from the second pair; Z12 and Z2i are the mutual impedances ' 1 he transfer impedances. Similarly Yn is called the admittance seen n I he lirst pair of terminals and Y22 the admittance seen from the second Bftii'i Yn and Y2i are the mutual admittances or the transfer admittances. If wc leave the second pair of terminals " open " so that I2 = 0, (1) be- Vi = ZnIu V2 = Z2Ji. (2-3) him if one ampere is passing through the first pair of terminals, Zn is the II ill ,i|'.r across this pair and Z2i is the voltage across the second pair. Simi-lailv if the second pair of terminals is " short-circuited " so that V2 = 0, then (2) becomes Ji - YnVi, h = Y2XVi. (2-4) I Irncc if a unit voltage is impressed on the first pair of terminals, then Yn In the current through this pair and Y2i is the current through the second Ipinr. Consider an w-mesh passive network of impedors with two pairs of acces-■ ullile terminals, one pair in the rath mesh and the other in the kth. Then It! (1-16) all the F's are zero except Fm and Vk. Eliminating all the 7's 1 1 1 j 11 /,„ and Tk, we obtain equations of the form (2) and hence the inined-[ flncc coefficients of the transducer. By the theory of determinants it may III hown that the transfer impedances of a transducer consisting of a network with two pairs of accessible terminals are equal. For each type of transducer the corresponding reciprocity theorem should be proved separately as there is no d priori reason why the matrix of the impedance t'ticfricients should be symmetric. In connection with cylindrical waves wc shall encounter generalized transducers whose impedance matrices are not symmetric. Kl«. ELECTROMAGNETIC WAVES ' Chap. IMPKDOUS, TRANS! HKT.US, NETWORKS 107 The admittancea can he expressed in terms of the Impedances and vice versa; thus solving (!) for the/'s ami comparing with (2), we have ril = > * 12---Jj~ > *2i - D Zu D ' D — Z11Z22 —'Z12Z21 — Z11Z22 — Z12. Similarly we obtain Zu - A i % - - — , Z21 - - a (2-5) 21 -22 Iii A ' A = YnYi2 - Y12Y2l = YnY22 - Fa|. (2-6) These equations show that if Z2i = Z12, then Y2i = Y12 and vice versa. If the expression for any of the Z's from (6) is substituted in the expression for the corresponding Y in (5), we obtain DA = 1. Multiplying the first equation of set (1) by If, the second by 7|, and taking half the sum, we obtain an expression for the complex power * = WJ$ + V2ID = WZ^IJX + Z12{hH + Ffh) + Z22I2H]. (2-7) If /1 and I2 ate the amplitudes of I\ and I2 and if # is the phase angle between them, then (7) becomes * = \{Zul\ + 2Z12lJa cos tf + Z2%1\). (2-8) The real part of * is the average power contributed by the impressed forces to the transducer. Multiplying the first equation of the set (2) by Vf, the second by F* and taking half the sum, we have the corresponding expressions for the conjugate complex power (2-9) ** = UYu^n + YX2RKS 109 ll in evident Otl physical grounds thai the input impedance of the chain ilia I the transfer ratio across each transducer are unique and we are laced with the pt'tihleni of choosing the proper sign for the square roots in (5) Mm I {(>). The product of the two values xi, X2 for the current transfer Hit n 1 is unity. Thus if the absolute value of Xi is less than unity, the ab-11< 111 value of x2 is greater than unity. In a dissipative chain the amplitude of the current must necessarily decrease and we must choose that sign ■ 1 iK. .pian- root in (6) which makes the absolute value of the current linieJei ratio less than unity. This choice determines uniquely the sign in 1 he expression for K. It is apparent from (6) that for small values of Z|.j the proper sign is positive. We may represent the current transfer ratio as an exponential function h h _ _ —r. Vt = Vie-™, Vt = K+Ii, Hie constant Y is the propagation constant or the transfer constant of the chain. The real part of Y is positive and is called the attenuation constant; 1 In imaginary part of Y is called the phase constant. Since one of the values of the current ratio in (6) is e~r and the other in us reciprocal ev, we have eT + e~T Z„ + Z22 i coshŠ—— Sgf*. (3-7) The current and voltage across the terminals An> Bn may now be ex-piessed in the form It = He-™ R here the superscript " plus " is used specifically to indicate a wave traveling from the Niiurce toward the right in an infinite chain [Fig. 5.11) of which the semi-infinite chain forms a part. For a wave traveling to the left in an infinite chain, we should have since in this case the amplitude should decrease as n decreases. Here the current ratio is represented by the second value in (6). For the voltage we have where — K~~ is the value of K in (5) other than the one designated by K+. The impedance K~ is the impedance of the semi-infinite chain extending to B-i A-i Ao Fig. 5.11. Two sections of a chain extending to infinity in both directions. no Kl.Kl TUoMACiNKTIC WAVKS ClIAl'. 5 In fact if K+ happens to (3-8) tlic left, as seen from any pair of terminals, correspond to the upper sign in (5), so that K+ = J(ZU - Z22) + Vl(Zn + Z22)a - Z?2; then, in accordance with the above definition, K~ = £(Za2 - Zxx) + V|(Zn + Z22)2 - Z22. (3-9) If the elements of the chain are symmetric, then Ztl — Z22 and K+ = K~. Since iC1" and K~ are impedances of passive networks, their real parts cannot be negative. These two impedances are called the characteristic impedances of the chain of transducers. The expressions for the current and voltage in a chain consisting of a finite number of transducers may now be written in the following form /„ = Ae~rn + Bern, Fn = K+Ae~Vn - ITSe% where A and B are constants obtainable in terms of the terminal conditions. For instance, let the total number of transducers in the chain be m and let an impedance Z be inserted across the m\\\ pair of terminals; then we have K+Ae~Tm - KTBeTm Z = — = + Be1 2 1 -5 5 Fig. 5.12. A symmetric 2"-network. From this equation we can express B in terms of A. The constant A can then be found in terms of the input voltage FQ or the input current I0. Similarly A and B can be expressed in terms of Fq and In, or in terms of and Fm, or in terms of Iq and Im. In the last two cases we obtain equations representing the chain of transducers as a single transducer. 5.4. Chains of Symmetric T-Networks A symmetric ^-network (Fig. 5.12) is a transducer whose impedances are Zu = Z22 = |Zi + Z2, Z12 = —Z2. (4-1) Substituting in (3-7) and (3-8), we have „ , _i Zj + 2Z2 _ r = cosh ——-- = cosh 2Z2 K+ = K- = ^Zi(Zi +4Z2). These are the constants for the iterated network shown in Fig. 5.13. (4-2) r [MPEDORS, TRANSDUCERS, NETWORKS ill E vidcnily any symmetric transducer may be represented by a symmetric 7' network; thus from (1) we have Z, = 2(ZU +Z12), Z2 = -Z12. • (4-3) I Km, expressions (2) may be used for any chain of symmetric transducers. Pi 2il Fig. 5.13. A chain of symmetric T-networks. s.s. Chains of Symmetric U-Networks Consider now a symmetric Il-network (Fig. 5.14). Starting with the following iiush equations ■2ZJ1 - 2Z-J + 0 = Vh -2ZJ1 + (Zi + 4Z2)7 - 2Z272 = 0, 0 - 2Z,7 + 2Z.J2 = V-i, mil eliminating the current 7 in the intermediate im ill, we obtain 2Z2(Z, + 2Z2) 7i - 4Z2 Zi + 4Z2 Zi + 4Z2 2Z2(Z, + 2Z2) 4Z| Z.4-4^ I lencc we have 72 = Vu 72 = V> {Er I 17) g] T) la) Zu. = z22 — Zi + 4Z2 2Z2(Zi + 2Z2) Fig. 5.14. Asymmetric Il-network. Zx + 4Z2 Zi2 — — 4Z| Zi + 4Z2 2Z-; Fig. 5.15. A chain of symmetric H-networks. Therefore K = VZU - Z\2 = V(Zu + Z12)(Zn-Z1;,) = 2Z2 ^ (5-1) is the characteristic impedance of a chain of H-nctworks (Fig. 5.15). 11 ! i i r t t lie >MAl fNETTC WAVES Our. 5 'I'lic propagation constant is evidently thi- silnic as in the disc of the chain «>f '/'-networks; A.' Iius a different value only because of terminal differences between tlu: two chains. In fact, we can obtain (I) from the characteristic impedance in (4 2), Designating the litter by K and the former by A!', we have , = (K + |Zi)2Z« 4Z2* + 2Z1Z2 K + \Z\ + 2Z2 IK + Zi + lZ* It may be shown that K' is identical with A' in (1). 5.6. Continuous Transmission Lines A continuous uniform transmission line may be regarded as a limiting case of a chain of transducers. If the distributed series impedance and shunt admittance per unit length of the line are respectively Z and Y, then 1 • Zi = Z dx, Z2 = Ydx where dx is an element of length. By (4—2) we have K = ^M, cosh r. - 1 -4- £r* H----= 1 + \ZYdx2, r = Z2 = iwL2, — =--T—7 ; the pass-band is determined by 03 > Uc, 01c ~ T ioůCi ' * ítoCa' Z2 2L,C2 + -^ (7-6- T X Fig. 5.18. A band-pass filter. and 0 < w2LtC2 Ci <4, co, <. W <- co, This is a band-pass filter. If however £1, Ci, C2 are continuously distributed, we have Z4 — Xj dx, Ci = ^, C2 = C2dx, (7-7) where Zj, Ci and C2 refer to unit length. The upper cut-off frequency becomes infinite and the transmission line has the characteristic of a high-pass filter with a cut-off given by 1 (7-8) We shall see that propagation of transverse magnetic waves is governed by equations of this type. -|—'W—f Tnnp— Fig. 5.19. A band-pass filter. For the structure shown in Fig. 5.19 we have 1 1 Z\ Zi = iuLu — = ioiC2 + —- , — = —w2LiC2 + Zj2 t(j}L,2 &2 (7-9) I lie. ■ lint line is also a band pass filter in which the lower and the upper 1 til oils are specified by I hrn such a structure becomes a continuous line, then Li = L\ dx, C2 = Č2 dx, Z,2 = "T1 > 'Á dx VLW' (7-11) 2^2 Mini the upper cut-off has receded to infinity. Propagation of transverse ^Htl'ic waves is governed by equations of this type. '1 II Forced Oscillations in a Simple Series Circuit < >ne of the simplest electric networks is a circuit consisting of a series ■rjinhination of a resistor, an inductor, and a capacitor (Fig. 5.20). The peihi nee of such a circuit is mi 1 Z = R + io>L + — = R + i iíoL flu* reactance component vanishes when 1 (8-1) to = to (8-2) VZc' I he frequency so defined is called the resonant jrc- , . . . , e riG. 5.20. A simple series iyt)icy of the circuit. At the resonant frequency circuit. (In* reactances of the inductor and capacitor are 1 111.11 except for sign; thus I čúL i- ft K. (8-3) The quantity K is called the characteristic impedance of the circuit. The Impedance of the circuit at any frequency can now be expressed in the form Z = R + i \OI «/ (8-4) At resonance the absolute value of the impedance is minimum and the I'll 11 ent is maximum. The sharpness of the resonance curve (current vs. In quency) is seen to depend on the ratio of the characteristic impedance of the circuit to the resistance, that is, the " 0 " of the circuit Q_K_wL__1_ R~ R ~~ uRC (8-5) Mr, ELECTROMAGNETIC WAVES CllAľ, ,1 [MPKIJOKS, TkANSIHJCFUS, NFTW< )KKS 117 This quantity can be defined in terms of the total energy S stored in the circuit :it resonance and the average power IV dissipated in A'. Startinp, with the definitions of the resistor, inductor, and capacitor (section 2.7) and obtaining the work done by the applied electromotive forces, we find that at any particular instant the power dissipated in the tesistor, the energy stored in the inductor, and the energy stored in the capacitor are respectively Rlf, \Ll\, \CV%,i, where Ii is the instantaneous current in the resistor or the inductor and Vc) is the slope of the input reactance plotted as a function of oi. 5.9. Natural Oscillations in a Simple Series Circuit The natural oscillation constants pi and p2 are the zeros of the impedance function (2.7-6). The impedance and admittance functions can be factored and thus expressed in terms of these zeros: zip) = r(P ~ ~ fa) Y(p) (9-1) p » ' L{p -pi)(p - fa) Depending on the relative values of the circuit constants the natural oscillation constants may be either real or complex. Thus if R 1 or R > 2K, 2L VLČ' the constants are real and the " oscillations " degenerate into an exponen- *The impedance of the capacitor is —//wC and the voltage is lagging behind the current by 90°. IMPEDORS, TRANSDUCERS, NETWORKS 119 I Jul decay. On the other hand if R < IK, the oscillation constants arc ■.....r.nr . omplex + *$>> fa = P* = I ~ 4 (9-2) where the amplitude constant £ and the natural frequency w are given by e = - R_ 2L 2& 2Q (9-3) Em high Q circuits the natural frequency is nearly equal to the resonant | frequency u = w. I 5.10. Forced Oscillations in a Simple Parallel Chxuit The theory of a parallel combination of an inductor, a capacitor, and a re.-istor (Fig. 5.22) is very similar to the theory of the series circuit. The I Input admittance of the circuit is 1 (io-i) Y = G + iuC + -r 1 .....paring this with the input impedance (8-1) pi the series circuit, we observe that the equations of section (8) can be adapted to parallel Circuits if we interchange £ and C, Z and Y, and FlG- s-22- A s™Ple parallel replace R by G. We should also replace the i huracteristic impedance K by the characteristic admittance M; but ftubsequently it may be more convenient to reintroduce K. Thus we have the following expression for the input impedance of the circuit 1 GZ = where Q is defined by * w 1 + iQ M G = \(3 tti/ oGL 1 KG (10-2) (10-3) If the input current is fixed, the voltage across the circuit varies with the frequency exactly as does the current in the case of the series circuit (Fig. 5.21). Another type of parallel circuit is that shown in Fig. 5.23. In general the frequency characteristics of this circuit are different from those of the 120 ELECTRi MAGNETIC WAVES Chap, circuit in Fig. 5.22. If, however, both circuits have high Q values, then their behavior in the neighborhood of resonance is approximately the m;iiiic. To prove this, consider a parallel combination of two impedances Zi and Z2 such that | Z2 | 3> | Z\ |. For i the input impedance Z of this combination we have (o)) I ci approximately' Zi + Z2 z,> Fic. 5.23. Another type If Zi is a pure reactance and Z2 a pure resistance of simple parallel cir- then the last term is positive real and a large resist! ance Z2 in parallel with Zi may be replaced by a small resistance in series with Z, or vice versa. Hence the circuits in Figs. 5.22 and 5.23 are approximately equivalent in the neighborhood of resonance if R w2L2G G = R oři} _R Kz Substituting in (2) and (3), we have Hence the maximum input impedance is K2 Zmax — — = KQ. If a generator is connected as shown in Fig. 5.24, then the maximum input impedance is K\ a>2Ll l-t (10-4) (10-5) (10-6) 7 - 1 A. R (10-7) L-L, X R Assuming that Jfi is still large compared FlG" S'24" A. tepped para!lcl . . circuit, with R, the effect of shifting the terminals of the circuit is to reduce the inductance to Li and to increase the effective capacity. The resonant frequency is evidently unchanged; but the maximum input impedance is reduced in the following ratio %^«á, (10-8) Of course, if L\ is so small that ZiLi is no longer large compared with R, the above formulae must be modified. For the circuit shown in Fig. 5.22 and approximately for the one shown IMPEDORS, TRANSDUCERS, NETWORKS 121 in hip,. 5.23 we can obtain an equation similar l<> (8 14); thus Sa - }/A2, (10-9) where V is the maximum voltage amplitude across the circuit. ''II. Expansion of the Input Impedance Function The input impedance (1 20) as seen from the terminals of a generator in a typical mi ';li of an electric network is a rational fraction when considered as a function of the M illation constant p. The numerator and the denominator are factorable and the Impedance may be represented as a ratio of two products Zip) = A tp _ p^(p - pt) (f ~ Pi)(p - Pi) (11-1) where is a constant. The zeros p\, pi, ■ ■ ■ of Zip) are infinities of Y(p); they represent the natural oscillation constants of the network when the voltage across the input terminals is zero and hence when the terminals are short-circuited. The Infinities pi, pi, • • ■ of Z(p) are the zeros of Y(p); they represent the natural oscilla- ......constants of the network when the current through the input terminals is zero and hence when the network is open at these terminals. A rational fraction can be expanded in partial fractions. If all the zeros of the admittance function are simple we may write Z{p) in the following form a\ a*. Z(i) =--+ p — pi P — Pi + ■■■ +IÍP), (H~2) where/(j>) is a polynomial in p. Multiplying by (p - Pi) and letting p approach pU we have «1 = lim ip - Pi)Z(p) = lim as p ^ ph Therefore Y'(pi) consequendy wc have (11-3) (11-4) where the summation is extended over all the zeros of Y(p). Similarly the admittance function may be represented as follows 1 Y{p) = i:iP-pm)z'(pm) + s(p)- (11-5) In network theory it is shown that f(p) and g(p) are polynomials of degree less than 2. We have seen that the complex zeros and poles of Z(p) occur in conjugate pairs; 122 ELECTRi (MAGNETIC WAVES Ciiai1. 5 i hits for typical pairs of zeros and infinities we have f>m = Jm + /Mm, J>„ = tm — tfft* pm = £m + MOm, pm = £m — /0>m. The real parts £TO and £m arc never positive since positive values would mean that the amplitudes of the currents and voltages in the network were steadily increasing. Then infinite power would be dissipated in the resistors and infinite energy stored in the inductors and capacitors without a continuous operation of an impressed force, that is, without a continuous supply of energy to the network. Let us now consider the values of Z(j>) and Y(p) on the imaginary axis Z(io>) =R(o>) + iX(u), (H-7) Y{$w) = G(w) + iB(u): R{h>) and G(o>) are never negative; if they were negative for some value of oj, then at this frequency power would be contributed to the generator by the supposedly passive network. If the network is only slightly dissipative, R() and G(a) are small. In this case the zeros of Z(j>) and Y(p) are given approximately by X(wm) = 0, />>,„) = 0. In order to obtain the second approximation we note that Z(i&m + Sm) = Z(mm) + tmZ'{i(bm) H----= 0, Solving, we obtain 0. j Z(/oim) Y(ioim) Om--. , 8m = — Z'(mm) Differentiating (7) iZ'(Uom) = R'r&m) + iX'(wm), iY'(icom) = G'(um) + iB'(a:m), and substituting in (10), we obtain 8m ™ — (11-8) (11-9) (11-10) en—n: R(wm) X'(dm) — iR'{Cim) B'(com) - iG'(o)m) '' X'{wm) ' [A"(im)]2 (fM _ ,G{wm)G' (11-12) Thus the approximate expressions for the real parts of the natural oscillation constants are R(&i> X'{&m) G(um) (11-13) IMPEDORS, TRANSDUCERS, NETWORKS >.i Since R tad G are positive and the £"s are ncgtitivc, wc have the following inequalities X'(i>m) >0, B'iUm) >0. (U-14) Substituting the above approximations in the general equations (4) and (5), we I.i v c • Z(iw) = T.—2 Y(i*>) = £ 2/w («2 — CD2 — 2iu£m)B' (ci)m) 2*'« (£2 - w2 - 2iw|m)A"(&„1) + #(/w). (11-15) Comparing the second of the above equations with (8-12), we find that a slightly dissipative network behaves like a parallel combination of simple series circuits whose inductances and Q's are given by £.-**(«*), Qm- 2^ 2R^m) (11-16) Likewise we can regard the network as approximately equivalent to a series com-bination of parallel resonant circuits whose capacitances and Q's are Cm = fJB (cdm), (?m — t„,) 2G(wm) (11-17) In view of (8-14) and (10-9), equations(15) can be expressed in terms of the energies stored in die circuit at the various resonant frequencies. Thus we have Z(m) = £ ^J^-n +-Q-) (11-18) where 6m is the energy stored in the circuit at the /nth resonant frequency when the input terminals are open and when the voltage amplitude at these terminals is unity. Similarly we have Y(m) = £ ■ 2& (11-19) where $m is the energy stored in the circuit at the mth resonant frequency when the input terminals are short-circuited and when the current amplitude at these terminals is unity. So far we have tacitly assumed that none of the natural oscillation constants falls on the real axis. Let us now suppose that p = po is a real simple zero of Y(p). * It should be recalled that X{o>) and B(w) are odd functions and therefore X'(w) and B'((i)) are even. We retain only the principal terms in the final approximations. 111 E LECTROM AGNETIC WAVES Then (lie corresponding term in (•!) is nor paired with any other; it is 1 /„(/•) If po is small, we have approximately (p - po)Y'(p0)' so that, for p = /to, equation (20) becomes ZoO'co) = 1 ClIAl'. 5 (11-20) G(0) + icoS'(O) ' (U 21) G(0) is the direct current conductance of the network and B'(0) the direct current capacitance; thus G(0) = C? = /F0, B'(0) = C = 2S„, (11-22) IMPEEM >US, TRANS! >l ll'KKS, N KTVV< >K KS 125 ttrl'ie* resonant circuit (or in a special case nonresonant) and a succession of parallel 11 onant circuits or as a parallel combination of a parallel resonant circuit and a suc-ii i. hi nt' scries resonant circuits. In the first case the series circuit is obtained when ........I (he parallel resonant circuits degenerates into an inductance and another into I i .ipacitance; the parallel circuit in the second case is obtained similarly. The ■ venerate circuits correspond to the zeros and pules of Z(p) at the origin and at Infinity. If the network is slighdy dissipative, then in the first approximation the qutvolent networks will differ from those in Fig. 5.25 only in that each series and parallel branch will contain a resistance element. -J Fig 5.25. Two equivalent representations of a general reactive network. where Wa is the power lost in the conductance and §o is the energy stored in the capacitance when a unit voltage is applied to the network. The term (21), which is to be added to (18), now becomes ^ G + iuC ///o+2ico§o ■ ( - ) Similarly if Z(p) has a simple zero on the real axis, we should add to (19) the following term 1 1 R (0) + mX' (0) R + ml fy0 + yog,' (11-24) where R and /. are respectively the d-c resistance and inductance of the network, JVq is the power lost in the resistance and S0 is the energy stored in the inductance when a unit current is passing through the input terminals. The foregoing results may be summarized graphically as shown in Fig. 5.25. A purely reactive finite network may be represented either as a series combination of a CHAPTER VI About Waves in General 6.0. Introduction If the impressed currents are known throughout an infinite homogeneous medium, the field can be calculated fairly easily; we need only obtain the field of a current element and then use the principle of superposition. The solution of this problem is useful even though most practical problems are concerned with fields in media composed of homogeneous parts and not in completely homogeneous media. Thus if the medium is homogeneous except for isolated islands, it is sometimes possible to obtain approximate polarization currents which can be used as virtual sources in an otherwise homogeneous medium (section 2.11). The first few sections of this chapter are devoted to this problem. The boundaries between media with different electromagnetic properties or the " discontinuities '•' may have a profound effect on wave propagation. In a homogeneous medium, for example, the energy from a given source will travel in all directions; but in the presence of parallel conducting wires at least a fraction of this energy will flow in the direction of the wires. The effects of such discontinuities will be studied in detail in subsequent chapters; but some general considerations are introduced in this chapter. A brief discussion of electrostatics and magnetostatics is also included in this chapter. These topics are of interest in wave theory for the following two reasons: (1) they furnish approximations to slowly varying fields, (2) they furnish exact solutions of certain two-dimensional wave problems. 6.1. The Field Produced by a Given Distribution of Currents in an Infinite Homogeneous Medium Our problem is to solve the electromagnetic equations (4.4-2) for harmonic fields. This is the most important case in practice; besides, the solution of the most general case can then be expressed in the form of a contour integral in the oscillation constant plane. The usual procedure for solving a simultaneous system of equations is to eliminate all dependent variables except one. In the present case this procedure would be unnecessarily restrictive since we should have to differentiate / and M and hence assume that they are continuous and differentiable functions. In practical problems f and M are localized and for all practical purposes 126 ABOUT WAVES IN GENERAL i he regions occupied by them have sharp boundaries. Thus it is convenient to introduce a set of auxiliary functions, generally called potential functions. To begin with let us write £ = £' + £", H = H' + H"> (1-1) where (£',#') and (E",H") are solutions of curl E' = -iuiiH', curl H' = J + (g + *'<*)£', (1-2) curl E" = —M - muH", curl H" = (g + iox)E". The field (E',H') is produced by electric currents and (E,f,H") by magnetic currents. Their sum satisfies (4.4-2). Taking the divergence of each equation in the set (2), we have div/ div H' = 0, div H" = - div E' = - g + io* div M - 3 (1-3) div E" = 0. The second and third of these equations require J and M to be continuous and differentiable; but one form of the solution of our problem is obtained without using these equations. In the other form of the solution which depends on them we may assume / and M differentiable to begin with and then extend the results to include discontinuous distributions. The first and last equations show that H' and E" can be represented as the curls of certain vector point functions H' = curl A, E" = -curl F. (1-4) Substituting from (4) into (2), we obtain E' = -m4 - grad V, H" = + iw)F - grad U, (1-5) where V and U are two new point functions which are introduced because the equality of the curls of two vectors does not imply that the vectors are identical. From (4) and (5) and the two remaining equations in (2), we obtain curl curl A — J — a2A — (g + iut) grad V, (1-6) curl curl F = M — o^F — ivy, grad U. Using (1.8-2) we have AA — grad div A — — / + a2A + (g + iwe) grad F, (1-7) AF — grad div F = — M + a2F + {&$ grad U. 128 ELECTROMAGNETIC WAVES Chap, h Thus we have expressed F. ami // in lei ins of I wo vectors .7 and /'and two scalars and t/, the new functions being connected by two vector equations. So far the vectors are somewhat arbitrary since equations (4) are unchanged if we acid to A and F the gradients of arbitrary functions. The functions Vand £7are completely arbitrary. Hence we have an opportunity to impose further conditions on these functions to suit our convenience For instance we may set w.-^4-,. BW-fltS. (1-8) so that equations (7) become AA = a2 A - /, AF = c2F - M. (1-9) When specified in the above manner, the functions A, F, V and U are called wave potentials, the first two being vector potentials and the last two scalar potentials. More specifically A is called the magnetic vector potential, F the electric vector potential, V the electric scalar potential and U the magnetic scalar potential. Lorentz was the first to introduce these wave potentials in dealing with nondissipative media and he called them retarded potentials for reasons that will soon become obvious. In general the wave potentials are not only " retarded " but also " attenuated," and the more general designation " wave potentials " is more appropriate. Thus we have the following expressions for the field produced by a given distribution of impressed currents E — —iwfiA — grad V — curl F, (1-10) H = curl A — grad U - (g + ioit)F, where V and V are defined by (8) and A and F are the solutions of (9). If / and M are differentiable functions, Fand Usatisfy equations similar to (9). Thus taking the divergence of (9) and substituting from (8), we have tJ?= qv, div M = —iwmv. (1-12) Consequently 1 1 AF = —frV--qv, AU = -p2U - - mv (1-13) ABOUT WAVES IN GENERAL [29 I'Yoiii |lie physical point of view the vector potentials can be obtained inn. h more satisfactorily by the method explained in the next section than i \ solving equations (9) formally. 6.2. The Field of an Electric Current Element i i insider a short current filament (Fig. 6.1) and assume that the current / In uniform and steady between the terminals A. and B so that the entire ■ in lent is forced to flow out of B into the external p medium and then back into A. If the medium is a peifeci dielectric this would mean a concentration of . 11 . i ric charge at B at the rate / amperes per second and an ever increasing electric field around the filament. The product II of the current and the length ol I he filament is called the moment of the electric i i incut element. Let us suppose that the current element is centered ^IG- An electric , i ■ i^ ■ . ^1 current element, at I lie origin along the z-axis. hrom a point source the i urrent would flow outwards uniformly in all directions; the density would ihen be 4^ dr \ 4irr/ I letice for two point sources separated by distance / the current density, at distances large compared with /, is the gradient of the following function II cos B dz\ Arrf 4*r2 ('onsequently H cos e lirr* II sin e iirr3 (2-1) The dotted lines in Fig. 6.2 are the flow lines. The magnetic lines of force are circles coaxial with the element and in order to obtain the magnetic intensity we need only calculate the magnetomotive force round the circumference of a circle PP' coaxial with the element (Fig. 6.2). This magnetomotive force is equal to the electric current 1(6) passing through any surface bounded by PP', Choosing the surface as a sphere concentric with the origin, we have 1(6) = ~2ir „0 = / / Jr r2 sin 6 dd dtp = II sin2 0 2r (2-2) I 10 hence 1.1 J X I U< >MA( jNI. I'H' WAVES v 2irrsin0 7/ain 9 4rr2 Chap, 6 (2-3) Equation (1-10) shows that H is expressible as the curl of a vector A. From (1-9) we conclude that each cartesian component of A depends only Fig. 6.2. Electric lines of force in the vicinity of an electric current element. on the corresponding component of the impressed current density. Thus in the present case A is parallel to the z-axis and we should have dAz dp Comparing with (3), we have 3At dr dr dp 11 Az ~ T ' 4«r dAz dr sin 6. (2-4) (2-5) Let us now suppose that the current is a harmonic function of time. As the frequency approaches zero the field must approach that given by the above equations where I, A, H are now complex amplitudes of the corresponding quantities and the time factor elat is omitted. From (1) we obtain the electric intensities II cos e 2r(g + iue)r' 3 > II sin 0 4ir(g 4- ioe)r3 (2-6) Next we seek that solution of Maxwell's equations which approaches (6) as oj —> 0. At all points external to the current element the magnetic vector potential A must satisfy (1-9) with / = 0 and by (5) it must be independ- ABOUT WAVES IN UENKUAL rut of 0 nnd v»; hence 131 I hid 18 equation (3.1-15) with k = 0 and its general solution is Pe~CT Qe°' /lz = 1--• r r III dissipative media the second term increases exponentially with the dis- .....e from the element and hence cannot represent the field produced by I lie element. The first term approaches (5) as w (and therefore cr) ap-I u ".idles zero if P — IJ/4r. Thus we have Nondissipative media may be regarded as limiting cases of dissipative media and then Az = 4irr 1 dAz 4irr \ cr/ lly (1-8) we have V= - g + dz The field intensities are now obtained from (1-10); thus (2-8) (2-9) £r = 01 1 + - I e"" cos 0, Hv = -—11+-J^sinO, 27rr" V or/ 4:nT \ or/ iwr \ cr 0, these expressions approach (3) and (6). Similarly the field of a magnetic current element of moment Kl is obtained from the following electric vector potential Kh~ST 1.12 Fl.FCTROMAG'NFTIC WAVES i II. us I, Any given ilisl rihulion ol applied currents may lie subdivided into ele merits, and the Held can be obtained by superposition of t lie fields of individ ual elements. Take an infinitesimal volume bounded by the lines of flow and two surfaces normal to them. The current in this element is / = J dS, where dS is the cross-section of the tube of flow; hence, the moment // i equal to / dv, where dv is the volume of the element. Thus we shall have ^ J SJ 4irr ^ J J J 4irr dv, (2-13) where r is the distance between a typical element and a typical point in space. The scalar potentials are then computed from (1-8). If, however, / and Mare differentiable functions, then Vand [/can also be computed from equations similar to the above. We note the similarity between equations (1-9), (1-11), and (1-12) and write V V = --/// -ni div fe~*r 4ir(g + iwt)r 4ire" ■ dv, dv, U U -IIP &vMe~"r 4-iriuftr -Iff mve dv, (2-14) 4x,ur ■dv. These equations can be extended to include the case of nondifTerentiable J and M by adding appropriate surface integrals, and, more generally, by adding line integrals and discrete terms representing the potentials of point charges. Thus in the case of an electric current element in a nondissipative medium (Fig. 6.1) we have two point charges at the terminals U = 10) and the scalar potential is then -i/SrA (2-15) (2-16) which leads to (9) when / is very small compared with r. For surface and line distributions of impressed currents, the expressions for A and F are similar to (13), the surface and line integrals appearing in place of the volume integrals. For any filament carrying current I(s) we have —'a-ds, 4irr where ds is a directed element of length. We shall now define the terms " large distance ' as used in wave theory. A given distance r is (2-17) and " small distance " large if I or ] » 1 and Ali< HIT WAVES IN GENERAL 133 i ill d | <>r | I. In perlect dielectrics this means thai /'is large if .'/i/'/X is large compared with unity; r is small if fir is small compared Willi unity. For example, r - 2\ is fairly large since 2fi\ = 4x = 12.57; mi i In oilier hand r = X/80 is small to about the same degree since j3X/80 a 10 = 1/12.7. The length r = X/2ir = 0.16X may be taken as the li'leicmc length. AI large distances from the element the field is particularly simple. I Im mi a nondissipative medium we have approximately sin 8, Eo = Er 0. (2-18) ft. 3. Radiation from an Electric Current Element Ihe flow of power across an infinitely large sphere concentric with the dement is* '-iff1 EgHlr2 sin 6 d6 d

MA(iNKTlC WAVES CHAPi 6 ABOUT WAVKS IN GLNLRAL 135 The work done by this force per second is seen to he equal to Win (1). ratio re(f) 2m,/2 R = 3X2 The (3-3) is called the radiation resistance of the element. The reactive forces in the vicinity of the element are very large. Assuming a finite radius for the element, we can compute these forces at the element itself; since, however, they depend on r, we should subdivide the element into smaller elements and then integrate the effects. In order to sustain a uniform current the impressed forces must be distributed along the entire element. It will be shown (section 6.8) that if the element is energized at the center (Fig. 6.3), the current distribution is approximately linear. Then the moment of the current distribution is Flo. 6.3. A short wire energized at the center. p = jI(s) ds = III, (3-4) where / is now the input current at the center. If the element is short the distant field and hence the radiated power will be determined by the moment. Thus from (1) we obtain ,„ 10x2/2/2 W <= -5-» I2 I2 R = 207r3-^ = 197-5 \i \i (3-5) It is sometimes convenient to express the field of the current element in terms of the radiated power. From (1) we obtain p - —-r~ = x\— - ~—7=; (3-6) bVti y 2xVio hence for the distant field in free space we have IIv I = VŽFsin 6 érrVlÔr ' VÍOW sin d. (3-7) 6.4. The Mutual Impedance between Two Current Elements and the Mutual Radiated Power Consider two infinitesimal current elements of moments A ds\ and I2 ds2, and let ip be the angle between the positive direction of one element and the E-vector of the other (Fig. 6.4). Let E\ be the electric intensity ihn in the first element and II. the intensity due ( tn I he second. The electromotive force of the I llelil ol the first element acting along the sec-iiihI element is Ex^{s2) ds2, where Eit,(s2) is the ......i.......m ol A', m the direction of tlf.y, hence llic electromotive force which should be im-|n t'Mscil on the second element in order to counter-M i the force of this field is r.ds, Fio. 6.4. Two current elements. -Ei,s(s2) ds2. (4-1) The ratio of this impressed force to the current in the first element is the mutual impedance between two current elements Zi2 — ExiS(s2) ds2 Ei{s2) cos ý ds2 E2,s(s\) dsx h E2(s{) cos ý dsx (4-2) The reciprocity implied by this equation follows immediately on writing explicit expressions for the forces involved. For example, the mutual impedance of two distant i2dz2 parallel elements, perpendicular to the line joining their centers (Fig. 6.5) is iidz, ha. 6.5. Two parallel current elements. lr\e 2\r d%i dz2. (4-3) If the reactive components of the self-impedances -Zn and Z22 are tuned out, then V\ - RaJi + zl2i2, (4-4) V2 — Zi2/i + R22I2, where V\ and V2 are the applied electromotive forces. If the elements are of equal length, then % = R + R22 = /?■+■ R2s R = 80tt: if" (4-5) where R\ and R2 are the internal resistances of the elements and R is the radiation resistance. If V2 = 0, then Zi2Ii R + R2' (4-6) 136 KUiCTKOMACi'NKTIC WAVICS Cma* 6 and the power dissipated in A'2 is This is the power " received " by the " load " resistance R2. If 7?2 = 0, the received power is zero; if R2 = °o, the received power is also zero. For some value of R2 the received power must be a maximum; this maximum value is obtained from dfP öR2 0. (4-8) Thus we find that for maximum reception the load resistance must " match" the radiation resistance n\2 R2 = R = 8GY2 The received power is then Substituting from (3) and (9), we obtain* 3t,|71/l2 45 SArr2 Sr2 In terms of the power radiated by the first clement, we have (4-9) (4-10) (4-11) |2 — R 5 hence the received power is Equation (12) gives the power received by the load resistance 7?2j the total power W received by the second element may be taken as 2(R + R2) (4-13) of which the following amount 2(22 + R2) * The first expression holds for any dielectric and the second is for free space. (4-14) ABOUT WAVES IN GENERAL 137 U " rcrndiated." When /?a ■ R, we have JVr = W = \W, (4-15) in.I i In- power absorbed by the load is equal to the reradiated power. If 0, the " received " power is completely reradiated and (4-16) The power radiated by the two elements is the real part of (5.2-8) W = \(7?„7f + 2R12hh cos t? + 7v22/2), (4-17) where T\ and /2 are the amplitudes of ij and 72 and & is the phase difFer-iii i . In this equation Ru and R22 are, of course, the radiation resistances 1(S|) dst I (s 2) lue. 6.6. Two current filaments. of the elements and do not include the internal resistances of the generators driving the currents. We have seen that the R's are proportional to the products of the lengths of the elements (4-18) 7?i2 = &i2 dsi ds2, Ru = ku ds\, R22 k22 ds\. It is also evident that *U = k22. (4-19) The total power radiated by any two current filaments of arbitrary shape and length (Fig. 6.6) can be expressed in the form l¥=Wu + 2^12 + W22, (4-20) where W\\ is the work done by the impressed forces in sustaining the current in the first filament against the forces produced by this current, with IV22 defined similarly for the second filament; W\% is the work done in sustaining the current in the first filament against the forces produced by the current in the second filament. While W\i and W22 are inherently positive, W\2 may be either positive or negative. For the mutual power radiated by two arbitrary filaments we have •2W12 = ff kuisi^Hsi)?^) cos 0 dsi ds2. (4-21) [38 I'l E< TR< >ma we #12 - #12 = v 2ir(zi — 2a) 27r(Z! - z2) [sin fl(Zl ~ Za) -, i L 8(Zl - 22) - C°S - &f j ^1 %i sin jfc7(z1 1 ^(21- - z2) z2) — cos j8(zx (4-24) WETJndin/; ' In'116 *StanCe Wee" the enters of the elements, expanding £12 in a power series we obtain 3X2|_ 5X2 35X4 (4-25) 6.5. Impressed Currents Varying Arbitrarily with Time 4-nr ■ da. (5-1) If the phase of / is at, the phase of the corresponding component of the vector potential is at — Br = a[t — (r/v)}, where v is the characteristic velocity of the medium. The time delay r/v is independent of the frequency; hence all frequency components of a general function J(x,y,z;t) are shifted equally on the time scale and A will depend on J[x,y,z;t — (r/v)]. Thus we have ^(W5') = J J J 4?rr dv. (5-2) A similar equation is obtained for the scalar potential. Then the field is obtained from ~ " grad V> H= curl A. (5-3) ABOUT WAVES IN GENERAL i:iy ruling / by a contour integral of the form (2.9-10), the proof can be III kit l< - nunc lormal. In the dissipntive case no simple formula analogous in (2) exists. I ,el us now consider an electric current filament of length / along the • nKin fit the origin and suppose that the current starts from zero at / = 0 llitil is .in arbitrary continuous function of time thereafter; thus dl 1(1) =0, / < 0; —is finite. dt (5-4) Tin- charge q(t) at the upper end is zero when / < 0 and q(t) = ( I(t) dt when t > 0. (5-5) At I he lower end the charge is —q(t). 1 imputing the field we find that it is composed of three parts. One of these parts (E',H') depends only on the time derivative of the current; Another (E",H") depends on the current alone; the remainder E'" de-peilds on the charges. Thus we write E = E' 4- E" + E'", II = Hr + H" + H'", W" = 0, li!i = vH'v, E'r = 0, H' =---— sm 9; 4irvr II E'T' = 2E',' cot e, H'l = iirtr sin 9, E"T' — 4xr2 <• - 0 sin 9; (5-6) 2tvtrA cos 9. To an observer moving radially with velocity v the first part (E',H') of the lotal field would appear varying inversely as the distance from the element, the second part (E",H") inversely as the square of the distance, and the third part inversely as the cube of the distance. At sufficiently great distances only (E',H') will be sensibly different from zero although this particular fraction of the field is very small unless the electric current is changing very rapidly. The entire field is zero outside the spherical surface of radius vt with its center at the element. This sphere is the wavefront of the wave emitted by the element and on it (E",H") and (E'",H"f) vanish. At the wavefront E and // are perpendicular to the radius. Ill Kl.IAH« (MAGNETIC WAVES ClIAl'. i ahoi'jt wavks in uknkral mi 6.6. Potential Distribution on Perfectly Conducting Straight I Fires Let the current I(z) on a perfectly conducting straight wire of rudius "a" (Fig. 6.8) be longitudinal and be distributed uniformly around the wire. This is substantially the case under any conditions if the wire is thin; if the " wire " is a cylindrical shell of large radius, circulating currents h.wcr wire and the other half is in '.eric, with the upper wire and acts in the opposite direction. Under these conditions the currents in the two Fig. 6.8. A cylindrical wire. will exist on it unless the electric intensity is impressed uniformly around the shell. Under the postulated conditions the vector potential is parallel to the axis of the wire. Let its value on the surface of the wire be II(z); then the corresponding value of the scalar electric potential V is iu>e dz (6-1) Since the electric intensity tangential to the surface of the wire is zero except in the region of impressed forces, we have Ez(a) ■ ioijil I dz (6-2) Thus we have obtained two equations connecting the values of V and II on the surface of the wire dV dz Eliminating either II or F, we find dn dz dz2 = -ß2n; (6-3) (6-4) hence V and II are sinusoidal functions of the distance along the wire and the velocity of propagation is equal to the characteristic velocity of the surrounding medium. The equations, however, do not show where the nodes and antinodes of V and II are located with respect to the ends of the wire. Since the radial component £p is determined solely by the gradient of F, F can be defined as the electromotive force acting along a radius from the surface of the wire to infinity; but V is not a quantity which can readily be measured. Consider now two parallel wires (Fig. 6.9) energized in " push-pull." Of the total impressed electromotive force F\ one half is in series with the lv/t Fig. 6.9. Two parallel wires energized in push-pull. wires are equal and opposite. Let Fu IT be the values of the potential functions on the surface of the lower wire and F2, n2 the corresponding values on the upper wire; then we.have dVx . _ dBt —— - — icoull!, —s~ ' —fweFii dz dz (6-5) dF2 . _ dJh ■ r/ dz dz everywhere on the wires except where the impressed forces are acting. Subtracting we obtain dV dz — —ZU nil, dU . — = -scoeF, dz diere V=VX-F2 = 2FU n = Hi - n2 = 2Hi. (6-6) (6-7) The potential difference F\s now the transverse electromotive force acting from the lower wire to the upper along any path between the wires, lying completely in the plane normal to them. When the distance between the wires is small, F is a measurable quantity. If the impressed forces acting on the wires are equal and in phase, the currents will also be equal and, then, V and II in (4) refer to either wire. If the generator is in series with one wire, we can replace it by two pairs of generators, one pair acting in push-pull and the other in phase (6-8) \F\ kFl 142 i:\.\xtk<).via(;ni-;tk: wavi-.s Equations (3) apply'only to thus, pans of the wire which are free I...... .mpressed forces. If £'(z) is the impressed intensity, then E{(z) = -£,(«) and dV — = -im-a. + E{(z). 6.7. Current and Charge Distribution on Infinitely Thin Perfectly Conducting Wires We shall now prove that on a perfectly conducting wire of vanishingly small radius the current and charge are sinusoidal functions of the distant I along the wire except in the immediate vicinity of the nodal points of the current and charge and in the vicinity of sudden bends. If I(z) and q(jz) are the current and charge in the wire per unit length, then no dt, (7-1) wire, 4irr ' ' J 4xer where the integration is extended over the length of the r = Va2 + (z - z)2, (7-2) and a is the radius of the wire. As a approaches zero the major contribution to n and /xis made by the current and charge in the vicinity of the point 2 = 2. Thus we shall have approximately dz ii(Z)=/(2) r_^.= where / is small and Calculating k we have 1 r*+l «(4 m~mf i-;,w, (7-3) (7-4) 1 dt + (z - 0 ~ 4x = iloB^ + v/?T^) = f loB^22±-'. I-/ 4tt Va2 + P - / Assuming that a is small compared with /, we obtain „2 Vp + d (7-5) (7-6) AHoliT WAVES IN < .IN IK AI. 143 he quantity /' increases indefinitely as the radius of the wire approaches /.....mil / is kept constant. The contributions to II and V from the rest of 1 lit* wire remain Hnite; heme equations (3) represent first approximations tu II and V, every where except in the neighborhoods of the nodes of / and q, when' 'he principal terms become small and contributions from more >li i .nit parts of the wire must be included in these first approximations. Substituting for II from (3) in (6-3), we have dV dz = —io>fikI, dl dz Mil i,uniting for V from (3), we have also dq . dl — = - Ucfxel, — = dz dz k ' Iwq. (7-7) (7-8) Flo. 6.10. A bent wire. The second equation in this set is really exact; it may be obtained directly limn the principle of conservation of charge. Thus we have proved the theorem stated at the beginning of this section for the case of Ht might wires. If the wire is curved (Fig. 6,10), our arguments are still valid except in the immediate vicinity of " angular points " where the wire suddenly changes its direction. In this case the exact equation connecting the scalar potential with the sector potential becomes dV ■ * — - —tconAs. ds The same approximations can be made as for straight wires except in the vicinity of angular points. Our conclusions are still valid if the radius of the wire is varying so long as the rate of change is finite. Let us now return to the case of two parallel wires energized in push-pull (Fig. 6.9). Here III, as defined in the preceding section, consists of two parts Hi = u[ + n{', each due to the current in one of the wires. If the distance d between the axes of the wires is small, the approximate expressions (3) «re particularly good since the contributions from distant portions of one wire are nearly canceled by contributions from similar portions of the other wire. For k we have AvJ-lWd' + X2 Vd2 + X2J dx 1 d — log-» lie a Ml li.!''.(' i'l'< )MA(; nI /11 c' w.AV i ■:; i'llAIS <\ as Icing as d and a arc hot li small compared with /. Thus k has become in dependent of the indefinite length /. liy (6-7) and (3) we now have dV . r. dl . — — —tuiLI, — = —twCF, dz dz /here H d L — - log - i tt a C = (7-9) (7-10) log : 6.8. Radiation from a Wire Energized at the Center Wre are now in a position to calculate the power radiated by an infinitely thin perfectly conducting wire energized at the center.* W'e have proved that the current distribution is sinusoidal; the ends of an infinitely thin wire must be current nodes; and the current I(z) must be an even function of the distance z from the center (Fig. 6.11). Therefore I(z) = I sin ß(l -z), z> 0; F:g. 6.11. Current distribution on a wire of finite length energized at the center. (8-1) = / sin 3{l + z), z < 0; where / is the maximum amplitude of the current. If the length 21 of the wire is equal to a half wavelength, (1) becomes I(z) = I cos ßz, (8-2) where the maximum amplitude is now at the center. The radiated power can be calculated by (4-22). Using only the first three terms of the power series for k\2, we obtain (for free space) W = \RP, R = 73.2 ohms. (8-3) More accurate calculation gives R = 73.129. In Chapter 11 we shall prove that the exact value of the input resistance depends on the radius of the wire, particularly for lengths greater than a half wavelength; there we shall obtain expressions for R as well as for the reactive component of the input impedance as functions of the radius of the antenna. 6.9. The Mutual Impedance between Two Current Loops; the Impedance of a Loop Let us now consider two current loops carrying uniform currents Ii and J2 in phase with each other (Fig. 6.12). The component of the vector * Or at any other point for that matter. The general formulae will be obtained in Chapter 9. ABOUT WAVES in GENERAL 145 itential due to 7a> along the element dsu is Li C e'^" cos Ý j2 r ť-.p.i, cos^ 4tt ,/ r12 (9-1) • here ip is 'he angle between the dements n r r sin ßr12 Riz = ~r~ j i -cos \p ds\ ds2, Air J J J'i2 ufi r r cos ßrl2 . , , Ai2 = — J J —--cos ý dsi ds2. (9-4) /\'i2 represents the mutual radiation resistance. The above expressions are exact if I\ and I2 are uniform as we have assumed; but uniform currents can be sustained only by properly distributed impressed forces. The usual method of energizing a loop is to impress an electromotive force between a pair of terminals (Fig. 5.1.) In section 7 it has been shown that the current distribution on an infinitely thin wire IF, ELECTROMAGNETIC WAVES Chap, o T is represented l>y a sinusoidal function o| the distance along the loop. Furthermore the current entering the loop trom the generator at ./ equals the current leaving the loop at B; hence the current is an even function of the distance s from the midpoint C of the loop. The even sinusoidal function of J" is cos fts, and this is nearly constant for small values of s. Thus the above equations should apply approximately to small loops energized by concentrated impressed forces. Furthermore for loops which are not too far apart we have approximately wju f f cos g 4?r J J ?*i2 (9-5) neglecting ^j32r12 and smaller terms in the integrand. This is seen to be proportional to the frequency and the coefficient of proportionality 4a jj^ r r co ~ J J t cos ý dsi ds<2 (9-6) is the mutual inductance between the two loops. For loops of small but finite radius the integration in these double integrals is performed along the axes of the wires, although for more accurate computation the wires must be divided into elementary filaments. If the loops are coincident the mutual impedance becomes the self impedance of the loop. Except at rather low frequencies the current is distributed near the surface of the wire. The vector potential of such a current distribution, at points external to the wire, can be computed by assuming that the current is distributed along the axis; but the second integration should be performed where the current actually happens to be and the corresponding curve of integration in the above double integrals must be taken on the surface of the wire. In particular these remarks should be kept in mind in computing the self-reactance or self-inductance of a loop; when calculating its radiation resistance, no great error is made if both integrations are taken along the axis of the loop. The reason for this is that the error involved in shifting the second path of integration from the surface to the axis is greatest for small values of 7*12, and for these values the integrand in /?i2 is nearly independent of r\2. On the other hand the greatest contribution to X%2 comes from small values of r\2- Even if the current is distributed throughout the cross-section of the wire, the " external " inductance of the loop is obtained from (6) by integrating once along the axis of the wire and once along a parallel curve on its surface. The " internal " inductance is computed separately. The mutual impedance between two closed uniform current filaments ABOUT WAVES IN GENERAL an also lie expressed as follows: 12 147 (9-7) here *is is the magnetic flux through the first loop due to the current in the second. Equation (6) shows that $2i = $12. The radiation resist- 11.1 appears through the component of i>12 in quadrature with I2. For the power radiated by two loops carrying currents differing in phase l)J) <\ the mutual power term contains the factor cos d; this factor appears 111 the expression for 4' for any transducer as shown by equation (5.2-8). II two current elements or two filaments are in quadrature they radiate Independently of each other. 6.1.0. Radiation from a Small Plane Loop Carrying Uniform Current 11 the loop is small, we may obtain an approximate value for the radiation resistance by retaining only the first two terms of the power series for Htit firi2 in (9-4); thus we have R=^ff^~ 4*W cos f dsx ds2 VP 4 ~ J* ft™ 4> dsi dtg?-~ >~ jf J* r?2 cos if/ ds, ds%. (10-1) Let Al 1 1 le 1 1 .IB = 1, we have VjlB + Vuc + Vco + Vda = (11-1) If the internal or surface impedances of the wires per unit length are Z\ I and Z->, then for the particular trans- Fig. 6.14. Two parallel wires. mission mode in which the currents in I he wires are equal and oppositely directed we have VAg = ZJ, Van = Z2I. I f V is the transverse voltage between the wires, then dV VBC + Vda = Vbc - Vad = -r • az Finally, $ is proportional to /; consequently equation (1) becomes (11-2) (H-3) dV — = - (Zi + Z2 + io>L)L az (11-1) ATT A J Fig. 6.15. A closed path on the surface of a wire. Applying Ampere's law to the circuit ABCDEFA on the surface of the lower wire (Fig. 6.15) we have Uab + Ubcd + Ude + Uefa. — It, or Ubcd + Uefa — h- The left side is the difference between the currents flowing in the wire at A and B; if AB = 1, then dl Ubcd + Uefa — —y — If dz (H-5) 150 ELECTROMAGNETIC WAVES < "ll.lľ. Ii On the other hand the transverse current is proportional to the voltage /, = (G + io>C)F, (H-6) and consequently — = -(G + iuC)r. (H-7) dz The transmission equations (4) and (7) apply equally well to coaxial cylinders (Fig. 6.16); only the expressions for L, G, C are different. If-the radii of the wires or coaxial cylinders vary slowly with the distance z along -1-7 I / I I J ' Si I Fig. 6.16. Two coaxial cylinders. Fig. 6.17. Illustrating a possible distribution of the longitudinal displacement current inside a metal tube. Fiq, 6.18. Electric lines inside a metal tube. the wires, then L, G, C are functions of z. Generally these transmission equations are approximate; but in Chapter 8 we shall find that under certain conditions they may be exact. Let us now remove the inner conductor of the coaxial, pair and see if wave transmission is still possible. The return path for the current in the metal tube is now the dielectric inside the -^-j—-!y-^- tube. If the tube is perfectly conduct v i ing, the longitudinal electric intensity __________|a___„„bj___must vanish on the boundary and ~~i*~ the longitudinal displacement current might be distributed as shown in "* Fig. 6.17. The lines of displacement Fig. 6.19. A metal tube and a rectan- current flow would then look like gular path formed by two radii and the ^ y 6 ,g jf the fiejd ig lines joining their ends. ■; . , symmetric, magnetic lines are circles coaxial with the tube. Let V be the transverse voltage from the axis of the tube to its periphery and / the total longitudinal displacement current (Fig. 6.19). Applying Faradayr's law to a rectangle ABCDA we obtain equation (1). The voltage FCd is given by (2); similarly, we have equation (3): but Fab is now equal to the longitudinal electric intensity on the axis Fab = Eo. (11-8) ABOUT WAVES in GENERAL 151 In the case of a coaxial pair this voltage is small, equal to zero, in fact, if the inner conductor is perfect; but with no inner conductor there is every reason to suppose that it will prove to be significant. Assuming that the longitudinal electric intensity is maximum on the axis, we have ■Bt = Eaf(p), /(0) = 1. (11-9) Since the total current is / = iuxEoJ* Jf(p)p dp dip, £° = ii? > (if **)s> wu have where S is the area ofithe cross-section of the tube. The quantity in paren-llicses is the average value of/(p) over the cross-section of the tube. * is proportional to I but the coefficient of proportionality L is naturally different from that for coaxial pairs. Equation (1) now assumes the following I.....i dV ( — m -tZa + iaL 4-^)1. (n_n) rtoC / This equation differs from the equations for coaxial pairs and parallel pairs in that it contains a term representing distributed series capacity C. This, of course, was to be expected. The second transmission equation is simply the equation of conservation 01 electric charge dl . dz * where q is the charge on the surface of the tube per unit length. This charge is proportional to the radial component Ep of the electric intensity and therefore to the transverse voltage V; taking into consideration our convention with regard to the positive direction of F, we have dz (11-12) Comparing (11) and (12) with (5.7-6) and (5.7-7) we find that the metal tube behaves as a high pass filter. The cutoff frequency is determined by (11-13) 'LC 152 ELECTROMAGNETIC WAVES Chap, n It is easy to make a rough estimate of this frequency. From the physical picture underlying the present transmission mode it is clear that the error will not be excessive if we assume a uniformly distributed longitudinal displacement current. Then f(p) = 1 and C = eS = tira'1. In this case the inductance per unit length is L ~ fi/itr. Substituting in (13), we have (H-14) = 2. Thus the cutoff wavelength is equal roughly to the circumference of the wave guide divided by 2.* Longer waves are not transmitted. The exact cutoff is determined if we use 2.40 instead of 2. More powerful methods for obtaining the cutoff frequencies will be described in later chapters. The above estimate has been made in order to show that, starting from a physical picture of a given field and applying the electromagnetic laws in their integral form, it is possible to obtain qualitative and even fairly satisfactory quantitative results. Fig. 6.20. Cross-sections of metal tubes of rectangular and semi-circular cross-section and magnetic lines. In tubes of noncircular cross-section magnetic lines will be deformed (Fig. 6.20), the numerical values of Z,, C, C will be altered, but the essential picture will remain the same. We can even make an estimate of the cutoff wavelength by expressing Xc for the circular guide in terms of the area of the cross-section instead of the radius. The direction of the conduction current in the tube is related to the direction of the magnetic lines of force. If the magnetic lines are counterclockwise, the current Fig. 6.21. Illustrating 'm ule tlIue uows away from the observer. Consider a transmission mode now a circular tube with an infinitely thin perfectly with two sets of conducting axial partition (Fig. 6.21) and assume closed magnetic lines. ,L r i ■ • r i 1 that waves of equal intensity or the type shown In Fig. 6.20 have been set up in such a way that at all times one set of magnetic lines is counterclockwise and the other clockwise. The total current in the axial partition is zero and the partition has no effect * On the energy basis L = p/Btt and \c = luaflZ. Much better results are obtained by taking/(/>) = ].—p-/a° or cos Wp/la) so that/(rt<) = 0. ABOUT WAVES IN GENERAL 153 i,n the field inside the circular tube. This partition can, therefore, be .......veil and we are I■ 11 with a new mode ol i rnnsmissi'm in a circular cube. In I his mode the conduction current flows in opposite directions in opposite halves of the tube; longitudinal displacement currents also flow in opposite directions. The cutoff frequency for this mode is higher than that for the first mode; the ratio of these frequencies is equal to the ratio of the eu toff frequencies for the first mode in the circular tube and in a semicircular tube of half the area. Hence, the approximate ratio of the two cutoff frequencies is V2 or 1.4; the exact ratiois nearly 1.6. The magnetic lines " avoid " the corners in the semicircular tube and this tendency makes i In i ffective area of the. tube smaller than the actual area. This synthetic method of construction of field configurations can be intended. The circular tube can be divided by radial planes into an even .....iiber of sectors. Assuming a wave in each sector, traveling in the first (©) (©) Fig. 6.22, A transmission mode with six sets of closed magnetic lines. Fig. 6.23. A possihle mode of transmission in a tube of triangular cross-section. mode, and assuming relative directions of magnetic lines so as to make radial planes current free and hence removable, we obtain a sectorial wave in the circular tube. Any rectangular tube can be divided into equal rectangular tubes of smaller cross-section (Fig. 6.22); assuming the first mode in each in such a way that the adjacent lines of magnetic force point in the same direction, we obtain a higher transmission mode in the original tube. All these field configurations can be constructed on the basis of symmetry. They furnish us with qualitative ideas when symmetry is no longer a guide. We feel certain, for example, that in the tube whose cross-section is shown in Fig. 6.23, there exists a mode with two sets of magnetic lines of force as shown in the figure; but without more complete analysis we do not know just how the available space is divided between these sets of lines. All we can say is that the area of the left sector will be larger than that of the right sector because the magnetic lines avoid corners, particularly sharp corners. The magnetic lines surround the longitudinal displacement current which is proportional to the longitudinal electric intensity; but the latter must vanish on the boundaries of the tube and it will approach zero more rapidly when two boundaries are close together. In ELECTROMAGNETIC WAVES t ' 11 A 1-, (i coaxial pairs similar waves can exist; thus in the circularly symmetric case tlie longitudinal displacement current may be distributed as shown in Fig. 6.24. All waves of the above type are called transverse magnetic waves or TM-waves because the magnetic vector is perpendicular to the direction of wave propagation. If the electric vector is perpendicular to the direction of wave propagation, then the waves are called transverse electric waves or TE-waves. Finally, if both vectors are perpendicular to the direction of wave Fig. 6.24. A possible distribution of the longitudinal displacement current between coaxial cylinders. Fig. 6.25. A rectangular metal tube. propagation, then the waves are transverse electromagnetic (TEM-waves). In general both field intensities have longitudinal and transverse components; such waves are called hybrid waves. There are no electromagnetic waves in which either the electric intensity or the magnetic intensity is totally longitudinal. A general idea of transverse electric waves may be obtained as follows. Consider two parallel metal strips, whose width is large compared with the distance between them. Such strips form a transmission line similar to a pair of parallel wires. Between the strips the electric field is almost uniform, except near the edges; the magnetic lines surround each strip and between the strips they are nearly parallel to them. The wave is transverse electromagnetic. The longitudinal currents in the strips flow in opposite directions and the circuit is made complete with the aid of transverse displacement currents. Let us now connect the edges of the strips metallically and form a rectangular tube (Fig. 6.25). The electric intensity, which we assume to be parallel to thejy-axis, must vanish at the boundaries to which it is parallel. Let us assume that E is maximum in the middle plane ABCD and that it is distributed as shown in Fig. 6.26. Magnetic lines cannot cross the conducting boundaries and must form loops (Fig. 6.27) surrounding the transverse displacement current. The longi- AKOI IT WAVES IN GENERAL 155 tudinal magnetic intensity is associated with the transverse conduction current in the tube. Now let I be the total longitudinal current in the lower face of the tube and —/ the corresponding current in the upper face.' Let V be the voltage from the lower face to the upper along a typical " central " line AB. This * 'I i age is equal to the total longitudinal magnetic current flowing through mm Fig. 6.27. Magnetic lines in planes normal to the E-vec tor. Kio. 6.26. A possible distribution of the transverse displacement current in a rectangular tube. i he rectangle ABEF in the negative z-direction, or the total current through ABGH in the positive direction. Assuming that the tube is perfectly conducting and applying the first law of induction to the rectangle ABCD, we obtain the first transmission equation. Expressing the variation of I with 2 in terms of the shunt displacement and conduction currents we obtain the second transmission equation. Thus we have dV_ dz — iíúLI. Thus in the case of transverse electric waves the tube also behaves as a high pass filter (Fig. 5.19). The constants can be calculated if more specific assumptions are made with regard to the distribution of the transverse displacement current. However, in Chapter 8 wc shall obtain the complete and exact solution of this problem. The purpose of the present discussion is to stimulate the development of physical ideas as we proceed with mathematical analysis. In Chapter 10 we shall solve rigorously the problem of cylindrical wave guides and confirm the existence of an infinite number of transmission modes. Kach mode is characterized by a definite field pattern in a typical plane normal to the guide. This field pattern determines completely the constants in the transmission equations (11) and (12) or (15), depending upon whether the wave is of transverse magnetic or transverse electric type. The propagation constant and the velocity in the guide depend upon the frequency and the particular transmission mode. The cutoff frequencies for various transmission modes may be arranged in ascending order of magnitude. The lowest of these frequencies is called the absolute cutoff frequency for the guide and the corresponding mode is the principal or the 156 ELECTKOMAUNETIt: WAVES Chap. 6 AHOUT WAVES IN (JEN.ERAL 157 dominant transmission mode. If the guide is energized tit some frequency lower than the absolute cutoff frequency, the propagation constants fat all modes are real (if there is no dissipation) and the field intensity ap. proaches zero with increasing distance from the generator. If, however, the frequency is above the absolute cutoff but below the next higher, then at a sufficient distance from the generator the wave will be traveling along the guide substantially in the dominant mode. AJ1 the other modes represent only the local field in the vicinity of the generator. As the frequency Increases and passes successive cutoff frequencies, the energy supplied by the generator will be transferred along the guide in an increasing number of transmission modes. Transmission modes are analogous to oscillation modes in electric networks. A simple series circuit has only one natural frequency and one oscillation mode; an w-mesh network has n oscillation modes; and a section of a transmission line has an infinite number of oscillation modes. Actual physical circuits are always multiple circuits, possessing in fact an infinite number of oscillation modes. The lowest natural frequency of some circuits, however, is so much lower than all the others that in a limited frequency range they may be approximated by simple circuits possessing only one natural frequency. Similarly all physical wave guides admit of an infinite number of transmission modes; but some wave guides, such as coaxial pairs, admit of one mode for which the cutoff frequency is zero and of other modes with very high cutoff frequencies. In a restricted frequency range such wave guides may be treated as simple transmission lines, possessing only one transmission mode. The absolute cutoff of metal tubes is high and the cutoff frequencies of higher modes are close to it (on the ratio basis); in such cases the existence of other transmission modes cannot be forgotten even if the operating frequency is such that only the dominant mode takes part in energy transmission. However, at frequencies between the first and second cutoffs, the higher transmission modes "represent only local fields in the vicinity of discontinuities such as generators, receivers, sudden bends or changes in the transverse dimensions of the guide. Under these conditions the wave guide acts as a simple transmission line in which the local fields associated with the discontinuities are represented by reactors either in series or in shunt with the line. 6.12. Reflection In sections 1 and 2 we have calculated the field produced by a given distribution of sources in an infinite homogeneous medium. Let us now suppose that the medium consists of two homogeneous regions separated by a surface (S). Without loss of generality we may assume that one of these regions is source-free. If the sources are distributed throughout both If the two Fig. 6.28. A surface enclosing the source of the field. ,, Ki(,iiH, we may regard flic hit il liehl as due in the superposition ol two lirhis, each produced by sources located in one region only. Tims let the sources be in region (1) as shown in Fig. 6.28 legions had the same electromagnetic properties, the field of these sources would be found from the equa-.....IS of sections 1 and 2. But when the electromagnetic properties are different the field (E\IP) i bus obtained is not the actual field. In region (1) It represents the primary field of the sources and is culled the impressed field. The field (Er,Hr) which must be added to give the actual field in region (1) is l ailed the reflected field. We may think of the reflected field as produced by polarization currents in region (2); in so far as these virtual* sources are concerned region (1) is source-free and the reflected field should satisfy the homogeneous form of Maxwell's equations curl m = -irf, curl UT = (ffi + i^i)F/. (12-1) i ,et the actual field in region (2) be {E^H1); this field is called the trans-milted (or "refracted") field and it also satisfies the homogeneous equations curl El = -i^2H\ curl W = % + Me2)El. (12-2) At the interface (S) of the two media the tangential components of £ and // are continuous Ei + El = 4 ^ + H< = ^ (12"3) This set of equations constitutes one formulation of the problem of determining the field of a given system of sources when the medium consists of two homogeneous regions. The method can be extended to any number of homogeneous regions. If the boundary (S) is a perfectly conducting sheet, then the tangential component of E should vanish on (S) Ej + Et - 0, or E\ = -E{. . (12-4) A perfectly conducting sheet can support finite electric current and the tangential component of H is no longer continuous across (S). In fact, energy cannot flow across a perfect conductor and the field in region (2) due to the sources in region (1) is equal to zero. The component of H tangential to (S) in region (1) represents the current density / on (S). By the second law of induction J is normal to the tangential component of H; hence if n is a unit normal to (S), regarded as positive when pointing * As distinct from true sources. 158 MM i Ui (MAGNETIC WAVES CttAfi 0 ABOUT WAVES IN GENERAL Into the source-free region (2), then J = 0t + //<) X n. (12-5) Since the vector product of n and the normal component of // is zero, wo can drop the subscript " tangential " and write J = (H{ + Hr) X n. (12-6) More generally if the impedance Z„ normal to the boundary is prescribed, then the relation between the tangential components in region (1) is B^ZMlXn, or E\ + Ej - Z»(H} + HI) X n. (12-7) Wave propagation in wave guides may be regarded as a case of reflection, We start with a certain system of sources inside a metal tube, for example; then the total field inside the tube is the sum of the impressed and reflected fields in the sense defined in this section. 6.13. The Induction Theorem Let us rewrite (12-3) as follows E\ - Et - Ei, H[-H\ = Hi, (13-1) and concentrate our attention on the " induced " field (E,H) consisting of the reflected field (Er,Hr) in region (1) and the transmitted field (£',#<) in region (2). This field satisfies the homogeneous equations (12-1) and (12-2) everywhere except on (S) and it may be obtained from a distribution of sources on (S) as well as from the original sources. It has been shown in section 4.5 that the discontinuities in E and H across (S) could be produced by current sheets on (S) of densities M = (El - ES) X n = E\ X 7i, (13-2) J = n X (Ht - HI) = k X HI- Since the vector product of n and a normal component of the field is zero, we have M = EiXn, J = n X H\ (13-3) Thus if we wish to determine the field whose only sources are the currents on (S) given by (3), we have to solve exactly the same equations as those used in the preceding section to obtain the induced field. In other words the induced field (E,H) could be produced by electric and magnetic current sheets of densities given by (3); this is the Induction Theorem. 6.14. The Equivalence Theorem Let us now suppose that (S) is a surface in a homogeneous medium, separating a source-free region (2) from the rest. In this case the " reflected " luld is evidently /.em and the transmit led field is the actual field in the ........(. (Vee region. Thus we obtain the following Equivalence Theorem: lb,- lichl in a source-free region bounded by a surface (S) could be produced by R distribution of electric and magnetic currents on this surface and in i lir sense the actual source distribution can be replaced by an "equivalent" 11 11 ibution (13-3). 6.15. Stationary Fields Stationary fields are fields independent of time and may be regarded as . iril cases of variable fields. For example from (1-10) and (2-14) we obtain the following expression for the electrostatic field produced by a Hi von distribution of electric charge in a perfect dielectric dq Aiver E = -grad V, V -h (15-1) where dq is a typical element of charge. The function V is now called the I lectrostatic potential. A similar expression may be obtained for the niiignetostatic field of a given distribution of magnetic charge H= -grade/, U = f^, (15-2) i J iirnr where dm is a typical element of magnetic charge. In this case the function /' is called the magnetostatic potential. These expressions are also the limits of the harmonic field when w approaches zero. In an infinite homogeneous conductor we have from (2-13) and (1-8) A CiL J Airr V - div A, t (15-3) where dp is the moment of a typical impressed current element. From (1-10) we have E = -grad V, H= curl A. (154) The field due to currents in conductors surrounded by a homogeneous dielectric medium can be obtained from curl II = gE + f (15-5) if we use (1.8-6) and recall that in the absence of magnetic charges div H = 0. Thus we have H = curl /// 4xr dv. (15-6) This formula can also be used, of course, for homogeneous media but it is more complicated than (3) and (4) which give H in terms of the impressed IM) ELECTROMAGNETIC WAVES i 11 ABOUT WAVES IN GENERAL 161 currents alone. Tims die latter equations give immediately the licld of ail impressed current element (equations 2-3, 2-5, 2-6) while this could onl) be obtained from (6) by integrating over the entire infinite medium. 6.16. Conditions in the Vicinities of Simple and Double Layers of Charge Consider a stationary distribution of electric charge on some surface ( ' | or a simple layer (Fig. 6.29). The normal component of E is discontinue mi across the layer; thus by (4,3-2) we have En. j? IS (16-1) where qs is the charge on the layer per unit area and « is the dielectric constant of the surrounding medium. The tangential component of E is continuous. In terms of the electrostatic potential V these boundary conditions become dV\ ds ev2 ds ' dV\ dn dn (16-2) From the expression for the potential in terms of the charge distribution it is evident that the potential is continuous across the simple layer; this condition implies the continuity of the tangential components of V. Fig. 6.29. A surface layer of charge. Fig. 6.30. A double layer of charge. Two close layers of equal and opposite charges constitute a double layer (Fig. 6.30). Such a layer may be subdivided into elementary doublets. If is the charge per unit area and / is the separation between the simple layers, then dp = qgl dS is the moment of an elementary doublet. The moment x per unit area is called the strength of the layer. For an ideal double layer / is vanishingly small and qs is infinitely large, while their product x is finite. We have seen that the potential of a doublet is 4xer2 (16-3) where ^ is the angle made with the axis of the doublet by the line joining doublet with a typical point; /' (Fig. 6.31). The potential of the entire luycr is therefore 1 C C X cos 4> , „ My the definition of the double layer and by (4.3-2) the normal com- l.....nt of the electric intensity is continuous; hence this is also true of the n.....ial derivative of the potential of a double layer in a homogeneous medium. On the other hand, the potential itself is discontinuous. Inside i he layer the electric intensity is — qs/t and the potential rise across the layer in the direction of the normal |l indicated in Fig. 6.30 is 781 (16-5) + + Vn = In terms of this potential discontinuity across the FlG_ 6 3L An eleraent layer, the potential outside the layer given by (4) be- of a double layer, comes If a conical surface is generated by sliding the radius from a fixed point ah aig a closed curve, the space enclosed is called a solid angle. The measure [) of the solid angle is the area intercepted by the angle on a unit sphere with its center at the apex of the solid angle. The solid angle at P subtended by an element of the double layer of area dS (Fig. 6.31) is dti = —s— dS, (16-7) 11 eos ip is positive. We shall regard this equation as defining the solid angle subtended by a " directed element of area " by permitting cos ^ to a time negative values as well as positive. This will make the solid angle subtended by a closed surface zero for an external point and ±47r for an internal point. Substituting from (7) in (4) and (6), we have I f the layer is uniform, then V(P) = ^ = r FoQ> 4ire 4jt (16-9) where fi is the solid angle subtended at P by the layer (Fig. 6.30). 162 laFCTIUlMAGNKTIC WAVF.S ClIAl', (J Similarly the potential of a magnetostatic double layer in an infinite homogeneous medium is (16-10) where x is the strength of the magnetic layer, defined as the magnetic moment per unit area, and U0 is the sudden rise of the magnetic potential in passing across the layer. If the layer is uniform, then •ZiTfi, 4ir (16-11) For an infinite homogeneous conductor the potential is given by (8) with g in place of e. 6.17. Equivalence of an Electric Current Loop and a Magnetic Double Layer Consider a, uniform magnetic double layer (Fig. 6.30) of strength x = pU0. The magnetomotive force along a path ABC leading from a point A on the positive side of the layer to an opposite point Con the negative side is Uq, since the total magnetomotive force round ABCA is zero. Imagine now an electric current loop along the edge of the double layer and let the current I in the loop be regarded as positive when it appears counterclockwise to an observer on the positive side of the layer. The magnetomotive force of the field produced by this current, round any contour such as ABC in Fig. 6.30 is I. Thus in so far as points external to the layer are concerned, the layer and the loop are equivalent if 1= U0. (17-1) Inside the layer the two fields are, of course, very different. Substituting from (1) in (16-11) we have the magnetic potential of an electric current loop Itt V{P) = 4xJ (17-2) at all points outside some surface (S) bounded by the loop. Let us now consider an infinitely small plane current loop of area S and the corresponding magnetic double layer. The total moment of this magnetic doublet is pIS. Assuming that / is variable, the magnetic doublet becomes a magnetic current element of moment P = Kl=pS^, (17-3) where K is the magnetic current and / is the length of the element.* * In dealing with magnetic current elements we are concerned only with the moment A7, and apart from this product neither K nor / need have definite values. ABOUT WAVES IN GENERAL 163 For harmonic currents we have p- Kl= iosfiSI. (17-4) This relationship between elementary current loops and magnetic cur-i. hi elements makes it very easy to obtain the field of the loop. We have already calculated the field of an electric current element of moment //. Wt: have also seen that the fields of magnetic currents are obtainable from an electric vector potential F which differs from the magnetic vector poten-n,il A only in that magnetic currents appear in the place of electric cur-tents. Thus for an electric current loop in a nondissipative medium we inive i^SIe~itiv F = Em — Kle~™ 4irr 10* SI 4-irr lap Sle 4flT 4xr ^T sin 6, ffr far \ i§r 0*r2] (17-5) 2xr2 V tjSrJ cos 6. At great distances from the loop the field is fSIe^ sin | irWr* sin 9 H8 = Ev = — ijflg, while near the loop it is far XV HT = 0, He = SI sin 0 Hr « SI cos 0 iwpSI sin 9 far2 (17-6) (17-7) A large loop carrying current I, uniform over the loop but varying with time, is also equivalent to a uniform double layer over a surface {S) bounded by the loop. In order to show this we need only imagine that (S) is divided into a large number of elementary loops filling the entire surface, each carrying current I in the same 6'32- Represcn- , ,.„ , rv a-..s cation or a large direction (big. 6.32). The electromagnetic effects of the loop ca|.ryi]lg ,]m_ currents in adjacent sections of the elementary loops form current by cancel our, and the system of loops is equivalent to subdivision into „ . . , . . , small loops, each the large loop. But each elementary loop is equivalent cat,.^llg the same to a magnetic doublet or to an element of the double current. IM MM 11\< >MAGNE'J li WAVES Chap, r> layer, ami hem r I lie loop as a whole will he equivalent lo a uniform double layer over (S). 6.18. Induction and Equivalence Theorems for Stationary Fields Next in simplicity to a homogeneous infinite medium is a medium which is homogeneous in each of two regions (1) and (2) separa.ed by a closed surface (S) (Fig. 6.28). Let us suppose that we have a distribution of electric charge in region (1) while region (2) is source-free. Let F{ be the potential of this distribution in an infinite medium with a dielectric constant «i, equal to the dielectric constant of region (1); we shall call this potential the impressed potential and the corresponding field the impressed field. Let the difference between the actual potential in region (1) and the impresse I potential be Fr; we shall call this the reflected potential and the corresponding field the reflected field. Finally let the actual field in region (2) be represented by the transmitted potential Fl. The reflected and the transmitted potentials satisfy Laplace's equation Af = 0, AF' = 0. (18-1) Assuming that there are no sources on the interface (S) between the two regions, we have the following conditions to be satisfied over (S): yi + yr = yt dn du «2 OF' dn (18-2) The first of these conditions states that there is no double layer of charge over (S) and the second that there is no simple layer. The above equations together with supplementary requirements of finiteness, continuity and proper behavior at infinity suffice for the calculation of Fr and F*. Equations (2) may be rewritten as follows yt _ yr F\ dF* dFr QF* ~— — «1 "~7 = é1 " (18-3) dn dn dn Suppose now we have a double layer in region (1) at the boundary (S), with potential discontinuity F\ and a simple layer of density 3Fl qs = £\^~. • (18-4) dn Furthermore let the rest of space be source-free. In order to obtain the field of these surface sources we have to satisfy equations (1) and (3) and supplementary requirements of finiteness, continuity, and proper behavior at infinity which are the same as in the previous problem. In other words the two problems are indistinguishable and the field consisting of the reflected field in region (1) and the transmitted field in region (2) may be produced by the postulated simple and double layers of electric charge. This is the electrostatic version of the Induction Theorem. If the dielectric constant of region (2) is equal to that of region (1), then there is no reflected field and the transmitted field is identical with the impressed field given by F\ The induction theorem becomes now an equivalence theorem which states that the simple and double layers defined by (3) produce a field which is equal to zero ABOUT WAVES IN OKNKRAl, IfiS Í.33. A system conductors. in region (1) and to the aeiual Held in region (2); that is, the field in the source-free region (2) produced by a system of sources distributed throughout region (1) may also be produced by a proper system of sources over the boundary (S) separating the two regions. In the situation contemplated in the above equivalence theorem the entire space is homogeneous and the expression for the potential of the simple and double layers over (S) can be written at once. Dropping the superscript i, we have í)F F cos Tp +—3 ^ 4xS \_r dn ^dn \r)_ {/{-P' ~ 4ir J Jm \rdn ' r2 }~" 4x. (18-5) The magnetostatic field can be treated similarly. f>. 19. Potential and Capacitance Coefficients of a System of Conductors Consider a system of n conductors Ki, Kz, ■ ■ ■, Kn (Fig. 6.33) with total charges respectively equal to q\, q2, • - ■ , qn- The potential F is a linear function of the total charges on the conductors F{P) = fm +fm + • •' +/«?», C19-1) where the coefficients fu/h "'■>/«■ are functions of position. The function/„, represents the potential due to a unit charge on the »3th conductor when the remaining conductors have zero charges. On a conductor electricity moves freely so that, when a steady state has been reached, the tangential component of the electric intensity vanishes and the surface of the conductor becomes an equipotential surface. Designating by Ft, F2,"-,Fn the potentials of the conductors, we have Fx = jpnyi + pi2?2 + pisqi +----h pi«?n> Fi = piiq\ + p2tq2 + pwqz 4- • ■ • + pinqn, .................................... (19-2) Fn = pniqi + pn-m + Pmq% + ■ ■ • + pnnqn, where the p's are the corresponding values of the f's. These coefficients are called the potential coefficients. The setplm, p■ ■ ■ , pnm represents the potentials of the conductors when a unit charge is placed on the mth conductor while the other conductors remain uncharged. Solving (2) for the q's we have + (z - h 4- Incf 4x« V7+ (a + A + 2«)5 * (22-7) where p is the distance from the line of charges. The method of images can be used to satisfy other boundary conditions. Thus if the image source in Fig. 6.35 is of the same sign as the given source, the normal derivative of the potential will vanish at the plane. This is the boundary condition at a perfect " magnetic conductor " in the case of electrostatic fields, at a perfect electric conductor in the case of magnetostatic fields, and at a perfect insulator in the case of steady electric current flow. In each case either the normal component of displacement or the normal component of current density is required to vanish. Thus if instead of a point charge q in Fig. 6.35, we have a point source of electric current 2, ABOUT WAVES IN GENERAL 171 then the Potentin! in the semi-infinite homogeneous conductor bounded by a perfectly insulating plane is V '.(' + wg\r (22-8) where g is the conductivity of the medium. To summarize: the image of a simple source in an infinite plane is equal in strength In the source but of opposite sign if the potential or the tangential component of the Held intensity is required to vanish at the plane; the image is equal in strength to (lie source and has the same sign if the normal derivative of the potential or the normal component of the field intensity has to vanish at the plane. +1 -ct» -a it ■f m -m We- +1 !-m im •—»—• it -It Kt +m Fig. 6.38. i ■m -Kt ■Kt, The images of various doublets and current elements in a perfectly conducting plane. This rule can be broadened to include doublets and current elements. Consider for example a perfectly conducting plane and a variety of doublets and current elements (Fig. 6.38). The images of the following sources have the same sign as the sources: the electric doublet and the electric current element normal to the plane; the magnetic doublet and the magnetic current element tangential to the plane. The remaining sources have images of opposite sign: the electric doublet and the electric current element parallel to the plane; the magnetic doublet and the magnetic current element normal to the plane. It is easy to verify that the above ride applies to electric and magnetic current elements with variable moments. The rule for the images of electric loops is identic.d, of course, with the rule for magnetic current elements. I 12 ELECTROMAGNETIC WAVES Chap, tj ABOUT WAVES IN GENERAL 17;! Wo shall now extend tin- mci Inn I of imagi-s in anudier direction. I.cl the plane In .in interface between two homogeneous dielectrics (Fig. 6.39). Consider a point charge q at point A in the upper medium, As has been explained in suction IN we may regard the total field in this medium as the sum of the impressed lield, defined as the field ol the point charge on the assumption that = «i, and the reflected field. We already know that in at least two cases the reflected field is equal to the one produced by nil image charge at point B, which is the geometric image of point A. Thus if e2 = 0, thru the displacement density in the lower medium is identically 'zero and therefore the normal component of the displacement density in the upper medium should vanish at the boundary; in this case the image charge "producing" the reflected field id qT = g. If ti = co, then the electric intensity in the lower medium must vanish, or else the displacement _£i_ density would be infinite; in this case the tangential C2 component of the electric intensity in the upper medium should vanish at the boundary and the reflected field could be produced by qr = —q. Further. Fiq. 6.39. Illustrating the im- more if ^ = eLj tue reflected field should vanish, ^theory for two dielectric Witli this information in mind, we assume tentatively that the reflected field in general could be produced by an image charge at B, having the following value *1 - «3 (22-9) €i + «2 The ratio qr/q defined in this manner reduces to 1, —1, and 0 in the three cases considered above; but naturally this does not mean that (9) is true in general. In fact, we do not even know that in the general case the reflected field could be produced by an isolated point charge; we merely start with (9) as a hypothesis which can be either proved or disproved. The potential of the pair of charges is (22-10) where ?.\ and r2 are respectively the distances from A and B. The potential along the boundary is P+Fr = this potential is equal to thai given by (11). Tims both boundary conditions are satisfied and we may finally say that the field of a point charge q, located at a point A in a semi-infinite homogeneous medium separated by a plane from another semi-infinite homogeneous medium, may be represented as follows; (1) on the same side of the boundary as point A, the field is the sum of the fields which Would be produced in an infinite medium by the original charge at A and by an image I barge qr at B, assuming that the dielectric constant of the medium is en (2) on the Other side of the boundary the field is the same as that which would be produced by a charge q', placed at A, in an infinite medium with the dielectric constant e2. For magnetic fields we have a similar theorem. In the above formulae electric i'lmrges are replaced by magnetic charges and e's are replaced by jx's. The rules for di millets can be formulated very readily since doublets are pairs of point charges. These theorems do not apply in general to variable fields. That this is the case in obvious when the intrinsic propagation constants of the two media are different; the fields of simple point sources cannot possibly be matched along the entire plane boundary. But it is conceivable that such fields could be matched when the propagation constants are the same; and this is actually found to be the case. 6.23. Two-Dimensional Stationary Fields A two-dimensional field is defined as a field depending on two coordinates and, in particular, as a field depending on two cartesian coordinates. While such fields are special cases of three-dimensional fields, the simplifications resulting from the decrease in the number of effective coordinates are so great that two-dimensional fields are usually studied separately. Assuming that tiie field is independent of the z-coordinate, we obtain the following equations for source-free homogeneous regions under different conditions. For an electrostatic field the potential satisfies the two-dimensional Laplace's equation aty tf-y .^_ + ^l = 0, . (23-1) dx% 3ja which in polar coordinates becomes 3 / g/A . bW dp \ dp / dip2 The electric intensity is equal to — grad V\ thus 0. CM Ev~ by'' (23-2) (23-3) E = E M „^_\ bp pdip (23-4) Exactly the same set of equations describes die steady current flow. For magneto- 171 ELECTROMAGNETIC WAVES static Ileitis the i'(|iiat.....ti an- similar, with the.....luetic potential U taking the |>la< <■ of the electric potential /' and // appearing in place of !>'.. Magnetic fields produced by electric currents ate derivable from the vector potential A. which, when the current is parallel to the z-axis, has only one component /-/,. Thus such fields depend essentially on one scalar function A, = U', usually called tin stream function. In terms of this function we have (23-5) W "x - — , ay a* pdip "dp" (23-6) In source-free regions the stream function satisfies equations (1) and (2). A two-dimensional electrostatic field is produced by a system of uniform filaments of electric charge parallel to the z-axis. These filaments may form either a discrete or a continuous set. A uniform line source is an elementary source of such a field in the same sense as a point source is an elementary source of a three-dimensional field. The potential of a line source is independent of the ^-coordinate. Hence from (2) we have Therefore, dp V = P log p + constant. E„=- (23-7) (23-8) If q is the electric charge per unit length of the filament, then by taking the radial displacement over the surface of a cylinder concentric with the filament we obtain litpcEf, = q, Substituting in (7), we have and P = - — '7 2xe y =--log - > 2tt£ a where a is a constant length which remains arbitrary, itself we have 2irep 0. (23-9) ■ (23-10) For the electric intensity (23-11) The last two equations can be derived without using Laplace's equation. Thus (11) follows directly from symmetry considerations and from the divergence equation (4.3-2). Furthermore it is evident that E can be expressed as the gradient of a function depending only on p and that this function is given, by (10). Laplace's equation becomes of real value, however, when only a part of the complete distribution of electric charge is known and the information regarding the remaining charges is replaced by boundary conditions. Consider for example a conducting cylindrical tube and a known line charge parallel to the axis of this tube. Instead ot being given the distribution of electric charge on the cylinder we are required to find it, using the boundary condition that the component of E tangential to the tube vanishes. This time our AIM HIT WAVES IN < JENEKAE 175 pinUoni is lo find ;t telle! led field :,aiiiifyiiii', ('.') and having si tangential component i ,|n d and opposite to the tangential component of the field which would be produced bj i h>' line charge in an infinite medium. In the csisc of magnetic fields produced by a distribution of parallel currents the i Iciiicntary line source is a uniform infinitely thin current filament carrying current I. Hy (4.6-1) wc have 0. (23-12) When the field is steady, there is no displacement current parallel to the filament and i I ') is valid at any distance from the element, not only in its immediate vicinity, i ', imputing (12) with (6), we find that H may be obtained from the following stream I u net ion 1 i P =--log - > 2ir "a (23-13) where a is an arbitrary constant. The value of the stream function becomes evident when we attempt to find the held of several current filaments. Such a field may be I il ulated directly from (12) by adding vectortally the magnetic intensities of the individual current fila-n nuts. On the other hand the stream function is a scalar and the addition of stream functions is much simpler. For example in the case of two filaments (Fig. 6,40) one passing through point (1/2,0) carrying Current / and the other through (—1/2,0) with current —I, we have y si -i 1 2 i z x * = — — log — . 2jt pi (23-14) Fig. 6.40. The cross-section of two infinitely long parallel wires carrying equal and opposite currents. When pi and pt are large compared with 1/2, we have P2 = P + - COS tp, Pi = P - £ cos MAi INETIC WAVES ChaPi d oilier distribution, for which (lie total ciirrcnl is zero, may In- subdivided into pairs nl oppositely directed current (iliirnents. Thus we have the general theorem. The method of images ran evidently he applied to two-dimensional fields. Tin rules for the magnitudes and the signs of the images of line sources are the same as l< h corresponding point sources. 6.24. The Inductance of a System of Parallel Currents Generally, in the case of steady parallel currents, the stream function satisfies the two-dimensional Poisson's equation (24-1) where / is the total current density. The energy of the magnetic field produced by this current distribution may be expressed in terms of ty\ thus the energy per unit length in the z-dircction is W ■ By where the integration is extended over any plane normal to the z-axis. To begin with let us consider the above integral extended over a finite area. Green's theorem we have w = yf t J|U -yjff ds> (24~3) where the line integral is taken over the periphery of the chosen area. Let this area increase indefinitely in both linear dimensions and assume that / is distributed over a finite area. If the total current in the z-direction is different from zero W will also increase indefinitely. On the other hand if the total current is zero, as is the case in practice, then varies ultimately as* 1/p and cM'/dw as 1/p2, consequently the line integral in (3) approaches zero. Substituting from (1) in (3), we thus obtain »'// (24-4) Effectively, this integration is extended only over the areas occupied by the current. Let us now consider a system of parallel wires and let the currents in these wires be uniformly distributed throughout their cross-sections. In this case / is constant for each wire and (4) becomes (24-5) where Si, Sz,-" are the cross-sections of the various wires. Introducing the average values of the stream function "ty over each cross-section (24-6) * ^ may contain a constant which does not affect the field and hence maybe taken as zero. ABOUT WAVES IN GENERAL 177 ml iiotiiiu thill the total current /„, in the will wire is /„, JmSm, we transform (5) ..... //-- - faZ**!*. (24-7) II there arc only two wires, carrying equal and opposite currents, then h - /, J,- -I, uiiI the energy of the field per unit length along the wires is IF = - *»)/. (24-8) Ii. t alues of and ^2 are proportional to I so that W = \U\ • Ii ii 11n eoellicient L is seen to be the inductance per unit length of the wires and its Value is L = n *l — *3 (24-9) For two pairs of wires let Ii = —Ii, Ii = —Iii then equation (7) becomes W = M*i - + |*(¥j - ¥«)/* (24-10) I In average values of the stream functions are linear functions of I\ and Ii: thus MOI'i - *3) = Lull + Li-J2) M(*2 - *4) = £21/1 + Z.22/2. ■ Substituting in (10), we have W = |LuU + |(Ca + L2l)hh + \Lnll. These formulae may be extended to n pairs of wires. Thus in order to compute the inductance coefficients of a system or parallel currents we have to compute first the stream function *te and then its average values over the 1 / I \ +I \ Fig. 6.41. The cross-section of two parallel cylinders. cross-Sections of the different wires. Two examples will illustrate the procedure, let us take a pair of wires of circular cross-section (Fig. 6.41). We have already umed that the current distribution is uniform throughout the cross-section of each I7H MM I K( (MAGNETIC' WAVES ( MM wire. From symmetry ■ < •ir.i.lrr:iii< mis wr cmu'liidr dial (lie si mini I.....linn Im . n |, w iiv, 111 I In' region external in il, is equal In I lie si ream function I ha I would lie oltluiiii i| il the entire current were concentrated along the axis of the wire. Thus for point! external to both wires the st ream function is given by (23-14). Inside the wire A till] magnetic intensity due to the current in the wire itself is rrf ™ to the interval (0,2ir), we assume in effect that the sources of the field are located on the positive »z-plane. Since the potential rise across this plane is q/t, there must be a double layer on this plane of strength q (Fig. 6.43). The electric intensity is Fio. 6.43. A half-plane double layer. * The algebraic sign of potential and stream functbns is a matter of convention. ABOUT WAVES IN GENERAL 181 Since Ea, is continuous, there are no simple layers of charge. Tin- electric lines are I Hi li s and ihr cquipotclltial lines are rtldli, '.him the radial planes are equipntential, we ran assume any pair of them to be in i lis i i onductors insulated from each other along the line passing through the origin (I'lH. 6.44). The displacement density along one plane, passing through the x-axis, is 1 2irp (25-11) Fio. 6.45. Illustrating the conformal transformation of a region bounded by a closed curve into the upper half-plane. (25-12) Kin. 6.44. A wedge funned by two half-planes. , I he displacement density at the other plane is the negative of this. Since the poten-lial difference between the planes is qů/2ire, where ů is the angle between them, the ■ ipacitan.ee between the planes per unit area at distance p from their adjacent edges is C = -L PŮ1 Let us now consider the general problem of a line charge in the presence of a perfectly conducting cylindrical boundary whose generators are parallel to the line charge. Let (C) be the contour of the conducting boundary (Fig. 6.45) and let the complex number Zo designate the position of the line charge. Suppose that we have found a function w = u + h = /(z), (25-13) such that the curve (C) goes point by point into the i/-axis and the interior of the region bounded by (C) goes into the upper half of the ic-plane. Let two be the point corresponding to zo and assume that a line charge of density q is passing through too-The complex potential of this charge is Wi = - ~— log (if — Wo)-2?re (25-14) If a perfectly conducting plane is assumed to pass through the a-axis, then its effect on the field in the upper half-plane may be represented by the potential of the image charge of density -q, located at the image point wfo this image potential is JVi m ^d°S (w - *»*)• (25-15) IB2 >M,\<;NI.Tk' WAVES The total potential is therefore W---«-|ogW-"» 2irt w — wf If now We substitute from (13) into (16), we obtain W = » , m-im — log 2« °/C0-/(#)' (25 u (25 17)1 which is the complex potential of the line charge passing through Zu in the presence of] the conducting boundary (C). The real part of (17) reduces to zero on (C) because til' our choice of/(z). In the neighborhood of z = Zo, we have /(z) - /(zo) - (z-z0y'(z0); hence in this neighborhood (17) becomes W- --*-log(Z 2ire zo) - — log 2xe D/(so) -MY (25 IH) (25 I'M The first term of this expression represents the potential of the line charge in tht infinite medium and the second term its modification due to the boundary. Tho second term is constant and does not affect the charge density on the source; h< m the real part of (17) satisfies all the requirements of our original problem. More generally (17) may be represented in the form T„ q - q f(z) - /(2p) W = _ (z _ 2o) _ log - - , 2rre 2tt£ (z - z0)\f{z) -/(zo) (25-20) in which the effect of the boundary on the potential is given explicitly. The second term in (20) has no singularities in the region bounded by (C). If instead of an infinitely thin filament, we assume a thin circular wire of radius a, the complex potential of the wire is given approximately by (19). In this approximation we ignore the redistribution of charge round the wire due to the fact that the reflected potential is not really constant but varies from point to point; actually the wire is in a transverse electric field which forces some positive charge from one side of the wire to the other. In fact, following this line of thought we may obtain a more accurate expression for the field of charge on a conducting wire of finite radius. However, if the radius is small compared with the shortest distance from the wire to the boundary, then (19) is a good approximation and in this case the capacitance of the wire per unit length is r = ? =__|f«__,,r,|, V logl/(z0)-/(zo*)|-log|/'(3l))|-loga' ' In the above problem the image source was the negative of the given source. If the boundary condition is such that the image source is of the same sign as the given source, then instead of (17) we have W = 2ire log im -/(zo)][/(z) -/&)]. (25-22) ABOUT WAVES IN GENERAL 1H! With a few appropriate modifications all the above formulae can be used for the J^'ie field produced by an electric current filament in the presence of a perfectly | ».,„„ I.icting cylinder. The real part of the complex function w= -Z-log (w- wo) (25_23) 27T .....w taken to represent the stream function. This stream function satisfies he 1,1a,,. condition as the electric potential. Hence our final result will be (17 \h replaced by /. The inductance per unit length of a th.n wtre of radtus * » I.......| in the same way as the capacitance; thus we have L = — [log l/(zo) -M) I - l«g \?W I - l0§ a] ■ 2tt (25-24) We shall now consider a few special problems. Figure 6-16 shows the cross-section „f a wedge formed by two perfectly conducting planes. I,, a line charge be at point z„. The function w = z» = p"e™v (25-25) |, positive real for

) • • • (z - «**-»*»), (25-30)1 where ů is the wedge angle. Similarly we can factorize the denominator of (27), This factorization leads to (2« — ]) image sources and V can be expressed in terms ■ .1 the logarithms of the distances from a typical point to the source and its images. Fig. 6.47. A charged filament and a conducting half-plane. When n is not an integer there is no system of image sources which could represent the effect of the wedge. For example, a half plane (Fig. 6.47) can be regarded as a wedge with d = 2x and n = \. In this case (27) and (28) become 2x6 and no factorization is possible. V = - -L. Ing -P ~ 2VVo cos \{o P - 2VpPo cos %(v + Vo) + po (25-31) y X Fio. 6.48. Illustrating the transformation of a region enclosed by a polygon into the upper half-plane. A general function transforming a polygon (Fig. 6.48) in the z-planc into the real axis of the tt'-plane was discovered by Schwarz. Let us set up the following integral /V) (w - Wi)*(a - - w3)n' ■•■dw (25-32) and examine the changes in 2 as we follow the real axis in the ta-plane, indented at is\, w2, etc. The indentations may be taken as infinitely small semicircles in the upper half-plane and are needed to make the integrand an unambiguous function of w when the m's are fractions. As we follow the «-axis in the positive direction, the phase of the integrand remains unchanged so long as we are on the straight part of the path; AHOIIT wavi'.s in (jkneuai. 185 In in e the increments of z are in phase and ■:. must follow a straight line. Let us suppose 1I1.11 we are Oil the left of w\. In the s-planc wc are on some straight line znzi and me moving toward the point corresponding to tt>i. As we go round the first infinitesi-111,il indentation, all the factors of the integrand except (w — Wi)"' are constant. I be absolute value of (w — twi)"1 dw is p"* \ where r is the infinitesimal radius; hence if ii\ 4- I > 0, the magnitude of the increment in z is infinitely small during the 1.......hi round the semicircle. But while 2 remains unchanged, its new increments beyond 101 will have a different phase. This is because the phase of m — w\ has decreased by x and therefore the phase of (w — W\)m has changed by i?i = — rt\ir. I lie new increments make an angle d\ with the old ones and we now follow a straight Inn- :.[z'< making the angle t?i with ZnZi. The second bending of the path takes place at v. Corresponding t0 w%> etc- The transformation (32) can now be expressed in the form (if - wi)-i,/*(«e - w2)-d'!"{w - ■wsf**/m ■ ■ ■ dw, (25-33) where d\, t>a, etc. are the external angles of the polygon. If from the point at infinity on the positive «-axts we follow round an infinite semicircle in the upper half of the ai-plane, the path in the to-plane becomes closed. The total change in z around this path is zero since there are no singularities in the upper hall-plane. Thus we shall return to the original value of z. The region enclosed by the polygon is transformed into the upper half-plane. The term " enclosed " is defined as follows: that region is enclosed by a curve which is on the left ot an observer following the boundary counterclockwise. As our first example let us take a wedge (Fig. 6.49). In order to transform the region (S) into the upper half-plane we follow its boundary in the counterclockwise direction and imagine that the contour is completed with an arc of an infinitely large circle. Choosing the vertex of the wedge to correspond to the point w seeing that the angle t?i is positive and equal to x — t?, we have Fig. 6.49. A wedge of angle &. Oand Ů .,; plementary wedge, hence it*-*) ^ dw »= Air 2x w(2ir-ií)/ir = w(2i—i»)/»_ (25-35) Since 2x — t? is the angle of the wedge (S') in the same sense as & is the angle of (S) the two results agree. 186 ELECTROMAGNETIC WAVES Ghat, A~K> Y~~a- (25-47) (25-48) The last equation defines the modulus k; then A may be calculated from either of the first two equations. The transformation becomes w = sn-■; (25-49) hence die complex potential of the line charge is 2Kz 2Kz0 sn- — sn- w=-^—z—m' sn ■ sn - a a (25-50) TRANSMISSION THEORY 189 CHAPTER VII Transmission Theory 7.0. Introduction In section 6.11 we have shown that in source free regions the approximate equations connecting the harmonic transverse electromotive force V between two parallel wires (Fig. 7.1) and the longitudinal current / in the lower wire (or the magnetomotive force round the wire) are dV dx (0-1) where the distributed series impedance Z and shunt admittance Y per unit length are complex constants Z = R + iosL, Y = G + mC, (0-2) b o— __g and * is the distance along the line. The (I) f v Fic. 7.1. section positive direction for V has been chosen from (2-> the lower wire to the upper and that for I -_0 in the direction of increasing ^-coordinate. - c If one of these positive directions is reversed, A diagram representing a the negative signs in equations (1) become of a transmission line. ■ . T,. , .. , v ' positive. It the distance between the wires is variable, Z and Y are functions of X and the equations are general linear differential equations of the first order. Equations (1) are not restricted to transmission lines alone but play an important role in the general theory of wave propagation. In the case of waves in three dimensions the field intensities E and H usually appear in place of V and 1. This difference is superficial since E is the electromotive force and H the magnetomotive force per unit length and V and / are the integrated values of E and H. It happens that at low frequencies it is easier to measure V and i", while at very high frequencies E and H are more readily measured. In the case of waves in three dimensions the field intensities are generally functions of three coordinates; nevertheless under certain conditions wave propagation along, let us say, all tf-lines is the same, and the remaining coordinates may be ignored in so far as wave transmission in the tf-direction is concerned. Moreover the more general types of waves may frequently be decomposed into simpler types traveling in 188 accordance with equations (1). For expository convenience however we shall discuss these equations as applied to a pair of parallel wires. 7.1. Impressed Forces and Currents Sources of energy may be of two types: (1) electric generators of zero impedance in series with the line, and (2) electric generators of infinite impedance in shunt with the line. The first type is represented by an impressed electromotive force E(x) per unit length of the line and the second by an impressed transverse current J(x), also per unit length of the line. The assumption that the internal impedances of the generators are respectively zero and infinite will not restrict the generality of our results since the actual internal impedances may be included in Z and Y. The transmission equations in regions with given source distributions may be obtained by the method used in section 6.11 for deriving (0-1). Let us assume that the impressed electromotive force E(x) per unit length is acting in series with the lower wire* and p c let the positive directions of E(x) and J(x) be as shown in Fig. 7.2. By taking the electromotive force round a rectangle . iBCDA in which AB = 1, we obtain equation (6.11-1) in which the electromotive force of the field along AB is now given by Vab = ZJ-E(x) and not by (6.11-2). Similarly the expression for the total transverse current per unit length is J, = (G + mC)V -\- ]{x) and not the one given by (6.11-6). transmission equations become Etxr b Fin. 7.2. The convention regarding the positive directions of the impressed scries e.m.f. per nni t length E(x) and the shunt current per unit length J(x). Thus the dV = ~ZI+ BM, dl dx (1-1) -YV - /(*). dx ' v" dx 7.2. Point Sources In practice the impressed sources are sometimes distributed over long sections of a transmission line and are sometimes highly concentrated in " the vicinity of a point. An example of one type of distribution is furnished by a radio wave impinging on an open wire telephone line and an example of the other type is an ordinary generator connected to the line. In theory * Strictly speaking the series impressed forces should be applied to both wires in a balanced " push-pull " manner, that is %E(x) in series with the lower wire and — %E(x) in series with the upper wire. Otherwise the longitudinal currents in the wires will not be equal and opposite (see section 6.6). Our assumption does not affect the results in. so far as the balanced mode of propagation is concerned. The unbalanced mode will be considered in Chapter 8 in connection with waves on a single wire. I'* I ELECTRl (MAGNETIC WAVES dun. 7 TRANSMISSION THEORY l'>l it' is amvcnlent to idealize enneent rated distrihiiti(ins and regard tliein us point sources. Let E(x) be distributed in the interval $ — s/2 < x < £ + s/2 and let J(x) = 0. Integrating (1-1) in this interval, we have (2-1) Assume that as s approaches zero, the applied electromotive force approaches a finite limit Jft+a/2 E{x)dx =!>(£) as s->0. (2-2) i-s/a If Z, Y, F, I are finite, the remaining integrals in (1) vanish in the limit and we obtain v{& + 0) - r« - 0) = % (2-3) 7(1 + 0) - 7(| - 0) = 0. Thus the current is continuous at x = £ while the transverse voltage rises by an amount A. Everywhere else V and 7 satisfy the homogeneous equations (0 1). These are the conditions for a point generator of zero impedance in series with the transmission line. Similarly if E(x) = 0 and J(x) is concentrated at x = £, we have j{x)d*=m (2-4) i B-s/2 and the conditions for a point generator of infinite impedance in shunt with the line at x = £ become F(% 4-0) - m - 0) = 0, (2-5) . /(I + 0) - 7| - 0) = In this case the voltage is continuous;and the longitudinal current drops by an amount /. Everywhere else F and 7 satisfy the homogeneous equations (0-1). In the above equations V and 1 may be regarded as general discontinuities in the transverse voltage and longitudinal current and not merely as applied voltage and current. Thus if an impedance Ž is inserted at x = £ in series with the line, the voltage drop across the impedance is ZI and the discontinuity f' in the transverse voltage across the line is /' -ZI. Similarly if an admittance >' is inserted in shunt with the line, the transverse voltage is continuous and the discontinuity in the longitudinal current is /- YF. If the solutions of the transmission equations, subject to whatever supplementary conditions may be necessary, are known for point sources, then the general solutions of (1-1), subject to the same supplementary conditions, may be found by integration. We need only superpose the waves of elementary sources E dx and / dx. 7.3. The Energy Theorem The method of obtaining the energy equations for transmission lines is the same as in the general case of three dimensional electromagnetic fields. Starting with the fundamental equations (1-1), multiplying the first by 7* and the conjugate of the second by V, and adding, we obtain dV dl* I*—+F—= -ZII* - Y*FV* + EI* - FJ*. dx dx The left side is the derivative of FT*; hence multiplying each side by \dx, integrating from X\ to x%, and rearranging the terms, we have | J* (EI* - FJ*) dx = \ (ZII* + Y*FF*) dx + IV(X2)I*(X2) - lF(x,)I*(x,). (3-1) The left-hand side represents the complex work done by the impressed forces and hence the power introduced into the transmission line; thus ^EI* dx is the complex work done by an elementary applied electromotive force in driving the current 7 while —\VJ* dx is the work done by an elementary shunt generator which introduces the current / dx against the transverse voltage V of the line. If we designate this total complex power by and replace Z and Y by their values from (0-2), we have I i?77* dx + f / GVV* dx + m j (|777* - \CFF*) dx + \n^)I*(x2) - 1^)7*^1). (3-2) The real part of # is the average power contributed to the line. The first two terms on the right represent the average power dissipated in the section (x\,x2). The difference between the power contributed to the section and that dissipated in it is represented by the real part of the last two terms. The power may be either entering or leaving the section at its ends; hence we may interpret the real part of ^F(x)T*(x) as the average power passing across the point x of the transmission line in the direction of l.1 it rROMAGNETIG WAVES increasing .v-coordinate. The imaginary term represents the fluctuating power. If the sources of power are located at points x < #1, then $ = 0 and (2) becomes }#$%P*f&) = I / RII* + GVV* dx + m C &LII* - \CVV*) dx + ptm**m. (3-3) In this case the complex power * = §F(xt )T* ) is entering the line at x = xi and it is accounted for by the various terms on the right. In the above interpretation of (1) we have implicitly assumed that the impressed intensity E acts on the total longitudinal current and that the impressed current J is acted upon by the total transverse voltage. This is necessarily true when the longitudinal current is localized as in conventional two conductor transmission lines. But if the longitudinal current is distributed, as in hollow metal tubes, for instance, then the left side of (1) will no longer represent the complex power contributed by the sources and the interpretation of other terms in the equation must be correspondingly modified. In these more general situations it is better to rely on the three-dimensional energy theorem (4.8-7) of which the present theorem, is a special case, than to try to obtain appropriate modifications of the foregoing equations. The various energy terms will usually differ from those in this section by a constant factor. 7.4. Fundamental Sets of Wave Functions for Uniform Lines Consider now a source-free section of a uniform transmission line. If either V or I is eliminated from (0-1), we obtain a second order linear differential equation with constant coefficients; thus (4-1) d2I „r d2F - — V2I - = V2V dx2 A dx2 1 % where the propagation constant T is defined by T = VlY = V(R + iuL)(G + iuC). (4-2) The general solutions for F and I may therefore be expressed either as exponential or as hyperbolic functions. Expressing / in terms of exponential functions, we have I(x) = I+e-r* + I~eTx, (4-3) where 7+ and I~~ are arbitrary constants. By (0-1) V is completely deter- i uansmissk )n THEORY IV3 mined by /; thus i dl F(x) * - ~ ~ Kr^-rx - KTVr*, 1 dx where the characteristic impedance K is defined by R + mL R + iuL G + iu>C (4-4) (4-5) G + io>C By definition I is a complex constant which lies in the first quadrant of the propagation constant plane or on its boundaries* and the equations i+(x) = i+e"Vx = j^y**-*** = Fr-r&tt = ki+(x) represent a progressive wave moving in the positive ^-direction, with an amplitude which is attenuated at the rate of a nepers per meter. Likewise the equations r(x) = reVx = rv**<* v~{x) = -i TKANSMISSK )N TI II il >KV '201 7.7. Transmission Lines as Transducers A section of a uniform line is a symmetric transducer. By (5.2-3) its se'f-itnpcdances are equal to the open-circuit impedance of the line and the transfer impedance may be found from (6-12); thus Zu m Z22 = K coth IV, Zi2 = -TCcsch 17. If the transmission line is represented by a T-network (see Fig. 5.12), then by (5-4-1) the impedances of the shunt arm and of each series arm are respectively Tl Z2 = K csch Tl, |Zj = K tanh — . Regardless of the length / of the section we have Z\\Z22 — Zf2 = K . 7.8. Waves Produced by Point Sources Let a section of the line of length / be terminated in impedances Zi and Z2 (Fig. 7.4). Let V\(x£) and h(x£) be the voltage and the current at point x = x when a unit electromotive force is impressed at x = £ in series with the l;ne.* To the left and to the right of the generator we have respectively It(x£) = Pi cosh Fx 4- Qi sinh Fx, Vi(x£) = —K[Pi sinh Fx + Qx cosh IV], x < £; 7, (x£) - P2 cosh F(l-x) + Q2 sinh F(l - x), ViixJi) = K[P2 sinh T(/ - x) + Q2 cosh T(l - x)], X > |. At x - 0, the voltage-current ratio is — Z1 and therefore Pi = PK, Qi = PZi, where P is a disposable constant. Similarly at x = I the voltage-current ratio is Z2 and therefore P2 = QK, Q2 = QZ2. At x = £ we have /i(í + 0,ř) -7i({- 0,0 =0, (8-1) Vx(& +.0,S) - ^(1-0,0 = 1. Making the necessary substitutions we obtain Q[K cosh F{1 - S)+ Z2 sinh T(/ - £>] = P[J? cosh + ZX sinh r& 0[X sinh rc7-£)+Za cosh T(/ - £)] + P[A sinh T£ + Zj cosh T£] = — The first of these equations is satisfied if we let DQ = TCcosh r| 4- Zi sinh r|, £»P = A cosh T(/ - £) + Z2 sinh T(/ - f). *The coordinate x is the distance from Zj. 202 ELECTROMAGNETIC WAVES t •„*,.. •/ Sulisliluling iti the second, we have D = K[K cosh Vi + X, sinh l'i.[[A' sinh F(/ - J) + Z2 cosh - ?)] + sinh r£ + Zi cosh l'£j[K cosh r(/ - £) + Z8 sinh F(Z - $)]. Multiplying and collecting terms, we have D = K[(K2 + ZiZaj sinh Tl + A(Za + Zi) cosh 17]. Thus all disposable constants are determined and we have = cosh Tx + % sinh Tx] X [A cosh r(/ - o + Z2 sinh T(/ - £)], # < £ = [X cosh T£ + Zt sinh rtf x [K cosh r(/ - *) + sinh r(/ - *)], x > % (8-2) ZW*,£) = -TO sinh Ftf + Z, cosh Tx] X [A' cosh r (/ - I) + Z2 sinh r(/ - £)], * < $, = ATA cosh r£ + Zi sinh 1^] X [A sinh r(/ - *) + Z2 cosh r(/ - jc)], x > i It is easy to see that /j ) is symmetric -řv>--1 '-r-J- Fig. 7.4. A section of a line energized by Fig. 7.5. A section of a line energized by a series generator of zero impedance. a shunt generator of infinite impedance. This proves the reciprocity theorem under the conditions stated at the beginning of the section. The arrows in Fig. 7.4 show that the current at the generator flows in the direction of the impressed electromotive force on both sides while the transverse voltages are in opposite directions on the two sides. If a unit current is impressed at x = % in shunt with the line (Fig. 7.5), then the voltage and current waves V2{x,%) and I2 (#,£) satisfy the following conditions at x — £ řift + o,o - v2(i - o,o = o, - /ad - o,j) = -i. (8-3) [HANSMISSION THEORY 203 In this case we obtain the following solutions Dh(x£) = K[K cosh Tx + Zi sinh l'.vl X I a sinh r(/ - 0 + cosh r(/ - m, * < % ■ m - K[K sinh T£ + Zt cosh rtf X [A' cosh r(/ - *) + Z2 sinh r(/ - *)], * > & DVa(x£) t= - A2[A' sinh Yx + ZY cosh Tx] X [K sinh r (1-0+ z2 cosh r(/ - £)], x < & N -K2[K sinh I"| + Zx cosh F£] X [A sinh T(/ - .v) + Z2 cosh r(/ - *)1 * > fc r-X (8-4) 1 Fig. 7.6. A series voltage and a shunt current applied at the same point of the line. The voltage wave function is now symmetric Vm F2(£,*)• The arrows in Fig. 7.5 show that at the generator the voltage is in the same direction on both sides while the currents are in opposite directions. In the special case when Zi = Z2 = K, we have mx,o = -K^, Jito) = 2Ke~r(^x)' x < ^ (8-5) _ l.-r(z-c) — 2e > J_ 2A x > I for a unit electromotive force impressed in series with the line. Similarly for a unit current impressed in shunt with the line, we obtain (8-6) K 2 _ _l„-r{z-f) — 2ť 3 x > & If a current —/is impressed in shunt with the line and a voltage V = KI in series at the same point X = f (Fig. 7.6) then from (5) and (6) we find that the wave to the left of * = £ vanishes and that to the right becomes V(x,0 = KIe~T <-T~i\ /(*,£) = /.-r(^. 204 I-11,c i u<>ma<;nktic waves Chap, 7 'This conclusion could easily lie reached from considerations of symmetry and of the relative directions of the voltages unci currents at the general ■ a as illustrated by the arrows in Figs. 7.4 and 7.5. Once we have come to the conclusion that, for given values of applied current and voltage, there is no wave to the left of x = £, it becomes evident that the impedance A' ai the left end of the line could be replaced by any other impedance and the left section could be completely removed. In practice these conditions can be realized only approximately since they demand generators of zero impedance (or " constant voltage generators ") and generators of infinite impedance (" constant current generators "). 7.9. Waves Produced by Arbitrary Distributions of Sources Knowing the wave functions corresponding to point sources we can immediately construct the wave functions corresponding to any given distribution of sources by proper superposition. Thus if a series electromotive force E{x) per unit length and a shunt current J{x) per unit length are distributed in the interval (xux2) then we have the solutions of (1-1) in the following form V(x) - CE^F^x,® dZ + P f{i)F,(x,^)d^ Emh(x£)a$+ / /(i)/2(*,£) 4-*1 . "XL (9-1) That these functions satisfy (1-1) can be proved by direct substitution. It should be recalled, however, that F\(x£) and h{x£) are discontinuous and hence nondifferentiable functions of #:at x *=» |; For this reason we break up the integrands as follows (x& d% + / E®Fi(*,{) di + / ]{t)V2{xJi) dk, (9-2) If*) = )E{i)It (x,0 dk f f J®h{x£) di 4- P fmUxj) d$. Each integral is now a differentiable function. In taking the derivatives of V(x) and l'(x) we use d_ dx d*~f £f(x& ±fM> Jxj /(*,£) 4 - j jJix&dZ-f^x), TUANSMISSION THEORY 2115 Thus we have dV dx EiD-^rdx&di fx' E^) dx + E(x)^ (x,x - 0) - E{x)F1(x,x + 0) + JT /(I) £ F$®M 4k Since F\ and V2 are solutions of (0-1) we may substitute — ZI\ and — ZI2 for the derivatives under the integral signs; by (2) the sum of the integrals is then equal to* — ZI(x). The remaining two terms are equal to E(x) in virtue of (8-1). Thus we have proved that the first equation in the set (1-1) is satisfied. Similarly we can show that the second equation is satisfied. Finally it can be verified that the boundary conditions are fulfilled. 7.10. Nonuniform Transmission Lines Let us now assume that Z and Y are functions of x. Eliminating first / and then V, we find that in source-free regions both variables satisfy general homogeneous linear differential equations of the second order d2F Z' dF d2I Y' dl yzv = q ^- _ - e _ YZI = 0. (10-1) dx2 Z dx ' dx2 Y dx A second order differential equation of this type possesses two linearly independent solutions and its general solution is a linear function of these solutions. Thus we havie I(x) - AI+(x) + BI-(x), where A and B are two disposable constants. The corresponding solution for F is then F(x) = AF"V (x) + BF-(x), where F+ and V are obtained from F+(x) = - I dl+(x) Y(x) dx F~(x) = - \_dl~{x) \x) dx (10-2) Alternatively a pair of fundamental voltage wave functions might be selected and the corresponding current wave functions then defined as follows 1 dF+(x) , 1 dV(x) I+(x) = Z(x) dx * This is true even if Z :s a function of x I~(x) = Z(x) dx (10-3) 206 M.ECIKOMAí.NKTK WAVES Ciiai'. 7 TKANNMISSION TIIEuKY Ml A w:ivc impedance may be associated with each fundamental pair of wave functions; thus K+(x) = I+(x) Yl+dx 1 d i r+ Zr' -iogr+' (10-4) * M = -Too = Yf-dV - y7>7 = 7^ (10-5) In the strict sense of the term there are no progressive waves in nonuniform transmission lines since any local nonuniformity in an otherwise uniform line will generate a reflected wave. However, in some instances wave functions may exist which bear considerable resemblance to the exponential wave functions and hence may be said to represent " progressive " waves in nonuniform lines. This is apt to happen when Z and Y are slowly varying functions of x. Even then it may be more convenient to select other sets of wave functions for the fundamental set. Thus in general we should look upon K+(x) and K~(x) as factors to be used in passing from a given current wave to the corresponding voltage wave and vice versa. Other ratios are useful in the general theory. Thus the voltage transfer ratios are defined by Similarly the current transfer ratios are defined by (10-6) (10-7) r>01 ""' v"r" r-(*0 Consider now a section of a nonuniform line extending from x = x± to x — x2. If the output impedance at x = x2 is Z(«2), then it is easy to show that the input impedance is V+{Xl) F+(x2) Z(x2) Z(xi) = 7+(*2) VW) r(xx) i-(x2) Z(x2) - r+{x2) i-(xi) ry (10-8) In order to obtain the impedance of the section at x = x2 when an impedance Z{xi) is across the line at x — %, we merely interchange *i and x2 in (8) and reverse the signs ofZ{X\) and Z(*a). The reversal of the sign corresponds to l lie change in the direction of I he impedance. 7.11. Calculation of Nonuniform Wave Functions by Successive Approximations Consider a section of a nonuniform line of length / and let Z = Zu-f-Z, Y=Y0 + t, (ll-l) where Zu and Yo are constants. These constants may be taken to represent the average values of Z and Yin the interval (0,/) Z0 = \ f' Z(x) dx, Y0 = 7 jT Y(x) dx. (11-2) I Jo t Jo Assuming that there are no sources in the chosen interval, the transmission equations are ~= -ZoI-ŽI, f - ~YoV dx dx ÝV. (11-3) We now seek that solution of these equations for which the initial values of the voltage and current are given F(0) = F0, 1(0) = /„. For this purpose we first find the solution of dZ.= M |s ^YoF, dx ' dx which has the following discontinuities in V and I at x = £ m + 0) = ?U - 0) - % J(| + 0) = Z(| - 0) - J. In the interval (0,£) we evidently have (see equation 4-10) F(x) = F0(x) = F0 cosh V0x — K0Io sinh r0*, F0 (11-4) (11-5) (11-6) (11-7) where I(x) = /()(•*) = — — sinh To^f 4-'7o cosh V0x} r0 = VZoFo, 7C0 = ^. (11-8) At * = £, F and 7 are decreased by ^and 1 and in order to obtain the solution in the interval (£,/) we need only add to equations (7) analogous expres- 208 El,ECTR< MAGNETIC WAVES I'llAI'. 7 sions in which the initial values are — V and —/; thus V(x) - Vn{x) - ^cosh V0(x - £) + A'u/Sinh r0(* - J), V I(x) = I0(x) + — sinh T0(x -£)-// cosh T0(x - £). (ii 9) We now consider a " continuous distribution of discontinuities " f = Z(0I(Z) dt, 1 = Y1£)V% ii (11-10) and construct the following solution of (3) V(x) = FB(x) - f Z$ )/(f) cosh T0(x-ft f f{£)V{£) sinh T0(x -J) /(.*) = /<,(*) 4--^- f 2(f)/(E)frinhr0(*f ?«)^(?)coshr0(*-J)4. (11-11) We have not really solved the original differential equations since the unknown functions appear under the integral signs; but we have converted the differential equations into integral equations. That these integral equations define functions satisfying the differential equations (3) and the initial conditions can be proved directly. At x = 0 the integrals vanish and V(0), 7(0) evidently reduce to I0. Differentiating V{x) we have dx ~ " r° S'z<£)m sinh r°(*ZMI{X) + z0 Ffimm cosh r0(* - 0 % The right-hand side of this equation is identical with the right-hand side of (3) if we take into account the expression (11) for I(x) and the following equation dF0(x) dx ■ZMx). Similarly it can be shown that the second equation of the set (3) is satisfied. From the integral equations (11) F(x) and I(x) can be calculated by successive approximations. Thus we set f(x) = r0(x) + rx{x) + v2(x) + ■■■ Kx) = I0(x) + +/«(*) + •••, (11-12) where (11-13) TRANSMISSION THEORY 209 Vn+iix) = - rz%In§) cosh r0(* - £) i\ Jo + K0 rV(f)^„tt) sinh r0(* - f) Jo 7n+i(-v) = t fXz(i;)in(0 sinh r0(^ - I) d\ Jo Evidently the differential equations (3) can be transformed into other integral equations in which Z0 and Y0 are not constants provided we can obtain solutions of the corresponding equations (4) and (5). Just as equations (11) are most useful when the transmission line is only slightly nonuniform, other integral equations may be particularly useful when a given nonuniform line deviates slighdy from another nonuniform line with known wave functions. It should, be noted that the solutions (12) are valid even if Z(x) and Y(x) are discontinuous functions. 7.12. Slightly Nonuniform Transmission Lines When Z and Y are nearly equal to their average values Z0 and Y0, so that.the relative deviations z/ZQ and Y/Yq are small, only the first corrections P~i(x) and Ii(x), or at most the second corrections F2 and 72, need be considered. When the deviations are large in a given section of the line, the section may be subdivided into smaller sections. Taking n = 0 in (11-13), substituting from (11-7), and rearranging the terms, we obtain the first corrections ¥0) = Vq[B(x) cosh V0x - a(x) sinh T0x + C(x) sinh r0*] — k0Io[a(x) cosh T0x — B(x) sinh TQx + C{x) cosh T0x], (12-1) Fn IiHx) = - tt [B(x) sinh r0* - a(x) cosh T0x + C(x) cosh r0*] An + h[A{x) sinh T0x - B(x) cosh T0x + C(x) sinh Tax], where AW = iJo S ~ K°*) C°Sh 2r°^ ^ °{X) = ^X* {To + Kot) Six) = | ~ K°Y) sinh 2T*t& (12-2) 210 kl.KiTIU(MAGNETIC WAVES ClIAl', 7 In sonic nonuniform transmission linos the product zy is constant) then we choose p0 — V!ZY, Zn — zivi, Yq — —^ . We now have as the first approximation Ý (12 3) (12 h Consequently C(x) = 0 and K0A(x) = r Ž(Ě) cosh 2r„ž n II 111 ()kv 21 and the corresponding expressions for the transmission coefficients Pi = Ť5 F 2K pv = F K + Z I + k ri~ K + z~ i + k - 1 + qi, (13-4) = 1 + qv- Thus the reflection and transmission coefficients depend on the ratio of the terminal impedance to the characteristic impedance. If this ratio is unity, the impedances are said to be " matched," and there is no reflection. If /' equals either zero or infinity, the reflection is complete; in the former case the current at the terminals is doubled and the voltage is annihilated while in the latter the current is annihilated and the voltage doubled. For all impedance ratios the voltage reflection coefficient is the negative of the current reflection coefficient. The voltage reflection and transmission coefficients have exactly the same form as the corresponding current coefficients if expressed in terms of admittances; thus m-y qv = m+~y' Pv 2M m + y (13-5) The expressions for the incident and reflected waves may then be written In the following form V*(x) = FV-r* F(x) = Fe~Tx; ťr(x) = qy-Jre?", F(x) = qiPeJ (13-6) assuming that Z is located at the origin. If Z is a semi-infinite transmission line whose characteristic impedance and propagation constant are Ki, Tj, then for the transmitted wave we have Vl{x) = pvF'e-^, F(x) = pxPe-f*. (13-7) 7 , * 1 The reflection coefficient depends on the ratio k = Ki/K of the characteristic impedances and is independent of the propagation constants of the two lines.__ Let us consider a special case of reflection A caused by an impedance Zi inserted in series with FlG- 7-s; An impedance in the line (Fig. 7.8). The impedance Z seen to the right from the terminals A, B is Z = Zi + K. Substituting in (3) and 212 (4), we have I'.I.IX II« )MAUNI'.IK. WAV KS ■Ii = Z, 2K + Zi' Pi = 2K 2K + Zi Chap, (1.1 R) Taking the reciprocals, we obtain 2K Zi 1 = - li 1, 2K l-i. Pi (13-9) Fig. 7.9. An admittance in T)ms the ratj0 z >K can be expressed quite simply shunt with a tine. . „ , ' ~ . . \J in terms or the reflection and transmission coefficients which in certain circumstances can be measured more readily than the ratio itself. If an admittance Y\ is inserted in parallel with the line (Fig. 7.9) we have Yi 2M Y = Yi + M, qi - -qv IM + Yi ' pv 2M + Fi' (13-10) 2M 1 _ J_ Yi qi qv 1, ' 2M pv Let us now consider reflection and transmission of power. The transmitted power Wl is W1 = |re(W*) = ^{pvpfV'P*). For an incident progressive wave in a nondissipative line V1 is in phase with P; and therefore for the incident power we have W1 = |re(F'7,:*) = JF*/**. The power transmission and reflection coefficients are then 'W , *, 4re(£) 9W w_ w1 \-Pw = \qi\% = W\ 7.14. Reflection Coefficients as Functions oj the Impedance Ratio The impedance ratio k and the voltage reflection coefficient qv are complex quantities k = R + iX = Aew, qv = aeia, where the amplitudes A and a are essentially positive. The phase of k lies in the interval (—ir/2,7r/2) while the phase of qv is in (-7r,7r). The reflection coefficient is the ratio of two complex quantities represented by the lines AP and BP (Fig. 7.10) TRANSMISSION THEORY 213 i'( I ) in I'ik). Tin- phase (V of i/y and the phase #/ of q( an' the angles formed by ihese lines as indicated in big. 7.10; the amplitude a is the ratio of the lengths AP and BP. Fig. 7.10. The complex plane for representing the impedance ratio. Taking the square of the amplitude of the reflection coefficient, we obtain (R-l)2 + X2 , fn 1_+ 1 (R + iy + x* + x2 = 4a* (1 (14-1) This equation represents a family of circles surrounding A and B as illustrated in Fig. 1.6. If X = 0, then 1 - R-s = 1 + a 1 - a 1 + a" are the real values of the impedance ratio for which the reflection coefficient is spa. The unit circle represents points for which the absolute value of the impedance ratio is unity. On this circle the phase of the reflection coefficient is ±90°. The points of intersection of this circle and (1) are 1 - a2 v la - , X = a--. 1 + a2' R = 1 -|- a2 The loci of points for which the phase of the reflection coefficient is constant are circles passing through A and B. The equation of this family of circles is X , X tan R — i — tan" R + 1 = Ů, or R2 + (X - cot ß)2 In making a chart (Fig, 7.11) showing the dependence of the reflection coefficient on the impedance ratio we could limit ourselves to one quadrant inside the unit circle. The absolute values of the reflection coefficient are equal for k, l/k, k* and \/k*. The phase of q for the impedance ratio l/k is different from that for k by f80° and the phase for k* is the negative of that for k. The amplitude of the reflection coefficient as a function of A and ip is am(7) 1A cos

q = qm> (19-6) i hen the series is T = pe~v* + pqe-™ + pq2e-™ + pfr™ + ■■■ (19-7) = p{\ - qe-2''°J)-1e-r*1, which agrees with (2). The values of the factors p and q depend on the type of wave we are Minsidering. It is also necessary to remember that while pi is the transmission coefficient for a wave passing from region (A) to region (B), qi is the reflection coefficient for a wave traveling in (B) toward (A). Thus we have (see section 13) m Ki + K2' K2 — Ki A2 + Ki Pl,2 ~ qi ,2 2A2 K2 + Z> K2-Z K2 + Z' Pv.i = qv.i = 2K2 Ki + K2' Af — K2 Ki 4- A*2' PV,2 = qv.2 = 2Z K2 + Z' (19-8) Z — K2 Z + K2 Hence for the voltage wave we have 4A2z (K1 + K2){K2 + Z)> qv~qi> and, using this value of p in (7), we obtain Ty. The apparent lack of symmetry in equations (2) is caused by the use of pi in Loth equations. As a reminder of the directions for which the partial reflection and transmission coefficients should be employed we may write (6) in the following form P = pipt, q = qlqt. (19-9) 226 ELECTROMAGNETIC WAVES cwai', 7 T i - The above calculations can In: extended to any number of sections lie tween the given line and X; thus for two sections, we have, instead of equation (1), I_r _ IrIqIp li ~ Iqlpli ' where R marks the end of the second section. Noting how (2) and (3) have been formed from the equations which precede them, it is easy to write a general formula for any number of inserted sections. Let there be n sections; let the constants of a typical section be Km, Vm, lm; let the impedance looking to the right at the begin ning of each section be Zm; let K be the characteristic impedance of the line in which the sections have been inserted; then the transmission coefficient is Ti - pňO - ?/,i*-2riíI)(l - qi.ze-™) • ■ • (1 - qi,ne-2l'»l»)]-i X -T.h-rtU-----r„i„ Pi = 2K - 2Kj ■ 2ÄV • ■ 2Kn (K + Ki)(Ki + K2)(K2 + XV) • • • (Kn + Zn+1)' _ (/(", - K)(KX-Z2) (Km - Km^)(Km - I1* ~ 71? i 17771? , r, s > 91,m = (19-10) ) (Ki + K) (JSTi 4- Z2)' - - {Km + km_0 (Km + 2^4) ' where Zn+i is the terminal impedance. If the attenuation in each section is high, then Zm is approximately equal to Km; in this case the expression for Ti becomes simply Ti = pie •Tth-Vth-----life 09-11) The voltage transmission coefficient is obtained if we multiply 7j by Zn+i/K. 7.20. Reflection in Nonuniform Lines The equations at an impedance discontinuity are the same for uniform and nonuniform lines (13-1). Equations (13-2) are replaced, however, by the following more general equations V* = K+I\ V> = ~jfp, V* = ZIl. (20-1) The impedances K+, K~~ associated with the incident and reflected waves need no longer be equal. Solving (13-1), subject to the above conditions, we have 91 qv K+ - Z K- + Z' M+-Y M~+ Y' K- + K+ pv K- + Z ' M~ + M+ M~+Y' (20-2) TRANSMISSION I IIIOKV 227 where the admittances A/1, M , Tare lbe reciprocals of the corresponding unpedanccs. The transmission coefficient across a section (.vi,.v2) of another line id between the given line and an impedance Z may be obtained at Once ni the form of an infinite series analogous to (19-7). The principal difference consists in replacing the exponential factors r(*2>*l), (20-3) and replace (19-7) by T = p(\ + qx + *V + • • • )x+(*i,*2) = x+(*i,*2). (20-4) l — qx 7.21. Formation of fVave Functions with the Aid of Reflection Coefficients In section 10 we have seen that the most general wave functions are linear combinations of pairs of the " fundamental " wave functions. If we wish to form wave functions F{x), I(x), whose ratio is prescribed at x = x2 by terminating the line in a given impedance, then we may proceed as follows. We choose one fundamental set / 1 (.v), I+(x) as a wave which is " incident " on the given impedance and the second . i as a reflected wave; then we write F~(x) F+(x2) F(x) = r+{x) + F+(x2)qy{x2) = F+(x) + gv(x2) —— F~(x). (21-1) Similarly we obtain /(*) = i+{x) + rxfcú Sil 1 {x2) The impedance at point x = x\ may then be expressed as follows Vjxx) _ 1 + qv(x2)xt{xitxi)xv(xi,xi) I(xi) ~ ( 1 + íí(*i)xí (X1,X2)XI(X2^1.) ' (21-2) (21-3) For example, in the case of uniform lines we may choose the following fundamental wave functions /+(*) = he~v*, I~(x) m Iie**, F0 = KIa, F+(x) = *V-ri, V-(x) = Fierx, Fi = -Kh, Then (1), (2) and (3) become (if x2 - Xi = /) F(x) = F0e-Vl + Fme-^le^, I(x) = he'** + hq lC-™ev*, (21-4) (21-5) 1 4- qve 1 + qie' -2rt* 228 ELECTROMAGNETIC WAVES Cha*. 7 TRANSMISSION THEORY 229 ll is important in observe thai there are mi restrictions on the choice of fundamental pairs of wave functions V*, 7+ and V~, l~ beyond the requirement of linear independence. In dealing with uniform lines the choice of progressive wave functions is, perhaps, more generally useful; in the case of nonuniform lines other choices are frequently preferable. Even in the case of uniform lines we may find it desirable in some problems to choose a different fundamental set. For example, let us choose the following set as the fundamental set r+(x) = Foe'**, /+(*) = 7V -r> V~(x) = Vi cosh Tx, I~(x) = Ii sinh Tx, Vx = -KIh (21-6) where V , I represent a stationary wave with the current equal to zero at x = 0. Substituting these expressions in (1) and (2), we have V{x) = *V-r*+ ?v(/)/V- cosh Vx ' cosh VI ' lilii) = hc~r* + qjWI^'^-T— . , sinh IV (21-7) The above choice of V~, I" is such that " reflection " does not affect the original current at x = 0 and is particularly suited to problems in which the original wave is generated by a fixed current generator at x = 0. Under these circumstances the choice of a progressive wave to represent reflection would necessitate a consideration of multiple reflections since the wave reflected from an impedance Z at x — I would be reflected again at the generator. The choice (6) takes care of these multiple reflections in one operation. The input impedance is now expressed as follows Z(0) = K + qvU) sh Vi K. (21-8) The first term represents the input impedance of a wave which does not see the discontinuity at x = /; the second term may be called the induced impedance or the impedance coupled to the generator in consequence of reflection from the far end of the line. The induced impedance represents the effect of the environment as changed by the terminal impedance and it may be called the " mutual " impedance. Even though the line is uniform, the reflection coefficients should now be calculated from the general formulae (20-2) since K~{1) is no longer equal to K+(l) = K. Thus V-(l) K coth VI, qr = (Z - K) cosh VI K cosh 17+ Z sinh Yl' The impedance is then Z(0) = K + (Z - K)e-Vl ,K. K cosh 17+ Z sinh Yl Naturally this expression gives the same total value for Z(0) as (6-2). Another choice ol fundament al lintel ions is | >.i......larly suitable in the case when the line is energized by I fixed voltage generator (zero internal impedance); here we have f-(x) = JT sinh Tx, /-(*) = /, cosh Vx, Fi- - Kh, _ „, sinh r# V{x)= Foe-^ + qy(l)rae-r'- I(x) = loe-^+qiiDhe -rt sinh Yl' cosh r* cosh Yl' In this case the voltage at the generator remains unaffected and the effect of the impedance at the far end of the line is represented by the " reflected current." The input admittance is ,-rt T(0) = M+gi(l) cosh Yl ' and the effect of reflection is represented by an induced admittance. In this case (Y - M) cosh Yl M~(l) = M coth Yl, qi(l) Y(0) = M + M cosh 17 + Y sinh Yl' (Y-M)e-Fl M cosh Yl + Y sinh Yl m. 7.22. Natural Oscillations in Uniform Transmission Lines In equations (0-2) for the distributed series impedance Z and shunt admittance Y we have explicitly assumed that the transverse voltage and the longitudinal current are harmonic; but throughout the greater part of the succeeding sections no use has been made of this assumption and the results obtained apply equally well to the general case in which the oscillation constant p is complex, that is, Z = R + pL, Y=G + pC. (22-1) In the circuit shown in Fig. 7.25, where Zi and Z2 are the impedances seen from the terminals A, B to the left and to the right, the current is V z, + z2 (22-2) An exceptional case arises when the oscillation constant is a root of the following equation Zi + Z2 = 0. (22-3) In Chapter 2 these roots have been named " natural oscillation constants." When p is a root of (3), Fic. 7.25. Illustrating the condition for natural oscillations. 230 lU.ECTROMAGNETIC WAVES Chap, 7 an electric current In1" niiiy flow in I lie cireuil without a continuous up-plied voltage /V. Such nalural oscillations may he started by an impul sive lorce and they may be calculated hy the methods outlined in section 2.9, when the roots of (3) are known. Let us now consider a uniform transmit sion line of length /, short-circuited at both ends (Fig. 7.26). Here Zx and Z2 are the impedances seen from any pair of terminals A, B and in particular, from terminals C, D at one of the short-circuited ends. In the latter case Zx = 0 and Z2 is given by (6-3); thus Fig. 7.26. A section of a line short-circuited at both ends. where tanh T/ = 0, or sinh 17 = 0, YJ = rnri, n = 0, ±1, ±2, • ■ • )±JT*7RG + ( 7 \ LC r V r = V(* + PL)(G+pC) From (5) we obtain p 1 \2L 2Cf M LC If R and G are small, we have approximately (R G\ 1 p = -\Yl + Tc)± Substituting from (4), we obtain (22-4) (22-5) R 27 vlc Mir iVlc (22-6) — , (22-7) where v is the wave velocity in the line. The phase constant and wavelength (in the line) corresponding to the natural frequency wn are /3n =m/J, Xn = 21/n. The lowest natural frequency corresponds to a wavelength twice as great as the length of the section. The above solution for p is based on the assumption that R and G are independent of p while in practice they are functions of p. However if R and G are small, their effect on p is small and equation (6) is approximately true if R and G are computed for p = /'&>„. From (7) and (5.11-16) we have an expression for Q ® 2£ Co«7 unc) transmission THKORl 231 SultKl-ititling for w„ from (7), we express (J in terms of the characteristic impedance and the attenuation constant 0 =»^ + GA"l = ^=^=^l. (22-8) L" I \K J 2at 2a a\n Exactly the same results are obtained for a line which is open at both i 1But if one end is open and the other short-circuited, then the equation for the natural oscillation constants is coth Tl — 0, or cosh 17 = 0, iml its roots are Ynl = i(n + |)x, n = 0, ±1, ±2, • • •. In this case X. 4//(2b + 1), and the lowest natural frequency corresponds to a wavelength (in the line) four times as great as the length of the section. More generally if the line is terminated into an impedance Z\ at one end and an impedance Z2 at the other, the equation for natural oscillation constants is Z2cosh 17+K sinh Yl Zi = Q K cosh Tl + Zi sinh Yl K (22-9) Frequently this equation can be solved by successive approximations. Suppose for example that Z\ and Z2 are pure resistances, small compared with K. Then as a first approximation we assume them equal to zero, and, if the line itself is nondissipative, we have plVpC = nivi. Then we write jf/VZc = niri + A , where A is a small quantity, substitute in (9), and retain only first order terms in A to obtain KA , K, K + RzA K 0, A K Thus the effect of small terminal resistances on the natural frequencies is at least i if the second order; the first order effect consists of damping. As another example let us take Z\ = 0, Z2 = R'i + pLi, and assume that the line itself is nondissipative. Equation (9) becomes (R« + Lip) coshyvTr + K sinh ph/Tc= 0. Let i— iu plv LC = iu, p (22-10) IVLC substituting in (10), we have KI —tR* cos u + -—u cos u + K sin u = 0. I f ^2 is small, the first approximation is a root of tan u _ l2 (22-11) (22-12) 232 i-1 E< Tl« MAGNETU WAVES C'nAi'. 7 Letting ;/ ft I A, substituting in (I I), anil neglecting powers of A nbove the first, wc have A = Hi /<•> cos * It + 57T' Roots of (12) may be obtained either graphically or numerically. If, however, the right-hand side is small compared with unity, then we have approximately it = nir + 5, where 5 is a small quantity. Hence S = — nwLz/lL, as long as n is not too large, and therefore tt = nw[\ — (Lv/IL)}. 7.23. Conditions for Impedance Matching and Natural Oscillations in Terms of the Reflection Coefficient Let one of the impedances in Fig. 7.25 be the characteristic impedance of a transmission line (Fig. 7.27). We assume that an incident wave is originated at infinity and that the reflected -o wave goes back to infinity without further reflection. The reflection coefficient (for the current for example) is 7j\ — Z2 Zx + Z2 (23-1) Fig. 7.27. Illustrating the condi- tions for impedance matching Tf th impedances are equal q vanishes (23-2) Zx - Z2 = 0, q = 0, and there is no reflected wave. On the other hand if Zx +Z2 - 0, q = oo, (23-3) then a reflected wave may exist without an incident wave. This is the condition for natural oscillations. In circuit theory the reflection coefficient is used as a measure of impedance mismatch for any pair of impedances (Fig. 7.25). If the impedances are equal the reflection coefficient is zero and the power is evenly distributed between them; if the sum of the impedances is zero, the reflection coefficient is infinite and oscillations may exist without a continuously applied electromotive force. 7.24. Expansions in Partial Fractions The input impedance and admittance can be expanded in partial fractions in terms of the natural oscillation constants. II the line is slightly dissipativc, such expansions can be obtained by computing the energies associated with different oscillation modes and using equations (5.11-18) and (5.11-19). In the present case this is unnecessary since the impedances have already been calculated and it is easy to obtain the expan- TRANSMISSION THE* »ky 233 mom. directly. For example, for short circuited and open-circuited lines wc use the well known expansions (24-1) tanh .v Hence for a line of length /, short-circuited at the output end, we have r/ i 2 * Tl /(()) = 8K Z gpr.I)y»Hr *r' KTl+Kn^nV+TV Since KT = Z and T/K = Y, we have Z(0) -Zit, 1 °° 2 (24-2) (24-3) These expressions show how the low frequency impedance Zl of the line is modified at high frequencies. The second equation, in particular, gives the effect as an impedance in parallel with the low frequency impedance. Substituting r = a + »j3, neglecting a1 in the denominator and a in the numerator we have ~3fa [(2„ _ 1)V _ 4/3V1 + S/«p72' If 1 l V m 1 (24-4) The approximate naturaljrequencies can be found from the above equations by inspection (since 0 = *VlC). Thus if the input terminals are short-aretnted, the natural frequencies are »ir ^^tVTc' and if the input terminals are open, then (In - 1)tt COn~ nVlc ■ In terms of the natural frequencies and corresponding Q's, wchave (24-5) (24-6) m=lá col - + , Y(0) = 2 , (R + iuL)lr „„ 2 , fob Wn — tů "t- Similarly for a line of length /, open at the output end, we obtain (24-7) 1 2 * > t >*» &l - co2 4- y(o) = f,z 0, „ are given by (5) and (6). 2.14 ELECTRl (MAGNETIC WAVES Cii/i Let us now consider a section of length /, short-circuited at both ends, with the input terminals nt distnncc x - d from one of the ends (Fig, 7.28). It is particularly easy to obtain the expansion lor (he input admittance Y(d) using the „,ethod explained in section 5.11. The infinities of Y(d) are the natural frequencies when (he input terminals are short-circuited and hence are given by (5). The current in the line (if the line is only slightly dissipative) is substantially proportional to cos 3nX, where x is the distance from one of the ends; hence, for a unit amplitude at the input terminals we have Fio. 7.28. A section of a line cos short-circuited at both ends. In(x) = -. cos pnd' For the nth oscillation mode the energy stored in the line is u (24-9) \L f ih{x))~dx = lb = Ml (24-10) Substituting in (5.11-19) and including the term corresponding to (5.11-24) we have t) cos2 3nd 1 2 M {R+iu>L)l LlK = 1 ú)„ — wz-|-' (24-11) On Similarly it is easy to find the expansion for the input impedance in the case shown in Fig. 7.29 because, if the terminals are open, the natural frequencies are the same as in the preceding case. The voltage in the line is proportional to sin dnX and, adjusting it for a unit amplitude at ,y = d, we have sin ar.a (24-12) Fic. 7.29. A section of a line short-circuited at both ends. The energy of a typical oscillation mode is then [Vn{x)Ydx=- CI Substituting in (5.11-18), we have z{d) = - £ ■ /to sin2 {ind to2 + (24-13) (24-14) In order to obtain the input impedance in the case shown in Fig. 7.28 we have to determine the natural frequencies when the terminals are open and then calculate the energies of the corresponding oscillations. In this case the two sections of the line are in series and the input impedance is the sum of two expansions similar to the expansion for Z(0) in (7). In the case shown in Fig. 7.29 the two sections of the line are in parallel and the input admittance is the sum of two expansions similar to the expansion for Y(0) in (7). TRANSMISSION THEORY 23S 7.25. Multiple Transmission Lines Kipiations (0 1 ) are said to describe a simple transmission line or a line admitting of only one transmission mode. In general transmission equations are of the following form —~ = — £ ZmlJks ~T~ = —■ E Yrnk^k. dx k I "X (25-1) For example, if two transmission lines rim parallel to each other waves in one may Influence waves in the other, since alternating longitudinal currents in one line induce .....-Minima! voltages in the other, and alternating transverse voltages induce transverse currents. Thus for a pair of interacting transmission lines we have dx dx (25-2) éEl = _ ZtlIi - Z22/2, ^r= - Y^V, - YnV%. dx dx Zlí is the distributed mutual impedance per unit length and Y12 is the distributed mutual admittance per unit length. Equations (1) are linear differential equations with constant coefficients and hence possess solutions of exponential form Vm(x) = rme~r*, /„(*) = Ime~T*. (25-3) Substituting in (1), we obtain vvn = E 2™A nm = E YnkVk- (25-4) Thus we have In linear homogeneous equations connecting 2n variables Vm, Im. These equations will possess solutions, not vanishing identically, only if the determinant of their coefficients is zero. This determinant is an equation of the »th degree in l'a and its solutions represent the natural propagation constants of the multiple transmission line. For each value of T we can determine the ratios of Vm, Im to some one variable. Thus there are In arbitrary constants at our disposal and these may be determined to satisfy assigned boundary or initial conditions. The line is said to possess n transmission modes. Each transmission mode is characterized by its pair ±Pm of propagation constants and by a relative distribution of voltages and currents peculiar to it. In practical applications the mutual coefficients are frequently very small. Then equations (1) may be solved by successive approximations, the first being the solution of n independent pairs of equations dVm _ * dim dx ax ■Y P. í nun' m (25-5) in which the interaction between the individual transmission lines has been neglected. The values of the voltages and currents obtained from (5) are now substituted in all >.u, ELECTROMAGNETIC WAVES Chat. 7 11n small If i.......I ( I ); thus w e obtain dx f'mmim XI '/'inkJky dl,„ dx -Ymmrm - Z'y^k, (25-6) where the primes indicate the omission in the summation of terms for which k = m. These equations are the equations of simple transmission lines with given distributions of applied voltages and currents. Solving these equations we obtain corrections to the previous solutions. The process can be repeated as often as may be necessary; but usually the first corrections are sufficient- In communication engineering the interference between neighboring lines is called crosstalk. The interference due to the mutual impedances Zm;. is called the impedance crosstalk; similarly the interference due to the mutual admittances Y,nk is called the admittance crosstalk. 7.26. Iterative Structures If a pair of sections (Fig. 7.30) of uniform transmission lines is repeated an indefinite number of times, an iterative structure is obtained which may have properties radically different from the properties of the original lines. Thus if the original lines were capable of transmitting all frequencies, the iterative structure might suppress some frequency bands. K,,r, 12 Fiq 7.30 A transducer formed by two line Fig. 7.31. A chain of transducers, sections in tandem. The equations for the present iterative structure may be obtained from the general equations of section 5.3 as soon as the constants of the transducer shown in Fig. 7.30 are calculated. This transducer consists of two transducers in tandem (Fig. 7.31). Using single primes for the constants of the first transducer and double primes for those of the second, we have Zuh + zUj = vx, z'Ai + zhh = v2, (26-0) ziJi + (z'22 + z'A)i + ziiit = o. Solving the last equation for / and substituting in the remaining equations, we obtain the constants of the combined transducer Z ii = Zu — z.22 Z22 = Z22 Zh ,71i 12^.21 Z22 + Z'l'l Z12 = — z2i = — ZÍ2Z12 Z22 + Z'{i' Z22 + Zi'i (26-1) TRANSMISSION THEORY 237 In the case of uniform transmission lines the transfer impedances of the constituent 11..inducers are equal and consequently Zu m Z21. „],•■■. t From (1) and from equations of section 7, wc obtain the following expressions for 1 be transducer shown in Fig. 7.30 Kx + KiKz coth Tih coth ^2 Zu = i.22 Ki coth Tih + Kt coth Tilt K\ + KiKz coth T1I1 coth T2h Ki coth Tih + K2 coth IV2 ' K1K1 csch Tih cschTj^ (26-2) 12 Ki coth T1/1 + Kt coth IV2 By (5.3-7) the propagation constant V per section of the iterative structure is K\ + K\ cosh r Uinh Tih sinh T2lt + cosh IVi cosh T2l2 = - [cosh (Tih + r«/s) - ?2 cosh (r,/i - T2h)], P where g is the reflection coefficient and? the product of the transmission coefficients 4KiK2 2_(Ki- K2)* (Ki+K2)iJ (Ki+Kt)*- Let us suppose that the transmission lines are nondissipative; then l\ - «V*i. 1'2 = iu/v2, and we have cosh r = - (cos toT — g2 cos wt), t Vl »2 h It T = —--• »1 Vt For some values of o>, cosh T will be in the interval (—1,+1) and T will be imaginary; for other values T will be either real or complex. Hence the structure will transmit some frequencies and suppress the remaining. Pass and stop bands may be determined by plotting cosh T as a function of coTor tor. 7.27. Resonance in Slightly Nonuniform Transmission Lines Consider a section of a slightly nonuniform nondissipative transmission line of length / and let this section be open at x = 0 and short-circuited at x = I. In the first approximation the longest resonant wavelength is X X = 4/, / = (27-1) The first correction may be obtained from the equations of sections 11 and 12 The current 70 at * = 0 and voltage V(l) « V0(l) + FAD at * = / '.MM ELECTROMAGNETIC WAVES ClUP. v vanish; thus tin- equation for resonance is approximately U + Hi)) COS fit - i/t{!) sin 81 + iC(/) sin # = 0. (27-2) Assuming that 4 5 and C are small compared with unity, we let X = 47(l-5), (27-3) where S is small compared with unity. Substituting in (2) and ignoring small quantities of the second order, we obtain 5 2 * -~- tMt) + iC® = 0, 8 = - [C(/> - <*(/)]. (27-4) The values of and C(I) are computed* from (12-2) M - I /J - «0^ cos y£ rf§ C(/) = | jT" + Kot) d$. (27-5) If the line is short-circuited at x = 0 and open at x — I, so that Vc, = 0 and /(/) = 0 then the first approximation to the principal resonant wavelength is (1) and the correction 5 is 2i 5 = - W(f) + C{1)\. IT (27-6) In many practical cases the product ZY is constant and (see 12-3) ft Z0 i^o (27-7) An is the average characteristic impedance which may be defined by Ko = ) fo K(x) dx, Ki*) = yj^y (27-8) and iX is the deviation of the " nominal " characteristic impedance K(x) from this average impedance fc(x) = K(x) — K0. In this case C(l) = 0 and Cl K{x) ttx , 7T r'f, K(x)~\ ttx Hence if the line is open at x = 0 and shorted at «• = /, then b = \ cl \*zp _, 1 cos« A = i r' xw cos ™ ^ (27_10) * In the integrand of A([) wc replace X by its first approximation 4/. TRANSMISSION THEORY 239 11 i he line is sliorur a I x 0 and open at ,v /, then J = | fTl - S£]»tB A = - 1 f' A'Wcos^, (27-11) /«/.o L A0 J / iK0J0 I The integrand in the last terms of the above equations is positive near x — 0 and negative near x = 1. If the capacitance per unit length varies more or less uniformly and if it is larger near the open end than near the shorted end, then the principal resonant wavelength is somewhat shorter than 4/. If the capacitance is larger near the shorted end, then the resonant wavelength is longer than 4/. If the line is shorted at both ends x = 0 and x = I, then the principal resonance occurs approximately when X = 11, I = X/2. It is left to the reader to show that the more accurate expressions are X = 2/(1 + x), / = - (1 - x), (27-12) where x = (l/^)[A(l) + C(l)] and A(l) = \^ (Jr - KQTJ cos ?f dx, C{1) = fj£ (M 4- Ko?) dx. (27-13) If ZY is constant, then we have cos-"*Jx. (27-14) If both ends are open, then the correction term is — x- The above approximate expressions give the two principal terms, one independent of and the other varying inversely as the average characteristic impedance. By continuing the process of successive approximations in the solution of nonuniform transmission lines more accurate expressions for the resonant lengths of such lines can be obtained; but usually the above formulae are satisfactory for practical purposes. The method applies also when the line is terminated in some reactance at either or both ends, as in this case we may start with the prescribed value of y(0)/I(0) and plot the ratio V(l)/I{l) as a function of / until we obtain the prescribed value of this ratio. In the simple cases when the ends of the line are either open or shorted, another method of treatment yields the above results and some similar approximations. Under these terminal conditions the energy equation (3-3) for nondissipative lines becomes f CVV*dx =■ (*LII*dx. (27-15) 240 r.u:ctkomaoni'tic waves Chap, 7 This equation simply reiterates the fact that at resonance the maximum electric and magnetic energies are equal. In the important special case in which LC = v = constant, the transmission equations may be expressed in the following form io dx ' u dx Substituting in (15), we obtain 032 4r2 X 1 \dl I dx dx X dx f Jo L\l\2dx f dx C\ V\2dx (27-16) If now the line is open at x = 0 and short-circuited at x = /, we assume the current and voltage distributions that would exist in a uniform line TfX /(*) = /sin ~, nx) = rcos^, and substitute in (16). Thus we obtain 1+5 1 - $ whe Jo 16/2 X2 TVX JL cos — dx 1-8 i+l? Cl r ** I L cos — ^0 / dx J Jo Z, i/.v J C dx (27-17) (27-18) The above value of 5 is the same as that given by equation (10). If the line is shorted at x = 0 and open at x = /, then IS/2 _ 1 - 5 = 1 4- £ X2" " 1 + 5 ~ 1 - $ ' If the line is shorted at both ends, we assume 1TX ... . xv (27-19) and find /(*)=/ COSy, F(*) = rsiny, 4Z2 .. 1 - x 1+i 1 + x 1-x* (27-20) TRANSMISSION TIII.OKY 241 i\ l-elc "(I 2t.v cos -j- dx f C cos -j C)F, dz dz T _ tib ga ta abb The characteristic impedance and propagation constant are b I loin a a \ I— } p = a = V iap.(g + *'we). / g + iux * Assuming I positive in the positive .-direction which is away from the reader in Fig. 8.2. QWií^tóv ES t ii,IP, mi ELECTROMAi If the guard plates nre removed, thtftfm "Idj A S near the edges of our parallel strips will bulge out as shown jw/f'^t >ywW the magnetic lines will bend round to enclose one conduct(j,'^/:BVH V Subsequent analysis will show that a wave of this modili&j •f;t) \ ^vtxist at all frequencies ______ and that the shape jj'/('f t(i ion of the electric and magnetic lines are inc/>j'| if the frequency. For such a wave the edgi,'/ ^'.l^ V tail if b is small compared to a since th,(r/V./d|', distributed largely between the plates. . If b is small con^|'|w ^#he strips can be bent into cylinders to f^Whe \\^inductors with nearly Fig. 8.3. Coaxial equal radii (Fig. lines will run along ^afradif near'y radii and inaSnet',c V^/rtJl^axial circles between eqU' ra the conductors. Thy/, / _. ^Wk8 slight " curvature effect" instead of the edge effect. ThM'{r\^ V^ect is comparatively small; thus if the radii of the conduct^If ^ \1 VI b {b > a), then by the parallel plane formula (using the av(f / v, mference for the approximate length of the magnetic line&)JWjm\ it/'.' wa.Hi ^ = y(b - a) _ |J (b + a) 'Iff ~. l.v /» 41.6. y \ *dius are added while . V lines, the character- If* = 2a, this gives A = 40 ohms; the^$'.e^' Since the voltages along various partsjff£ the magnetomotive force is the same forinductors might be subdivided. Thus if b — a is divided int„ . »i ^rts, the exact value of K may be expressed in the following f<^ | , r i I &w \ A = lim 120(J-_) --rr—7 + J=s0 /'fl % L(2w -1)«+* (2»-' ^'f1 as « J', h *M2: J_1 :»-i)*J choosing » - 2tlifjjrtyJ\ X = 41.1; this value te by about 1 p^'J^M^ \^ Let us now return to waves in an at&W ffipu With transverse dimensions fading out of the picture, we/ '"' lA Av—^- Taking again b = 2a and differs from the exact value intensities E and#, ^^^^f| W^lfdf* the ratio E/H as the mt* impedance in tfff \ \V« d a^ A uniform plane wave can be generated U? \VnTI! ProP^on. \ ent sheet of uniform : generated by/,r * When a numerical value is ascribed to the intj assumed. 1 ent sheet of uniform ^e, free space is usually A WA VILS, WA Vl<. GUIDES AND K ESC 1NATORS — 1 245 density. Consider such a sheet in the xy-plane and let its density be /«. 'in. i the elect ric intensity is continuous at the sheet while the magnetic intensity is discontinuous, we have £,(+0) = £,(-0), Hv(+0) - Hv(-0) = -/* The current sheet acts as a shunt generator and sends out plane waves in both directions E$(z) = -hJ* 0, £r(sO = -hJ*e", H-(z) = \jxe°°, z < 0. The complex power (per unit area) contributed to the field by the impressed forces is * = -\EMJt = bJ.Ji> II the medium is nondissipati ve, then the power carried by each wave per unit area in an equiphase plane is _<+ = %Et(z)[Hi(z)\* = Uj,Jt> V = -hE~(z)[H-(z)}* - hJ*Jt- The sum is equal to the power contributed to the field. The total power carried by a uniform plane wave in an unlimited medium is infinite and the wave cannot possibly be started by an ordinary generator. The principal reason for considering such waves at all is their simplicity, combined with the fact that at great distances from any antenna and in a sufficiently limited region the wave is nearly plane. If the medium is nondissipative it is possible to send all the energy in one direction only. Consider two parallel equal current sheets (1) and (2), a quarter wavelength apart, and let the currents be in quadrature. If the current in the left-hand sheet (2) is 90 degrees ahead, then the right-hand wave generated by it will be in phase with the right-hand wave generated by the sheet (1); the two waves will reinforce each other. The left-hand wave from (1) will be 180 degrees out-of-pbase with the left-hand wave from (2); the two waves will destroy each other to the left of the plane (2). The electric intensity of the wave produced by the sheet (1) will directly oppose the electric intensity of the second sheet and reduce the total intensity at that sheet to zero; hence the second sheet contributes no power and may be taken to be a perfect conductor. The electric intensities of the two waves reinforce each other at the sheet (1). Assuming that this sheet is in the plane z = 0, we have therefore E$(z) = -nJS*\ H+{z) = -Jxe-**, z > 0. The power emitted by the sheet is twice that which would be emitted by an isolated sheet. 246 ELECTROMAGNETIC WAVES ' Chap. 8 Let \is look al (lie situation in another way aiul assume at the start that the plane (2) is a perfect conductor. Hy (7.6-3) the impedance a;, seen from plane (1) leftward is Z~z = r\ tanh ifil = irj tan /3/, where / is the distance between the planes. If / = X/4, this impedance is infinite and no power will flow to the left of plane (1). (2) CD (3) (2) (i) (31 lAi Fig. 8.4. Passing of waves through and reflection from resistance sheets. Fig. 8.5. Transmission diagram representing the case in Fig. 8.4. The wave to the right of sheet (1) can be completely absorbed by a thin conducting sheet (3), with surface resistance equal to ij, if the sheet has a perfectly conducting sheet (4) a quarter wavelength behind (Fig. 8.4). To facilitate the use of the transmission theory of the preceding chapter we construct a transmission line diagram (Fig. 8.5) in which the current sheet (1) is shown as a shunt generator, the resistance sheet (3) as a shunt resistance and the perfect conductors (2) and (4) as zero resistances at the ends of the line. Without the reflector (4) the impedance of the sheet (3) would be in parallel with the intrinsic impedance of the medium behind it, the impedance presented to the incoming wave would be only |r?, and some of the wave would be reflected. It should be noted that the absorber (3) will function just as well even if the medium between the resistance sheet and the reflector is different from that between the resistance sheet and the generator. The impedance normal to a plate of thickness / (Fig. 8.6) is in general Fic. 8.6. A cross-section of an infinite metal plate. Zž(0) = C°Sh ~*~ V S'"k ^ v cosh al 4- Zt(l) sinh al' (1-3) where Zz(l) is the impedance looking to the right of the plane z = /. If the latter plane is a perfect conductor, then ZZ{1) = 0 and we have 2S(0) = i) tanh al. If 2 = / is a sheet of infinite impedance, then Z»(0) m v coth al. (1-4) As we have already pointed out, in practice an infinite impedance sheet at Z — 1 can be provided by placing a zero impedance sheet at z = / + X/4. WAVES, WAVE GUIDES AND RESONATORS 1 247 II the plate is a good conductor its intrinsic impedance jj is very small i veil al very hi^h frequencies. II the medium to the right of z — I is free I ice, then Z,(l) = 377. This impedance is so large compared with r\ ih.il equation (4) represents an excellent approximation to the impedance imnual to a plate ol high conductivity provided / is not too small. If j; ■'i much smaller than Zs(l), then, regardless of the thickness of the plate, wc can ignore the second term in the numerator of (3) and obtain the following approximation ri coth al Zz(l) Thus the input admittance is represented as equivalent to two admittances in parallel, the admittance of the plate on the assumption that ZZ(T) = oo and the admittance Yt(l) itself. When / is very small the " open-circuit " impedance (4) for any quasi-i onductor becomes approximately 2.(0) _,(J. + iW+...)_i-I. 1/377*. A n,,o-, This impedance is equal to the free-space impedance if / sheet of this thickness with a reflector behind it to provide an open-circuit condition will completely absorb a plane wave incident normally to the plate. It should be noted however that for very thin films the value of g is different from that for the substance in bulk. The formulae for the reflection of uniform plane waves from a plane interface between two homogeneous media (Fig. 8.7), when the incidence is normal to the interface, fol low immediately from (7.13-3) and (7.13^1); thus we have 2tj2 2vi Pe = -;-, Ph = Fig. 8.7. Reflection at normal incidence. V2 — 'it m + n v\ — 172 I?l + 12 IJl + 1)2 ' i?i + m (1-5) At a metal surface the reflection is almost complete, E practically vanishes and H is doubled; almost pure standing waves are formed with nodal planes for E parallel to the metal surface at distances 0, X/2, X, ■ « ■ from it and nodal planes for H at distances X/4, 3X/4, 5X/4, ■ • The planes of maximum H coincide with the nodal planes for E and the planes for maximum E are the nodal planes for H. The shielding effectiveness of metals is great; it can be judged by using (7.6-10) to obtain the ratio of magnetic intensities at the two surfaces of 248 ELECTROMAGNET! C WAVES Chap, h ii plate in hr.c space //(/) //«)) V cosh al4- 377 sinh al' Even for quite thin plates sinh tf/}8 approximately equal to \cal and (I r,) H(0) 2rj_ 377 //(0)| 377 where £R is the intrinsic resistance of the plate. As the frequency dimin ishes, tj and 8. Elliptic polarization. (Fig. 8.8) whose semi axes are E\ and /?•>■ The wave is said to be elliptically polarized; it is circularly polarized if Ei = E2. If = 7r/2, the ellipse of polarization is exactly the same but the vector rotates clockwise instead of counterclockwise. This polarization is said to be left-handed as distinct from the right-handed polarization in the preceding example. If in the right-handed polarization the electric vector is represented by the handle of a corkscrew, then as the vector rotates the screw advances in the direction of wave propagation. For values of 1? other than ±90°, the wave is still elliptically polarized but the axes of the ellipse do not coincide with the coordinate axes. So far we have considered the electric vector in the plane 2 = 0. For % > 0 the amplitudes of both components of E are multiplied by e~0,1 and the ellipse becomes smaller; the phases of both components are retarded by @z but this simultaneous retardation does not affect the orientation of the ellipse. The magnetic vector describes another ellipse. In nondissipative media // is evidently perpendicular to E at all times; but in dissipative media this is not the case. When t) = 0 or n, the E-eliipse and the //-ellipse degenerate into straight lines and the wave becomes linearly polarized. 8.3. Wave Impedances at a Point In an orthogonal system of coordinates the components of the complex Poynting vector are Pu = §(£,//* - EwHf), P, = ^(EWH* - EUHZ), Pw = %(EUH* - EM*). !50 FJ.jjctto 'Magnetic waves Chap. 8 The real pail ol each component rep resents the average power per unit area flowing parallel to the corresponding axis. The following ratios are defined as the wave impedances at a typical point looking in the directions of increasing coordinates Jtl y ±1 uj 11 ,L 7+ _ Ev_ 11 u *Cúz — p 7+ = — The waveImpedances looking in the directions of decreasing coordinates are denned by a similar set of equations 7~ — — 7~ = _ ^1 v- -Eju "tt jfÍM The ^-component of the Poynting vector becomes Pu — ziZ^HyH* -pZ^/f^/f*) = Tj,(AimHvH* -f- Zm,HwH*); the remaining components are obtained by cyclic permutations of «, y, ot. The algebraic signs in the definitions of the wave impedances have been so chosen that, if the real part of any given impedance is positive, the corresponding average power flow is in the direction of the impedance. We have seen that the impedance concept plays an important part in transmission theory, but the general formulae of the preceding chapter have been obtained for simple transmission lines having only one impedance in a given»direction. At a junction between two simple transmission lines two variables V and I must be continuous and the reflection coefficients depend on their ratio. Transmission theory of this type can be extended to a transmission line with two transmission modes, when there are four variables V\, Ii and V2,12 which must satisfy continuity requirements at a junction. The resulting formulae are so complicated that it is doubtful if they would actually save labor in solving problems. At any rate, until a sufficiently large number of problems involving such two-mode transmission lines arises, it is preferable to treat each individual problem by itself, particularly since in many practical problems double mode lines can be approximated by two nearly independent single mode lines. In considering waves in three dimensions the situation is in general vastly more complex. For a general wave the wave impedances are point functions and no advantage is derived from their introduction; one might just as well waves, wavI1'. GUIDES and RESONATORS 1 .'SI deal with the n hkiI field intensities. Hut lei us suppose that two impedances Z„„ and Z„u, for example, associated with a given wave, are independent of the // and v coordinates; then in effect we have a double mode transmission line. If at any surface w = Wq the properties of the medium are suddenly altered, the four tangential components have to satisfy continuity conditions at one point only — the continuity conditions elsewhere are automatically satisfied as soon as they are satisfied at this point. The amplitudes of the reflected and transmitted waves will depend on the associated wave impedances. If, furthermore, the two wave impedances in the same direction are equal zt = 7+ _ = Eg Hi _j_ _ Eu _ _Ev with the corresponding set for the impedances looking in the opposite directions, then the transmission theory of the preceding chapter can be applied in full. E„, Hv, Z,t and E„, — Hu, Zt form right-handed triplets. It is not necessary that all the wave impedances should satisfy these equations. If we are concerned with reflection of waves at the surface w = w0, only Z,| and Z„ need exist and be independent of the « and v coordinates. Likewise, we are concerned only with Z„ and ZZ when considering reflection at the surface it = «o- 8.4. Reflection of Uniform Plane Waves at Oblique Incidence In considering reflection of uniform plane waves falling at an arbitrary angle on a plane interface between two homogeneous media (Fig. 8.9) it becomes necessary to distinguish i 2 between two orientations of the field vectors: (1) the case in which H is parallel to the interface, (2) the case in which E is parallel to the interface. The impedances normal to the inter- 0) face are different in the two cases.* If neither E nor H is parallel to the FiQ- 8.9. Reflection of uniform plane waves interface, the wave is resolved into !ncidcn.[ «bIi1uel>'at a/«" boundary; H , . - is parallel to the boundary. two waves, one having the first or the above properties and the other the second. This resolution is always possible since the E-vector for example can be resolved into two components, one parallel to the interface and the other in the plane of incidence, that is in the plane determined by the wave normal and the normal to the interface. The second component is along the line of intersection of the equiphase plane and the plane of incidence. The component of H associ-* For a general orientation the impedance normal to the interface does not exist. ) i,' > PXECTRi (MAGNETIC WAVES Chap. 8 ated with this componenl of E is perpendicular to it ami to the wave normal; therefore this // is parallel to the interface. First we shall assume that // is perpendicular to the plane of incidence and hence parallel to the boundary between the two media; in Fig. 8.9 the positive direction of H is in the positive .v-direction (toward the reader). The angle # between the wave normal and the normal to the boundary is called the angle of incidence. In radio engineering its complement \-w — t? is frequently used. The equations for the incident wave are E = Eoe-", H = Hoe-", E0 * VH where E0, Ho are the field intensities at 0 and s is the distance from 0 along the wave normal. In cartesian coordinates we have s = y sin § — z cos »!>, and Hx = ttoT", Ev = E0 cos distinguish between the two cases and shall refer to either angle as the Brewster angle. If the medium is nondissipative and if the .vy-plane is a perfect conductor, the total magnetic intensity is Hx = 9in 9 cos' + jfi» cos *) = 2/Yo cos Ote cos )*-**,in' . The equiphase planes are normal to the .vy-plane and they travel parallel to it with the phase velocity vv = p/sin «?. The components of £ are obtained cither by adding the incident and the reflected components or directly from (4.12-16); thus Eu = 2/7,//0 cos d sin (j33 cos v)e~ii3v*in*, Ez = 2r,Ha sin # cos (8z cos d)c~Vv si" * . The wave impedances associated with the total wave are Zj = 77 sin d, Z% = — i-q cos § tan (Bz cos 1?). The impedance looking in the z-direction is imaginary and on the average there is no flow of power in this direction. O Y Fig. 8.10. Oblique incidence, E is parallel to the boundary. If the electric vector is parallel to the .vy-plane and the magnetic vector is in the plane of incidence (Fig. 8 10). then for the incident wave we have Ex = Eoe-1^", H„= - — cos 0 e~r**T", Hz=--- sin & f-r^+''^j Z- = j? sec «?. The impedance associated with the reflected wave is also 77 sec 1?. Thus we obtain the following reflection and transmission coefficients for the tangential components of E and H Pe = Z — 77 sec ?? ■ 77 SI 2Z qE =-2^r£_- v sec t? - Z Z+7,secV q,l> ^JZf^> Z + 77 sec 1? 2tj sec # 17 sec t? -f- Z' waves, wave (.hides and resonators i 2SS for the normal component of // we have Jjy„ - //„ = />;,;. It is now evident that the reflection coefficient depends on the state of polarization. Tlie impedance Z,z is never greater than 77 when H is parallel to tin- .yv plane and it is never smaller than 77 when E is parallel to the xy-plane. When the angle of incidence d is nearly 90 degrees, the component of /'.' parallel to the ,ry-pJane is very small for the polarization in Fig. 8.9 and hence the impedance is also very small. No matter how small Z may he, for angles sufficiently near 90 degrees the impedance associated with the incident wave will he much smaller than Z so that the total H and the lota! normal component of E will nearly vanish at the .vy-plane, while for most values of»? these quantities are nearly doubled. On the other hand for the state of polarization shown in Fig. 8.10, it is the component of H which is small when # is near 90 degrees; the wave impedance is then very large. If Z is smaller than 17, then Z is smaller than Zz for all angles of incidence; and as the angle of incidence increases, reflection only becomes more nearly complete. The preceding equations apply either to the special case in which the medium below the plane is homogeneous or to the more general case in which the medium below consists of homogeneous layers with their boundaries parallel to the .ry-plane. Let us now consider the special case in detail. For the wave below the .ry-plane the propagation constant y„ in the direction parallel to the j'-axis must be equal to the corresponding propagation constant in the upper medium or else the tangential E and H cannot possibly be continuous everywhere; thus yy = Ty = a sin «3. (4-3) Since jx = Tx = 0 and since yx + 7j + 7z = °"2> where a is the propagation constant characteristic of the lower medium, we have Vi3 — cr2 sin2 ■&. (4-4) If both media are nondissipative, equation (3) may have another interpretation besides the obvious one that the velocities along the ■y-axis of the wave above the Ary-plane and of that below it are the same. Let us assume that the transmitted wave (or the refracted wave in the terminology of optics) is a uniform plane wave and that the angle of refraction is 1? (F'ig. 8.11); then just as in the case of the incident wave we have Fig. 8.11. Angles of incidence, reflection, and refraction. iß sin 7« = iß cos i?, (4-5) 256 ELECTROMAGNETIC WAVES ■ Chap, a and equation (.1) becomes I . - sin I) 0 6 V/Jii ... J p sin t)» = B sin 0, or-- ».»■.*■ —i== : (4-6) sin i? 0 v V^e that is, the sines of the angles of incidence and of refraction are proportional to the characteristic phase velocities of the media. When a wave passes from a medium with higher characteristic velocity into a medium with lower velocity, the equiphase planes tend to become Fig. 8.12. Refraction of waves passing from a medium with high characteristic velocity into a medium with low velocity. Fig. 8.13. Conditions existing when the angle of incidence is equal to the angle of total internal reflection. more nearly parallel to the interface (Fig. 8.12); in passing the other way, they tend to become more nearly perpendicular to the interface. This means that if v < v or "\//i7 > V/i£, there will exist an angle of incidence 0 for which the angle of refraction is equal to 90 degrees and the equiphase planes in the lower medium are normal to the interface (Fig. 8.13). This critical angle of incidence is called the angle of total internal reflection be-i cause, as we shall presently see, the wave is completely reflected. Setting $ <= 90° in (6), we obtain . I :' *9 $M sin y = - = - = - . For this angle the propagation constant yz in the lower medium vanishes since in the present nondissipative case we have "fz ~ ip cos 0, cos sin3 0. For 0 < 3, yz is imaginary and the angle of refraction is real; but for 0 > 0, y? becomes real and the eauiphase planes become normal to the WAVES, WAVE GUIDES AND RESONATORS I 257 miniate. The field in the lower medium is I hen attenuated exponentially with the distance from the interface, which indicates that the average How of power across the interface is zero. Since the free space wave velocity is higher that) the wave velocity in any other dielectric, the phenomenon of total internal reflection can always occur at a boundary between free space and a dielectric when the incident wave is in the dielectric. In the case of water and free space p = 9$ and 0 - 6° 23'. When 0 is sufficiently greater than y-. is given approximately by It sin 0 7z = P sin 0 =--, yz\ = 2* sin 0. * For short waves the attenuation becomes substantial. We have seen that if the incident wave is uniform, the reflected wave is also uniform; on the other hand, the transmitted wave is not necessarily uniform even in the special case of nondissipative media. The foregoing properties of transmitted waves are independent of the state of polarization. This state has to be specified if the values of the transmission and reflection coefficients are sought. Let us start with the case in which H is parallel to the boundary. Inasmuch as the wave is generally nonuniform we shall write our equations in terms of the propagation constants yy and yt and use the " angle " of refraction 0 only for the sake of attaining formal symmetry in the results. We simply define the complex angle 0 by the following equations yy = « These expressions are of the same form as for uniform waves except that 0 is no longer real. We now let Z = Z7 and obtain 1j cos 0 — ij cos 0 qn ij cos 0 + y cos 0 2r) cos 0 PH ~ 17 cos & + v cos $ 2r] COS $ . 77 cos 0 + >) cos 0 1 - k l + k' m, = —la, 2 l+k' 2k l + k' ps> (4-7) 25« ELECTROMAGNETIC WAVES W,,elC * ia t,,B fi'««'win« impedance ratio r* . V cos " brom (3) we have Chap. « cr sin t> = J- s-m $t Z_ m sjW sin $ u sin if If both media are nonmagnetic (or, more generally, if they have the same permeabilities), then ij k is well inside the unit circle and H as well as the normal component of E may be almost doubled; but for low angle waves (d 2> 0>o), k is well outside the unit circle and these components are nearly annihilated. In the latter case the tangential component of E is nearly doubled; but this component is small to begin with. Thus near grazing incidence the entire field at the ground is nearly annihilated by the reflected wave. 8.5. Uniform Cylindrical Waves A wave is cylindrical if its equiphase surfaces form a family of coaxial cylinders; it is uniform if the amplitude is the same at all points of a given equiphase surface. Choosing the axis of such waves as the z-axis, and assuming d/Bip = 0, d/dz = 0 in the general equations, we have Ep = 0, Hp = 0, and dEz — (pffv) = (g -f io>e)pEz; |- (pE^) = -iwfipHt, ~ = ~(g+ icM)Ev. (5-1) (5-2) Thus uniform cylindrical waves are transverse electromagnetic and they may be of two types: (1) waves with the £-vector parallel to the axis, (2) waves with the //-vector parallel to the axis. If Hv is eliminated from (1), we obtain dzEz ^dEz 2 P ~JY * 1--) represents an outward bound wave, hoi large values of p it is very similar tp a plane wave except that the amplitude is steadily decreasing, as indicated by the factor p~1/2. While the hrst solution is infinite at p — 0, the second is finite for all finite values ol />; hence it is appropriate for source-free regions for which p < a. The //-wave functions corresponding to (3) are obtained immediately from (1); thus :(p) = - - lUiap), H-(P) = -Map). II j (5-4) From (3) and (4) we obtain the radial impedances Ki}(ap) 7+ - Kdap)' zz - /'n(trp) (5-5) z o p Equations (1) apply either to cylindrical waves in an unlimited medium Or to waves between two perfectly eon ducting planes perpendicular to the E-vector. Let one of these planes be z — 0 and the other z = h. If V is the transverse voltage from the lower plane to the upper (Fig. 8.14) at a distance p from the cylindrical axis and if I is the total radial current in the lower plane, then V — hEz, I = —lirpH^. The second equation can he derived in several ways. Thus the outward radial current I(p) is equal to the downward transverse current inside the cylinder of radius p; since this transverse current is equal to the magnetomotive force, we have the desired equation. Substituting in (1), we have FlO. 8.14. Two parallel conducting planes supporting uniform cylindrical waves whose axis is OZ. dV . ,7 dl —■ = —iwL.1, ~r~ dp dp (G+mC)F, where the distributed constants per unit length of the " disc transmission line " are ph 2vgp L = 27P' °- h C == 2-rrep Since the electric lines are straight lines normal to the two planes, we could have obtained G and C directly by considering the conductance and the capacitance between annular rings of width dp, one in each plane, and dividing the result bv dp. The inductance per unit length along a radius 262 ELECTROMAGNETIC WAVES could be similarly obtained from the inai-nelic flux passing through the rectangle A BCD shown in lug. 8.15 and lolling III -■■ ,{p. For the wave impedances of the " disc transmission line '* we have = — 7+ K- - A 7- The complex power carried by a progressive wave traveling outward is then = ^K+II*. The wave £j, H# is not progressive; it is strictly stationary when the medium is nondissipative. If the medium is homogeneous within the cylinder of radius p, the radial current in the planes must vanish at p — 0; hence the disc line must behave" as electrically open at p = 0 and the energy will be completely reflected. Some energy will travel Fig. 8.15. Illustrating elementary derivation of transmission equations for uniform cylindrical waves. Fio. 8.16. A section of an infinitely long wire. inward only if the medium is dissipative or at least if a dissipative wire connects the two planes along the axis. The wave functions (3), each corresponding to a homogeneous region seem to be more suitable for practical purposes than other possible sets. If the region is homogeneous and source-free only between two cylindrical surfaces p = a and p — H>, then the field is expressed in terms of both wave functions. The complex power flow in the stationary wave is ■ff" = ^K~II*. These expressions for the complex power represent the total power flow across a cylindrical surface between two parallel planes. The radial flow per unit area depends on the radial impedances (5); thus There is another aspect to K+ and K~. Consider an infinitely long wire and let an electric intensity E'{ be applied uniformly round the surface of the wire (Fig. 8.16). Let / be the-current in the wire in response to If Fr(p) and E+ip) are respectively the field intensities in the wire and outside it, then Ei = E~(a) - E+(a). The intensity driving the return current, external to the wire, acts of course WAVES, WAVE GUIDES AND RESONATORS I 263 in the direction opposite to the intensity driving the current in the wire. Dividing by /, we have (5-6) where Z,- and Z„ are respectively the internal and the external impedances of the wire per unit length; they are equal respectively to K~fk and K+/h. We shall now consider the numerical magnitudes of the impedances under various conditions. If the frequency is so low that | aa \ t)a + \itůiia, K — (g + 2£jje)irtf2 Sir [ft = 0 or if o>e can be neglected in comparison with g, then h K~ R + i«4 R = gita 87T Thus we have the low frequency resistance and internal inductance of a wire of radius a and of length h. If on the other hand g = 0, then 1 ÍTO K - — + iwL, C- — , (5-7) and we have an expression for the low frequency capacitance of a capacitor formed by two parallel circular discs of radius a separated by distance Inasmuch as these expressions have been obtained on the assumption that the electric lines are normal to the metal discs, in practical applications of (7) we must assume that h is small compared with «; then the formula furnishes an approximate value for the " internal capacitance " of the two discs. The external capacitance between the outer surfaces of the discs is more difficult to calculate; but it is, of course, considerably smaller than the internal capacitance. When | a-a | -C 1, the outward looking impedances (in a nondissipative medium) become Z% = imixa (log ^ + 0.116) + ^ , 2A 2» V B 2m» The external inductance depends on the frequency and becomes infinite at / = 0; but the external impedance vanishes at/ = 0. '.'(,1 ELECTRí MAGNETIC WAVES Chap, h At high frequencies in good conductors we obtain from the asymptotic expanses of the modified BesseJ functions the following expressions K+ t}h 2wa égra 2 ' 2ra 4g-n Since in this case v = 9.(1 + the high frequency resistances of a wire of radius a and of length h and of a metallic conductor extending to infinity in the radial direction are respectively 2wa igira R 2-wa Awga2 The inductive reactances are equal in magnitude to the first terms in these formulae. The exact expression for the internal impedance per unit length of a conducting wire is 7 _Zrp T}I0(va) 2wa 2iral\ (aa) (5-8) The phase of a is 45° and, in order to separate the real and imaginary parts, the following auxiliary functions are introduced /o(aV*) = ber a + i bei a. The power series for these functions can be readily obtained from the power series for the /-functions; thus ber a = Y, ,„ : aj, bei a = 23 n=o24"[(2w)!]5 ,l=o24"+2[(2« + l)!]2' Hence if we let a = ds/ujxg in equation (8) and separate the real and imaginary parts, we obtain Zi(f) a[ber u bei' a — bei u ber' a] . «[ber a ber' a + bei a bei' a] Zt(Q) = 2[(ber' a)2 + (bei' a)2] ' 2[(ber' uf + (bei' a)2] ' where 2,(0) is the d-c resistance of the wire. The ratio of a-c to d-c resistance is represented by the solid curve in Fig. 8.17; the dotted curve represents the ratio of a-c reactance to d-c resistance. Let us consider the field external to an electric current filament of radius a. For the magnetic intensity at distance p we have H*(p)=2.aK1() = t + 2* (log*- 0.222) where £0 i-s the intensity of the incident wave. This is the field scattered by a narrow perfectly conducting strip. In optics this field is called the field diffracted by the strip. If the strip is not perfectly conducting we should add its internal impedance to Ze and then compute the induced current. * Remembering that this means 8s is small compared with unity. WAVES, WAVE GUIDES AND RESONATORS 1 267 H.6 Cylindrical Cavity Resonators Consider a perfectly conducting cylindrical box of radius a (Fig. 8.18) and assume that the medium inside is nondissipative. An electric disturbance, once started inside this box, will continue indefinitely since no energy can escape through the conducting walls. Thus there may exist free oscillations similar to those in a simple circuit containing an inductor and a capacitor or to those in a transmission line short-circuited at both ends. Even in the latter case there are infinitely many oscillation modes and corresponding natural frequencies; the box, having three dimensions, may be expected to have a triple infinity of oscillation modes. In this section we shall confine our attention to the particular oscillation modes in which the £-vector is parallel to the axis of the box and is independent of the ). Since Ez must vanish on the boundary p = a, Ba must be a zero of fo(x) and 27Tfl T pa =-= 2.40, 5.52, 8.65, 11.79, The consecutive values differ approximately by jr. For the mode corresponding to the lowest natural frequency we have 2.40 d = la = - -X = 0.764X, X = 1.3Id, where X is the wavelength characteristic of the medium. The corresponding frequency/ is then v/\ = l/XV^e. On the axis the electric intensity has the greatest amplitude and the magnetic intensity is zero at all times. In general Ez and Hp are in quadrature. The charge density on the bottom face of the cavity is qs = zEfo(Pp), and the total charge is q = lirtE CpMPp) dp = ^±) **(i+! <1-|) In dealing with uniform cylindrical waves in nondissipative media bounded by two cylinders p = a and p = b, where b > a, it is more convenient to employ the following wave functions E7(p) = Jo(0p), Et(p) = /VoOJp); t,//-(p) = ijitfp), vHtif>) = /Ari(/3p). The /C-function which is more suitable for waves traveling to infinity is now replaced by the A/'-function which represents a stationary wave with a singularity at p = 0. The radial impedance looking from p = a to p = b may be obtained from (7.10-8); thus for a perfect conductor at p = b we have htfaWoim - N0(fia)Mfib) *W ■ 7,(fia)NoW) ~ N^Joifib) r This impedance either vanishes or becomes infinite, according as aw = /#) Jitfa) = hi&) (6-1) The first case corresponds to the natural oscillations when there is another perfectly conducting cylinder at p = a (Fig. 8.19) so that we liave a toroidal cavity. The second case corresponds to a screen of infinite impedance at p = a; it approximates a cavity with a small hole through the center of each of its flat faces (Fig. 8.20), When a and b are large, the roots of the above equations are easy to calculate since the disc line becomes nearly uniform and in the first approximation b — a = X/2 (for the gravest mode, of course) in the case of the two conducting cylinders and b — a = A/4 in the Fig. 8.19. A toroidal cavity bounded by two coaxial cylinders and two parallel planes. 270 ELECTROMAGNETIC WAVES Chap, h case of the one perforated cylinder. In this case only the small deviations from these values need to he calculated. For this purpose the Bessel functions are replaced by their asymptotic expansions Approximate formulae for the roots of (1) are available in books on Bessel functions. When a and b are fairly small which is the case for the principal oscillation mode, it is more expedient to compute the roots graphically. The ratio Jo(x)/No(x) is plotted as a function of x. This graph consists of an infinite number of branches and it cuts the .v-axis when x is a zero of J0 and goes off to infinity when xis a zero of Nq. Then we pick pairs (x1}x2) corresponding to the same ordinates, and thus obtain pairs of values of fia and fib which satisfy the equation. Starting with fia = 0 and selecting the smallest corresponding value of fib, we plot the latter against fia. Such a curve makes it possible to compute the dimensions of the resonator or the resonant wavelength, as may be seen from Fig. 8.21. In this figure Fig, 8.20. A per-forarcd cylindrical cavity. a = ~b~—-A k = lira X A a and the curve is the locus of W) = h{a)NQ{l) - N0(a)JQ(b) = 0. Similarly the curve in Fig. 8.22 is the locus of U{a,b) - N,{á)JoCb) - JMN*$) = 0. From this curve we can obtain data on the approximate resonant frequencies of the cylindrical cavity shown in Fig. 8.20. Thus the small holes do not affect appreciably the principal resonant frequency; on the other hand a perfectly conducting cylinder, even if quite thin, changes the resonant frequency by a substantial percentage. However an infinitely thin wire does not affect the resonance conditions. The radial impedance Z„{a) is positive imaginary if b is sufficiently small. Assuming a capacitance sheet over p = a (Fig. 8.23), whose radial capacitance is Cfi, we shall have resonance when the sum of the two impedances vanishes ZM + t^t = 0- This condition may also be expressed as follows Yp(a) + iosCp = 0, or iYp(a) = uCp. Plotting iYp(a) for different values of k = b/a, we obtain the family of WAVES, WAVE GUIDES AND RESONATORS 1 271 E 2 01 0.2 Imo. 8.21. Curves pertaining to resonance in the cavity shown in Fig. 8.19 when the walls of the cavity are of zero impedance. / / T / / / u (a ,b) fl H = 20/ As 10/ 1 / h Fig. 8.22. Curves pertaining to resonance in the cavity shown in Fig. 8.19 when the inner cylinder is of infinite impedance and the remaining walls are of zero impedance. 272 TKi MAGNETIC WAVES curves shown in \:iy,. K.'.M and limn these the dimensions of the resonant cavity can be determined for various values <>l Cp. The radial capacitance of the sheet may lie expressed in terms ol the total internal capacitance Ct as follows r 'M " ~ 2ra ' since the capacitances of unit areas round the cylinder admit more current for the same voltage and hence are in parallel while the capacitances of unit areas stacked longitudinally along the cylinder are in series. After all types of cylindrical waves have been examined it will be obvious that, if h < X/2, the cavity shown in Fig. 8.23 represents correctly a cylindrical cavity with a coaxial plunger (Fig. 8.25). The value of Ci is determined by the capacitance between the base of the plunger and the base of the cavity, including the " fringing capacitance." The latter may comprise a substantial fraction of the total capacitance C,-. Fig. 8.23. A cylindrical cavity in which the inner cylinder is a capacitance sheet and the remaining walls are of zero impedance. Fig. 8.24. The radial admittance seen from the capacitance sheet of the cavity she Fig. 8.23. WAVES, WAVE GUIDES AND RESONATORS 273 II Solenoids anil II'edge Transmission Lines We shall now turn our attention to the converse type of uniform cylindrical waves in which the magnetic lines are parallel to a given axis while the electric lines are i miliar. For this type the transmission equations are (5 2); these equations are similar to (5-1) and their ■iiiliition can he obtained by analogy. Thus for the field intensities we have //+(p) = AKnbtp), Hj(p) = BhQrp), E%(p) = AvKi&p), Ej'p) = -Brih(o-P), mid for the radial impedances (7-1) Ko(o-p) 27 (P) = ylijap) hiflp) Fig. 8.25. A cross-section of a cylindrical cavity with a coaxial plunger. (7-2) Comparing these expressions with (5-5), we find that the products of the corresponding radial impedances for the two types of waves arc equal to the square of the intrinsic impedance; the radial admittances of the present waves are obtained if we divide the impedances of the other type by ij2. In nondissipative media, for small values of p = a, we have approximately Z~{a) = i iospa, Y\(a) = iuta (log — + 0.116) + — . For large values of p the outward looking impedance approaches r, while the inward looking impedance fluctuates between — and + °o if the medium is nondissipative and approaches tj otherwise. Consider now a circulating current sheet of density Jv = / per unit length on a cylinder of radius a; that is, a " coil " with one turn per unit length. The electric intensity is continuous across the sheet but the magnetic intensity increases by an amount /; thus « E+(a) = Ej{a) = -E, IIj(a) - H+{a) = /, where E is the driving electric intensity. Dividing the second equation by the corresponding terms of the first, we have Thus the internal and external media are in parallel. For small values of a, Y~ is very much larger than Y+ and we have substantially E = Z~ J = \icopa J. For a solenoid wound on a cylinder of radius a, with closely spaced turns of fine wire, n Uirns per unit length, we have / = nl. If V is the voltage applied to a portion of the solenoid of length /, then the voltage E per unit length of the wire is V/litanl; hence V = itoLI, L = pirahi1!. Thus we have the inductance of a solenoid of length / when it is a part of an infinitely long solenoid, or when the end effects are eliminated by bending the solenoid into a toroidal coil. In the latter case there exists 274 El i1 11\< >ma«;ni-:ti(' waves Chap. « some curvature effrri, of <.....sr.; but one may expect il in lie small il the radius nl ilie cross-section of (lie coil is small compa nil with the mean radius of the torus itself, The equations of this section may he used in the. solution of problems which have no apparent connection with solenoids. One such problem is that of the diffraction of plane waves by a narrow slit in an infinite perfectly conducting plane (Fig. K.2o). We shall approach this problem by considering first a " wedge transmission line " formed by two half planes issuing from the same axis or nearly so (Fig. 8.27). Let / Fig, 8.26. A'uniform plane wave incident on a perfectly conducting plane with an infinitely long, narrow slit. Fig. 8.27. A cross-section of a wedge by a plane normal to the axis of the wedge. be the voltage impressed on this line and let / be the input radial current in the planes (per untt length along the axis). If the half planes terminate at distance p = a then the input voltage is given by v = where ^ is the wedge angle. Hence the approximate input admittance per unit length is (7-3) In Fig. 8.26 tbe input admittance of each wedge line, one looking to the left and the other to the right is given by the above expression with \p = it. The two lines are in parallel and the total input admittance is y = {2io}€/ir)Ka{ifia). This expres-, sion becomes more accurate as a becomes smaller. For a finite slit of width s a better approximation to the input admittance is obtained by assuming a = \x — x \ and averaging the admittance over the slit just as we have done in the case of a metal strip.* Comparing the above expression with (5-11), and using (5-12), we obtain the following average value y = ioe + -(log-- 0.222) = - + -(Io^_ 0.222) Consider now a uniform plane wave incident on a screen with a narrow slit through it and let E be perpendicular to the slit. If there were no slit, the wave would be completely reflected and //would be doubled; electric current of density 2H would flow upward in the plane (Fig. 8.26). With the slit present we should still have complete reflection except in the region surrounding the slit. The electric current between the edges of the slit is reduced to zero and the voltage necessary to reduce the *See equation (5-10). WAVES, WAVE GUIDES and RESONATORS 275 ■unv.iil dctisily 111 to /.em is III ijX/7 TV + 2/(fog ^- 0.222) (7-4) This is the counter-electromotive force produced by charge concentrations on the edges of the slit. Expressions (1) for E£, h\ to the right of the screen may be found in terms of the voltage v applied over half the circumference near p = 0; for in this neighborhood £i(pi) — — f/vpi. Hence at all distances ^(p) ~ xpi KidM . hup) = y+{p)eup) = - ~ mm®. Substituting from (4), we have Ht(p) = i ^log 1 _ 0.222) * + 2i (log I - 0.222) as p- 00. 8.8. Wave Propagation along Coaxial Cylinders Consider a pair of perfectly conducting coaxial cylinders. If we apply a transverse voltage between these conductors we expect that longitudinal currents will be generated. If the voltage is so applied that circular symmetry is preserved, we expect that the resultant field will be independent of the ^-coordinate. One such field is described by equations (4.12-8) and the other by (4.12-9). The first of these has no radial electric intensity; hence we need consider only equations (4.12-9). In section 6.11 we have found that if Ez vanishes on the surfaces of the cylinders but not between them, waves will travel along the coaxial pair only if the wavelength is comparable to the transverse dimensions. In Chapter 10 we shall deal with such waves in detail; but at present we are concerned with a tyqDe of wave propagation which is possible at low frequencies as well as at high frequencies. This leads us to assume that the wave is transverse electromagnetic (Ez = 0). Our equations now become ~* = - k + S = T WW = 0- (3-1) dz dp The last equation implies that the magnetomotive force round any circle coaxial with the cylinders is independent of the distance from the axis. This is natural since in the absence of longitudinal displacement currents this magnetomotive force must equal the conduction current I{z) in the 276 ELECTROMAGNETIC WAVES inner cylinder 2*pff9 - /(a), ffpm /(,) 2wp CKAPi 8 (8-2) Substituting fi-om (2) in (1), integrating from , = * to , . b and intro y = Mg + me) ju/l £ h > Z = Jog - log - 2,r # a (8-3) Except at very high frequencies it is easier to measure voltages and currents than field intensities, and equations (3) are preferable to the original equations (1). The propagation constant V and the characteristic impedance K are r = ř is iw/j, dz (9-4) (9-5) (9-6) WAVES, WAVE CilJIDES AND RESONATORS — 1 283 Except for a coefficient depending on z the Stream function »1» has the same hum as the potential V. Substituting from (5) in the last two equations of the set (2) and comparing with (4), we obtain rs-tX-f. (9-7) g 4- loie dz This and the preceding equation form the familiar set of transmission equations. Potential and stream functions are also solutions of the two dimensional Laplace's equation and to any solution of this equation there corresponds a transverse electromagnetic wave. 8.10. Transverse Electromagnetic Waves on Parallel Wires Equation (6.23-14) represents the stream function of two parallel current filaments carrying steady currents. In accordance with the preceding section it is also the stream function associated with transverse electromagnetic waves along the wires. Thus substituting from (6.23-14) in (9-6) and (9-7) we have Pi log dV tup, pi- ft. _Pa dl dz 2tt P2 27r(£ 4- Me) dz (lo-i) where pi and p2 are the distances from the filaments carrying currents / and —I respectively. The cylinders u = log P2/P1 — constant are equipo-tential surfaces. In cartesian coordinates their equation is (x — c coth u)2 + y2 = c2 csch2 a. (10-2) Let two perfectly conducting cylinders be introduced along two of these surfaces, u = «1 and it — it2. We can remove the original filaments without disturbing the wave between the cylinders and will be left with a transverse electromagnetic wave along a pair of cylinders, either external to each other or one inside the other. Equations (1) apply to any line parallel to the cylinders. Let V\ and ¥% be the potentials on the cylinders; then the difference V — V\ — P% will satisfy the transmission equations in which L = — («1 - u2), C ~-Itt 2ire G = lirg III — "2 (10-3) Ml — «2 Let the distance between the axes of the cylindrical conductors be / and let a and b be their radii. Then from (2) we have a2 = c2 csch2 «1, b2 = c2 csch2 u2, (10-4) /2 = c2 (coth U\ — coth «2)2- .'hi ELECTROMAGNETIC WAVES Chat, h Expanding tl)c last equation and substituting coth2 « = csch2 « + 1, wc obtain I2 = c2 csch2 «1 + c2 csch2 «2 + 2c2 (1 - coth U\ coth a2) = c2 csch2 «i+ c2 csch2 «2 — 2c2 csch U\ csch «2 cosh — k2). In substituting the radii a and * from (4) into this equation we should bear in mind that U\ and u2 rnay be either positive or negative whiJe the radii are essentially positive; thus we have I2 = a2+ b2± lab cosh (k, - u2). The upper sign corresponds to the case in which U\ and u2 have opposite signs and the cylinders are external to each other (see Fig. 1.6); the lower sign corresponds to the case in which U\ and #2 have like signs and one cylinder is inside the other. Thus, depending upon whether the cylinders are external to each other or one inside the other, we have «i — u2 = cosh 1 ř- a2- b2 lab or cosh a* + b2 - I2 lab Substituting in (3) we obtain L, C, G. The foregoing formulae have been obtained on the assumption of perfectly conducting cylinders and if the conductivity is finite the above results have to be modified. At very low frequencies, for example, for the case of two solid cylindrical wires, the magnetic flux penetrates the wires and the inductance is L = - log —== + — . 7r V ab 4ir This inductance is larger than that given by (3); the capacity on the other hand remains the same. Hence the wave velocity on the wires is smaller at low frequencies than at high frequencies. Furthermore we should include the resistance of the wires in series with the inductance. At very high frequencies the flux is largely forced out of the conductors and the resistance per unit length is given by R = £R. j'HSH% ds, where //„ is the component of H tangential to the wires and the integration is taken round both wires. The field H is that produced by a unit current in each wire. If the radii of the conductors are changed by an infinitesimal amount Sn in such a way as to increase the inductance, then the increment in the inductance is SL = pZn J'HsHt ds. Hence we obtain the following WAVES, WAVE GUIDES AND RESONATORS I 28S simple principle* ft hn' where hLjhn is the variational derivative of the inductance. For two wires external to each other we have therefore _ 9_/_ dJL _ d£\ H \ da db / Calculating the derivatives, we obtain 2(a + b) + (^ + ^(l2~a2-b2) 9. St i la2' fb = a. Similarly when one wire is inside a cylindrical shell, then (for b > a) dL\ + T7 ; hence R = mí m * = da^ db) g V[(7+ bf ^i2m - *)2 - ^ The high frequency resistance of parallel wires increases as the wires approach each other. This phenomenon is called the " proximity effect." If a wire is inside a cylindrical shell, the resistance is minimum when they are coaxial; in this case the proximity effect is sometimes called the " eccentricity effect." 8.11. Transvirse Electromagnetic Spherical Waves (TEM-waves) For transverse electromagnetic spherical waves we have Er = Hr = 0. The theory of these waves is similar to the theory of transverse electromagnetic plane waves. We have two divergence equations £ (sin dEo) + ~Ev = 0, |- (sin 0tf,)+ — 0, (11-1) * Harold A. Wheeler, " Formulas for the Skin Effect," I/R.E. Proc., 30, pp. 412- 424, Sept. 1942. ELECTROMAGNETIC WAVES Cnai-. h WAVES, WAVE GUIDES AND bth;s< )nat< )1 — dV dd ' r sir. 0H$ d_a r sin f? £p = - — , Oip dA 01-4) If is regarded as a radial vector, then H = curl a. Our choice of auxiliary functions satisfies the second equation of the set (1) and the first equation of the set (2). Substituting from (4) in the remaining equations of these two sets, we find that v and a satisfy equation (3.6-14). Equations (3) show that r£s and r//p vary as Ex and Hy in a uniform plane wave; rEv and rat behave as Ev and Hx. Thus the propagation constant of all transverse electromagnetic spherical waves is equal to the intrinsic propagation constant of the medium and the wave impedance is equal to the intrinsic impedance. Substituting from (4) in (3) and integrating with respect to 6 and (! + esc ý) The (5 is maximum when ^ = 24°. 1 and then „ 104 The input impedance at resonance is VV* K2!2 Zi = Hence we obtain 7200tt ft 2(#i + #2) 2(#t + ^2) " [log (cot I tan I)] log (cot ^ tan ^0 + 2>(csc 0i + esc 02) For two equal and oppositely directed cones we have / A2 I log cot -1 14400* log cot — ~\~ p esc FJJECTROMAGNETIC WAVES TJiis impedance is ma\iimim when \p 9°.2 and then - 3.74 X 10* . Zi =-^-ohms. 91 For a cone in a hemispherical cavity we have ( Zt = 7200t, I cot 0 log cot - + p(\ + CSC Tp) This is maximum when \f/ = 7°.5 and then 1.70 X 104 Zi = ihms. When conical conductors such as those shown in Fig. 8.29 are terminated at some distance / from the apex, the transmission problem is complicated by a sudden change in the physical character of the transmitting medium. Transverse electromagnetic waves require longitudinal conductors and when these are absent such waves are no longer possible. Thus the discontinuity at the " boundary sphere r = / is more than just a discontinuity in the characteristic impedance of the radial transmission line; the set of transmission modes for r > / is different from the set for r < /. The theory of wave transmission on such terminated cones and on wires of other shapes will be considered in Chapter 11. 8.13. Transverse Electromagnetic Waves on a Cylindrical Wire The theory of cone transmission lines can be extended to cylindrical wires (Fig. 8.31) of sufficiently small radius a since such wires will support nearly spherical waves. In this case the inductance and capacitance vary 8 a Fig. 8.31. A cylindrical wire energized between points A arid B. with the distance r from the origin and by (12-6) may be expressed approximately as follows L = - log 2r C = log 2r' (13-1) When L and C are varying slowly with r, we have approximately K(r) « Jk - log - = 120 log - . (13-2) V C x a a (13-3) WAVES, WAVE GUIDES AND KKSONATOKS I 271 Hltu-c Z(r)Y(r) is constant, we can use (7.12-5) for obtaining the functions ,lvrd m the second approximation to the voltage and current in the i.i. ..hi transmission line; thus 1 2/ K0 = - f K(r) dr = 120 log - - 120, /Jo a Mr) - $■ fW) - 7C0] cos 23rdr, B(r) =-!• f [Ko - K(r)] sin 26rdr. Kor further convenience we introduce the following functions M(r) = K0B(r), N(r) = iK0J(r). Then we have X20T sin .v log * dx = 60 (1 - cos 28r) (log l- - l) + 60(log 2Br - Ci 28r + C), A/(r) = 60 (log 281 - 1) sin 28r - 60J cos x log x dx = 60 (log -r - 1J sin 28r + 60 Si 23r; M(l) = 60 (log 281 - Ci 281 + C - 1 + cos 281), NO) = 60 (Si 281 - sin 281). The voltage-current equations become r M(r) „ N(r) . 1 ytr) = F(0) cos Br + cos 0r - — sin /3r j F M(r) . „ iV» 1 -iK0I(0) sin jSr - sin Br - cos 0rj , 10) /(0)f • M(r) JV(r) jSr + -|p sin 0r + -^r- cos ] (13-4) T M(r) , N(r) . 1 +7(0) cos Br - cos Br + — sin (5f J . We shall now calculate the approximate input impedance of an infinitely long wire. Taking I = X/2 we obtain from (2) the impedance looking !92 1 1 •' 1 i R< (MAGNETIC WAVES Chap. B outward from r - l\ tlnis Z{\/2) - K(X/2) - 120 log \/a. On the other hand from (4) we obtain approximated = 120 flog ~ + Ci 2x - C - / Si 2*-^ . For a spherical or a hemispherical resonator with a cylinder running from the center to the periphery (Fig. 8.32) we can find the resonant length by setting 1(0) and V(l) equal to zero. Thus at resonance we have Fig. 8.32. A hemispherical cavity with a conducting cylinder inside. cot ßl From this we obtain approximately 1.85 Kq + M(l) 2 2 1og^ la 0.352 X 1 4 0.295 2 log — - 0.352 In practice there is usually some capacitance C0 at the center and the resonance conditions become m = o, tm --iwC0 = —ißvCo, where v is the characteristic velocity. Substituting these values in (4) we obtain equations from which l/\ can be determined. 8.14. Waves on Inclined Wires The results of the preceding sections can be generalized to cover the case of wires diverging from a common point and making an angle less than 180 degrees with each other. We start with two diverging conical conductors (Fig. 8.33). Using (12-1) and (3.6-15), we construct the following stream function for two infinitely thin wires along arbitrary radii 9 = 9U ip = ipi and 6 = 62, tp = 2 coť - 2 cot - cot — cos (

- m_ 2 {fira2 - 1) sin fia + fia cos fia ()n the axis of the antenna near its center, we have E7(r) = ^ (1 " 0.13V). (15-2) I lence the mutual impedance between the antenna and the sphere is approximately Er(0)/_ vfi2lP Adding this mutual impedance to the self-impedance of the antenna, we obtain the total input impedance of the antenna inside a perfectly conducting metal sphere. The input impedance must be a pure reactance since under the assumed conditions no power is either dissipated or radiated. The radiation resistance of the antenna in free space must be canceled by an equal but negative resistance component of Zm- Separating the real and the imaginary parts of P, we obtain (fi2a2 - 1) cos fia — fia sin fia p = hni (fi2a2 - 1) sin fia + fia cos fia + hll. Hence the real component of ZM is RM = - W™$WH as we have already anticipated. The self-reactance X0 of the antenna is largely capacitive and it may be obtained from (13-3); thus* where b is the radius of the antenna. Hence the total input impedance of the antenna is Z = iUfiPřfia sm fia - (fi2a2 - 1) cos fi 6ir fia cos fia + (p2a2 - 1) sin fia This input impedance becomes infinite when fia Ki-K-1)- (15-3) tan fia f= 1 - era .2 „2 cot fia —---fia. fia (15-4) *Thc distributed impedance and admittance per unit length of the line are ial = ifiK and iaC = ifi/K. 296 II Kl !TR< )iM A( i'NKTIC WAVES ' ClUF. K This equation determines the frequencies for resonance and also for natural oscillations inside a hollow metal sphere; at these frequencies a field of type (1) can exist without a continuous impressed force. The smallest root of (4) and the corresponding natural wavelength are ßa = 2.744, X = 2.290«. (15-5) The larger roots of (4) are nearly equal to mr and hence are approximately given by k7t ßa = rnr--2~ä-T > »V - 1 2a n - = M--5—Ö--- X »V - 1 (15-6) The Q of the spherical cavity might be calculated by the method we have used so often in preceding sections. However, there exists another method which we shall now illustrate. If there is no antenna in the cavity the field must be of the form (1). At the surface of the sphere the sum of the inward looking radial impedance and the surface impedance of the sphere must vanish; thus Z~(a) -f- ij =0, where the surface impedance of the sphere has been assumed equal to the intrinsic impedance 77 = ik(l + i). In view of (1) this condition becomes (ß2a2 - 1) sin ßa + ßa cos ßa ßa(sm ßa — ßa cos ßa) IT) V (15-7) When 77 = 0, the principal solution of dris equation is given by (5) and in general by (6). Since 17/77 is very small, an excellent approximation to any solution of (7) can be obtained by assuming &a = k + A, (15-8) where k is the corresponding solution of (4) and A is a small quantity. Retaining only the first powers of A, we reduce (7) to AN\k) _ _ij P{k) " V ' where N'(k) is the derivative of the numerator and D(k) is the denominator; thus A ~ ~ vN'(k) ™ i,N'(k) 1 vN'(k) * The natural frequency is no longer real and it is better to introduce the natural oscillation constant p = m in (8); thus we have ik + iA WAVES, WAVE GUIDES AND RESONATORS I 297 On the other hand the oscillation constant may be expressed in terms of Q mil the new real natural frequency co h torn the above equations we observe that the natural frequency is slightly ii I breed by the imperfect conductivity of the sphere. For Q we obtain 1 9lD(k) _ hN'{k) 3 M 2Q kvN'(k) lRD(k) Since N'(k) = k sin k + k2 cos k, we have in view of (4) N'(k) _ sin k + k cos k _ 2 - k2 D(k) ~~ sin k - k cos k k2 hoi- the principal resonance we have 7) 77 , . . „ 380 0 = 1.0076^^, and in air 0 = -^ At resonance the electric intensity is maximum at the center of the sphere. The voltage Valong the diameter can be expressed in terms of this intensity /''n; thus I'Vir the principal resonance we have V — 0.686£nX. At the boundary of the sphere the field intensities are EM) = 3£0 Hv(a) = i cos (?, ■5£, Tjk sin Ö. For the principal resonance we obtain ET(a) = 0.423£0 cos 9. Returning to the impedance function (3), we shall now calculate its zeros. Since the second term (the self-impedance of the antenna) is rather large and the mutual impedance is rather small except near resonant points of the spherical resonator, Z should vanish at frequencies not very different from resonant frequencies of the sphere. The equation which we have to solve is (ßaflßa sin ßa - (ß2a2 - 1) cos ßa] ßa cos ßa + (ß2a2 - 1) sin ßa r.I.ECTm iMAÍíNKTK' WAVlíS < ■ i i i ■ . a Writing fia <=> k 4- S, where k is a solution of (4), substituting in the above equation, and retaining only the first powers of S, we obtain 5 = - H&* - *2-i-1 )ř 6(k2-2y(\ogj- l) This approximation is not good when k becomes so large that 5 is no longer small. The large zeros of Z would seem to be nearer to the zeros of the numerator than to those of the denominator. The smallest zero of Z is somewhat smaller than the smallest infinity (except for that at the origin). This was to be expected because the zeros and infinities of a reactance function separate each other. The fractional deviation of the first zero from the nearest infinity is S/k. If / = 0.1« and b = 0.1/, the fractional deviation is 0.0007576 or 0.076 per cent. 8.16. Circular Electric Waves Inside a Hollow Sphere Circular electric waves are waves with circular electric lines of force. Such waves can be generated by a uniform circular current filament. The current distribution in a small electric current loop is approximately uniform even if the loop is energized at one point only; thus small loops can be used to generate circular electric waves. In free space the field of a loop of area S carrying current I is given by equa-\ / tions (6.17-5). Nsk^__^/ Consider now a loop concentric with a sphere of radius a Fio. 8.38. A sphcri- (F't>r- 8-38)- If wechoose the plane of the loop as the equa-cal cavity with a torial plane of our system of coordinates, the field reflected loop inside it. from the sphere can be obtained from the following electric vector potential: Fg = sin fir/r. Thus we have TT P ( . „ cos fir sin fir\ IIJ - ~ ^\sm Pr + -pT~ ~w)5111 e> (16-1) (sin fir \ cos fir I cos 8, Ez, iP( Yr\c0Sfir- sin fir fir sin 6. If the sphere is a perfect conductor, then the total tangential electric intensity vanishes at the surface and P = hfiSIe^" 1 4- ifia fia cos fia — sin fia i n ct $a sin @a + cos fia pa cos fia — sin fia WAVES, WAVE GUIDES AND RESONATORS 1 299 Near the center the electric inlcnsily ill the plane of the loop is I I. in i the mutual impedance between the loop and the sphere is 2w&Ey(l>) _ ifiWP Zm= ~ J ~ 31 ' m lure h is the radius of the loop. The resistance component of Zm is the negative (jf the radiation resistance of the loop. Thus the input impedance of the loop is a pure reactance in „ fia sin fia.+ cos fia . b Z = ~r- J3'1.?2 -— . + tr,fib log - , Sit fia cos fia — sin fia c (16-2) where the second term represents the reactance of the loop in free space (c is the radius nf the wire). The input impedance becomes infinite when t*nfia = fia, fiasco. (16-3) When the frequency is zero, Z is zero and not infinite; hence the exception. The lowest root of this equation is fia = 4.493 ■ ■ and X = 1.398a, a = 0.715X. The lii st zero of the input impedance is at/ = 0. The remaining zeros are obtained from the following equation fi*a3(fiasm fia + cos fia) _ _ 6a3 b fia cos fia — sin fia c For small loops the right side of the equation is large and the zeros of Z are near its infinities. It is not surprising that the zeros and infinities of Z should be close to each other; the former are the natural frequencies of the spherical cavity with a small perfectly conducting ring at the center while the latter are the natural frequencies oi the cavity when the ring is open. If the ring is small it should make little difference whether it is open or shorted. We leave it to the reader to show that the approximate solution of (4) is 7r(*24- I)*3" i + --r fia = k 6as log where k is a solution of (3). It is also left to the reader to show that the equation for natural oscillations inside an imperfectly conducting sphere and the Q of the cavity fia [fia cos fia — sin fia) 03 V - 1) sin fia + fia cos fia B, Q 29. 8.17. Two-Dimensional Fields In this section we shall obtain basic wave functions independent of the z-coordinate. Such functions are solutions of (4.12-6) and (4.12-7). The first set represents trans- 300 FXKctuomagnktk: waves Chap, h verse electric cylindrical waves with i lie /'.'-vector parallel and the //vector normal to the axis of the waves; the second set represents transverse magnetic cylindrical waves with the //-vector parallel and the /'.'-vector normal to the axis. Instead of solving the equations directly, we may obtain the complete set of wave functions from the wave functions considered in sections 5 and 7. Let us start with a uniform infinitely thin electric current filament parallel to the z-axis and passing _x through the point J(p0, Evidently E, is a periodic function of p and we should be able to represent it as a Fourier series in p0. « = 1 (17-2) Each term of the first expansion represents a cylindrical wave of the stationary type and the second expansion is composed of progressive waves traveling outward. The magnetic intensity in terms of die new coordinates may be obtained from Thus let t'here " tbiup dip tap. dp Ez(po, po- n=0 KnVfpa) (17-6) waves, wave (.-UII )F.s an I > l< F.St )nat( )us - I 301 lioni (3) and ((>) we now obtain the radial component of the magnetic intensity 2 ^ nE»„nr Un \ sm "W ~ Po), P < PO, Vn-l apl,x(crp{)) 1 " Kn( po; Uld the circular component Hv{p,)» P < Po, 1 £ /C4(<7p) = Z 2- ■£» y i COS »(«?- po), P > PO, where the prime indicates differentiation with respect to v) = __ £„ — cos w(p - po), P < Po, " p; . w/xf = iZ En — cos w(p - po) 4- ^7 (log p0. (17-11) n=l P For the radial component of H we have 7/P(p,p) = -Tj — sin n(

po. *•11 n = 1 P And finally for the circular component of //, we obtain I w pn_1 ^(P#) = - 7Z z —5T cos - Po), P < Po, *t»=-1 Po 7 00 Po = 7Z z -rpi cos "(

po-z?r«=oP (17-12) (17-13) klkctuomaonktic waves ClW. I WAVES, WAVE GUIDES AND RESONATORS —1 303 The approximate values of the radial impedances for ti > 0 are then (17 -14) Let 1m current filaments, carrying alternately equal and opposite currents, be equispaced on the surface of the cylinder p = po, and let the first filament carrying current I/m pass through the point (po,0). Since 2m —t m (-)"exp\initp--) =0, if» ^ (2£ + \)m, = 2me^, if » = (2£+ the electric intensity of this set of filaments becomes Ez{p, po we have approximately ivpj Km{op) Emm =---*t? }—i cos mip. (17-17) If p is also small while remaining substantially larger than po, we have the quasistatic case icop/plj LApsp) =--—cos rutp. m-xp™ (17-18) Thus just as equations (1) represent a progressive cylindrical wave generated by a single electric current filament, various terms in the second line of equation (6) represent progressive cylindrical waves generated by groups of 2n current filaments, carrying alternately equal and opposite currents, equispaced on the surface of a cylinder of an infinitely small radius. Equation (6) shows that the field of a simple line source at p = pa, at distance p > po, may be regarded as the resultant of an infinite number of multiple line sources at p = 0. In the special case of two filaments separated by distance / and carrying equal and opposite currents, we have iapall &t{P,e dp In the vicinity of p = 0 we have Js cos tp E" = 2ir(g + i is the permeability of the dielectric. If the allel current fila- thickness of the shield is small compared with the ments and a cylin- radius, the divergence of the wave in passing through dncal shield. the shield may be ignored and the wave propagation in the shield may be regarded as substantially plane. Hence in the shield we have Z+ = Z7 = ^ = J—, ji0a. In decibels the reflection * We ignore the slight variation in the radial impedance (1) in passing from one surface to the other. I , R bet oines U + 1 la R = 20 logio 1 4|^jl \\ hen k is'either small or large, we have respectively 1 I 305 (18-5) R - 20 log io m, « = 20 1og10Li I he lotal shielding improvement is the sum of the reflection and attenuation losses; thus S = R + A. The reflection loss may be considerable. For a shield whose radius I cm the radial impedance in air for / = 10° is Zp = 0.079/ ohms. I >n the other hand for copper we have Zp = 0.000369V7 ohms, Hence I '. | k [== 214, and the reflection loss is 214 R = 20 logio -£ 35 db. We have seen that at this frequency the attenuation loss in a 0.2 mm shield is about 26 db; thus the reflection loss is the larger part of the total shielding efficiency of 61 db. For magnetic shields the reflection loss may be small. Consider for i ample, an iron shield with relative permeability 100 and with conductivity only a fifth of that for copper. The radial impedance in the shield is then increased by a factor V500 — 22 which brings it up to 0.008 Sri. The impedance ratio becomes nearly 0.1 and the reflection loss is comparatively small. In air the radial impedance is proportional to/ but in metals it is proportional to V/: hence the magnitude of the foregoing impedance ratio becomes unity at about / = 104 and the reflection loss, due only to a 45° phase difference in impedances, is negligible from the shielding point of view. But the attenuation constant is larger in magnetic shields than in nonmagnetic. Thus for the iron shield just considered the attenuation constant is increased by a factor V20 = 4.47. When the shield is thick this will more than compensate for the lack of reflection loss. However, if the shield is thin, the reflection loss may become more important than the attenuation loss and the nonmagnetic shield will be more effective than the magnetic shield- For two metals the impedance ratio , = is independent of the frequency. The impedance mismatch, as between copper and iron, may be considerable and it is advantageous to use a shield M)(, EI.ECTl« MAGNETIC WAVES consisting of composite layers of copper and iron, with copper layers on tht outside, to utilize tlie substantial reflection losses at air-copper surfaces. The complete formula for the shielding efficiency is obtained from equa tion (7.19-2); in decibels it becomes i| = -20 logu, ] T I = R + A + 20 log 10 (i+i)2 (18-6) This represents the difference in the amplitude levels of the incident wave just inside the shield and the transmitted wave just Outside the shield and it is substantially equal to the difference in field levels before and after introducing the shield. In order to obtain the latter difference in levels we should multiply T by bja for the electric intensity and by b2jd2 for the magnetic intensity. If the line source is not along the axis, the shielding efficiency varies with position round the shield. The actual source may be replaced by an infinite number of virtual sources along the axis in accordance with the results obtained in the preceding section. One of the sources is the actual source translated to the axis. The remaining sources emit higher order cylindrical waves whose radial impedances vary somewhat with the order of the wave in air and are nearly constant in metals. Hence tire field distribution is altered by the shield, though not very greatly. Except for this unevenness in the field reduction, the shielding efficiency is the same as for an axial source. Let us now consider a spherical shield with a current loop at the center. The radial impedance in the shield is die same as for a cylindrical shield; but the radial impedances in air are (18-7) The external wave impedance is calculated from (6.17-5) and the internal from (16-1) assuming that r is small. The product of two transmission coefficients is now obtained from (7.20-2); thus (Z- + Z+)2r, 3k (zrT + V)(V + z+) (i + A)(i + }*)» where k = tj/ioifioa. When k is small, then p = 3* and the reflection loss becomes 1 1 R = 20 logic 3 \k = 20 logio 4|*| + 2.5 db. The difference between the formulae for cylindrical and spherical shields consists in a 2.5 db improvement in shielding efficiency for the latter. A cubical shield may be replaced by a spherical shield in estimates of the shielding efficiency. Since, in practice, high shielding efficiencies are required, one is not justified in looking for exact solutions, especially as such solutions show that in general WAVES, WAV L GUIDES AND KES< )NAT< )KS I 307 In. Id hasa Mingle figure of merit, independent of I he position of the point at which tin shielding (-•llicicticy is measured. In the two preceding examples the fields are largely " magnetic " since the wave Impedances are very small quantities and, in elementary discussions of such fields, pi' on intensities are usually excluded from explicit consideration. Another type nl li. Id is the so ealled " electrostatic " field which is produced, for example, by electric i Ii.ii. en mi [In- pi.ill's ol a capacitor. Such "electrostatic " fields may vary at the ......if one million cycles per second and perhaps a different name would be more Appropriate. What really matters is that some fields in the neighborhood of small i ures are low impedance fields so that magnetic components are large and electric . iiinponents small and other fields are high impedance fields with large electric and mill magnetic components. The terms " low impedance " and " high impedance " no n .nl here with reference to die intrinsic impedance of the dielectric in which the In Id happens to reside. Thus the field (17-23) of two charged filaments is a high impedance field since in a good dielectric i I n nX 2+ = z~ =_■ = —— = —-_ p p io>ep ipp 2-Ktp' which is large compared with rj. A simple calculation shows that at a metal boundary ,i high impedance field is reflected so completely that there is no need to compute i lie shielding efficiency. The reflection loss tends to infinity as the frequency approaches zero. The theory of shielding thus far developed may be called the" transmission theory ■ 'I shielding " since our attention has been fixed on cylindrical waves passing through i he shield. A different physical picture of shielding is possible. The fields are pro-iIm ed by electric currents, the original field by currents in the source along the axis and the final field by the currents induced in the shield. It may be said that die shielding action is caused by currents in the two halves of the shield flowing in directions opposite to the currents in the source. In Fig. 8.40 the current in the right half of the shield is negative and in the left half positive. The principal weakness in this theory is that frequently it is difficult to compute the induced currents widiout fust solving the shielding problem itself. In order to find the induced currents we have to calculate the tangential magnetic intensity at both surfaces of the shield; but i he shielding efficiency is then determined. On the other hand the second method suggests some deductions not immediately indicated by the first; thus from the second picture of the way in which the shielding is effected we conclude at once diat a greater impairment in shielding will occur if the shield is cut so .is to interfere with the induced current flow than if it is cut along the lines of flow. For example, if a plane wave is falling normally on a perfectly conducting screen with an infinitely long slit, more power will be transmitted when the Elector is perpendicular to the slit than when it is parallel to it. The two physical pictures of shielding are complementary. In some cases it is more natural to use the second theory even in analytical work. An example is furnished by a " two-wire shield " (Fig. 8.41), for which it is easy to obtain the currents induced in the wires A and B and to show that these currents tend to weaken the original field at great distances, o • • o B -I +1 A Fio. 8.41. Normal cross-section of two parallel wires at A and B which act as a shield for two current filaments. 308 ELECTROMAGNETIC waves Cn/vi'. 8 Wc now turn our attention fo mn.u.iH'tostaii. shielding. At / = 0 the electric field vanishes and we lose our principal tool, the radial impedance. The transmission theory of shielding could he reformulated in tern is "I properly defined " radial reluctances "; but it is more convenient to regard steady magnetic fields as fields of vanishingly small frequency. The field of two closely spaced equal and opposite current filaments is then given by (17-20). The field reflected from the shield must be of the form El = Qp cos ami <} of t lie I lansmission and reflection . in 1I1. miiI , across the two boundaries of the shield we have 4* 4MlM2 _ (1 E k)2 _ (mi - M2)2 ^(1 + k)2 (Ml + p2? ' q (1 + k)2 (mi + M2)2 ' Thus we have determined all factors in the shielding ratio s. When the difference mi — is large, then q is approximately equal to ily and .f = pi(1 — x)- The effectiveness of the shield depends largely mi the ratio of the permeabilities. The thickness of the shield when in-ri ised beyond a certain value has relatively little effect. Thus if x be-■ 1 nns small, we have simply s — p. When shielding depends on the reflection loss the shielding is improved b) using less, not more, shielding material. Thus if the shielding material I tends from p = 1 to p = 1000, then x = 10-6 and S is practically equal in p. Suppose now that the material between p = 10 and p = 100 is n moved so that in effect we have two shields. For each of these shields >" 0.01 and the shielding ratio is still practically equal to p; hence the shielding ratio for both shields is p2. Assuming shielding material extend- II ir. Irom p = 1 to p = 102""1"1 in layers, each layer extending from p = 102m in /1 = 102™"'"1, we obtain the shielding ratio p'l+l. If the space between I be above layers is also filled with the shielding material the shielding ratio becomes p. These figures illustrate the principle. It is not practicable in use shields with radii whose ratio is so large that the full benefit of reflec-tion is obtained from each layer. A formula for a practical case may be obtained from (7.19-10) in which the exponential factors should be replaced by the transfer ratios. For a shield with only two layers calculations are not too laborious but for a large number of regularly arranged layers it: would be preferable to consider first the problem of wave transmission 1 hrough the corresponding iterated structure. In the case of spherical shields equations (6.17-7) give the field of a current loop at distances which are large compared with the radius of the loop as long as the frequency is so-small that the product Br is small, and from (16-1) we obtain the following form of the field reflected from a spherical shield concentric with the loop* = icapPr sin 8, Hj = IP sin 8, II~ = -2P cos 8. (18-12) The magnetic intensity of the reflected field is parallel to the z-axis since Hp = 0, *The numerical factor P has been changed. Once Ev is obtained, Ho and Hr can be calculated from (4.12-10). :tio Pn = M.r.i TR( (MAGNETIC WAVES ' <•„*... lie radial impedances arc given by (7) unci the transfer ratios arc r\ ft x£(n,rj) ■ J » Xb(>Vi) - - , XaC'V*) = p, x# (>Vi) = 1-The product which enters the shielding ratio (11) is then XE = mfafahGifati} = 4 = X//. For the products of the transmission and reflection coefficients across the two boundaries of the shield we have (with k = /u2/m1) ft+!)(** + *)_ 9* 3 + k)& +1) (1 + 2t)(2 + *) " pB-_ (\k-\){k-l) 2(k - l)2 qu 9E (k + JKp +1) (1 + i*)(2 + A) * If A is large, then qOt 1; and the shielding ratio becomes /> 9 ^tig l-X~2A(l-x)_2M2(^-^)' where a is the inner radius of the shield and b the outer radius. Comparing this with the corresponding shielding ratio for cylindrical shields, we find that when b is much larger than a, the spherical shield is somewhat less effective than the cylindrical shield; but in practical cases, in which the thickness of the shields is moderate, the spherical shield may be the more effective. The preceding specific formulae for the shielding ratio apply to the case when the permeability of the medium inside the shield is the same at all points. If the current loop is surrounded by some magnetic material (Fig. 8.42) as is the case for coils wound on magnetic cores, then the radial impedance looking inward from the inner surface of the shield is modified and the shielding ratio is altered. In order to obtain more general expressions for p and q in (11) we have to distinguish between radial impedances for waves in homogeneous media and radial impedances affected by reflection. We shall now designate the former by the letter K and the latter by Z. From equations (7.20-2) and (7.20-3), paying due attention to Fig. 7.24 which explains successive reflections within the shield, we obtain _ [Zj(a) + K+(a)][Kz(b) + K+(b)] P [Zj(a) + Ki(a)][K^{b) + K\(b)]' (18-13) q [K+(a) + Zl(a)]{Kz(b) + K+(b)]' where the subscript 2 refers to the shield. Fio. 8.42. A current loop surrounded by a magnetic core and a spherical shield. WAVES, WAVE GUIDES AND RESONATORS I 311 i c i ilie radius of the spherical " erne " surrounding the current loop be c. The hidiiiI impedance Z\~((t) may be obtained from (7.10-8) if we let xi = a, xi = c and \iu>HaC, where fia is the permeability of the core; thus Z, (a) = J/oJ/uitf 2(1 -*) + *(! +2X) X = (18-14) (2 + X) + J(l - x) a- m i In i..... netic core has altered the impedance looking inward at the inner surface of ili. shield in the following ratio Zt(a) = 2(1 - X) + HI + 2X) Kj(a) (2 + x) + t(1 - X) " In | ii act ice JU3 is usually much larger than (x\ since the purpose of the core is to increase i In inductance of the coil. Hence we have approximately 1 + 2X *" 1 -x' thus the impedance Z\~{a) is increased by the presence of the core. Thi in i him, however, should not be used when X is near unity. Expressing (13) in terms of various ratios, we obtain is approxi- P = 1 - m + k)(U+D (k + 2)(2k + l)' 20- !)(*-*) (* + P)i*+l) (k + 2)(2k+k)' where k and J retain their previous meanings. Without the magnetic core % = 1 and the core makes J larger. Since k is large for an effective shield, an increase in ~k~ makes p larger while q is affected only slightly; hence the shield is less effective for a Coil wound on a magnetic core than for a coil wound on a nonmagnetic core. If the core is made of the same material as the shield and If* = «, the shield loses its effectiveness completely. The above examples show that transmission theory provides a method for solving problems on cylindrical and spherical shields under a variety of circumstances. This method is particularly useful from the computational point of view when die structure is fairly complex. Even in the case of a single layer shield for a coil with a magnetic core the solution of the boundary value problem would require writing expressions for the fields in the core, in the region between the core and the shield, in the shield and outside the shield. These equations would contain six arbitrary constants and we should have to use the six equations of continuity of the tangential components of E and H to determine these constants; and the labor involved in solving these equations is considerable except when the method used is essentially equivalent to the transmission method of handling the problem. From the mathematical point of view the transmission method consists in solving the boundary value problem for a single section, then applying the results to each additional section. Besides, die transmission theory enables us to make qualitative appraisals of the effectiveness of shields once we have mastered the simple but very fundamental ideas concerning the effect of impedance mismatch on reflection. 9 312 ELECTROMAGNETIC WAVES I'll a We .shall now examine ihe effect of I ho presence ol (he shield on llie son roe itself. For example, for the enrrenl loop at the ten (er of a spherical shield the incident held is expressed by (6.17-7); the reflected electric intensity is then cw \ n+, ^r i<*HlSIr sin 6 E$m = 9*mw - = -is——, where qg is the ^-reflection coefficient at r = a which takes into account the shield and the medium beyond it. Hence by (7.20-2) we have* = K-(a)[Z+(a) - K+(a)] = Z+(a) - i^a qB K+{a)[Z+(a) + K~(a)] 2[Z» + fa^g]' The reflected field is impressed on the current loop and introduces an impedance 7ircE^,{c) ioipiSc* tufaS* ZM =--f- = W"2^T» where c is the radius of the loop. In the case of nonmagnetic metal shields at frequencies which are not too low, Z+(tf) is small compared with ia>p\a and for perfect conductors 7J~T{a) is always zero; then ?b = —1 and the impedance added in series with the loop becomes Zm = Lm = — s* 2tm3 ' * 2to3' Thus the inductance of the loop is made smaller by the presence of the shield. 8.19 Theory of Laminated Shields In the preceding section we have shown that a shield made of alternate layers with different permeabilities may be more effective than a solid shield. Laminated shields made of regularly spaced layers may be treated as special cases of iterative structures. As an illustration we shall consider a cylindrical shield made of coaxial layers with the permeabilities of the adjacent layers equal to pi and p.^. Let the common ratio of the inner radius of a layer to its outer radius be Vx and let the permeability of the layer between pi — a and p2 = b = «/V% be jt\. First of all we calculate the transducer impedances —Ei = Z[fL\ + Z,'2//2) Et(a) = Ei cos pd> J— , Z2'2 = 10>mb y—^ . Ihe next layer from pi — b = «/Vjc to p2 = b/\Z% = xd - x«) Z11//1 + Zia^s = -£l> (19-6) Z2,//1 + (z22 + ^)^ + Y//3 = 0-Let dre //-transfer ratio be 7/2 77i * (19-7) Substituting from (5) and (7) in (6), we obtain ■ n+x)2 X (l-x)a Uk + l\ = lJ^ + ^\, Solving, we have X Mil El TRI (MAGNETIC WAVES Our. i If n\ =■ juj, then v — 1 and * 1 + X2 , 1 -X2 1 . , , X 2X 2x X The first value is the //-transfer ratio for the converging cylindrical wave in a homogeneous medium and the second is that for the diverging wave. Thus in the laminated medium we have X+ = x(A - yfA^X) = *~ = X(^+ Va^T) = a+Va^i' X ^ tranSfer rati°S by the ending ratios for a homogeneous medium 9ssiP A- VsP - i * v2---- X v X_ = X__I A - VA^l ~ X The quantity x represents the shielding attenuation ratio for each pair of layers; for n pairs the attenuation ratio will be x". If the layers are thin, x is near unity and we may write x = 1 ~ 5. Making the necessary substitutions, we obtain a~ i +!(„+i)s2, Va*- i = o-^p^, Supposing that the shielding space between pi = a and p% = b has been divided into In layers, we have 5=1 — %fa]i. Let a b - = tf-6, 6-log-, b a and substitute in the preceding equation; then 5= i - direction in the laminated liiedn.....nay he obtained from (6) and (7); thus K If K+m- 77i~Zu + Z„7r-Zii + Jc+Zi,, or A'1" = Zn 4- x2xZi2. tli "i Inrly t he characteristic impedance in the negative p-direction is obtained from the intlog of equations (7.26-0); thus K = Zii + = Z22 + xZ2i. X III the limit as the thickness of each layer approaches zero we have K+ 7 K~ — /wpVp. 11^2. If a shield with infinitely thin laminations is used in a medium with permeability p,, 1 In shielding ratio is pe 4k (1 - *)« 1 _ ?ř-2(r+Do » t (1 + £}* > * (1 4. Vpip2 Pi + P-2 p 2-> P1P2 h.20. Diffraction Problem In section 5 we have calculated the field scattered by a wire of radius a when the /'.'-vector of the incident plane wave is parallel to the wire. Let us now suppose that the A'-vector is perpendicular to the wire (Fig. 8.43). If the radius of the wire is small, the electric intensity impressed on the wire is E% = £0 cos . wtp Since at the surface of a perfectly conducting cylinder we must have Ev + = 0, the coefficient A must be A = - v = P2a*Ho. vK[(il3a) Thus the reflected field is proportional to the square of E the radius of the wire in wavelengths and is considerably smaller than in the case in which E is parallel to the wire. In the latter case the impedance to the current flow in the wire is small whereas when the jE-vector is perpendicular to the wire the impedance is large. Fio. 8.43. A plane wave incident on a cylinder; E is normal to the axis. ELECTROMAGNETIC WAVES 3lfi Chap. 8 8.21. Dominant Waves in Wave Guides of Rectangular Cross-Section (TEi,o-mode*) Consider a wave guide of rectangular cross-section (Fig. 8.44) and let the electric vector be parallel to the shorter side. That this assumption is consistent with the electromagnetic equations is shpwn by assuming Ex = Ez = 0, and substituting in (4 12-1 in (4.12-1) to obtain u 1 dEu 1 dE„ Fig. 8.44. Cross-section of a rectangular wave guide supporting a wave in which E is parallel to thej'2-p.lane C-fEi,o-wave). ios/i dz ' . , Hy = 0, dv dj% dy (21-1) - 0. solution is 0Dt„„cd J^ris; sslszsz fif-°nd ** we have eliminating trie magnetic intensities, dx2 * dz2 (21-2) It has been shown in section 3.1 that this equation possesses solutions which are exponential functions of z and which represent, therefore, waves traveling along the tube. Assuming Ey(x,z) = Eu(x)e-T> (21-3) and substituting in (2), we have = -x%, x2 = r2 -*2, r = VTT^. (21-4) The general solution is Ey = A sin \x + B cos x*; but if the tube is perfectly conducting, B must be equal to zero since Ev must vanish at the face x — 0; furthermore we must have X'i = nir, n = 1, 2, • • •, (21-5) to ensure that Ev vanishes at the opposite face. Thus we have Ey = E sin nitxja, where E is the maximum amplitude. Substituting in (3), and then in (1), we obtain 77 TP ■ v™ _T, Ey = E sin-e 1 z, a lh = nirE nirx -.-cos-é where the wave impedance in the z-direction is Kz = im/T. * The meaning of subscripts is explained in Chapter 10. (2.-6) WAVES, WAVE (.HIDES AND RESONATORS I 317 n nondissipative media there is a sharp separation between the fre--lu.anirs at which the power is carried by the wave along the tube and the npit'iit'ies at which the wave is attenuated. This may be seen from the l i .mu:; for the propagation constant and for the impedance /l/V 47T1 ~ \ a2 a2 ' T (21-7) When the wavelength is large, V is real and the wave is attenuated; the wave impedance is a pure reactance and energy merely fluctuates back and forth across any given cross-section of the tube. On the other hand, when \ is small, F is imaginary and the amplitude of the wave is independent of jj the wave impedance is real and energy is carried by the wave along the tube. The cut-off frequency and wavelength are obtained from the condition r » 0; thus _ la v _ nv _ n (21-8) The lowest cut-off frequency corresponds to n = 1 and the corresponding wavelength (characteristic of the medium) is twice the length of the side to which E is perpendicular \ = 2a. , (21-9) This wave is called the dominant wave because, having the lowest cut-off frequency, it is the only wave which will exist at a distance from the source of energy when the frequency is in the interval between the absolute cut-off defined by (9) and the next higher cut-off. Let the ratio of the operating wavelength to the cut-off wavelength be X fr o)c n\ X0 / co 2a Introducing this ratio in (7) we have v (21-10) r = iS„ 32 = bVi - v2, kz = Vl - „3 a \ v" Mx \ v / X < X„ X > xc. (21-11) The wavelength XE and the wave velocity vz in the tube are X v _ 2x _ vT (21-12) ELECTROMAGNETIC WAVES J7Ú .25 0 Fig. 8.45. ■f C 2fC 3ffc Ratio of the charac- Ckap, it At; I In cut nil (In- wavelength ami tlu- wave velocity are infinite; an the frequency increases, they approach the value.1-: characteristic of the medium. _i In Fig. 8.45 the ratio v/vz is shown as a function of the frequency. The wave im pedance is infinite at the cut-off and approaches the intrinsic impedance as the frequency increases. Below the cut-off the wave impedance is a positive reactance and sufficiently .below the cut-off it is substantially an inductance. At low frequencies the propagation constant tends to a con-terisvic velocity to the velocity i / ,. . a . i ., * stent value nur/a so that the attenuation in the wave guide. per length equal to a is m nepers. For dominant waves this attenuation is tt nepers or about 27 db. In dissipative dielectrics no sharp cut-off exists since power is dissipated at all frequencies wherever there is a field and this power must be carried by the wave to the place of dissipation. However, in high Q dielectrics, the conditions approximate those in nondissipative media and if we define the frequency ratio v as above, by ignoring g, we may express the propagation constant as follows r = IB J (I - "2) - (21-13) \ 0)6 Since we have assumed that Q = iae/fg is large, we have approximately + «, = fed; (] - *2r1/2> (2i-i4) except near the cut-off. The effect of dissipation on the phase constant is of the second order. We now turn to a more detailed discussion of the dominant wave; its field is r = a, Ey = E sin — s — " —- cos ■- * lff3 where $ is the longitudinal phase constant. The e maximum voltage F is bEé-^. 2£SV0ngí,tudÍnai ™t / in the lower face of the tube (Fig. 8.44) is obtained by integrating ~HX; thus * &#*J-M I = dx 2a$E •qB-w 7 7~ C jrA", (21-17) WAVES, WAVE CHIDES AND RESONATORS I 319 The power W curried by the wave is abE2 I" E„E* Jx a&E (21-18) The " integrated " characteristic impedance may be defined in several ways. Thus we may define it either as the ratio of the maximum voltage in the total longitudinal current or in such a way that the formulae for the power in terms of the longitudinal current or in terms of the maximum voltage remain the same as the corresponding formulae for transmission lines consisting of parallel wires; thus Kv.i V 7 5 2W FF* Kw,i = -jfl ■> Kw,v = |pr - (21-19) These integrated characteristic impedances satisfy the following equation kv,i = VkwjKw,y. (21-20) Other definitions are possible; but at present these satisfy all practical requirements. From these definitions and from the expressions for F, I, and W, we obtain the following expressions for dominant waves KVJ=^KS> Kw,7 = -Ks, K^r^^lQ. (21-21) 2a a 8a All these integrated impedances are proportional to the wave impedance at a point and involve the dimensions of the guide in the same way. They are proportional to b, the dimension parallel to the electric lines, and inversely proportional to a, the widdi of the faces which carry the longitudinal current. These expressions do not differ greatly from the approximate characteristic impedance [bja~)-q for transverse electromagnetic waves in a pair of parallel metal strips. The integrals in (17) and (18) may have obscured simple considerations by which the values of the integrated impedances may be obtained from the wave impedance Ks. The longitudinal power flow per unit area is proportional to Kz; if the transverse field were uniform throughout the cross-section of the guide all the integrated impedances would be equal to (b/a)Kz. Since the longitudinal current distribution is sinusoidal the total current is only 2/ir times the maximum current density and the voltage/current ratio becomes (-7rb/2a)Ke. Similarly, for a sinusoidal distribution of the field over the cross-section, the power flow Is only half what it would be in the case of uniform distribution with the same maximum voltage along electric lines; hence the impedance on the power-voltage basis is twice that for the uniform, field. As we have already seen, the impedance concept plays an important role in die theory of reflection. When the field distributions in equiphase 320 RLECTRi (MAGNETIC WAV1 S Chap, h WAVES, WAVE CJIIIDKS AND RESONATORS—1 321 surhucs arc I lie same lor (he incident, reflected and t ransiiiit led waves, the reflection and transmission coefficients depend solely on the ratio of two wave impedances, taken in the direction of wave propagation. This is the case when the boundary conditions over the entire interface between two media are satisfied automatically as soon as the boundary conditions at any one point of the interface are satisfied. In some cases the incident wave may be resolved into components which are thus simply reflected. Examples of this type of reflection have been encountered in the theory of cylindrical and spherical shields when the sources were not axially or centrally disposed. In such cases we have to consider an infinite number of wave impedances, one for each wave component, and the reflected wave will depend on an infinite number of impedance ratios. In wave guides the exact theory of reflection is often even more complicated. Nevertheless there are instances in which the ratio of integrated impedances may be expected to give a satisfactory indication of the amount of reflection. For example, in two conductor wave guides (parallel wires or coaxial pairs) at frequencies so low that only transverse electromagnetic waves are transmitted while other waves are rapidly attenuated and contribute only negligible end effects, the integrated characteristic impedances* determine quite accurately the reflection and transmission coefficients at a junction between two lines. In Chapter 12 we shall prove that for frequencies between the lowest cut-off and the next higher, the above mentioned end effects may be represented by proper reactances either in shunt or in series with the wave guides. If a and b are varied simultaneously and fairly slowly, it is possible to keep the characteristic impedances constant and thus eliminate reflections which would ordinarily occur when the dimensions are altered. Reflection losses may be avoided even when the change in the dimensions is abrupt provided we introduce compensating discontinuities. If the conductivity of the tube is finite but large, the tangential component of E is very small. The above expressions for the field become first approximations to the exact expressions. The magnetic field tangential to the tube is large and is not appreciably affected by the change in the boundary condition; hence the tangential electric intensity is obtained if we multiply the magnetic intensity by ijc = 91(1 + i), where 9t is the intrinsic resistance of the tube. The power absorption of the two faces parallel to the £-vector is then W-S. (21-22) o ri fi a * For transverse electromagnetic waves the integrated impedances Kv. i, Kw.v and Kir, i are equal. (21-23) The power absorbed by the faces perpendicular to /',' is o 2if i he power absorbed by these faces depends on the frequency only through !/{ and hence is proportional to the square root of the frequency. On the Other hand, lPy has the square of the frequency in the denominator and In mi e becomes small at sufficiently high frequencies. This is not surprising mi i- with increasing frequency the displacement current between the faces carrying the longitudinal current will predominate and the transverse ■ i induction current will become smaller. In the_)'2-plane the transverse conduction current is _IE_ . tjs* _ ttijLuzb An equal current flows in the opposite face. For the transverse displace- ment current we have Jd = im I 0 2iuea En dx = - io>taEe lt" = —— V- trb bo mi these equations we obtain the shunt capacitance per unit length and the shunt inductance on the voltage current basis tiab 2^ ' The longitudinal inductance L per unit length may be obtained from its definition dV . r 4 T___ViV tool dz ' lea Cv,i = ~~7 > *-v,i Tit? - = —iooLyjI, dz These expressions provide and from (16) and (17); thus Lyj = irnb/la. another method of obtaining KVj. The integrated distributed constants of the guide may also be defined in terms of the energies associated with currents and voltages. But from the practical point of view the primary constants of wave guides are of lesser importance than the secondary constants. The total power dissipation per unit length of the guide is From this equation and from (18) we obtain the attenuation constant °%77r^(,+-)%vrT70+T)- <2I-25> I I I c I U< JMAC.-NI'TIC" WAITS ClIAC fl In the case of imperfect dielectrics the total attenuation constant is obtained by addinn, ( U ) an.l [IS). I 'In- a 11 einial inn c< uist aril in an air-lil Id I copper tube with a = 20 cms and b = 10 cms varies as shown in big. 8.46. The wavelength is expressed in centimeters and the attenuation in decibels per meter. In order to produce waves of the type considered in this section the electric field impressed over die cross-section of the guide must conform to the .04 .03 a .02 .01 7.-20 22 24 26 28 30 32 34 36 3B 40 Fto. 8.46. The attenuation constant for the dominant wave (TEi.o-wave) in a rectangulnr wave guide; a = 20 cms, b m 10 cms, copper walls. electric lines and to the intensity distribution; thus the impressed intensity must be independent of the ^-coordinate and be a sinusoidal function of the x-coordinate. However, from general equations of Chapter 10 it will be apparent that if b < a and a < X < 2a, then the field of any source will differ from the one here considered only in the vicinity of the source. 8.22. Dominant Waves in Circular Wave Guides (TEi,i-mode) A wave similar to the one considered in the preceding section may exist in tubes of circular cross-section. Electric lines cannot be parallel, of course, since they must end at right angles to the tube; but they show an unmistakable resemblance (Fig. 8.47) to electric lines for the corresponding transmission mode in rectangular guides. From the equations of section 21 we find that for the dominant wave the longitudinal magnetic current flows in one direction in one half of the guide and in the opposite direction in the other half. This suggests that in the present case we should seek a solution of the following type H, = H;(P) cos)- The second solution becomes infinite at p = 0 and should be 111< buled only when the axis is excluded from the region under consideration. Thus (1) becomes HM = HMxp) cos^-r*. (22-2) The remaining field components are now obtained from the general . h i tr<(magnetic equations by setting Ez = 0 and using (2); thus TH H„=--/i(xp) cos Ev = -AV/p, x TH Hv = -o- /i(xp) sin

\i \<;ni i k waves Chap. 8 WAVES, WAVE (illIDES AN'I) RESONATORS I 325 Calculating the maximum transverse voltage V and dividing by 7, obtain KY,i we = '^J-^Jo ****** = 1J0 ^(*> * - *<*>J . % = 7 = /it*) The power flow above the cut-off is W = 4*«-*C*" - l)7!W/32«^//2; therefore %r = | (*2 - 1)/C For the dominant mode the above expressions become 520 „ 354 VT - „5 Vl - v2 764 The expressions for the attenuation constant and the power absorbed by the tube are For dominant waves in air-filled guides the attenuation constant becomes g> ___ a = - [3.76(1 - p2)-^2 - 2.65%/1 - «2]10~~3. a ■ 8.23. The Effect of Curvature on Wave Propagation Heretofore we have considered only straight wave guides. In this section we shall study the effect of bending the wave guide by solving four simple problems; two concerning the bending of parallel strips, and two similar ones for rectangular wave guides. Imagine two parallel metal strips of width a, separated by distance b a and assume that after bending they form portions of two coaxial circular cylinders. We shall consider that mode of propagation which was transverse electromagnetic before bending, with electric lines running normally from one strip to the other. After bending these lines become approximately radial as shown in Fig. 8.49. If the strips were straight we would use cartesian coordinates and call the field intensities perhaps ET, and Hv for a wave moving in the z-direction. In the present case, however, cylindrical coordinates are more suitable. The principal component of the electric intensity is Ep and the magnetic intensity is Hz; the wave propagation is in the direction of the ^-coordinate. We shall ignore the edge effect. It is possible to state a clear-cut mathematical problem in which the edge effect is absent. We need only con- 8.49. Bending of waves. Id......uiii'.iiI.h wuvi guide in which lln impedance ol two opposite faces is zero run I tin' impedance of the two remaining faces infinite. The tangential electric inten- |f|||y niul the normal magnetic intensity vanish at a surface ol /1 io impedance; similarly the tangential magnetic intensity tiliil the minimi electric intensity vanish at a surface of in- i...... impedance. Thus in the present ease we assume zero Impedance boundaries at p = pi and p = p2 where pa — pi — b .....I minute impedance boundaries at z = 0 and z = a, fin field under consideration has at least two components E„ and Uz. If we were In ii'i'iiimc only these two components we should find from (4.12-2) that the assump-\W ill is i riconsistent with the equations. We then make a less stringent assumption 11 J () which leads us to equations (4.12-7), connecting E„, Hz, and Ef. It appears ili n lore that one consequence of bending as indicated in Fig. 8.49 is to introduce an ileel lie intensity in the direction of wave propagation. For the type of wave under i niuiideration this intensity must be small, of course, butitis required by the induction lllW.H. For progressive waves traveling in the negative >i)" One method of solving this equation would be to plot both sides against q and obtain the points of intersection. Another method is to use the series expansions for the Bessel functions and solve for q by successive approximations, this method suggests itself when j3p2 and q are small. In any case it is evident that the solution 326 EI.Et II« >l\l At !NETI( WA\ ES CmAi>. 8 of (2) is more difficult Iliau that of (he cc.iiespi.mliii^, < 111i;iti.wi lor a strair.ht guide, and it is more practical in look at the problem from a physical point ol view. Thus if Pi = Pi = oo so that the guide is straight, the wave under consideration has very simple properties; the transmission equations are the equations of a uniform line with distributed inductance and capacity given by L = pb/a, C = ea/b; Ef and H, arc uniformly distributed and is zero. Assuming therefore that in the first approximation the bending does not alter the uniformity of IIZ and that the displacement current density iceeE^ may be ignored, we can calculate the new values of the distributed inductance and capacity. The inductance L and the capacitance C per radian are then la p\) _ pbc a C = Pi + Pi log — Pi Hence the characteristic impedance, the phase constant, and the wave velocity become v4 \Č a V P2 b pi Vbc \ Pi log Vbc If b is small compared with r we have approximately Thus the characteristic impedance has been increased by bending; likewise the wave velocity along the mean circle has been increased. The characteristic impedance can be reduced to the former value either by decreasing the distance between the parallel strips or by making them wider. The foregoing extremely elementary calculations may have created an erroneous impression that the fundamental electromagnetic equations have been ignored. In reality these calculations are based on equations (4.J2-7). Thus dividing the first of these equations by p and integrating the result as well as the remaining equation from p = pi to p = pi, we obtain 3 1 pi>2 dip J, p Hz dp = iueF' ~ = J E* Jp' dV pn (23-3) pEv(p) The first term in the third equation vanishes on account of the boundary conditions. The second equation shows that the variation in H is small because pi — p2 is small compared with the wavelength and E9 is small for the wave under consideration; hence H2(p) = — I/a, where I is the conduction current in the strip of radius pi. WAVES, WAVE (ilHDES AND RESONATORS —1 327 i it ml 11111 i Ii)' in (3) and inlcgrnting, we have iL _ »°m(pü - Pi) j (H = _ iwa dip la ' dq> log Pi Pi (23-4) II...... following physical intuition we have not strayed very far from the funda- iiii ni al equations. 0.2 is i \ V \ / / Fig. 8.50. Returning to the exact formulation ni the boundary value problem and to equation (2), we note a curious fact i hat this fundamental equation has been derived from the boundary conditions imposed on E9, that is, on a function which nearly vanishes for small curva-turcs. Equation (2) is not easy to handle numerically. However some values satisfying this equation can be obtained indirectly; thus if q = 1, we ran plot j[/N\ against /3p from the available tables (Fig. 8.50), select pairs of values /3pi and (ipz satisfying (2), and plot Be = |/3(pi-f pt) against Pi/fli (Fig. 8.51). In this special case the phase q

0 and £2(p2) should vanish we must have _ R = Aftfri) _ AQ3p2) CHAPTER IX Radiation and Diffraction 'ii Introduction II I he current distribution producing an electromagnetic field is known, ili> |lower radiated in a nondissipative medium can be calculated by ob-i lining i n her I he power contributed to the field by the sources or the power 1 nig through any closed surface surrounding the sources. The second hid hod is based on the energy theorem (4.8-7) which states that when H ' 0 the average power contributed to the field is equal to the real part ■ •l i - • i lain surface integral (4.8—8). The surface (S) of integration may In chosen to suit our convenience. Since in general the distant field is linn h simpler than the local field, a sphere of infinite radius is a particu-litily suitable surface of integration. For the same reason the second Diet hod is usually simpler than the first from the computational point of view. On the other hand from the theoretical point or view the first mi i hod is more direct and fundamental than the second; and in some in- inces, when the local field can be obtained in a simple form, this method is ah. i preferable from the computational point of view. Furthermore the first 1111■ t In id makes it possible to determine the reactive power while the second method is useless for this purpose. In computing the power radiated in ili sipative media tire second method does not apply; in this case we must add to the surface integral the volume integral which represents the dissipated power. 9,1. The Distant Field Equations (6.1—10) represent the intensities of the field produced by a given distribution of electric and magnetic currents in terms of two vector potentials A and F and two scalar potentials V and U. In nondissipative media the vector potentials are F-f „—iff? 4'JT?- dpn (1-1) where dpe and dpm are the moments of typical electric and magnetic current elements, f is the distance between an element at P(x,y,z) or P(r,0,(?)■ (M>1 Similarly we define the electric radiation vector L and then express the electric vector potential at great distances as follows *-/*m CM * dpm, F = fc- + o(£S. (1-9) Equations (6.2-11) show that at great distances from a current element the radial component of the field intensity varies inversely as the square of the distance and that the principal components of the field are therefore R VDIATK in and FRACTION 333 lnim.il i.. i lie radius, lamu 1 he foregoing discussion it is evident that, ' i any distribution of current sources which can be enclosed within a [mil line ol finite area, the principal field components are normal to the Indue, drawn from some origin in the region of the sources. These principal i ni oponent s can be expressed in terms of the radiation vectors, using the i......al formula (6.1-10). The divergence of A varies at least as 1/r and tin 0 and ^-components of the gradient contain a factor 1/r; hence the ulily contributions to the principal terms come from the vector potentials. Willi this consideration in mind we obtain the following expressions for llie intensities of the distant field E0 = ijHp - 2\r {r,N0 + (1-10) Ev = -vHe = — (-VNV + Le)e-^. '>.2. A General Radiation Formula I he complex flow of power (4.8-8) across a sphere of radius r is * = \r j (E6H% - EVH%) díl, where dQ is an elementary solid angle, dü = sin dddd&. Substituting from (1-10) and assuming that r becomes infinite, we find that * is real and we obtain the following expression for the radiated power W II = J *dil, 3> = 4>u + 24>12 + #22, $u = (A/tfArt + iV*N*), (2-1) *£2 = ^2 (LoL*e + LVL%), $12 = —z re(N0L% - NVF*). The integrand $ is called the radiation intensity in th_e direction (0, f- cos \f/ = ?q cos i/'i + r2 cos ty2. (3-6) 11 we have an array of several identical and similarly oriented radiators, ■ill-itted at a number of points Pi(ri,8i,i, S = ] zZ^nei0rnCOS ** |. (3-9) B.4. Directivity The radiation pattern or the directive pattern of a source or an array of Hues is represented by <3> or V#. An overall measure of directivity In defined as follows. A uniform radiator is taken as a standard. In the ■ use of acoustic waves this standard is represented by a sphere pulsating i iii li.illy. In the case of electromagnetic waves such a standard cannot be ii ili/ed; but if figures of merit relative .to this standard are known, other relative figures of merit can be immediately obtained. Secondary standards may be varied as convenience demands. Let the strengths of the (riven source and of the standard be adjusted for a unit power output W <= I QdQ = 1, Wi 0 = J $0 da = I; (4-1) 11 icn the directive gain or the directivity of the given source is defined as I be ratio of the maximum radiation intensity to the radiation intensity of KI.KITI« >MAi;NETIC WAVES' 1 Ii vi- I the standard. This gain is expressed also in logarithmic units; thus -l>„ J mux - , a S = -r- > 0 = 10 logio g. = sin3- 0, In ine I he gain is W = J $ dQ = lit J sin8 o do = 1.76 db. (5-2) g = 1.5j, I In above power fjf could also have been obtained from equation (6.3—1); in make *,„„ unity we let 72/2/X2 = l/15ir. Next we shall consider the effect of perfectly conducting ground (the ■ <|u,itorial plane) on an element at height z = h above ground; the axis Dl the clement is supposed to be perpendicular to the ground. The image • ■I I he element is positive and therefore the radiation vector and are* Nz = cos 9 + *H»CM ») = 211 cos 0k cos 0), Ne = -211 cos (fih cos 0) sin 0, 2vT~ : cos (ßh cos 0) sin 0. The " horizontal "pattern is still a circle, but the " vertical " pattern is Affected by the height. If h is sufficiently large, the radiation intensity may vanish for values of the angle 0 other than zero;, these angles of the cones of silence are obtained from (2m + 1)X ßh cos 0 = 7/7T + 2' cos 6 4A The height must be greater than a quarter wavelength before other nulls than in the direction 0 = 0 make their appearance. Cones of silence are produced because in some directions the direct wave from the source is canceled by the ground reflected wave. The power radiated by the element in the presence of ground may be obtained from the equations of section 6.4. The mutual radiation resistance of the element and its image is obtained from (6.4-24) if we assume Zi — z2 = 2A; thus '2 'sin 2f3h ~~ h? \ 2ßh — cos 2ßh and the total radiated power is (SM * For the upper element 6 = 0 in equation (1-4) and for the lower element B = ir; for both elements f = h. ELECTROMAGNETIC WAVES til Al'. 9 We remind the reader that this is the power radiated by the source and its image in free space; only one-half of this power is radiated above ground. The maximum radiation intensity is unity when 72/2/X = 1/60tt, and the corresponding radiated power and directivity become _ Ißh 4/3z " 3 £=3 Ißh sin 2ßh Aß2fi2 \^2ßh~ — cos 2ßk If the element is just above ground, g becomes equal to 1.5 and the ground has no effect on the gain. If the element is very high above ground, then g = 3 and the ground adds 3 db gain. When sin 28k 28h cos 2ßh = 0, h = .357X, Fig. 9.3. The gain in decibels of a current element normal to a perfectly conducting ground. the added gain is also 3 db. The added gain is greater than 3 db when the mutual radiation resistance i?i2 is negative; otherwise the gain is less than 3 db. Figure 9.3 shows how the gain in decibels varies widi the height in wavelengths. 9.6. Directive Properties of a Small Electric Current Loop In free space the radiation pattern of a small loop carrying substantially uniform current is the same as that of a current element. In the plane of the loop the diagram is a circle with its center at the center of the loop; in any perpendicular half plane the diagram is a circle tangential to the axis of the loop. This is not surprising since the loop is equivalent to a magnetic current element. If the plane of the loop is parallel to ground, the image source is negative and the radiation intensity in the ground plane vanishes. If the loop is near ground and the current is kept constant, the radiated power is considerably reduced. If the plane of the loop is perpendicular to ground, the image is positive. Let the center of the loop be at height z = h above ground and let the axis of the loop be parallel to the .v-axis. If 5 is the area of the loop, then the moment Kl of the equivalent magnetic current element equals iwfxSI and the magnetic radiation vector of the loop and its image is Lx = 2icopSI cos (8h cos 6). RADIATION AND DIFFRACTION 339 L I he radiation intensity we obtain * - 7^2 l*l* (cos2 9 cos2 * + s5n2 ^ = 24Qx;92/2 cos2 ißh cos 0)(cos2 6 cos2 *> + sin2 g=sa9 ab- The gain of a vertical current element, a quarter wavelength above ground is only 3.62 db; hence the loop has a gain of 1.87 db over the element. 9.7. Directive Properties of a Vertical Antenna In the oase of a vertical antenna above a perfectly conducting ground the ground may again be replaced bv an image antenna (Fig. 9.4).> As 340 ELECTROMAGNETIC WAVES ClIAP, 9 iii preceding sections we suppose- the ground removed and consider thill the antenna and its image constitute a wire in free space energized from i In- center. II the wire is |icrfect Ij conducting ami if its radius is vanishingly small, the current distribu tion is given by (6.8-1). In the next section we shall study the effect of finite radius on the assumption that the current distribution remains sinusoidal and in Chapter 11 we shall determine the extent to which t In-actual current distribution deviates from the assumed Fig. 9.4. Vertical an- distribution. The finite radius affects principally the tenna above a perfect jnput impcdance and the radiation pattern in those ground and its image. ... ... • • directions in which the radiation intensity is small. The radiation vector in the present case is sin fiil — z) cos (fiz cos 0) dz _ 27[cos (ff/ cos 8) - cos ff/] p sin 0 where / is the length of the wire above ground and 2/ is the total length of the antenna in free space. Hence for the radiation intensity we have 60/2 sin3 | - (1 - cos 0)1 sin2 | — (1 + cos 0)1 -^-4—^-+. (7-2) it sin 0 In the special case when / = X/4, 01 = 7t/2 and the radiation intensity becomes 15/2 cos2 (^x cos 0) _ it sin 0 The radiated power may be obtained by integrating is given by (7-2). If0a 1, jail and the radius has a negligible effect on the radiated power. 342 MM, TK< MAGNETIC WAVES' Chap. 9 9/J. Linear Arrays with Uniform Amplitude Distribution Consider an array of » like radiators equispaced on a straight lino (Fig. 9 6). Let the amplitudes of the sources be equal and let «? be the phase lag as we pass from one source to the next from left to right. If o nl I nl In this case the width of the major radiation lobe is 2\ A = nl For the broadside array the major lobe is narrower than for the end-fire array. 344 '.i.i'.c'ruoMA(;Nici'iť waves' Between successive null direct inns there will exist secondary of the spaee factor. These maxima coincide approximately with the diree tions for which the numerator in (3) is maximum; thus (Ik + 1 W Č = ± if (2* + 1 ),r where A is an integer. The amplitudes of the successive maxima, beginning with the second, vary approximately as 11 sin (2k + l)r' In For large » and small k the maximum amplitudes beginning with the prin cipal maximum, are In In In Jjt J>7r /;r or 1 : 0.212 : 0.127 : 0.091 The level of the second maximum is about 13.5 db below the principal maximum, independently of the actual value of n as long as n is large. Figure 9.7 illustrates how the space factor varies with £ for the case n = 10. As the angle \b varies from 0° to 180°, which is the maximum span for ^, £ varies from Bl - d to 81 + t?; if this range is within (-ir,+ir), 144 iso 216 252 288 324 360 Fig. 9.7. Universal radiation pattern for linear arrays of ten sources with equal amplitudes. then the space factor contains only one major lobe and the minor lobes diminish with increasing angle from the direction of maximum S. For a broadside array the span of £ is (—81,81) and so long as / does not exceed one-half wavelength, the above condition prevails. As soon as / becomes larger than X/2, however, the secondary maxima beyond a certain point begin to grow more prominent; for / = X, for instance, diere will be just as much radiation in directions parallel to the array as at right angles to it. RADIATION AND DIFFRACTION 345 |"• due1 ol vanes from 0 to 2(1 L In ho, ').S. Radiation pattern of a broadside array of two elements; / = -• Fig. 9.9. Radiation pattern of an end-fire array of two elements; / = - • We shall now consider two special cases. The first is a broadside array of two elements, one-half wavelength apart. The space factor of this array is (reducing the principal maximum to unity) 5 = cos 2 (gcQS^ $ vanishes along the line joining the sources; the polar diagram is shown in Fig. 9.8. The second case is the end-fire array of two sources, one-ipiarter wavelength apart. In this case i S = cos Th , -go- polar diagram is heart-shaped (Fig. 9.9). cos (?) ] 9.10. The Gain of End-Fire Arrays of Current Elements First we shall consider an end-fire array of n elements, perpendicular to the line of the array, a quarter wavelength apart. Let us assume that the z-axis is the line of the array and that the elements are parallel to the * Only the absolute value is important in the expression for S, 346 ELECTROMAGNETIC waves • a axis; then I In- span factor and the radiation intensity become . (n-K . „ (A n ^ stn J - 1 • (10-1) o1 = sin 15,r/2/2 -73— (cos2 0 cos2 p + sin2 | >< >it i< >t l a I to the square root of the length of the array in wavelengths. Figure 9.10 represents the radiation pattern (.V) of an end-fire array of eight nondirective element s, spaced one-quarter wavelength apart. 9.11. The Gain of Broadside Arrays of Current Elements For a continuous broadside array of electric current elements parallel to the .v-axis, the radiation intensity with its maximum reduced to unity is 4 sin2 {\Bl cos (?) ß2l2 cos2 0 (1 - sin20cosV). (11-1) The first two null directions, one on each side of the maximum direction are obtained from ^ßl cos 9 — rbrr. The total width of the major lobe is „ fa A 4tt 2X A = 2\2-0) = 37 = T- (11-2) The directivity of the array is calculated to be S = Fig. 9.11. Radia- (SÍA/ i **^ß sin ßiv-1 \ ßl + ß2p ' fjßj ' (11_3) tion pattern of A , . , a continuous As ' lncreases, we have asymptotically broadside array, two wavelengths 2/3/ 41 / long. &■ = — = V G = 10 JoSio r + 6 db. (11-4) Figure 9.11 represents the radiation pattern of a continuous broadside array of nondirective elements, two wavelengths long. 9.12. Radiation from Progressive Current Waves on a Wire Let us now suppose that a progressive wave is traveling in a wire of length / with velocity u, in general different from the velocity v characteristic of the medium. If the wire extends from z = 0 to 2 = /, the radiation vector and the radiation intensity are n, = / f:^:*m* m t dz M 'lĚ^Mt - ? cos g)/ Jo ß - ß cos 8 oos 0)1/2 = 30wl2[l - cos (ß - ß cos 0)1] \26S - ß cos Ö)2 sin2 0. RADIATION AND DIFFRACTION \ 10 Winn integrating «I> in order to obtnin W, wc introduce a new variable mm (/) — 8 cos 0)1; thus we obtain IM H x 1-cos/ 20 f^+^l-cos/ i i-cos* i r (1—cos/) (ř-/s)i 4 11 i he phase velocity along the wire is equal to the phase velocity charac-haeiiie of the surrounding medium, then /3 = 8 and w = 30/2 (log 281 -G2BI+C-1+ . (12-1) The direction of maximum radiation is found by equating to zero the d'e-iu alive of 4> with respect to 6; thus we obtain the following equation tan u ^ A u \ / u u ~ \ 81) ' X ~~ x(l - cos 0) ' The greatest maximum corresponds to the smallest root of this equation. When the wire is long, this root is substantially independent of / and it in i urs for a value of u about halfway between 1.16 and 1.17. The maximum radiation intensity is unity if 72 1J6X o„^X I 1 = 30/ sin2 1.16 = °-046 7 • Substituting this in (1) and calculating the gain, we obtain G = 10 logic, | + 5.97 - 10 log10 (log10 l- + 0.915^ . (12-2) 9.13. Arrays with Nonuniform Amplitude Distribution In all arrays with uniform amplitude distribution the shape of the axial cross-section (by a plane passing through the line of radiators) of the space tactors is more or less the same, regardless of the number of elements, so long as this number is not too small. If, however, the amplitudes of the individual elements are varied, the shape of the major lobe may be altered. Consider, for example, an array in which the amplitudes are proportional to the coefficients in the binomial expansion 1, (» - i)(„ - 2) (« - !)(» - 2){n - 3) . _ , j 1 • 2 1-2-3 350 (•'.I I .c t'i<< )m a< ;n l/l k WAVES Chat, 0 In I he iiol ill ion (il seel inn '), we have S = I 1 H-^l"-1 = 2"-1 cos"-1 |. Thus the space factor of an array of n elements with binomial amplitude distribution is the (w — l)th power of the space factor of a pair of elements of equal amplitude. While in the case of uniform distribution the numbei of radiation lobes increases with the number of elements, in the " binomial " array the number of lobes remains the same as for a pair of elements. Similarly we can start with an array of three elements with uniform amplitude distribution so that the space factor is S0= 1 1 + + 0 | , and design another array with the space factor equal to tire square of Sq S = Sl= | 1 + 2# + 3e2i* + 2e^ + e4i* | . (13-1) This array will contain five elements with amplitudes proportional to the coefficients in equation (1). We have seen that in a uniform array the levels of the secondary radiation lobes are substantially independent of the number of elements if this number is fairly large. In an array of type (1) the secondary lobes are considerably reduced in size. On the other hand the width of the major lobe is larger than in the case of a uniform array of the same number of elements. It is possible to assign the disposition of null directions at will, thus let S = I ( o might In- used instead of (.1). Kor instance, from (lie- liirm of (10 2) for small values of 0 we might assuml 6 g 0 g tt. Integrating, we find „, 8Q 1 W = Y$> G = 10 log10-+ 13.72. This value is very close to that found in (5), but the simplicity of the ap proximation (3) may at times outweigh the advantage of greater accural ) obtained by using this particular form of $. 9.15. Broadside Arrays of Highly Directive Elements Consider a pair of highly directive radiators located at (—d/2,0,0) and (y/2,0,0). Assume that the radiation intensity of each radiator is *o = I, 0 < 8 < 8, 0 <

needing section, where d = 2h and h is the height of the radiator nlmve ground. If the image is negative, then the sign of the mutual radia-Ii,in intensity becomes negative. In this case the maximum * is sin2 fth IIn long as h does not exceed a quarter wavelength; for larger values of h lite maximum intensity is unity. When the image is negative, the radiation intensity in the original maximum direction (parallel to ground) is . ,niivied. If the ground is not a perfect conductor, then its approximate effect tuny be obtained by assuming a reflection coefficient equal to that for uniform plane waves; thus $ = | l 4- q exp(2/0A sin 8 cos j8«i sin 8 cos y) £ exp(/?0£i sin 8 sin / — l)/,j of the array remain constantj let the amplitude of the elementary radii..... be AdS( = Aa\h) where dS is an element of area; then in the limit have S = AS tea . — sm 0 cos co a — sin 0 sm tp a (17 .1) where S is the area of the array. The space factor is maximum and equal to AS when 0 = 0. 9.18. Radiation from Plane Electric and Magnetic Current Sheets Consider a plane electric current sheet of density Jx and a plane magnei § current sheet of density Mv. The radiation vectors are Nx = J f M*j)em 008 * dS, Ly = jjMv(*J)&i* dS, where p is the distance of a typical current element from the origin and cos \p — sin 0 cos (tp — tp), f> cos $ = (x cos

f the couplet is g 2 and • 11n< i livily of the I liiygens soun e is ,r, 3. H Transmission through a Rectangular A pa-lure in an Absorbing Screen \ nine i hat the .vy-plane is the plane of a screen and let the boundaries \\ the aperture be x = 0, x — a, y — 0, y = b. Let a uniform plane m i impinge on the screen normally and let its field in the tfy-plane be Ex = Eo, Hy = Ho, E0 = yHo. Thi • licet of the screen is expressed by an added field which, by the In, Im. nun Theorem of section 6.13, may be produced by electric and mag-iii hi i in rent sheets of densities fx — — Ho, My = —Soil the screen is a perfect absorber and if the aperture is large, we may HBniime that the waves emitted by elementary sources directly in front ..I ill si reen arc completely absorbed and that the field of the sources over is in the direction normal to the aperture and 1 HIS2 _ ElS2 (19-1) (19-2) z X2 2r;X2 ' where S is the area of the aperture. In the #z-plane, # = 0 and (1) becomes * = (1 + cos 0)2 If 8 is small, we have approximately sin -m 0) sin2 0 sin2 u iraO Figures 9.13 and 9.14 represent V* and * as functions of a. In the optical frequency range the pattern observed on a remote screen parallel to the aperture would consist of alternate bands of high and low illumination. The distribution of $ in the yz-plane is similar to that in the .vz-plane. In optics the field transmitted through the aperture is called the diffracted field. ELECTROMAGNETIC waves ClIAl' Pia. 9.13. Vi vs. u = ™t>/\ for n rectangular aperture in an absorbi nig screen. Flo. 9.14. * vs. u = TraO/X fat 2rr u=3n ' a rectangular aperture in an absorbing screen. ^lo^ZT^ tkrmgh: Cimdar APemre md • Circular Plate for a circular aperture of radius a equation (18-1) becomes ??7r2//j I na , -jjjr o + cos e)2 jn /„(^ sin ^ ^ The radiation intensity vanishes when & sin 6 = ^ sin ^ = (20-1) /!(*») = 0, radiation and DIFFRACTION .157 mi. mot being excluded. In the ease ol liujii waves we should see on a distant n alternate light and dark rinus. u ill lie remembered that equation (I ) has been obtained on the assumption that ii mi iloes not ailed the livid of the secondary sources over the aperture; to in an idea of how large the aperture should be to make this approximation satis-H v, we shall compute the ratio ol the power//'radiated by the electric and mag- •......it sheet.', iii free space to the power//',>- .JT/ji«r//<','delivered to the aperture II Incident wave. For this purpose we integrate $ over the unit sphere. Tn the OJ a power series we have W_ _ " (-)n(ßa)»>+2 ^o~n=o(2» + 3)[(« + l)!]2; (20-2) lint when 8a is large, the following asymptotic expansion is more convenient for nu-Iim i ii a I computations IF _ 1 + /i(2ft.) /,(2/3*) J ma) H\ 2ßa + (2ßa)> (2ßaf + 1-37,(2/3«) + (20-3) '1 (2/&)4 ler to obtain this expansion we multiply (2) by 8a, differentiate the result with :t to j3«, compare the series with the power series for Ja(x), and then integrate "espect to fia; thus we have mm)=1 ■ ***** m>=1 - Wai Mt) dt- 1.0 w 7 .01 A d(8, Integrating by parts, we obtain (3). I'igure 9.15 shows how W/Wq varies wiih the size of the aperture. When the diameter equals one wavelength, (la = ir and the power ratio is about 0.9. It may be expected that for this .md larger apertures, the radiation pat-lern will not differ very much from the yja ^ Actual radiation pattern, particularly in directions in which the radiation is large. The radiation pattern near the null directions is likely to he affected to a much greater extent by the approximations. Huygens was the first to suggest that any wavefront could be regarded as an array of secondary sources and he postulated that the field of the secondary sources should be entirely in the direction of the advancing wavefront. He offered no explanation of this property and no suggestion of the physical nature of these secondary sources. The Induction Theorem states that each secondary source is a combination of electric .1 1.0 /3a 1Q Fig. 9.15. Power ratio vs. ßa = 2ra/\; fV9 is the power incident on a circular aperture and //'is the power scattered by the aperture if the edge effect is ignored. 358 ltf.KCTKUIYIA(;Nľ,TI(J WAVKS Cham, flj T and .....gnetic current elements whose ......iieiiln ale respectively | nop< irl inna I In thn magnetic arid electric internal ie;; tin mentis I In I lie u'iivcfrnnl. 'I'lie Indiietinii Them ■->>> is, of course, not restricted to the wavel'ront; hut its practical usefulness is In - i] limited either to situations in which we have reason to believe that the field ot llu secondary sources is not appreciably affected by the surroundings or to situation which the surroundings arc such that the exact field of a typical secondary source may be determined. The approximate solution of the problem of reflection from a large conduetinu plate is quite similar to the above. Thus on reflection from an infinite plate ilia I tangential component Ht of the incident field is substantially doubled and the linciir current density in the plate will be 2rlt. In the case of a large plate the current if ie il is assumed to be 2Ht in the first approximation. The component of the current density normal to the edge of the plate must vanish, of course, and our assumption I is at its worst in the vicinity of the edge; but this " edge effect " depends on the cil'- ' cumference of the plate while the main effect (for large plates) depends on the areil If a plane wave of the type considered in the preceding section is incident normaII) on a circular plate, the radiation intensity of the field reflected from the plate is given by (1) provided we replace the factor (1 + cos 6)- by 4(cos2 9 cos2 tp + sin2^)- 9.21. Transmission tftrough a Rectangular Aperture: Oblique Incidence -Let us now consider the case in which the incident wave strikes the screen obliquely, Let the angle \j/ between the wave Rectangular aperture in an infinite screen. normal and the normal to the screen hi small (Fig. 9.16) and assume that the wave normal is parallel to the #2-plane. Tin components of E and H parallel to l ht screen are nearly equal to the total E and / / and the principal difference between thin case and the case of normal incidence is in the progressive lag (Shp of secondary soun in the positive .v-direction. Hence the r.uli ation intensity is Vflo * = (1 + cos ,ißx (sin 6 nos f-ŕ)+fpí sin e s la t fiy- ■na'Wm —(1 + cos BY |«A) t/q ■ 9 \"7fa 1 f ^ \ m I y (sin 0 cos

. Figures 9.13 and 9.14 represent the variation in V* and * if we take u = W.(Q - ^)/A. The radiation pattern in the^planv is independent of ib. RADIATION AND DIFFRACTION 359 ^■2. Radiation from an Open End of a Rectangular Wave Guide |>,u the dominant wave in a rectangular wave guide open in the plane 11 U we have Ev = £0 sink = Vl d wt: assume that in the first approximation the field is not altered by ■lie sudden discontinuity. On this basis we calculate the radiation ■ . , i,,r, ;uul the radiation intensity si j 0 sin ^ 7t.V eiß$ sos iŕ+p sin m \< ;ni. i k WAVES' i IhaPi i1i1lx 2a2lrE\ If we also assume that there is no reflection of power in the wave guide at the aperture, then we have the radiated power W from (8.21-18); thu W = abE^/A-q. If the electric intensity is adjusted for unit power output, then $„iax = 8ab/r~\2, and the directivity of the wave guide as a radialoi is g=~, G = 10 log10^+ 10.08. (22-1) 7TA A If the field were uniform over the aperture of the wave guide, then we should have ab = X2"' for unit power output and the gain would be Axah _ £ = ~~Z5 ) G 10 logI0 Ě + 10.99. a (22-2) Hence the effective area of the aperture is 8/tt2 times the actual area or about four-fifths of the actual area. 9.23. Electric Horns Ordinarily the cross-section of a wave guide supporting a dominant wave is comparatively small; the larger dimension will be, perhaps, greater than X/2 and less than X; the smaller dimension may be about one-half the larger dimension. If an open end of such a wave guide is used to radiate energy, the radiation pattern will be comparatively broad. In order to increase the directivity the wave guide is flared out into an "electric horn" (Fig. 9.17); it may be flared out either in the direction of electric lines of force or in the direction of magnetic lines of force or in both directions. Let / as shown in Fig. 9.17 be defined as the length of the horn and let 2\b be the angle of the horn. We shall assume that \b is so small that the ratio of the area of the aperture to the area of the wavefront at the aperture is nearly unity. Since Fig. 9.17. Electric horn, flared out in one plane. sin \p ** 24/2 . RADIATION and DIFFRACTION 361 thi' ratio is nearly unity even for a fairly large tingle \b. We shall assume tin laid distribution over the wavcln nit is that which would exist if (In I.....i were continued. The approximate gain can then be obtained by hp method used in the preceding section. Hfll'Ht let us consider a sectorial horn flared out in the magnetic plane (lli plane). The radiation intensity in the forward direction (the ■Onitive z-axis) is --O I _!./>> _ „ /o ttx *<0>=|S r r^e^dxdy IK J-b/2l/ -a/2 a 12 (23-1) a/2 wlnir (.v,_y,z) is a typical point in the wavefront iinil hence £ is the distance of a point in the \\a\i hunt from the plane of the aperture (Fig. p,IH). This distance is approximately «-/(cosf-coS|)=g-|. (23-2) If the horn flares out in the electric plane we r , ,. . . . , r r it; 9.18. Distances connected 1 limy express the radiation intensity in the for- with the horn in Fig. 9.17. ■ h'I direction in the following form „Tj2 nb/2 „a/2 ZA o _t,/2 «* -a/! cos -7- e p dx dy (23-3) -b/2 «* -a/2 where £ has the value denned by (2). In both cases the radiated power is approximately abE2 w An = lyabHo, (23-4) (23-5) which is equal to unity when ■qabHl = 4. Expression (3) is simpler to evaluate than (1). Substituting from (2) in (3), we obtain 72 I nh/2 _A |3 &2\ ^ 2 dx . The second factor may be expressed in terms of Fresnel integrals as de-lined by equations (3.7-32). Thus we find Avb2lHl[ Ví* + S' 362 ELECTROMAGNETIC WAVES Chap. 9 I Substituting for //,'! from (5) and using (4-3), wc have (23 (>) s f. « N ~; Fig. 9.19. Cornu's spiral. Since Ify = a, I = «/2^, this may be written 32^ g (23-7) RADIATION AND DIITKACTION Mi.' plot of S(x) against C(.v) is called ('onmi's spiral (lug. 9.19). The dis- ........I ii point on the spiral from the origin is Vt?2 4- S2 and the distance lirltvrni any two points is V(C3 — Ci)a + (S2 — Si)2. Since C(x) and K#) arc odd functions, the sums C(xi) -\- C{x-,) and S(xi) + o'(aV) can Bvtiys he regarded as differences C(xi) — C(—#2) and S[X\) — S(-x^). I ii, I. n,-ili .>• of the spiral between the origin and a point corresponding to ■ ~ x is = J*V[dC(x)f + [d~S(x)]2 = £ dx - Fig. 9.20. Directive gain of a horn flared out in the electric plane. Thus we have a simple geometric interpretation for the independent variable. Cornu's spiral helps to interpret formulae involving Fresnel integrals. Titus it is evident at once that if ip is kept constant and a is increased, a point is reached at which g of (7) is maximum; for larger values of a the gain in the forward direction will never be as great. What happens is that as the area of the aperture increases, some of the secondary sources on the wavefront at the aperture are out of phase with others and interfere destructively in the forward direction. 361 ELECTROMAGNETIC WAVES Chap, 9, Figure 9.20 shows how ,1; varies with the size of the aperture arid the length of the horn when b = X. WTe now turn our attention to the ease in which the horn flares out in the magnetic plane. Integrating (1) with respect to y and expressing the cosine in terms of exponentials we have *(0) = ir% Lexp{-)+ expr t)Jexp{-nrl- Introducing new variables we reduce this to Fresnel integrals and obtain *(0) 4X \[C(u) - C(v)]2 + [S(u) V2\ a VX// V2\ * V\l/ Substituting from (5) and using (4-3), we obtain g = u- \[C(u) - C(v)f 4- [S(u) - S(d)]2). Figure 9.21 shows how g varies with a and / when b = X. If the horn flares out in both planes, then . = a2 + b2 _ J + f 2 8/ 2/ ' Substituting in (1) and integrating, we have (23-8) X \[C(u) b C(v)f+ [S(u) - S(v)]2\, (23-9) [C(u) - C(v)]2 + [S(u) - S(v)]^ Tlie directivity of such a horn can be obtained from Figs. 9.20 and 9.21 if we multiply the g's corresponding to the two sides of the aperture and divide the result by 32/jr = 10.02. For example let / = 100, then if the side normal to E is 5X, from Fig. 9.21 we find^i =51; if the side parallel to E is 4X, then from Fig. 9.20, we find g2 — 41. The directivity of the horn is then g m O.lgi,^ = 209. Of course, when / is large and a and b small, it does not make much difference whether the horn flares out one way or the other; the curves in Figs. 9.20 and 9.21 begin to differ as they approach the maximum points. RADIATION AND DIFFRACTION 365 I'lie directivity can be expressed in the following form ■v.ii X;: g = 4ir- (23-10) where E0 dS .„ Ev = - e~*T (1 + cos 0) sin )2 + z2, where (x,y,0) is a typical point in the plane of the aperture. When z is large compared with the dimensions of the aperture this is approximately 2z If we now assume that z is in the range in which the amplitude factor 1/r is substantially constant but the phase factor e~'lBr is not, then equation (1) becomes (x - *y + (y Ex = ~kTJ J £oexp -i3 2z dS. (24-2) * Assuming that the incident wave is normal to the screen. RADIATION AND DIFFRACTION \t normal incidence Eo is constant over the aperture; hence for a rectangular Apia lure bounded by x = — a/2, x - a/2, y -= —b/2,y = b/2, we have On I he z-axis the square of the amplitude is I I'"4i- [*" (yk)+ s' (vis)] [c' (vk)+ s° (M\ ■ \ . the size of the aperture increases the electric intensity increases until it reaches ih< first maximum (see Cornu's spiral). II is easier to follow the variation in the intensity when the aperture is circular. In iIns case we have, when x = y = 0, E* ~ liE»sin H exP [ - $ (2 + ■ The maximum amplitude of Ex is 2Ea and it occurs when a = VnXz, n = 1, 3, 5, ■ • -. The minimum value of Ex is zero and it occurs when n is an even integer. The above points correspond to the radii for which the difference Vz2 + a"- - z = 2z between the distances from the point (0,0,z) to the edge of the aperture and to its renter equals an integral number of hall wavelengths. The plane of the aperture is divided into Fremel zones. The neighboring zones produce equal and opposite intensities at (0,0,z). The first zone produces an intensity which is twice the intensity which would have existed at the point if the screen were removed. The aperture has a focusing effect on the field at distances of the order a"j\, provided a is large compared with X. The intensity at (0,0,z) is increased still more if alternate zones are blocked out; but the successive increments will eventually become smaller.as the distance between (0,0,z) and the zones increases. The case of a rectangular aperture can be discussed qualitatively by the use of Cornu's spiral. Consider, for example, the variation in the electric intensity along a line parallel to the .v-axis.' In this connection we need to fix our attention only on the factor On the Cornu spiral this factor is represented by the chord joining two points separated by distance 2a/V2Xz measured along the curve; the midpoint of the arc is at distance 368 KLECTRnMAGNKTIC WAVES Cut from the origin (along the curve), hirst let the position of the reccivor ffl fixed; then the midpoint of the arc is fixed. As a increases, the ends of the nre nan i outward and, up to a certain width of the aperture, the amplitude of the field ul ihi> receiver increases. Beyond this point the arc begins to wind itself on the coil ,1 I, spiral and the amplitude starts to decrease and finally reaches a minimum, fi'miil there on the intensity passes through successive maxima and minima but the fiuH tuations gradually diminish. On the other hand, if the size of the aperture and lh»< distance between the receiver and the screen are fixed, then the arc is of fixed longl h while its midpoint will move as the receiver is moved parallel to the ,v-axis. Tin spiral is least curved at the origin; hence when the receiver is on the ^axis, the length of the chord is maximum As the receiver is moved away from the k-axis, the length of the chord decreases and therefore also the received voltage. If the arc is very short (narrow aperture) the diminution is gradual; but if the arc is long enouull (a wide aperture), it will wind itself around one of the coils of the spiral and the re ceived field will fluctuate. Consider now a source at some finite distance. 5 from the screen (Fig. 9.22), If the source and -—( Siuusuidatly Distributed Currents I An ti rule the calculation of the complete field produced by a given current hltiiloinoii involves difficult integrations. One important exception is m\e case of a sinusoidal distribution on a thin straight wire. In section 6.7 11 i■. |i i :,ln)wn i hai such distribution is obtained on an infinitely chip fcei'lcctty conducting wire and therefore may be taken as the first approxi- ........ii to the distribution on a thin wire. I .el: the current filament be along the z-axis between z = % and i = z2 ami let the current I(z) satisfy the following differential equation mm di2 -{pirn- (25-1) Id the current and its derivative be continuous in the interval Zi < z < z%. The vector potential of a typical element of the filament is dJ^UI(i)dz, n = ^, r = \/P2+(s-z)2 (25-2) lly (6.1-10) the corresponding electric intensity parallel to the filament is dEz H ~(-^ + P2n)l(z)dz, u»f. \ dz / From (2), we have (25-3) d? dz df an di' dz an di dz2 a2n as3 Substituting in (3), we have %$*4~l /(^iri^ + — / Jimdi. to>iJ3, ■ dz iMA< íNKTlC WAVI'.S < iui', Let n and n be (lie distances from the end of the filament to a typffl point P(p, + (z - z2)2; (25-then (4) becomes —~ 7'(zi)-- -+ 7(zi) r--- /(z2)-- . (25 4tt/ü>€ |_ n r2 oz n az r2 the electric intensity parallel to the filament is expressed in ten of the current and its derivatives at the ends of the fl ment. lil By (4.12-9) the magnetic intensity is expressed terms of Ez as follows Hp (pHv)"MtpEs, pHr = iue J PEtdp+F{z), (25 where F(z) may be determined from the condition 2irp7/„ = 7(z), z, < z < z2, as P -* 0, 2 < Zi or z > z2. = 0, From (5) we have pdp = n dru pdp^ r2 dr2. (25-8) Fio. 9.23. Distances involved in the p dp = r2 dr2. (25-9) expressions for the curreVmTwire Substituting from (6) and (9) in (7) and integrating, wi of finite length.' have -47TiBpHv = I'(2l- I'(z2)e-^ + 7(zj) ~ ^ - 7(z2) »H»r, (25-10) oz dz except, perhaps, for a function of z alone. In order to evaluate the derivatives we note that W**^ coJ dz 6Vi 3z ri where 0i is defined in Fig. 9.23. A similar expression is obtained for the derivative in the last term. Thus (10) becomes 4xP77p = -If(z2)e-^ - i/'(2l),-^ to tB + /(zi)e~ipr' cos 0] - 7(z2) M. (ď-«^i + ř-#S _ 2e-ißr cos ßl), 4irp (25-14) 30/7 (e~ißTi cos 0i + e~ißr* cos 02 - 2 cos ßl cos 0). In the case of a perfectly conducting wire Ez should vanish on the surface except in the vicinity of the center where it should be equal and opposite in the applied electric intensity. Hence on a wire of finite radius, no matter how small, the current will deviate from the sinusoidal distribution although the deviation diminishes as the radius of the wire approaches zero. In Chapter 11 we shall obtain a quantitative idea of the magnitude ol this deviation. electromagnetic waves 1 MM 9.26. The Mutual Tower Radiated by Two Parallel Wires Consider two pant!lei wires (Fig. 9.24) of equal length 21. Let eno3 wire be energized at the center so that h{z) - h sin 8(1- z), I2(z) - I2 sin B(l - 2), 0 < z < I, (26- =■ h sm 8(1 + z), = /jj sin 8(1 + z), -/ < z < 0, 2_I I__£ where 7i and 72 are the maximum ..... plitudes. Without loss of generality, we may assume 7r and 72 to be in pluHH The complex power contributed to (In1 ~j--1 1-j ^n v'rtue °f ^ electromotiv. o force in one wire sustaining the cupJ Fit;. 9.24, Parallel wires, each energized reEt ^n ^ against the field of the Otlll I at the center' wire IS *i2 = -hf^EUzIÍÍz)dz= -±J'e2,zI*(z) dz. (26-2) Substituting from (25-14), we have %g =>. 30(7x72 p' />—iffTi «■9 \ n + p—ißr* „—ißro --2- cosM) sin (3(7 - z) dz, (26-3) where ra is the distance from the center of one wire to a typical point on the second wire. Defining the mutual impedance Z12 with reference to current antinodes 2*12 JI2 /?13 + tX12 ~ (26-41 and integrating* (3),_we obtain % = 60[2 Ci /% - Ci + /) - Ci (3(^4 - /)] + 30[2 Ci ßP - 2 Ciß(r0i + I) - 2 Ci ß(r0i - I) + Ci ß(ru + 21) + Ci/S(r,4 - 2/)] cos 2jS/ 4-3012 Sif^pl - 0 - 2 Si p>C4 4- | +Si p>i4 + 21) ~ Si /3(i-M - 21)} sin 2ßl, X12 = 60[Si ß(r0i 4- /) 4- Si ß(r0i - /) - 2 Si ßp] ' +30[2 Si ß(r0i + I) + 2 Si ^ - /) - 2 Si ft> -Si flf>M + 21) - Si ß(r14 - 21)] cos 2/3/ 4-30[2 Ci ß(r0i - I) - 2 Ci j3(r04 + | 4-Ci ß(r14 + 21) - Ci ß(n4 - 2/)] sin 2ßl. * See equations (3.7-39) to (3.7-42). (26-5) (26-6) RADIATION AND DIFFRACTION 373 1 he principal advantage of the present method of calculating the radiated bower over the method based on the power flow across an infinite sphere 1 1 hat it enables us to obtain the reactive power. However, in Chapter 11 wr .hall show that, in practical situations in which the radii of the wires 11. anall but not infinitely small, the reactance as given by (6) can be u 1 d only for a pair of parallel wires so energized that they carry equal ami opposite currents, and that in the case of an isolated wire certain modifications are needed. It should also be noted that in the above expulsions Z12 refers to maximum current amplitudes and not to input Currents; just how these expressions can be used to obtain the input Impedance will be explained in Chapter 11. |9„27. Power Radiated by a Straight Antenna Energized at the Center The self-impedance (referred to the maximum current amplitude) of an isolated wire of radius a is obtained from the mutual impedance formula 1 if I he preceding section if we assume that the distance p between the axes of the filaments is equal to the radius of the wire. When the radius is very small we have approximately VP 4- a2 1 + 2V vV 4-^ = 2/-)-^ (27-1) 4/ -30 Substituting in (26-5) and (26-6) and evaluating, we have R = 60(C 4- log 267 - Ci 20/) + 30(Si iß! - 2 Si 2p7) sin 2p7 +30(C + log fil-2 Ci 2/3/ 4- Ci 4/3/) cos 2p7, (27-2) X = 60 Si 2ßl 4- 30(2 Si 2ßl - Si iß!) cos 2/3/ i ^log ~ - C - log 2tt - Ci 4(3/ 4- 2 Ci 2ßlJ sin 231, (27-3) If 7 is the maximum current amplitude, the radiated power is W = ^RI2. When expressed in this form the radiated power is independent of the radius of the wire. If 7(0) is the input current, then as the radius of the wire approaches zero we have 7(0) - 7 sin ßl; hence the asymptotic expression for the input resistance is £t' = -^. (27-4) sin' ßl Tf we plot this R as a function of 81, we obtain a curve which is approached by the input resistance curves for finite radii as the latter approach zero. 9.28. Power Radiated by a Pair of Parallel Wires Consider now two parallel wires of length 21 (Fig. 9.24) carrying equal and opposite currents. The complex power * is ¥ = (Z-Z12)I2, (28-1) 37» ELECTRi (MAGNETIC WAVES Cham where /. is I lie self impedance of cither wire ^aluating /,,„ we find that when the interaxial separation,/ is ,...... R - 2407T2,/2 iii, r,at 3 sin 2(3/ I ' + 2 cos 2Bl--— -i--1 (28 .fl 48GW "n W ,1 , 7 cos 2Bl\ 4/3/ 2W2 + ~16^F)> <28 | X = ~>T~ (1 + * cos ^0-120 Jog | sin 2/3/, (28. and the power radiated by the two wires is IV = It shou|d J CHAPTER X Waves, Wave Guides, and Resonators — 2 III I Transverse Magnetic Plane Waves (TM—waves) Assuming that transverse magnetic waves are traveling in the z-direc-[tlnn, we have by definition Hx = 0. Since there is no longitudinal mag-lirlii' current, the transverse electric intensity can be expressed as the grail H ui of a scalar potential Et = -grad V. (1-1) The divergence equation for H becomes c)x dy mil the magnetic intensity can be derived from a stream function II Hx an m = - an dx I lence we may write H = curl A, Ax- Ay = 0, A, = II, where II satisfies the wave equation erg a'n jsffn a*2 + a/ + 322 For the electric intensity we have o-2n. (1-2) [ (1-3) (1-4) (1-5) curl i7 grad div ^ — AA _ grad div A — tr2// ,g" + iote g + /cue J + toe Since has only a z-component the transverse electric intensity is grad div A 1 an g + io>£ and in equation (1) we may assume V = — grad g + toe 02 l an g + toe 5z 37S (1-6) 376 ELECTROMAGNETIC waves chai I The longitudinal elcctrit inlensil ) is tin n J_/c^n 2 \ 1_/a2n , a2n\ Thus the entire field has been expressed in terms of one scalar wave him tion II. The foregoing expressions are general for any field in which // For waves in which the field pattern in planes parallel to the .vy-plani the same, we have n = T{xj)t(z). In general T(x,y) may be complex T(x,y) = T^ixj) + iT2(x,y); (I (1- but for plane waves there should be no phase change in any directid parallel to the tfy-plane (except for a complete reversal) and T(x,y) tin be real except for a possibly complex constant factor. This factoi i I always be included in T(z), and hence for plane waves T{x,y) may be taheti as real. Substituting from (8) in (5) and dividing by II, we have f\dx 2 + d2T\ 1 d2T dy2) + T dz2 (I Id) The first term is independent of z and the second is independent of x ami y, while the sum is a constant; hence each term is a constant and d2T d2T dx2 + dy2 ~ ~ri' (l-i I; For plane waves x2 is real and x itself is either real or a pure imaginary, Substituting from (11) in (10), we have dz2 = T2t, r V7T x2; t(z) = Pe~T* + QeT* (1 I ', Substituting from (8) and (11) in (7), we find that the longitudinal electric intensity and the electric current density differ from the stream function II by constant factors ' Et = g + f»€ n, (J + iux)E, = x2n. WAVES, wave GUIDES, and RESONATORS 2 377 In cartesian coordinates the transverse field intensities are dy dx g f icoe dz I df (1-14) ig 4- io>e) Tdz Ex = Z\~ Hyy Ev — —Z\ Hx, Z\ — — lii cylindrical coordinates we have //„ = & T, H,= -d-fT, EP = Z+ H„ E„ = -Zt H„. (1-15) pdip dp For progressive waves traveling in the positive z-direction, we obtain f = Pe-r*, Z+ = K, = g + mm Then in cartesian and cylindrical coordinates we have (1-16) Ex = KJIm Ey = — KZHX; (1-17) F Pd Ep = Ktfte, Ev = -KZHP- In nondissipative media the propagation constant V and the wave impedance K, for progressive waves become zoie tp (1-18) When the frequency is such that \ 03 2t ß = - = — = X, v X then r == 0. The frequency so defined is the cut-off frequency, since for lower frequencies T is real and on the average no energy is transmitted in the z-direction. The cut-off frequency and the corresponding wavelength are X 2.TT = 2ir/c = xv = —7= j "Ks = V |ie X (1-19) 378 II I* I K< (MAGNETIC WAVES CiiAi>, |Q WAVES, WAVE GUIDES AND RESONATORS — 2 379 In terms of the frequency ratio x0 (I !0 / K a' equations (18) become, above the cut-off, r = ioVl - v2, K, = ijVl - v2. (I 21) Below the cut-off T is positive real and the wave impedance is a negative reactance. Sufficiently below the cut-off, we have approximately K, x toe (1-22) I and the wave impedance is substantially a capacitance e/x. Equations (6) and (12) imply that V ami II satisfy the following trans mission equations : T - -(g+me)V. dz \ g + to*/ an dz These are the equations of a transmission line with series distributed con stants per unit length equivalent to an inductance p plus a capacitance e/x2 in parallel with a conductance gfx2, and with shunt distributed constants per unit length consisting of a conductance g and a capacitance «, Equipotential lines in transverse planes are defined by the following family of curves T(x,y) = const, or T(p, Ey — dy dx (2-2) (2-3) (2-4) and the electric intensity can be derived from a stream function * Ek Hence E — — curl F, where Fx = Fy = 0, Fz = M>, and * satisfies the wave equation. As in the preceding section we find U--+.&. (2-5) top: dz The longitudinal magnetic intensity is then ^2 top,\d2? ) top,\dx'2 dy2) Thus the entire field has been expressed in terms of one scalar wave function *. The foregoing expressions are general for any field in which Ez = 0. For waves in which the field pattern in planes parallel to the ^y-plane is the same we have * = T(xj)T(z). In general T(xj) may be complex as in (1-9); but for plane waves it is real. The functions T and T satisfy the same equations as in the case of transverse magnetic waves. Substituting in (6), we find that the longitudinal magnetic intensity and magnetic current density are constant multiples of* 2 toft In cartesian coordinates the transverse field intensities are dy dx topdz Hx * -Y+Ey, Hy = YtEx Tt=-= - — Z~\~ tojiT dz (2-7) (2-8) WAVES, WAVE guides, AND resonators -2 381 In cylindrical coordinates we have !)T 3T E„---f-T, Ev = --t, H„= -Y+Ey, H„=YtEp. (2-9) po

)]. In an unlimited medium progressive waves can be generated by an infinite plane current sheet. If the generating current sheet is in tin *y-plane, then the current density is ]m = -2Hy(x,y,+0) = 2PM,~, Jy = 2Hx(xj,+0) = -2PMZ^, dy dx dx dy 0. Let us now suppose that we have an arbitrary current sheet. If we can find two functions T\ and T2 satisfying the following equations dTt | BT% dx dy dT\ dy dx we shall be able to separate that part of the total field for which Hz = 0 from the other part for which Ez = 0. Eliminating either Tz or T\ from the above equations, we have a2Ti , d2Tx dx1 + df dx dy ' d2T2 , d2T2 dx2 dy2 dy dx Thus we have a pair of partial differential equations for the unknown functions T\ and T2 10.3. General Expressions for Electromagnetic Fields in Terms of Two Scalar Wave Functions The most general electromagnetic field in a source-free region can be expressed in terms of two scalar wave functions 13 and SI'. Suppose we WAVES, WAVE (a ill)ES, AN I) R ES< )NAT< >KS — 2 383 ■,i,n t with a given general field and determine a particular ii from öS J j — - an dn (I mi where n is the outward normal to the cylinder. In the case of transverse electric waves the longitudinal density is till tangential derivative _ dU 1 d2* 1 dTdf /2 = - = - —— = - —Ty (Til) ds iojjx dsoz iwp as dz of the magnetic potential, taken in the counterclockwise direction as seen] from the positive side of the #y-plane; the counterclockwise transvet'M( conduction current density is 2 2 jt = iL * = iL tT. For progressive T"£-waves traveling in the z-direction, we have /.-TT r dT 1 dT ioiji ds Ki ds Jt = iL Te-Tz (4-12)1 (l n, Below the cut-off /z and Jt are in phase; above the cut-off they are in quadrature. If Jz is kept constant as the frequency increases, Jt approaches zero. The total longitudinal current is proportional to the total change in T around the periphery; hence this total current equals zero. In the case of progressive transverse magnetic waves the power absorbed by an imperfectly conducting cylinder is obtained as usual by integrating the square of the tangential magnetic intensity or the square of the conduction current density; thus & = mj hJ\ds = J(^jds = fsP, (4-14) where s is the length of the periphery and I is the root mean square conduction current. The attenuation constant (due to the losses in the conductor) is therefore p\2 J ds 2W 2vx (l - ^r1'2. (4-15) dS waves, wave i.'impes, and RESONATORS - 2 387 i in ihe other hand, for progressive transverse electric waves we have tin following expressions for the dissipated power (if the dielectric is nor-ili iip.itive) //, WJjJUs = \rrJT2ds. (4-16) I lent e the general expression for che attenuation constant is 2r, Mil ds dS Vi (4-17) 10.5. Natural Waves in Rectangular Wave Guides Let a rectangular wave guide be bounded by the following planes x = 0, x = a, y = 0, y = b. Pol transverse magnetic waves the boundary conditions are then T(0j) = T(a,y) = T(*,0) = T(x,b) = 0. Particular solutions satisfying these conditions and the corresponding values of x are rrnrx . mry 2 (WsY (»A2 T(xj) = sin — sin — , x. = {—) + [j) , where m$ = 1,2,3, - - •. To each pair of nonvanishing integers there Corresponds a definite field pattern and a definite propagation constant. The cut-off wavelength of the " TMm,„-wave " is 2tt Xm,n 2ab 1 (5-1) + b2 The longest wavelength corresponds to m = n - 1 2ab V> + b2 (5-2) 'Thus the cut-off wavelength for the dominant transverse magnetic wave is twice the distance from a vertex of the cross-section to the opposite 388 ELE( TU< (MAGNETIC WAVES l ItAP diagonal (Fig. 10.2). A typical higher mode is the dominant inodit guide with dimensions a/m and b/n. The cut-off frequencies air mv proportional to the corresponding wavelengths; hence these freipi are proportional to the radii from the .....[i points (na,mb). The attenuation constant < tained from the general formula >,f i he |...... section; thus 2&(m2b3 + n2aa). r,ab(m2b2 + n2a2) d-^.n)"1'8. ( Fig. 10.2. Relation of the cutoff wavelength for the dominant trans- For wave guides of square cross-section this he. . ... verse magnetic wave (T.Wi.i-wave) to the _ 2Ji 2 \—1/2 dimensions of the guide. a „„ V vm,n) rju The field of a TMm>n-wave is then obtained in the form Hx (S mr . mirx niry ~ — sin--cos —- rm,„(z), o a b r//„= - mir mirx . niry * — cos-sin —— Tm,n (z), a a b Ex _1_dtn. Xm.n ■ miTX . —■—;— sin —— sin g + 2ctie a Tm.n — A, 1 m,nz b 'xLn + °2. In the case of transverse electric waves dT/dx must vanish at x = and x = a and dT/dy must vanish at y = 0 and y = ^; therefore mirx niry I (x>y) = cos-cos —— , where ?w and n are integers, not equal to zero simultaneously. Hence tie cut-off frequencies are given by (1), but since either m or n may be zero, there exist more transverse electric modes than transverse magnetic. The cut-off wavelengths of the additional modes are 2a m 2b n (5-5) For these modes the electric lines are parallel to one of the faces of the wave guide. F rom the general formula the attenuation constant of trans- WAVES, WAVE GUIDES, AND RESONATORS - 2 j. rie electric waves is found to be vblp2m 2 + »2 Vl-iiBJ q \b a / -(- + 7.0(1 « = 0, **0, r; \« ^ / |H here p = b/a. In square tubes this reduces to 389 (5-6) 29i(l + ,2) .f , n 9^(1 + 2"3) •, „ . -,. , if m, n =j= 0; a = -, , it tn nay/[ - v2 1 - " For the field of a TEm>n-vrave we obtain 0, n = 0. (5-7) nir mirx . niry ~ - cos-sin —— jfm,rii,z), b mr . mirx mry - Ev = — — sin -cos —r" Im.nV-), a a b 1 dTm,n ioipTm,n dz ' Hx — ~Ym,nEy> 7/„ ~ ^m,nEx, Ym,n z rr xl.n mirx niry - //, = - COS - COS —Y 1 m,« (Z) ■ soop a b 10.6. Natural Waves in Circular Wave Guides In a circular wave guide solutions of (1-11) in cylindrical coordinates are T(p, where a is the radius of the tube. If x« = *n,« is the Wth nonvanishing root of this equation, then the values of kn,m for small values of n and m are *0tl = 2.40, *0,a = 5.52, *0,3 = 8.65, *0,4 = J 1.79, • • • jfcM = 3.83, *i,a = 7.02, kU3 - 10.17, *M = 13.32, • • • *2,i = 5.14, *a.a - 8.42, k2,3 = 11.62, A-2,4 = 14.80, • ■ ■ *3>1 = 6.38, £3.2 = 9.76, *3,3 = 13.02, k3A = 16.22, • ■ • (6-2) 390 H.iaTUOMAciNKTIt: WAVES' Ciur. The boundary condition lor ira nsverse electric waves is /,'(x<0 ■ ■ In this case if x<* =■ *n.m is the wth nonvanishing root of this equatfl then the first few k values are *0,1 = 3.83, ^0,2 = 7.02, *0.3 = 10.17, kl.l = 1.84, *1,2 = 5.33, *M = 8.54, = 3.05, ^2,2 = 6.71, £2,3 = 9.97, *3.i = 4.20, h,2 = 8.02, £3,3 = 11.35, (6 ;») The cut-off wavelength is lira (6 I. Inspecting (2) and (3) we find that there is one transverse electric wuvil the T^i.i-wave, which has a lower cut-off frequency than any transverm magnetic wave; this ZjEi^-wave is the dominant wave in a circular wavfl] guide. Substituting from (1) into the general formula for the attenuation constant, we obtain for transverse magnetic waves « = - (1 - ^,m)~1/2. rja Similarly the attenuation constant for transverse electric waves is &( "2 1 , \n 2 v 2 x-l/2 Kk2 - n- For circular electric waves n = 0 and a becomes (6-S) (6-6) (6-7) As the frequency increases the attenuation constant of circular electric waves approaches zero. 10.7. Natural Waves between Coaxial Cylinders When we consider waves between two coaxial cylinders we have no reason for excluding the second Bessel function and the value of T to be considered is T(p,v) = 0, (7-2) WAVES, WAVE (.'HIDES, AND kliSONATORS — 2 391 In ir is the radius of I he inner conductor and b the radius of the outer. I lence _ Q = hM = M**) P NnUu) Nn(xb) I or transverse electric waves the boundary condition is dT(a,P = mr, p = -r, ft = 1,2,3 • • •• (Hi)] In order to satisfy the boundary conditions at the cylindrical surfaces, v must be it root of (7-3) with p in place of n. If a = 0, then fp(xb) = 0. Similarly for transverse electric waves we have T(p,aiop) . (M) r£Jirm0d=, T = yj(U6) 395 Fig 10.7. The ff-hnes for transverse magnetic waves. The density of the lines indicates the amplitudes ol the transverse field components. The transverse component of F is perpendicULir to the /Wines. The numbers above the figures represent the cut-off frequencies in terms of the cut-off lor the transmission mode shown in Fi) + 0.fi3ft(2.4Op)cos 2d the equation of the boundary Ts T = 0; WAVES, WAVE GUIDES, and RESONATORS —2 397 relative directions as shown in the figures correspond to waves moving toward the reader; for waves moving away from the reader the directions ol the longitudinal currents are reversed. 10.9. Slightly Nancircular Wave Guides The effect on wave propagation of a small change in the shape of the cross-section 01 i wave guide can be obtained very simply by " perturbing " the field distribution function T. For example, in considering the effect of a slight compression of a circular wave guide on the propagation of waves with circular lines of force we might assume the following solution of (1-11) T{p,, — = -2uJ2(x) Sin 2tp, ■—■ = 0. Clip Op (9 6) Thus at the surface of the metal tube dT/ds is proportional to « and the first term in (4-17) is nearly proportional to «2. For small departures from perfect circularity the attenuation constant is dominated by the second term of (4—17) in the neighbor hood of the cut-off frequency, and as the tube becomes more nearly circular the second term remains dominant within an increasingly large frequency range. In this range the attenuation constant diminishes with increasing frequency. Substituting from (6) in (4—17), we obtain the following approximate formula for the attenuation constant in a deformed circular wave guide* a = --\?'0 - „=)-"2. V pV-?m . = ^l^=L835A. l/3(3.83)f Minimizing a with respect to the frequency, we find that, to the same order of approximation, the minimum occurs when v = Vp/3 = 0.782A. If for example the largest * The coefficient of the first term in the brackets is also affected by the departure from perfect circularity and a term proportional to A" should be added to unity. However, this term is of less importance than the correction term involving p. WAVES, WAVE GUIDES, AND RESONATORS 2 399 illiiincter exceeds the smallest by one per cent, the minimum occurs at a frequency «lín li is 128 times ns great as the cut-off. tO. 10. Transverse Magnetic Spherical Waves The theory of spherical waves is in all respects similar to the theory of plane waves. Thus there exist transverse magnetic waves for which I /, 0 and transverse electric waves for which Er = 0. In the former iiiHC the divergence equation for H becomes 4 (sin 0 Hi) + S = 0; 30 dtp (10-1) .....sequently the magnetic intensity may be expressed in terms of a stream function II i lence we have ön TJ dJl rsinö//^ — , rH¥= - - H = curl A, Ar = n, A9 ■ = - 0. (10-2) (10-3) Since the radial magnetic current vanishes, the transverse electric intensity may be expressed as the gradient of a potential function; thus dV dV In addition to (1) we have the following field equations rE0 = 1 d(rHv) g + icoe dr rE„ = 1 d(rHe) * g + iox dr d (sin d Hu) dHo . . . ■ n e -1-^---= (g + io>t)r sin 0 Er dd dip dEr . . d{rEp) . . „ —- — sin 0-— = —iojp.r sin 0 He, dip dr dr dd Substituting from (2) and (4) in (5), we obtain an dr = - (g + iose)V. (10-4) (10-5) (10-6) (10-7) (10-8) (10-9) •KM) ELECTKi uviacnetic waves The radinl elect ric intensity may he , ,1)1 aims] iVi.in (,,) mi i l,r and from (7) or (H) on the oilier; thus I (ff + UM)t •2 Lsin 9 30 \Sin tW/ + sin2 0 VJ' £r = -to^n - —- = , , (—T - a2n j. or g + toe \ 3^ / ClIAC. lffl our li in, (10 10)1 Comparing, we have 2 a2n , i 3 /. an\ l a2n The complete field has been expressed in terms of a single scalar function II. In the narrow sense of the term equation (11) for II is not a wavi equation; that is, it is not equation (1-5-) in spherical coordinates. We shall now consider fields for which H(r,9, and n\n -f- ij k{n -\- 1) WAVES, WAVE (.-hides, and kesoNatous 2 101 whose distributed shunt constants per unit length are a capacitance 6 and a conductance g. Sufficiently far from the origin the series capacitance and conductance are very large. When H is of the form (12), the transverse field intensities are aT A r sin 0 He = — T, rH9 = 3tp 39 Ť, (10-17) (10-18) Eq = Z*HV, E

t)Ťdr ' In terms of the functions defined in section 3.5, we have Ť « Aln{?r) + Bkn{or). (10-19) b'or progressive waves traveling outward we must have A — 0 since 2 becomes infinite at r = oo. For stationary waves having no singularity at the origin we must have 5 = 0. Thus the radial impedances for progressive waves and for stationary waves become respectively KJ = — ij-xry-r;, KT = n In nondissipative media the latter becomes Tn(or) K- = MM (10-20) (10-21) The T-function is a spherical harmonic and is briefly discussed in section 3.6; in general T is either a series of terms of the following type T(8#>) = [CP£(c.os e) + DF:(- cos 0)](P cos m

n, P™ vanishes; hence we may assume m < n. 402 KUÍCrJíOMAlíNimc WAVES Cil ai'. 10 The following are the Legendre functions of low order P0(cos 0) = 1, Pj(cos 0) = cos 0, PÍCcos 0) = - sin 0, i(3 cos20 - 1), P2(cos 0) = -3 sin 0 cos „ \ = - (g + mm - -T- = r- ( tt - * * J • The stream function * satisfies equation (10-11). ELECTS (MAGNETIC WAVES Chap, I0„ Whan * - T(0,v)t(r), the field pattern in spherical surfaces concentii with the origin is independent of r. In this ease the expressions foi ih< field become r sin d Eg = — — T, dip Bg = ~.Y+EV> n(n 4- 1} 77, dT de 1 ^ ;uju7 dr ' "77 > (II- m = lOlpT *, iuy,r2HT =» »(» -)- 1)*. Tand f satisfy equations (10-13) and (10-14). The magnetic potential /' and the electric stream function ^ satisfy 5* • rr — - — *«/xt7, (f 4- toe) 4- iwfir (H-3) Thus * and 17 vary with r as the voltage and current in a transmission line with series inductance ju, shunt conductance g, shunt capacitance «, and shunt inductance ^ + ■ all per unit length. Sufficiently far from thi origin, the shunt inductance becomes very large. The radial impedances for progressive waves and for stationary waves without singularities at the origin are respectively K: fnXo-r) mm* "r "rn{*r) In nondissipative media the second impedance becomes (11-4) (11-5) Comparing with (10-20) and (10-21), we find that the product of the corresponding impedances for transverse electric and transverse magnetic waves is equal to the square of the intrinsic impedance. The T-function is, in general, a series of terms of the form (10-22) or it may be expressed in the form of an integral. In free space m and n are integers and m < n. In a region 0 < 6 < \b bounded by a perfectly conducting cone 6 = % E^ and therefore dT/dd must vanish on the boundary; thus n must be a root of the following equation % P"'(cos W (11-6) WAVES, WAVE GUIDES, AND RESONATORS 10', Eor a region between two pet lia tly ■ oiulm ling cones 0 = G\ and 6 = H> u must satisfy d0\ _ dO'2_ — p:(- cos «y j- p:(- cos e2) ad\ at)2 (H-7) If perfectly conducting half-planes

e dp dz (g + iu>e)p dtp dz Ez= - r2n_ g 4- ioie (13-6) a > - —. In general E and H are elliptically polarized. The plane of the //-ellipse is always perpendicular to the axis of the wave but the plane of orientation of the £-ellipse varies from point to point; thus the wave may be called a " magnetically oriented wave." In nondissipative media T is either real or imaginary and consequently Ev and Ex are in phase. In this case the plane of the £-ellipse passes through the wave normal. In free space q must be an integer; but inside a wedge formed by two perfectly conducting half planes

»*-- these boundaries and consequently ' ^ Sl,0uld Vi,ni-1, " »7T 0 = 0, ? = —, „ = 0,1,2,.... (13 mi In free space £ may assume any real or imaginary value; but in a region boumle, I I perfectly conducting planes 2=0 and z = h,T must vanish on the boundaries and therefore tmir C=0, m= 1,2,3,.... (13-11) In the case of circular symmetry T( til waves T(p) = ^/B(rP) + Mm), T(z) = c cos $ + D sin f2. (13-12) The radial impedances for outward bound progressive waves and for stationary waves having no singularity on the axis are rwp) r/„(rp) ' (ir + ^A'^rp)' te+/«e)/,(rp)- UJ 1JJ The field of transverse electric waves is obtained from (9); thus dp im ' ' iwßdP~dz~> TKo(Tp) If T is in the form r/o(rP) (13-14) (13-15) WAVES, WAVE (JUIDES, AND RESONATORS 2 409 DM the axis of the v\ lives, except in the neighborhood of the axis itself where the cone in " blunted." Thür, if the medium is nondissipativc, and if Vp is large we have ff, 'VI' (13-16) When £ > 8, this function determines a wave traveling parallel to the z-axis; but when £ < 8> we write (16) as 1 ff' The cone of constant phase is pV/32 - £2 + £z = constant; the wave normals make an angle t? = tan" with the z-axis, and £ = ß cos t?, V- £2 = /3 sin (13-17) (13-18) (13-19) (13-20) Near p = 0, the phase of Ko(Tp) is nearly independent of p and the equiphase surfaces are normal to the z-axis. One exception is the case in which £ = 0; then if = 90°, the waves are traveling radially and the equiphase surfaces are cylinders coaxial with the z-axis. 10.14. Circulating Waves A wave is circulating if its equiphase surfaces are half-planes issuing from an axis called the axis of circulation. Such waves are possible only in nondissipative media. The field of magnetically oriented circulating waves may be obtained from the following stream function n = UA(xp) 4- BNq(xp)\(Ccos &+D sin &)&*>, x2 + S2 = 02- (14-1) This function is of the same general form as that given by (13-4); but T(p) must be real and it must satisfy certain boundary conditions. The form (1) is more suitable for the latter purpose than (13^1). Similarly the field of electrically oriented circulating waves is obtained from a potential which is of the same form as (1). Except when q is an integer, circulating waves can exist only within some ^-interval {tf>o,14.1.1 r■ins ,iikI wr iinisl icsiul in approximate methods. 10.15. Relations between Plane, Cylindrical, and Spherical Waves From the mathematical point of view the difference between pi,urn, cylindrical, and spherical waves is merely the difference in the coordinate used in the expressions for the wave functions. By transforming conn I i nates it should be possible to express a spherical wave function in term of cylindrical wave functions or in terms of plane wave functions. I In choice of the particular coordinate system and the particular type of win i function is dictated by the nature of the source and of the boundaries di viding the medium into homogeneous regions. Each current element in a homogeneous medium emits a spherical wave; hence if we know the complete current distribution it is natural to regard the field as the resultant of spherical waves emitted by the current elements. On the other hand, if an element is in a medium consisting of two homogeneous semi-infinite media, separated by a plane interface, we express the spherical wave emit led by the element in terms of plane waves " moving " toward the plane bourn I ary and use the relatively simple formulae for the reflection and trans mission coefficients of such waves in order to obtain the field of the element as modified by the change in the environment. Similarly if a plane wave is striking a spherical obstacle, we express it as a resultant of spherical waves incident on the obstacle and use the formulae for the reflection and transmission coefficients of spherical waves at a spherical boundary. In either case the reflection and transmission coefficients depend on the ini pedance ratio for a typical elementary wave. At times a mathematical transformation of a wave function throws a new light on a physical phenomenon. In section 8.21 we have considered dominant waves in a rectangular wave guide. Above the cut-off these waves travel along the guide with a velocity greater than that of light. This high phase velocity and the filterlike characteristics of the guide are the properties of all transmission lines having a shunt inductance in parallel with the shunt capacitance. On the other hand if we express the sines and the cosines in (8.21-15) in terms of exponential functions we obtain Ev = -\iE\e — e ■i[(tt/aM-/3z] I, ß = yjl: and corresponding expressions for the //-components. The amplitudes of the individual terms are independent of the coordinates; hence these terms represent uniform plane waves. The sum of the squares of the coefficients of x and z in the exponents equals B2 and a real angle C can be found to satisfy ji = 8 cos C, ir/a = B sin C. Comparing with the equations of section 4.10, we find that our equations represent two uniform WAVES, WAVE GUIDES, AND RESONATORS 2 ill plane waves moving in directions making ibe following angles with the i oordinate axes A = I + C, B = |, C = C; and A =\- C, B = ?, C=C. For each wave the corresponding total magnetic vector is perpendicular to i he electric vector and the ratio of the intensities is ??. Either wave is obtained from the other by reflection at one of the faces of the wave guide (Fig. 10.10). This picture also helps to explain why the velocity in the direction of the guide is higher than the velocity of uniform plane waves in free pace. The directions in which the uniform components are moving depend on the frequency. When the frequency is sufficiently high B is nearly equal to B Flo. 10.10. Directions of uniform plane and C is nearly zero; then the waves are waves mto which euided waves in . , ,. r , rectangular tubes may be resolved. moving almost in the direction or the guide. At the cut-off (3 = 0 and C = 90°; in this case the waves cease to advance in the direction of the guide. Below the cut-off this picture fails idtogether. If instead of dominant waves we consider waves whose electric intensity varies as sin rnrx/a, we obtain the following expression for the angle C _ nir n\ sin C = — = — • pa 2a If X is small compared with 2a, there will be several real values of C satisfying this equation, and these will be the " permissible " angles for uniform plane waves between two parallel perfectly conducting planes. Each permissible angle corresponds to a particular transmission mode. Even inside circular wave guides it is possible to express guided waves above the cutoff as bundles of plane waves repeatedly reflected from the cylindrical boundary. Consider for instance a bundle of Cone of directions of elementary uni form plane waves. Fro, 10.11 I------- "■■ hlLl^L ---- ~ " " • u u -1U to the *v-plane and let the wave uniform plane waves with H parallel to thexy■ J» •112 ELECTROMAGNETIC WAVES (iui' 10 typical elementary wave i.s A(>f>) dip, tlien Et - sin Cexp( — i/Szcos C) j A(, •TO (154 Since * cos ^ + J sin

: 2iir2(g + icoe) J id 7 I /*__., 7/ Kft(Tp)e'"dy, (15-4) f r sinh^ Kx(Tp)eV jy. J ten 2 M II I ( IK' (MAGNETIC WAVES" If /is infinitely small, t lie above expressions become ('mac. 10 E, = -- // f T2K0(Tp)e-" dy, to (15 £ =__ 4*V2(x-f toe) J(C) — f yTKippW dy, toe) J (C) •(C) On the other hand, using spherical coordinates, the field of the current, elem ;nt may be derived from II = Az in (6.2-7). Comparing with (5), wc have e~'r 1 C /- - =- / K0(pV, 2' and T is again positive real. In the case of nondissipative media I" is either real or imaginary on (C) except on the two infinitely small indentations;* it is real outside the interval 7 = —ifi, y => i0 and imaginary inside it. When r is real, the * There is no singularity at the origin and therefore (C) need not be indented there. WAVES, WAVE GUIDES, AND RESONATORS—2 115 equiphase surfaces of the elementary waves are normal to the z-axis and the waves are plane; when V is imaginary, the equiphase surfaces are Ini lined to the z-axis (and curved near it) and the waves are conical. "i-plane Fig. 10.13. Fio. 10.14. Separating the integral into two parts, one taken along the positive and the other along the negative imaginary axis, and reversing the sign of the variable of integration in the second part, we obtain — - I Kq(pV- f - 7*)eosh yzdy = - K0 (pvf - tf) cos ^ (15-7) The second integral is taken along the positive real axis indented above £ = /3. The equiphase surfaces of elementary waves are cylinders coaxial with the z-axis. The above expressions are valid either for positive or negative values of z. If z > 0, the exponential function in the integrand vanishes on the circle of infinite radius in the second quadrant and (C) can be deformed into the contour (Ci) shown in Fig. 10.15; on the negative real axis ph T = ir/2. The exponential function vanishes also in the third quadrant and (Ci) can be deformed into (C2) of Fig. 10.16. On the lower side of the negative real axis ph V = — r/2. Between 7 = 0 and y = —ij3, ph T = x/2 on the right side of the contour and ph T = — ir/2 on the left side. By (3.4-11) we have - K0(p^/^^^72) = -N0(p^+7) ~//o(p^32 + 7*) , ph T = \ > it z = -n0 (p^2+72)+ijo(p^T?) , phr - - (15-8) where the arguments of the /0 and 7Y0 functions are real on (C2) excepting the infinitely small circle round y = —ifi. The integral round this circle IW, ELECTROMAGNETIC WAVES • Chap. 10 is zero because: the A',, function becomes infinite as log | 7 -f ifi | while the length of the circumference vanishes as | 7 4- 18 |. On the two halves ol {Ci), 0, (15-9) (c3i where (Cy) is the positive real axis indented above x — & (Fig- 10.17). Unlike (6) this integral converges when p = 0; it converges also when Z = 0 as long as p 4= 0. In this representation all elementary waves are plane waves traveling in the positive z-direction. When x > B, the propagation constant in the z-direction has all real values between zero and infinity; when x < 8, the propaga-Fig. 10.17. tion constant is imaginary and the phase velocity assumes all values in the interval (t>, °°) where v is the characteristic velocity. This second group of waves can be expressed as a group of waves traveling with constant velocity v but in different directions; we need only express Joixp) as an integral of type (1). When z < 0, we have ~0r x dx £_ _ f r * (cs> Mxpyv^>2 Vx2 - ß2 = T f K0 (pV-8* _ yjS ey*dy> (15-10) WAVES, WAVE GUIDES, AND RESONATORS—2 417 where the contour of integration (C4) is shown in Fig. 10.18. On the real axis the elementary waves are plane and their propagation constants are leal; on the imaginary axis above 7 = 18, V is M al and the waves tire plane, traveling in the negative z-direction with velocities smaller than the 1 haracteristic velocity; in the region between 7 = 0 and 7 = i@, T and 7 are imaginary, with the sum of their squares equal to — B2, and the waves are conical, traveling with velocity fin directions making acute angles with the negative z-axis. iß CC4) Fig. 10.18. 10.16. Waves on an Infinitely Long Wire Consider an infinitely long cylindrical wire of radius a and assume that an electromotive force V is applied uniformly over a section of length s; then, assuming that the wire is along the z-axis, (16-1) V s s EM = - , - 2 < z < 2 ; outside the interval the applied intensity is zero. Representing this impulse function as a contour integral we have v f sinh? tTTSJlC) 7 '(C) t Consequentiy the magnetic intensity at the surface of the wire is (16-2) / sinh ~ _T / _2_ iirsJ (C]y[KJ(a) + ) + K-{a)\ dy. (16-3) In order to obtain the magnetic intensity at any distance p from the axis of the wire we multiply each element of the integrand in the above expression by the //-transfer ratio; thus vi _t sinh^XiCrp)«*« a) + K-ia^KiiTa) dy. (16-4) The corresponding Et is obtained if we multiply the integrand by —Kf(p)\ hence / EM ~ i*s(g + im)/orlKfia) + K~[a)\K,(To) H' (1^S) rsinhy K0(Tp)e^ •UK EI.Et TU< MAGNETIC WAVES ' Chac, 10 In tin- special case when the wire is a perfect conductor ami the medium is nondissipati ve these equations reduce to Ez{p,z) = - — I. sinh^ K0(TP)eT ft dy, r = V-y _ ft rs »/ «v (16 6) sinh^XiCW The current 7(z) in the wire is then 7(z) sinh^ Kl{Ya)eyz Mi ten yVK0(Ya) ■ dy. (16-7) If the input admittance Y; is defined as the ratio of the current I(s/2) through the upper terminal of the " generator " to the applied voltage, then Yi = 2wea j - sinh y Ki{Ya) s J(c) yYKG(Ya) dy. (16-8) If j" is small, the real part of the input admittance is nearly independent of s; but the reactive part of the admittance is positive and approaches infinity as j approaches zero. This is natural since the capacitance between two infinitely close cylindrical wires must be infinite. For thin wires the above integrals yield the results of section 8.13. 10.17. Waves on Coaxial Conductors Consider now two perfectly conducting coaxial cylinders and assume that an electromotive force Va, defined by equation (16-1), is applied to the inner cylinder. The longitudinal electric intensity is then the negative of the impressed intensity (16-2) and Ez(a,z) - - ■4 rs J in sinh ys nrs *s (C) y Between the coaxial cylinders we have ■ if* dy. (17-1) E*(p,z) = fits J (C) y ys 2 (17-2) WAVES, WAVE. CHIMES. AND RESONATORS sinh -fr xn,(a,p) <*■ dy, 119 (17-2) I sinh — XeM'P) E^>z) = i?\Jv)Kt~yWpT~ if* dy. Por each elementary wave we may assume Ez = [AMxp) + BNo{xpW, x2 - y2 = P\ x (17-3) EP = [AJdxp) + BNitxpW- The longitudinal impedance of a typical wave is then A'2+ - - y/^e. Since E,(b) = 0, we have A = PAW), B = -P/o(x*); hence XuMyp) = } zt(p) = - Aro(x*)/o(xp) - Mxfi)N0(xp) JVo(x*)/o(x«) - /o(x^)ATo(x«) ' yJNo(x&)Mxp) ~ /o(xiVV (xp)1 (17-4) MJVo(x*)/i(xp) - /o(x*)A .) x«?(x«»x*) S(xp,x*) = /n(xp)A7o(x/^) ~ tVoMMx*), U(xp,xt>) = N1(XP)Mxb" - /i(xp)AW). The transverse voltage V(z) is obtained by integrating Ep along a radius and the longitudinal current 7(z) in the inner cylinder by multiplying 420 IT it n« (MAGNETIC WAVES i Hoi' //„(«) by 2ir«; thus wc have FW = - / dy, I{z) = SOU f sinh y U{xa>xl>) The first of these integrals can be calculated at once by the method n| residues. Thus if z> s/2, (C) can be closed in the left half of the plum The pointy = —i3 is the only pole enclosed and the value of the in tegi 8s 8s 2 2>2' Thus the transverse voltage is given by the principal wave alone. To evaluate I(z) we note that the integrand is a single-valued function of y and consequently we may again use the method of residues. Phi poles of the integrand are the roots of S{Xa,xb) = 0 or Z+(«) =0, (17 8) and thus represent the natural propagation constants. The contrast be tvveen this case and the case of an isolated wire is worth noting. In tht latter case the radial impedance function is multiple-valued and has no poles. If Xn is the »th root of (8), then 7,, = ±Tn> F„ = v'xn — B*> When the frequency is such that 8 < xi, all these poles are on the real axis. As the frequency increases, some poles move to the imaginary axis. In addition there are two poles 70 = dtid which are always on the imaginary axis. Evaluating I(z) for z > s/2, we have It*) = z In(z), In(z) = n =0 MoxaFo sinh -— U(Xna,xJ) .-Tni dS dXn IaW - —jT, # = — log-. A 2x a (17 S Í When 8 < Xi, all current waves except the dominant wave IQ(z) are attenuated. The impedance to the dominant wave is a pure resistance; the additional current waves add a reactance in parallel with this resistance. If we define the input admittance as the ratio of I(s/2) to the applied WAVES, WAVE (.Mihl:,, AND RESONATORS— 2 -111 voltage Vq, then we have . 6s sin , I 2 In y,'"„?oy'" Y°~2K (is „->'ft»/2 , Yn (17-10) When j is vanishingly small, the resistive component of the input admit-tance is 1/2AT. In this case, however, the reactance in parallel with 2A" is infinitely large; an infinitely small gap provides an infinitely large capacitance and effectively short-circuits the gap. Nevertheless the capacitance approaches infinity so slowly that for the " small gap " encountered in practice the capacitance is usually small. If the applied electromotive force were not distributed uniformly round the circumference of the inner conductor, all possible natural waves might be generated in the region between the two coaxial conductors. A study of the roots of the equations in section 10.7 indicates that if the wavelength is greater than the circumference of the outer conductor, all natural waves, except the principal, are attenuated. 10.18. Waves on Parallel Wires Consider now two thin perfectly conducting parallel wires, with their axes separated by distance /, and let the impressed voltage Vn be distributed along wire 1 in accordance with equations (16-1) and (16-2). The longitudinal electric intensities due to the currents in each wire are respectively £*(pi,z)= f J(y)K0(Tpi)e^dy, £,(p2,z) = f B(y)Ka{XPi)e^ dy, (18-1) where p\ and ps are the distances from the axes of the wires. At the surface of each wire the boundary conditions are ys V0 sinh — A{y)Ka(Ta) + B(y)K»{Tl) = - trsy A{y)Ka(Tl) + B(y)K0(Ta) = 0. Taking one half of the sum and of the difference of these equations, we have (18-2) \[A{y) + B(y)\ = - yt Vq sinh — \{A{y) - B(y)] 2Í7rsy[Ka(Ta) + K9(Tl)] sinh — 2iTrsy[Kn(Vl) - Kt(Ta)] = = C(t), DM. (18-3) ELI t I l\< (magnetic waves ( IIA.I' III Adding iiiul suhtriii ting we obtain //(7) ('(y) + IHy), /%) - C(y) - D(y). (18-4)1 Hence the total longitudinal intensity may be expressed as E, =» E, + P.g, where E,= f C{y)[K0(rPi) + A'u(I>,)k*' s/2, we deform (C) into (C2)ofFig. 10.16. The contribution from the infinitely small circle (Co) round y = —ifi is then h{z) . 8s r0 sin ~2 2K 8s 2 —ifiz The quantity K is seen to be the characteristic impedance of two parallel wires to a transverse electromagnetic plane wave moving parallel to the wires. As the length of the section over which the impressed voltage is distributed approaches zero, the second factor in the current formula approaches unity. WAVES, WAVE GUIDES, an 11 RESONATORS— 2 423 The integral over the straight portions ,,l the contour (C.) may be tram.....in.si with the aid of (15-8) and (.VI II) tn obtain the following expressions ITS J -at 0 8Ínhf 1 + jirálfVi(ž)/ + y\ I i(z) is in quadrature with Va for all values of z and hence represents a stationary wave; Ii(s/2) is positive imaginary and the corresponding part of the input admittance is positive and tends to infinity as s approaches zero. I?(z) represents a group of progressive current waves moving with velocities greater dian the characteristic velocity; ^2(2) approaches a limit as s approaches zero and /•.;(()) is in phase with the applied voltage Vn- consequently the corresponding admittance is a conductance. Thus I he total input admittance of a parallel pair energized symmetrically in push-pull consists of three admittances in parallel as shown in Irig. 10.19. The first admittance is the admittance to the dominant transmission mode ol the two halves of the parallel wire transmission line connected in scries and it accounts for the power guided by the line. The second term is the capacitance representing the stationary field in the vicinity of the generator; in this local field the electric lines connect the two halves of each wire on the opposite sides of the generators. The third term is the radiation conductance; it accounts for power radiated in directions other than the direction parallel to the wires. In order to prove the last statement we should show that ^2(2) approaches zero as z increases. Fig. 10.19. Admittance seen by a generator in series with a parallel pair; k is the characteristic impedance to the principal wave, Gs is the radiation conductance, and b\ is the capacilive susceptance representing the local reactive field. 10.19. Forced Waves in Metal Tubes Consider a perfectly conducting cylinder of radius a, coaxial with the z-axis. Let the source be an infinitely thin line source of finite length, parallel to the axis of the tube and passing through the point (£,0) in the equatorial plane (Fig. 10.20). If the source consists of electric current elements, the longitudinal electric intensity will be of the following form Et= f A{y)K,(iXp)e-» dy, ** * 7* + 0*, C19"1) where p is the distance from the axis of the source and x is the radial phase constant. Thus in the case of an infinitely short current element in nondissipative dielectric, we have A(y) = —//xV^coctt2. In the case of an infinitely thin antenna energized at 424 I i it IK( (MAGNETIC WAV l..% Chap. 10 the center we ohtnin from (V.2S 14) unci (15 6) (assuming the center at z - 0) 30/ A(y)--(2 cos 01 - «*' - r*1), 7T An antenna of finite radius may be regarded as consisting of infinitely thin filaments and the corresponding expressions for A(y) can easily be calculated. Fig. 10.20. Cross-section of a metal tube and a line source inside it. In the region p > b the ^Co-function in (1) can be expressed in terms of the coordinates p and

(, ELECTROMAGNETIC WAVES Kmni (1) wc obtain the p components of the field Ciur. 1(1 sm up, (20 .1' [«r iuii\ , ■i - jjxp) -i- « y»(xp) x-p X J //„ = - j^tf ^ /»(XP) + B /»(XP) J cos we-i . nV /'o)6o , «P ^— J'JX") + S— JM = -C ~- KL(.ka) - D -r Kn(ka). X X« * * a To solve these equations we write A=SKn{ka), C=SJn(Xa), B = TKn{ka), D = TJn(xa), (20-5) and substitute in (4) to obtain Eliminating i1 and T, and letting xa = P, ka = q, we have tmj'nip) (wj + jUp)K'*(q) , m*K'n-(q) (20 (,) PqJn(p)Kn(q) 12Kl(q) Since P2J«2(P) = -P*jn-l(p)j»+l(p) + »V5(*0, q2K2(g) = ?2Jv-n_,(?)A'„+i(?) + rPKUq), (20-7) equation(7) becomes PltlJn-l(p)Jn+l(p) 0*1*2 + jU2«l)/n(p)^n(?) P-2*2Kn-l (?)ff«+l(g) *7S(P) PqJ«(p)Kn(q) q'KUq) , Ml*i + P2*2 (20-8) WAVES, WAVE GUIDES, AND l\l\s( >NAT< >us 2 427 ''nun this equation wc obtain graphically or numerically pairs of values p, q in terms il which to ami 1' are given by P-i*i Equation (8) has only real roots. When q = 0, we have « = —/—r = */92. (20-9) (20-10) 'The wave is traveling with the velocity characteristic of the medium external to the wire. The frequency given by the above expression is the lowest frequency for which there exist plane waves (with a particular field configuration) traveling in the direction of the wire; for this reason it may be called the cut-off frequency. When q becomes infinite, the frequency is also infinite and the wave is traveling with the velocity characteristic of the wire. For large values of q the field outside the wire varies exponentially with the distance from the surface; thus the field is confined to a relatively thin film of the external medium and is largely inside the wire. In order to obtain the cut-off frequencies for various transmission modes we let q approach zero; thus for n > 1 we obtain , , ,Pjn-l(p) 0*1*2 + P-2*l) -r / \ = "(6l 62) 0*2 - Ml) + *2P-2 n-V In the special case p.i = pi, this equation becomes Jn~i(p) *2 pjn(p) (k - l)(tl+ e2) * If the dielectric constant of the guide is very much higher than that of the surrounding medium, the first few roots of the above equation are in the vicinities of the zeros of /n_i(p). As q increases indefinitely equation (8) becomes J„-l(p)Jn+l(p) jl(P) 0. (20-11) Thus in the limit the roots of (8) are exactly equal to the zeros of /„_i(p). In other words as q varies from 0 to =0 p does not change much in any assigned transmission mode. From (11) it might appear that the limiting values of p could be the zeros of /«+i(p); but this is impossible because in the process of transition p would have to pass through the intermediate zero of Jn(p) and no value of q is consistent with such zero. The case n = 1 requires special examination; in this case equation (8) may be written PieiJ0(p)J-z(p) 0*1*2 + /*2*i)/i(/>) p'Jiip) + p'Jiip) [1 K0(q)l L?2+fab)} _ p&iKaiq) yKa{q) ^ 2K'i( x K = -i f H*, = -m f - Ato)Kx{fxp)S» dy, J foil x (21-1) WAVES, wave GUIDES, and RESONATORS 2 429 where the contour (C'r,) is the one shown in Fig. 10.22 and is obtained from {(.\) of Fig. 10.18 in the same way as (C2) of Fig. 10.16 was obtained from (Ci) of Fig. 10.15. This contour may be used only for values of z such that z — h < 0, where ft is the height of a typical current element of the source, and it can be transformed into (C3) of Fig. 10.17 which represents an elementary spherical wave function as a group of plane waves traveling downward with different propagation constants. The wave impedance in the z-direction of each elementary wave is K = y/iiae. Hence if the impedance looking into the plate is 2, the reflection coefficients for the various field intensities and the reflected field are 0Lf (Cs) Fig. 10.22. tn = ?e, = q, 4ep = — q, q K — Z y — íwíZ K + Z ~~ y + iuxŽ £1 = f qA(y)K»{ixp)e-^dy, Erp X, (21-2) q^J(y)Kí(iXp)e-y'dy, > X Ha — (C„) X A(y)Ki(iXP)e-^ dy. In the dielectric plate the propagation constant 7 and the wave impedance in the z-direction are given by 7 = £2 (21-3) Hence the impedance Z normal to the plate is = ?/Ccosh 7/+£ sinh 7/ £ cosh 7/+ # sinh 7/' 1 ; where / is the thickness of the plate. The natural waves in the dielectric plate and the poles of the integrands in (2) are the roots of q = co, K + Z = 0. (21-5) Substituting from (4), we have K cosh 7/ + K sinh 7/ K , ,, 2KK K cosh yl + K sinh yl ~~ K1 7 ~ K2 (21-6) In order to solve these equations we make the following substitutions yl = il^fi2 - x2 = ip, yl = /v^ 4.10 Ei.ia;TROMAc;Nimc WAVES Cil a l>. í0 WAVES, WAVE CHIDES, AND RESONATORS 2 431 Substituting in (6) and solving we have tan p Pi = -p tan -, e 2 Pa = - : P cot - , (21-7) Both p and p are positive, and for each such pair of values satisfying (7), we have the following values of the frequency and the radial phase constant /ie — fit 12 ,p2 w. = yjune + -p=^u»e--p Natural waves in dielectric plates are cylindrical and in nondissipative plates the amplitudes of their field intensities vary Inversely as the square root of the distance. On the other hand, in free space, waves generated by sources in a finite region are spherical and the amplitudes of these waves vary inversely as the distance. Some energy is abstracted from the sources and guided in directions parallel to the plate. The plate acts analogously to parallel wires or dielectric wires which tend to guide waves by converting portions of spherical waves into plane waves. There is this difference between plane waves In free space and those in presence of wave guides; the former carry finite power per unit area (of an equiphase plane) and infinite total power, while the latter carry finite total power. Similarly, cylindrical waves in free space carry finite power per unit length (along the axis of the waves) and infinite total power, while in the presence of dielectric plates there may exist cylindrical waves carrying finite total power. Hence in free space no system of sources in a finite region can possibly generate either plane or cylindrical waves but they may do so in presence of ' wave guides." Cylindrical guided waves conform to the physical idea of surface waves introduced by Zenneck, Sommerfeld, and other early writers on this subject. In the case of a single interface between two semi-infinite media we have Z = K and the condition (5) for the existence of natural waves becomes K+K = 0. (21-8) For transverse magnetic waves the characteristic wave impedance is either a resistance or a negative reactance; hence, this equation can have no roots, the integrands can have no poles, and there are no surface waves. This conclusion is contrary to that reached by early writers on the subject. Inadvertently the condition for matching impedances was substituted for the equation of natural waves. The impedance concept has only i' ently been introduced into held theory and no intuitive check on formal manipulations was previously available, Equation (8) for the poles was usually written in its explicit form Vx2 + &2 0, and then was rationalized and solved 4(tit — Vx2 + S-3 instead of However this value of x is a solution of K — R. = 0 and not of equation (8). 10.22. Waves over a Plane- Earth In the actual problem of wave propagation over the earth the conductivity of the earth plays an important role arid the propagation constant y in the direction normal to the ground surface is y (21-3). Assuming an electric current element of moment II at height h above ground (Fig. 10.23) as our source, we find from (15-5) the following expression for A{y) in (21—1) The impedance normal to the ground is g + ÍM r -\~82 f&\ g + ÍWť Flo. 10.23. Electric current element ?■■£ Pi over a plane earth and image element at iV The intrinsic propagation constant a of the ground is greater than that of the air above, especially in the frequency range in which the ground i.. a quasi conductor (see section 4.9). When a is large, then we have approximately Ř = ij over a substantial part of the path of integration. In this case the ground behaves like an impedance sheet whose surface impedance is equal to the intrinsic impedance of the ground. The incident and reflected components of Es in the present case are 3- -A~f (ts + mo(/p^/7TěV(I-*) dy, m = - 4ir3we II 4tt3we *s wo where (C±) is shown in Fig. 1Q.Í8 f M íy2 + B2)Ko(Wy2 + Mtimm ^y, i mine, this contour back to (C<). In fact the following physical consideration show that it would have been permissible to use (C4) to begin with. Con sider E\ along the real axis of (C4); this part of the integral represent cylindrical waves traveling parallel to the ground surface with velocitit less than the characteristic velocity (since the radial phase constant V72 + 02 is greater than 8). Along the imaginary axis from 7 = 0 to 7 = id we may write 7 = 10 cos i?, V72 + B2 = 0 sin 1?, where lies in the interval (0,7r/2). The equation of the equiphase surfaces is then* p cos & + (z — h) sin 6 = constant. The corresponding wave normals make an angle # with the p-lines and an angle tt/2 -j- # with the z-axis. Thus this part of the integral represents a group of conical waves traveling toward the ground with the character!:;! it velocity. Finally between 7 = 10 and 7 = the radial propagation constant is real and we have a group of plane waves traveling normally to the ground with velocities smaller than the characteristic velocity (since y/i > 0). Thus the integral represents the spherical wave as a group of waves traveling either toward the ground or parallel to it. The only question which could be raised against using this group of waves for the purpose of calculating waves reflected from the ground would concern that wave-group which is traveling parallel to the ground. It is reasonable to suppose, however, that the reflection coefficient which applies for waves at neat-grazing incidence would also apply in the limit to waves at grazing incidence. It was to remove the above mentioned objection that contour (C&) was chosen. The original contour (C) is not permissible in the present case since the corresponding integral includes a group of waves traveling from the ground as well as toward it. This care in the choice of the correct contour of integration for a particular problem is essential. Thus for the reflected wave in the present problem the reflection coefficient 5(7) adds two branch points at 7 = ±V — 02 — a-2 to the integrand in addition to those at 7 = ±*/3 already present in the incident wave. The new branch points are in the second and fourth quadrants; and because of the branch point in the fourth quadrant contour (C) would give a different value for the reflected fields * Except in the vicinity of the z-axis where the equiphase surfaces depart from the conical shape. WAVES, WAVE GUIDES, AND RESONATORS —2 433 from the one given by (C«) or (C'r,). If the ground conductivity is permitted to approach /<•.....nc of the branch points of dy, nr=/jf a(y)K0(iPVy2 + 02)e~^+h) dy. 4«T J tpj If the reflection coefficient q(y) were independent of 7, the reflected field would be equal to the field of an image current element of moment qll (Eig. 10.23) rr = 4irr2 where the reflection coefficient q then corresponds to plane waves whose wave normals make the angle # with the ground plane. Consider next the field along some particular line P20 whose equation is (z -f- h) cos t? — p sin # = 0. For the distance r2 measured along this line from P2, we have r2 = (z + h) sin t? + p cos A pencil of elementary waves traveling in directions making an angle 1? with the ground plane corresponds to the values of 7 in the vicinity of 7 = 7 = i0 sin 1?. Let us now expand the reflection coefficient in the following power series q(y) = ?(t) + (7 - y)q(7) + 2(7 — t)V'(t") -i---- and substitute in IIr. Then we have it = nr0 + ni + rr2 + • • ■, q(j0 sin ii)7/g-'gr2 47ir2 ' M f (7 _ 7)K0(/pvV + 02)e^U+h) dy, i J (A) = f (7 - y)2K0{iPVy^T7)^+,l) dL 2 8<7t2 J (c.) The first term K represents the reflected wave that would be obtained from the incident wave by assuming that the latter ts reflected as tf * were •Ill BLECTKl >MA(.W,TIC WAVKS Chap, io a uniform wave traveling in the direction making the angle d with the ground plane. In order to compute the following terms we take 4irr2 wrv^] and differentiate it with respect to z to obtain and Next we evaluate álT/dz and substitute in IIJ an _ dŘ dr\ dz dr2 dz g'jiB sin ů) sin ŮI!e~iĚ>T1 . í/fi # sin ůlle-^ i_ j--— — sin i? — = - ■--- dr2 4irr2 0 + ^)> IT, = 47rrl The amplitude of this term of the reflected IF varies inversely as the square of the distance from the image source while the amplitude of the first term vanes inversely as the first power of the distance. The reflection coefficient is q = K~ & _ (j + ice)? - iweVy2 + s2 + Z2 K+£ Q + iut)y + /o»eV73 + B2 + P " When a- is sufficiently large, then we have approximately ^ = T ~ Koeg = l _ 2/cod) = _ 2T 7 + i'wej) 7 + iutr) y + *we0 ' When Ů is small, is nearly equal to -1. In the ground plane, where f\ = r2, we have then Since ?'(y) &Ž 2/i'we^, we have rrj at —;-- 2x/d)£^rf In this instance the total field is approximately proportional to the height of the source above ground and inversely proportional to the square of the distance from it. However, the total reflected field does not vanish WAVKS, WAV I'', (JlHDKS, AND RESONATORS 4,15 with /;; ll'. contain:; ,i term independent of /; and varies as the cube of the distance from the source. The above expressions for the components of IF" are asymptotic and, taking into consideration the physical picture to which they correspond, it may be expected that they are more suitable for numerical calculations when either the transmitter or the receiver or both are fairly high above pround than when they are both near the earth's surface. A practical rule to follow is to use the above expressions in that range of the variables lor which the first and second terms are much larger than the succeeding terms. 10.23. Wave Propagation between Concentric Spheres The curvature effect on wave propagation has already been considered in section K.23 where several problems concerning waves in wave guides bent into toroids have been formulated exactly and then solved by approximate methods. We shall now consider another problem in which bending takes place in two perpendicular planes. A study of wave propagation between two concentric spheres of nearly equal radii (Fig. 10.24) should give us an indication of the magnitude of the curvature effect on (he propagation of cylindrical waves between parallel planes. We have seen that in the latter case the principal wave is a uniform cylindrical wave for which the electric lines are straight lines normal to the planes and the magnetic lines circles coaxial with the axis of the wave. The field of the corresponding wave between two concentric spheres should then be independent of from which n can be determined. The solution depends on properties of /„ and regarded as functions of n. Let us now return to the equations (4.12-11) for our field. Assuming that Ee is small, we find from the second equation that rHv is approximately independent of r. The total radial current flowing toward the center, 1(6) = —1-xr sin 6HV, is then also approximately independent of r. Introducing 1(0) in the first equation of the set (4.12-11), we have 1 81 r1 80 2 -fl = —Iviat sin 6 ET. Integrating along the radius and introducing the transverse voltage V(0) we obtain f Er dr, (23-9) V a 81 . _ _ lireab sin 0 80--'^ c = -^-> rm where t? is the capacity per radian. If C is integrated from 0 = 0 to 0 = ir, the total capacity between the two spheres is found to agree with the exact value at zero frequency. The second transmission equation is obtained from the third equation of the set (4.12-11); thus introducing 1(0) and integrating from r = a to r = b, we have 8V . ß(b-a) = —tuLI, L = 80 2-K sin 0 (23-10) WAVES, WAVE GUIDES AND RESONATORS —2 437 Replacing the angle 0 by the distance .f =» cO along the circumference of the circle Ol radius c = Vab in the transmission1 equations, we have 8 V 81 — = -mLl,----mCV, 8s ' 8s (23-11) L = P.(b - a) 2wc sin 0 * C = 1-wu sin 0 II Ej is eliminated from the second and third equations of the set (4.12-11), we obtain (23-12) For each transmission mode rHv is proportional to t and therefore satisfies equation (3). Hence (12) becomes 8Er n(n 4- 1' Introducing 1(6) into this equation and into the first equation of the set (4.12-11] we have »,,,, _■ »(»+!) W dOrr> '2irwesin0 ' 80 -lirioie sin 0(r2Er). For any fixed value of r, Er and I and hence V and / satisfy nonuniform transmission line equations. For the principal transmission mode as r approaches infinity the series reactance tends to become an inductance independent of the frequency; for finite values of r this inductance is slightly modified by the longitudinal displacement currents proportional to E». 10.24. Natural Oscillations in Cylindrical Cavity Resonators A cylindrical cavity resonator of arbitrary cross-section may be regarded as a wave guide. Assuming that the generators of the cylindrical boundary are parallel to the z-axis and that the flat faces are Z = 0 and z = /, we may derive the fields of various natural oscillations from the following two wave functions Since IT = T(x,y) cos Zj-, p = 0,1,2, 4> = T(x,y)sm?j, p = 1,2,3,- 2 , A*2_^_3_i!l2 (24-1) (24-2) (24-3) 438 Kl TK< )MA< ."MP IIC WAVKS Chap, io where x I8 the transverse phase constant, we have the following expressions for the natural frequencies and wavelengths 2tt / I 7 P v (24-4) For transverse magnetic waves in a cavity of rectangular cross-section, using T(x,y) as given in section 5, we have . m-KX . nwy pirz U = sin-sin —— cos —— . a b I Hence the natural wavelength of the TMmiIli3,-oscillation mode is 9 (24-5) (24-6) \ a2 + b2+ I2 Similarly the field of the T^a^.p-oscillation mode may be obtained from niTX my . pirz yf = cos-cos —— sin —— . a b I (24-7) The natural wavelength is again given by equation (6). In the present case the designations TM and TE are arbitrary since the same parallel-opipedal resonator can be regarded as a section of three different wave guides and the same oscillation mode may be obtained from transverse magnetic waves in one of these guides and from transverse electric waves in another. In the case of cylindrical resonators of circular cross-section the natural wavelength is given by (4), where x = kn>m/a and kn/i) sin pirz/l; and the current densities in the faces z = 0 and z = I are respectively (_)*+*; ^—■ grad T and — ipir grad T. Hence the power absorbed by the walls is SPh-.Jfc+ JV2y where fa = j f dS, From the above expressions for the stored energy and the dissipated power we can express the 0 of the resonator in a general form. Thus in the case of TM-oscillations we have 0- _x 2X)C«) f f T2dS 4Mb___ifp =i=o, ds + u-1 f fj2 ds] 440 ELECTROMAGNETIC WAVES Cum: 10 Q - J.(g) and in the case of 77i-oscillations , if> = 0; 2 r r t2,av For a circular cylinder of radius d and length / the Q of the tm-oscilla-tions becomes 2 = capa 29? , if>=f=(V j? = 291 (1+0 For T£„,mi„-oscillations we have 29f[?2«W + + 2/>Wr1(*2,m - «2)] • In the above calculations it has been assumed that there is no dielectric loss in the resonator. If there is such loss but no loss in the walls of the cavity, then instead of (3) we have *,2 2 X + —p - = -a = —Ufl(g+ ««), where u is the oscillation constant. Solving for u, we obtain By (5.11-16) we have that is, the Q of the resonator is equal to the Q of the dielectric. The Q of the resonator having both metal and dielectric losses may be obtained from the general formula 1 = 1+1 Q Qi &' This formula follows immediately from the definition of Q in terms of stored energy and absorbed power provided the losses are not so large as to affect appreciably the natural frequency. CHAPTER XI Antenna Theory L1.1t. Biconical Antenna The antenna theory developed in the following pages is based fundamentally on the conception that the antenna and the space surrounding it are two wave guides. Thus in the case of an antenna consisting of two equal coaxial cones (Fig. 11.1) in free space the two wave guides are: Fig. 11.1. The cross-section of a conical antenna of length / and of the " boundary sphere " S. (1) the antenna region bounded by the conical surfaces and the boundary sphere (S) concentric with the center of the antenna and passing through the outer ends of the cones, (2) the space external to (S). There are infinitely many transmission modes appropriate to either wave guide. If the voltage is applied between the apices of the cones, then the waves possess circular symmetry; the magnetic lines are circles coaxial with the cones and the electric lines lie in axial planes; but the number of transmission modes is still infinite. Circular magnetic waves in free space are described by the equations 441 442 KLECTRí MAGNETIC WAVES ClIAl'. II of section 10.10. Hecnuse of circular symmetry we use m = 0 in the expression (10.10-22) for T; since the field must be finite for all value of 8, n must be an integer; consequently for progressive waves traveling outward in region (2) we have T(0) = Pn(cos 0) and the field of the wtli " zonal wave " is n(r,0) - Kn(ißr)Pn{cos 9), rEt = (#r)i* (cos 0), rfff = - Kn (ißr) jQ P„ (cos 0) = - Ku (ißr) Pi (cos 0), (1 -1) ?W2£r = »(» + l)Kn(ißr)Pn(cos 0). When n = 0 the field vanishes identically and the principal or dominant wave corresponds to ri = 1. For this wave Px (cos 0) = cos 0, P}(cos 0) = - sin 0, ß2r2 = 1 1 + 82r2 ' i8r(l + 82r2) The radial electric intensity vanishes in the equatorial plane and is maximum on the axis; the meridian electric intensity vanishes on the axis and is maximum in the equatorial plane; and the electric lines have the form (a) Fig. 11.2. Electric lines for the first order transverse magnetic spherical waves: (a) lines in free space; (b) lines in the presence of two coaxial conical conductors. shown in Fig. 11.2(a). For large values of r, the radial impedance is substantially equal to the intrinsic impedance; for small values of r the radial impedance is largely reactive and is approximately equal to l/iwtr. For small values of r the radial displacement current is forced to flow across ANTENNA THEORY •in a small area, the series capacitance is small, and the input impedance is large. For the zonal wave of the second order we have P2(cos 6) = \(2 cos2 8 - 1), Pi (cos 6) = -3 sin 8 cos 8, Uißr) = e-^(l+-^-^r2), K+, K2(ißr) The radial electric intensity vanishes when cos 6 = ±1/V3, while the meridian intensity vanishes on the axis and in the equatorial plane; the electric lines are shown approximately in Fig. 11.3(a). For large values lib in (a) 0>) Fig. 11.3. Flectric lines for the second order transverse magnetic spherical waves: (a) lines in free space; (b) lines in the presence of two coaxial conical conductors. of r the radial impedance is nearly equal to tj and for small values it is nearly 2/ioier. As the order of the zonal wave increases the'number of different sets of closed loops in the field also increases. The radial admittance of the núv zonal wave may be expressed in the following form 1 — i(u„u'n + vnv'n) i\Mn = 2 - (ui + vi) + w: + O ' n + \) 2-4,3V (» - !)»(»+ !)(» +2) (» -3)(«-2)---(» + 4) «n "= 1 — ň ä fl2„2 _nyn + 1) (» - 2)(n -!)■•• Vn " " 2ßr 2 ■ 4 ■ 6ß*rs 2 ■ 4 ■ 6 • 8 ßSA (a+ 3) (1-2) (n -4)(n - 3)- ■■(» + 5) 2-4 5„s 10/3V + -I.I.I 1.11 t I R< >i\ I \c; Ní I. I n VVAVES- Chap. II Figures 11.4 nnd 11.5 show respectively the products ?jGn and r\Bn of the intrinsic impedance with the radial conductance and the radial suseeptatn i For small values of r the ratio of conductance to susceptance diminishi very rapidly with the increasing order of the wave; hence a given averagl Fig. 11.4. The product of the intrinsic impedance 7j and the radial conductance G„ as a function of the phase distance jSr from the wave origin for each of the first seven transmission modes. Fig. 11.5. The product of the intrinsic impedance and the radial susceptance. radial power flow is associated with increasingly strong reactive fields in the vicinity of the center of the wave. In the antenna region we have corresponding transmission modes except that the electric lines in the vicinity of the conical conductors form small ANTENNA THEORY 145 half-loops terminating on the conductors as shown in Figs. 11.2(b) and I 1.3(b). As the cone angle becomes vanishingly small, these added half-loops become vanishingly small and the field configurations become nearly the same as in free space. In addition to these transmission modes there exists a mode in which the electric lines coincide with the meridians (Fig. 11.6); this is the principal mode in the antenna region and its theory has been developed in section 8.12. (a) (b) (1-3) Fig. 11.6. Cross-sections of infinitely long conical conductors and electric lines of force for principal waves. Let us now consider more closely the higher modes in the antenna region. Since the radii 0 = 0 and 6 = tt are excluded from this region by the conical conductors we have T(6) - APn(- cos 6) + 5P„(cos 6). This function is proportional to ET and must vanish on the conical conductors, assuming that they are perfect conductors. Thus if the cone angle is \p, then APn(- cos t) + PP„(cos *) = 0, APn(cos *) + BPn(- cos $1 = 0. These equations require A2 = B2, A = ±B. Hence equation* (3) are reduced to P„(- cos *) = Pn(cos *), A = -B; (1-4) Pn{- cos VO = -P*(cos *), A = B. (1-5) In the case represented by equation (4) we have (setting B = §) T(6) = i[P„(cos B) - Pn(- cos 0% rHv = ~ Ť{8r) ~ , rEe = - k V (Br) ||, (1-6) mer2Er = n(n + l)Ť(Br)T(B), Ť(8r) = A }*(&■) + BŇn(8r). 446 ELECTROMAGNETIC WAVES 'MAI. Jl Replacing 0 by it — 8 in the above expression for T, we see that dT(r - 8) dT(8) T{t-6) = -T{8), do d(K - 0) Hence, along the radii making angles 0 and it — 0 with the axis, the radial electric intensities are in opposite directions and the magnetic intcnsiti, are in the same direction. The current in the upper cone at distance r frt im the apex is 7(r,v0 = Iter sin ^ H9 = -2*f(pr) sin ft (1 B similarly the current in the lower cone, if regarded as positive when flowing upward is I(r,ir — ip) = f(r,\P). Thus the currents at points equidistant from the center are equal in magnitude and flow in the same direction. If ^4? is small compared with unity, equation (4) becomes approxi mately P„(- cos i) = 1. (1-9) From (3.6-10) we have as a first approximation P„(- cos ý) = % sin nv j^log I + if/(n) - ^(0)^J + cos wtt; log y + #(Q) - *(»)] ~ - ? log ~. hence (9) becomes »ir 2 tan — =-- 2 7t Thus the approximate values of n are » = 2m + 1 + —^, ?« = 0, 1, 2, • ■ -. log T Since for small cone angles the characteristic impedance of the cone to the principal ^vave is K = 120 log 2/>P, we may write w - 2m + 1+ ■ a. (1-10) Thus as the characteristic impedance tends to infinity, the roots of (4) approach the odd integers. From equations (3.7-43) and (3.7-44) wi find that í*2ní-i-i (cos 0) as A—»0. (1-11) Thus all the transmission modes in the antenna region, except the principal, approach the corresponding modes in free space. AMI WW THEORY 447 In the case represented by equation (5), we have no) = ilJVcos 9) 4 />,.(- cos 0)], T(v - 0) = Tie), dT(t - 8) diyc - 8) dT(8) d8 (1-12) Along the radii making angles 0 and it — 8 with the axis the radial electric intensities are in the same direction and the magnetic intensities are in opposite directions. The currents in the cones at points equidistant from •the center are equal and flow in opposite directions. For small cone angles the approximate solution of (5) is ff=2m + W. (1-13) K In this case T2m-|_a(0) —> P2m(cos 8) as A —> 0. The modes for values of m > 0 approach the corresponding free space transmission modes. The mode corresponding to m = 0 is the principal " anti-symmetric mode " in the antenna region; in iree space there is no corresponding mode because P0(cos 8) — 1 and the field vanishes identically. In particular for small values of A we have PA(- cos 8) 1 + 2A log sin - , (- cos 0) ta - (l + 2A log sin 0 cos 9. Figure 11.7 illustrates the behavior of these functions. The corresponding T-functions are TA(9) = 1 + A log sin-, r1+A(0) = ^1 + A log sin 0 cos 8. The behavior of these functions is illustrated in Figs- 11.8 and 11.9. For » > 0 the function Nn(8r) becomes infinite as r~n at r = 0, and its derivative becomes infinite as r~n~1, hence B in (6) must be zero; then Tn(j3r) = A/„(/3r). In the vicinity of r = 0 the function Jn{6r) is proportional to r"; thus the current associated with the higher order waves vanishes at the apices of the conical conductors In(0) = 0. The voltage Vn'r) along a typical meridian vanishes at all distances K(r) = f 'mm = -ivf'mm-K - $ - rm = o. Hence the input voltage and current depend only on the principal wave. sequently the input impedance depends only on the principal wave. Con- EI.ECTRt MAGNETIC WAVES so loo 9 IN DEGREES Pig. 11.8. Legendre functions of fractional order. ANTENNA THEORY 449 The total transverse voltage /'(/) and llie total longitudinal current I(r) in tlie upper cone may thus he written in I lie linllowing form, V{r) = n Kr) = /0(r) + l{r), 7(0) = 0, (1-14) where the principal voltage and current are n(T) = *W) cos 8(1 - r) + iKI0(l) sin 8(1- r), 70(r) = —£M sin 8(1 - r) + 1,(1) cos 8(1 - r), and / is the length of the cone. The total " complementary " current wave I(r) consists of an infinite number of current waves associated with higher order transmission modes l{r\ m h(r) + 73(r) + 7s(r) + ■ • -. (1-16) As implied by this equation only the odd order waves, for which T(0) is given by equation (11), appear when the cones are equal and when they are 10 0.8 o.e MA( ÍNI/I K' WAVES Chap. II Thus in so far as the input impedance, tin- total voltage dist rihut ion, unci the principal current distribution arc concerned llie effect of the complementary current waves is equivalent to a terminal impedance given by -id) id) V(l) •V(l) z £bw no 1 /o(/) /(/) - /(/) Taking the reciprocal, we have /(/) no F(l) F(l) (2-2) (2-3) Fig. 11.10. The terminal or Thus the terminal admittance consists of two radiation admittance of an admittances in parallel and the conditions at antenna* r — I may be represented diagrammatically as in Fig. 11.10. Since/(/) is the current flowing into one spherical cap of the conical antenna and out of the other we may interpret 1(0/^(0 as the admittance between the two caps. When the caps are small we have approximately /(/) = 0 and consequently MO F(l) i® F(iy v® MO EE no (2-4) The admittance between small caps is given at the end of section 11.6. Thus we shall represent the input impedance and admittance of conical antennas in the following form Z( cos ßl + iK sin ßl Zi = A. —-'.———:—r.. r« xi Yt cos ßl + iM sin ßl (2_5^ K cos 81 + iZt sin 81' M cos 81 + iYt sin 81' where the characteristic impedance and admittance are ft K = 120 log cot 120 log - M=~ 11.3. Current Distribution in the Antenna and the Terminal Impedance In section 1 we have seen that the radial electric intensity in the antenna region can be expressed as follows 2x*W2£r = £ *» #B Tn(B), Jn(ß0 (3-1) where Tn(6) is defined by (1-6) and the summation is extended over the set (1-10). The complementary' current in the antenna is then Ť7 x v- (ßr) ■ i T / r \ I(r) = - E , , n i~nA\ sin ^ 77 T"(^)- (3-2) ANTENNA THEORY 451 When f is small, we have approximately JT»W) • A 120 . 120 an)n(8r) ., M n>~ k r»(» + i)JnW In free space the radial electric intensity may be written in the form 2«W2£r - ih §7^7 P*(cos 6). (3-4) At r = /, Er should be continuous, hence 00 E anTn(0) = £ W*(cos 0). n A -= 1 As K increases indefinitely and \b—>0,n approaches 1m + 1 and T„(0) approaches P2m+i(cos 0); thus the limiting value of an is lim an = lim «2m+i+A = ^2?n+i as A —> 0. Hence for high impedance antennas we have approximately 120 , I(r)---F Ä C">m+1 (3-5) K MoMm + 1)(2>« + l)}sm+i(P0 We have seen that for infinitely thin wires the current distribution is sinusoidal, with current nodes at the ends. In the present case the conclusion follows also from (5). Hence as K—» °c, the current distribution in the antenna approaches the following value 70(r) = h sin 8(1- r), I0 = . (3-6) The field of this distribution has been obtained in section 9.25 and the values of the coefficients ^m+i w'h' he determined if we expand r2ET as a series of zonal harmonics. For this purpoa. ;*h the following expression C = lirr sin 0H9 - lil0(e~il3ri + - 1e-^r cos p7), (3-7) obtained from (9.25-14). At great distances from the antenna this becomes C = *70[cos (81 cos d) - cos 8l\e~ifiT. Then from (10.10-6) we have Iviuer^Er = ~--= iBIlo sin (81 cos e)e~^T. sin 9 d9 452 n.l'X'TliOMAONIiTIC WAV res Cuav, 11 The expansion ■ if sin (ßl cos 0) in terms of zonal harmonics is known to I" sin (ßl cos 0) = -j £ (-)m(4»/ + 3)/2m+1(/30/W(coS fl). P< m=0 Thus we have expressed the distant field in terms of zonal harmonics. < >fl the other hand at great distances from the antenna etpiation (4) becomi n ■ 2F £ (-r+1W^+i(cos 9)e-V , Comparing the alternative expressions for Er we have 'km+i = -//o(4;» + 3)f2m+l(ßl)[f2^1(ßl) - iN2m+l(ß/)}, (3-8) From (2-4), (6) and (8) we now obtain the terminal admittance Zo,{L) «u 4™ q_ 3 " ",go(m+l)(t» + l) ^+l(£)> «i=o fm + l)(2m +1) where L is the " phase length " of each cone defined by L = 2ir//X. Thus for cones of small angles the terminal impedance and its inverse an Zt 4 K2 K2 , — = Ra(L) 4- /^(i). (3-10) ^;(t:i + ixa{L)' Zí In computing the input impedance of the antenna we may replace the terminal impedance Zt by its inverse inserted in series with the transmission line representing the antenna, one quarter wavelength from its end. 11.4. Calculation of the Inverse of the Terminal Impedance The series (3-9) for the inverse terminal reactance converges slowl) and is not practical for numerical calculations unless L is small. A simple expression for Za(L) can be obtained as follows. The input impedance is . Za sin L — iK cos L ~p •—-T-^---jr. (4-1) A sin L — iLa cos L Zi = K ANTENNA THEORY As K increases indefinitely wc have Z Z{ —► -r-ö' z — iK cot L. sin' L 45.1 (1 2) Since the input current approaches the value 7o sin j3l, the power input becomes ST/ = \Zdl sm2L = \\_Za - liK sin 2L]H (4-3) On the other hand the power flow for a sinusoidal distribution may be calculated by the method described in sections 9.26 and 9.27. The integral of Hie product of the current and the tangential electric intensity should be calculated on the surface of an infinitely thin cone. In the limit the real part of this integral is independent of the shape of the longitudinal cross-section of the antenna and Ra — R as given by equation (9.27—2); but the reactive part is still a function of the shape of the longitudinal cross-section and must be calculated for the cone. In the limit the tangential component of the electric intensity is the sum of Ez and £„-sin \p, where Ez and E„ are given by (9.25-14). In this way the following expressions are obtained Ra{L) = 60(C + log 21 - Ci 2L) + 30(C + log L - 2 Ci 2L + Ci 4L) cos 2L + 30(Si 4L - 2 Si 2L) sin 2L, (4-4) Xa(L) = 60 Si 2L + 30(G AL - log L - C) sin 2L - 30 Si 4L cos 2L. The inverse of the terminal impedance is shown in Fig. 11.11. 350 r i 300 o z 2 150 i- < o < ee 100 > 50 1 \ ye I \ *• - f / / / j \ r 1 // I \ 1 \ - // h \ ■ ■ ; M,j t I t j t / i / I \ \ \ V 1 1 1 / \ V ' / f t ! \ \ t I I 1 i ■l \ \ 1 I t / y \ \ i \ \ \ 1 1 ■I 1 \ \ PHASE LENGTH, L, IN RADIANS Fiq. 11,11, Curves for the resistive and reactive components of the inverse terminal or radiation impedance of an antenna. I', I Electromagnetic: waves Chap. 11.5. The Input Impedance and Admittance of a Conical Antenna Separating t lie real ami imaginary parts of Z{ and its reciprocal, we have ra - i £|K sin 2L + Xa cos 2L - SÍ" ^1 sin2 L + — sin 2L + K Rl + Xl K2 cos" L Ra + Í r2 + x2 K sin 2L + Xa cos 2L--—jz—- sin 2L 2K cos2 I - y sin 21 + a j" -^a . 2 r~l -^3—sin ^J (5 (5-2) 100 1000 RECIPROCAL RADIANS (^r) Flo. 11.12. The characteristic impedance of a conical antenna. 10,000 The characteristic impedance is shown in Fig. 11.12, as a function of the reciprocal of the cone angle. For conical antennas of small angles the reciprocal of the cone angle is approximately equal to the ratio of the length of each cone to the maximum radius l/ý = l/a. Figure 11.13 shows the input impedance as a function of the phase length of each cone for different values of the characteristic impedance; the solid curves represent the real component and the dotted curves the imaginary. Figures 11.14 and 11.15 show the input resistance as a function of the length of each cone in wavelengths; similarly Fig. 11.16 shows the input reactance. The terminal impedance affects the resonant lengths of the antennas. As the characteristic impedance approaches infinity the input reactance ANTENNA CRY 455 5000 4000 i o II z •Í o III dl 2 2000 1000 1" l"' II II II II II -11- 1, 1 ,1000 y i' looo c-) w 1 1 \ 1000 (+) JJ ví 750 1 \ A 1000 (-) w -—r '—*- 1 I l750<-> m i \ \ w \ \ \ \ v 75 ' / 1 t i J MX j U \ it 1 \a 1 IA \\ xv Li i \ \\ \ \ r o(-) -u~~v \ A v 500 (-)\\ \ 450(-) i* Á / J/sooM ,^450(+) \^ TrL \ \\ > ■ ny ,500 (-) 2tt1 IN RADIANS Fig. 11.13. The input impedance of hollow conical antennas (without spherical caps) as a function of 2ir//X and K. Solid curves represent the real component and the dotted curves the imaginary. 456 I I It IK* >MA<;NF.TK' \\ WIS ClIAl'. II vanishes lor values of/, approaching (hose given by sin 2L = 0, 2L = kr, 21 = ^ , * = 1,2,3, ■ • I the exact values of the resonant lengths are denned by the following equatii in 2KXa 10 000 10 I o 1000 20 tan 2L = — y-2 r>2 y-2 : 1 K = I200—>y- 1000-fp 1 / 700-#-*7— 60 0-//y-- Fig. 11.14. The input resistance of hollow conical antennas (without spherical caps), which for large values of K becomes tan IL =--— , LL, = kic — ■-—- K K TkK 1 - 120 Si far + 60(-)k+1 Si 2far (5-3) Figure 11.17 shows the deviation of the resonant length of the antenna from k\/2. The input impedance of the antenna when the length of the cone is equal AN I INN \ Tlll.nKY to an odd multiple of X/4 is of course the inverse terminal imped Zi = Za. When / = X/4, this impedance is Zi = 73.129 + /'l 53.66. 457 lam C aooo t ooo' 0 36 0.52 0.56 0.40 0 44 0.48 Fig. 11.15. The input resistance of hollow conical antennas in the neighborhood of the second resonance. When / is a multiple of X/2, then the input impedance is equal to the terminal impedance. If Z„ were independent of /, the above two impedances would be represented by inverse points in the impedance plane. In fact the whole impedance diagram would be a circle inverse to itself with respect to a circle of radius K. As it is the impedance diagram is not quite a circle but surprisingly close to it considering the variation in Za- Figure 11.18 shows such a diagram for K = 750. I o ř o < o 3 04 0,7 os o e lA Fig. 11.16. The input reactance of hollow conical 0,3 300 600 1000 2000 3000 CHARACTERISTIC IMPEDANCE IN OHMS Fig, 11.17. Deviation of the resonant length of conical antennas (without spherical caps) from 11 = k\/2. (4S(!) ANTI-.NNA Till.OKY 459 t.o o.a 0.6 . 0.4 0. 2 K\ o -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 - 1.4 - 1.8 X 10* K-750 A 1 // l Fig. 11.18. The impedance diagram for a conical antenna. 11.6. The Input Impedance of Antennas of Arbitrary Shape and End Effects In the preceding sections we have shown that a biconical antenna may be regarded as a wave guide with infinitely many transmission modes. Only the principal mode is generated if two infinitely long cones are energized at their common apex; but if the cones are of finite length there is a discontinuity at the boundary sphere which separates the antenna region from the surrounding space and reflection takes place. Then higher transmission modes are generated, and the effect of these modes on the amplitudes of the total voltage wave, the principal current wave, and the input impedance can be represented exactly by an appropriate terminal impedance. This property we represent diagrammatically as in Fig. 11.19. We have also determined an approximate value of the terminal impedance for cones of small angles. When a single cone is placed normally to a per- •lul) ECl'ROMAGNETIC WAVES Cil a i'. II fectly conducting plane sheet, the characteristic impedance of the antenna and the terminal impedance are halved. If the shape of the antenna is other than conical, then the characterist i< impedance K is no longer constant. In the case of antennas whose trims verse dimensions are small the waves are nearly spherical and we may t real such antennas as nonuniform transmission lines whose inductances and capacitances per unit length are given approximately by the equations developed in section 8.13. For the terminal impedance we may take lbe expression (3-10) where K is replaced by the average characteristic imped ance. This treatment is based on analogy with conical antennas; i Indirect approach based on Maxwell's equations is theoretically possible but (i) (z) (3) (4) (5) (6) {7) = CHARACTERISTIC IMPEDANCE S77777777777777777777777777777777777777777777 _k 2 CHARACTERISTIC IMPEDANCE Fig. 11.19. The input impedance of a conical antenna of any size is equal to the input impedance of a uniform transmission line with a certain " output " impedance Zt. The input impedance of a chin antenna of any shape is similarly represented, except that the characteristic impedance is variable. at present is not practicable because no convenient method is available for computing the appropriate wave functions. In order to obtain such functions exactly it is necessary to use coordinate systems appropriate to the boundaries and to the physical phenomena. Spheroidal coordinates, for instance, fit the boundary of a spheroidal antenna but not the problem of the transmission of waves on it; and spheroidal wave functions are much more suitable for defining waves traveling away from the spheroid than waves traveling along it. From the theory of nonuniform transmission lines we obtain the following approximate ;:q;iression for the input impedance Ra sin L + i[(Xa - N) sin L - (Ka — M) cos L] * a[(Ka + M)s\nL + (Xa + AO cos L] - iRa cos L ' ( ' ANTENNA THEORY where M'and A' are the functions defined by M(L) = 8 f! [Ka - K(r,P)\ sin 2Brdry "o N{L) = 8 f1 [Ka - A>,P)] cos 28r dr. Separating the real and imaginary parts, we have KaRa(Ka + /V sin 2L - M cos 2L) _ 461 (6-2) Rl cos2 L + [(Ka + M) sin L + (Xa + N) cos L]'z' KJ&(R*+X*+M*-N*-Kpsm2L + (MN-KaXa) cos 2L+ (MXa-K„N)} R'i cos1 L + [{Ka + M) sin L + (Xa + N) cos Lf where Ka is the average impedance. For cylindrical antennas we find M(L) - 60(log 2L - Ci 2L + C - 1 + cos 2L), N(L) = 60 (Si 2L - sin 2L), Ka = 120 ^log j - 1V In free space, for antennas whose longitudinal cross-section is rhombic, we have M(L) = 60(C + log 2L - Ci 2L)(] -|- cos 2L) - 60 Si 2L sin 2L, . N(L) = 60 Si 2L(l - cos 2L) - 60 (C + log 2L - Ci 2L) sin 2L, >Ka = 120 log-, a where a is the maximum radius. For vertical antennas of triangular shape with base of radius a, above a perfecdy conducting ground, the above values are halved. For spheroidal antennas the corresponding formulas are M(L) = Ra{L) - 60(1 - cos 2L) log 2, N(L) = Xa(L) - 60 log 2 sin 2L, Ka = 120 log - , a where a is the maximum radius. In the case of antennas of rhombic cross-section above a perfectly conducting ground (cross-section (5) in Fig. 11.19), the first half of the antenna is uniform and the second half nonuniform. The input impedance of the 462 second half is Zi — Ka KU'X'TKI IMACJNETIC WAVES Chapi II L - (Ka - M) cos Äa sin —+ t (A^u - N) sin - - L {Xa + AO 1, L \(Ka + M) sin - + cos-_ — iRa cos — M(L) - 60 log 4 + 60(C + log \L - Ci 2Z.) cos L - 60 Si 2L sin Z,, Af(Z) = 60(Si2Z-2Si£)cosZ-60(CH-log|Z. + Ci 2Z,-'2CiZ) sin/,, 4/ a"0 = 120 log - . a It should be noted that in the above equation for Zj the quantities R„, Xa, M, and N are functions of L not of L/2. Using this impedance as the terminal impedance of a uniform transmission line of phase length L/2 and characteristic impedance equal to that of the first half of the antenna, wc obtain the input impedance of the entire antenna. 1200 400 200 Fig. 11.20. The average characteristic impedance: (1) cylindrical antenna, (2) spheroidal antenna, (3) antenna of rhombic cross-section. In Fig. 11.20 the average characteristic impedances are shown for several antenna shapes: curve (1) is for a cylindrical antenna, curve (2) for a spheroidal antenna, and curve (3) for an antenna of rhombic cross-section. In Figs. 11.21 and 11.22, the input resistance and reactance of cylindrical antennas are shown as functions of //X for different values of Ka. For i........ n..... nuo. .|o,in „:,-'in,, X O I 000 Z 600 . 600 400 100 eo 60 40 0.2 1 1200 /iooo> / 800 ___BOO) 50 Oj^" k Fig. 11.21. The input resistance of hollow cylindrical antennas in free space. For vertical antennas over a perfectly conducting ground divide the ordinates and Ka by 2. Fig. 11.22. The input reactance of hollow cylindrical antennas in free epace. For vertical antennas over a perfectly conducting ground divid" the ordinates and Ka by 2, 463 electromagnetic: waves Cm antennas above a perfectly conducting ground tlie values of A„ and of the ordinate* should he halved, figures 11.23 :md 11.24 show the resonant impedance of cylmdrical antennas for the first and second resonance, respectively.* 70r 1200 500 600 TOO BOO" 900 1000 1100 CHARACTERISTIC IMPEDANCE IN OHMS Fm, 11.23. The resonant impedance of hollow cylindrical antennas as a function of K when I is in the vicinity of X/4. 5°° 60ČT 700 800 900 Í000 iToo 1200-BOO CHARACTERISTIC IMPEDANCE IN OHMS Fig. 11.24. The resonant impedance of hollow cylindrical antennas as a function of K when / is in the vicinity of X/2. When / = \/4, the input impedance is *-^[*© + *©^©]- As Ka increases, this impedance approaches the following value * The points in Fig. 11.24 are experimental. ANTENNA THEORY l/,5 The limiting value of the input resistance is nearly 73.13 ohms. Eor any finite value of A,„ the input resistance depends also on Af(ir/2), that is, on the shape of the longitudinal cross-section as well as on the mean size of the transverse cross-section. For example, for cylindrical and spheroidal antennas M(tt/2) is equal respectively to —21 and —10; hence the input resistances are somewhat higher than 73.13. Even when Ka is infinite, the input reactance depends on the shape of the longitudinal cross-section. Thus for cylindrical antennas the reactance is 30 Si 2ir — 42.5 ohms and for spheroidal antennas it is zero as compared with 154 ohms for conical antennas. The effect of capacitance between spherical caps at the ends of the antennas may be included as follows. For two caps of small radius a the admittance between them is substantially equal to iwC where C is the electrostatic capacitance. Thus the value of this admittance is ia/3Q\ and it should be included, when necessary, in parallel with the terminal admittance Yt of (2-4). Introducing this correction in (3-9) we find Yt = + i K2 SXa(L) 1 so that the effect of the cap capacitance is obtained if we replace Xa by Xa + wCK2. Consider for instance formula (5-3) for the resonant lengths of antennas. The cap capacitance causes a shortening of the resonant length which may be expressed as iaCK/L. For conical antennas this may be written Ki/30if exp ( — AC/120), while for cylindrical antennas its value is K/30* exp (-AC/120 - 1). This effect will be negligible when K is sufficiently large, but for K = 600 we find that the cap capacitance decreases the resonant length by 4.3 per cent for conical antennas and by 1.6 per cent for cylindrical aptennas. If K is increased to 1200 these percentages are reduced to 0.058 and 0.021 respectively. The explicit expressions for the M and N functions given in this section have been calculated from (6-2) on the assumption that the distance between terminals is negligibly small. Under practical conditions one may assume that M and N are independent of this distance so long as / is measured from the end of the antenna to its nearest input terminal. In more accurate work the effect of finite separation can be obtained by recomputing M and N from equations (6—2). The distance r should be taken from the input terminals. As the distance between the input terminals becomes larger and larger the field distortion in their vicinity becomes substantial and will effectively introduce an impedance across these terminals in parallel with that of the rest of the antenna. ■Ion ELECTROMAGNETIC WAVES t'llAI-. II 11.7. Current Distribution in Antennas Tlie current distribution in a conical antenna is given by equation', (1-14), (1-15), (1-16) and (3-8). In an antenna with nonuniform clini acteristic impedance equations (1-15) should he modified in accordance with the theory of nonuniform transmission lines. Thus the current distribution depends on the distribution of inductance and capacitance in tin antenna as well as on the complementary current waves due to reflet tion at the boundary sphere. A typical complementary current wave is given by (r) = 60 (4m + 3)/0 (m + l)(2m + 1) [Ň.2m+l(al) + iJ2m+i(Bl)]J2^l(Br). 0.8 0- 0.2 < -0.4 m = o m = i m = 2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 DISTANCE IN WAVELENGTHS (r/X) Fig. 11.25. The amplitudes of the first, third, and fifth (m = 0,1,2) secondary waves m functions of the distance from the center of the antenna. Figure 11.25 shows the variation of the amplitudes of the first, third and fifth (m = 0,1,2) secondary waves as functions of the distance from i In-center of the antenna. In the vicinity of r = 0 the amplitude of the (2m + l)th current wave varies nearly as r2m+1; thus the secondary curtcni waves affect the current distribution mainly near the ends of the antenna, The maximum amplitudes of the secondary current waves depend on tin characteristic impedance and on the length of the antenna. Figure 1 1.2(5 shows the secondary current waves when K = 1000 and / = X/2. Thi solid curves show the components in phase with the principal current ANTENNA T1IBMY W,/ 0.20 0.16 0.12 0.08 0.04 -0.04 — i m = o \ —■- m -- ~~"~v *** " Fa TTi= 2^ Í t 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fig. 11.26. Secondary current waves for K = 1000 and / = X/2. The solid curves show the components in phase with the dominant current la sin 0(1 — r), and the dotted curves show the quadrature components. 1.2 0.8 0.6 0.4 0.2 -0.2 id? y J // // // 7 i / / — - ■—— — __,__ f.....— 0.5 r/l 0.7 0.8 Fig. 11.27. The total current in the antenna of length 11 = X; K = 1000. The solid curve represents the amplitude of the total current; the dash curve represents the amplitude of the component in phase with la and the dash-dot curve is the amplitude of the quadrature component. 468 ELECTROMAGNETIC WAVES ClIAIS It t.o 0.6 ID 0.6 < a. 0,2 -0.2 0.2 Ý - - ^> -* fa i(r) A / / 10(1") la / / f- l A / / / \ 0.1 ■0.[ -0.2 1.2 1.0 0.8 0.6 0.4 0.2 * V ICr) s H. M S J Kr) I0Ct)' r/l Fig. 11.28. Curves for the total current and the principal current. ANTFNNA TIIKoKY component lu sill B(l — r); only the first of these is important, except in the immediate vicinity of r = /. The dotted curves show the quadrature components of which only the first two need be considered. In Fig. 11.27 the solid curve, the dash curve, and the dash-dot curve represent respectively the amplitude of the total current, the component in phase with I0, and the quadrature component.* In Fig. 11.28 the total current is compared with the principal current. The difference between the real parts is quite small but the difference between the imaginary parts is relatively large except at the center. ' The current distribution determines the shape of the radiation pattern. The quadrature components radiate independently. Since the radiation intensity is proportional to the square of the moment, the radiation pattern is affected but little by the current in quadrature with 70 except in those directions in which the radiation is small, where a small absolute difference may contribute a large percentage deviation. Rewriting the expressions (1-15) for the voltage and the principal current in terms of the amplitude [0 as defined by equation (3-6), we have V(f) = -iKIo cos 8(1 - r) + (Ra 4- iXa)IQ sin 8(1 - r), I0(r) = Io sin 8(1 - r) + a ^ a I0 cos 8(1 - r). The real part of the principal current is practically unaltered except in the vicinity of the current nodes. These equations are in agreement with the general theory concerning the current distribution on thin wires. The imperfect conductivity of the ground has a much more marked effect on the radiation pattern than the deviations in the current distribution from the limiting distribution I0 sin 8(1 — r). Thus the radiation intensity of a quarter-wave antenna in the ground plane is zero for an imperfectly conducting ground; an effect in comparison with which the effect of finite K is altogether negligible. 11.8. Inclined Wires and Wires Energized Unsymmelrically The principal waves on inclined wires [see Figs. 11.19(6) and 11.19(7)] have been considered in section 8.14. The functions Ra and Xa may be calculated by the method outlined in section 4; the M and N functions are obtained from the theory of nonuniform transmission lines; in terms of these functions the impedance is given by equation (6-1). * In this figure the current does not qtnte vanish at the end of the antenna because only the first two complementary waves were included; the higher order waves reduce the end current to zero but do not affect the current appreciably at any distance from the end. 170 I'll.i no (MAGNETIC WAVES Chap, II Tilt: calm la tiim of the complementary current waves is nunc compl'i cated. If the angles 1^1 and ^2 of the conical wires are small compared] with the aftgle d between their axes, the proximity effect is small and 11n-current distribution in each wire is substantially uniform round its axis. Then the radial electric intensity is proportional to the following function T(0U62) = APa(~ cos By) + BPn{ - cos 0a), where Bi and 0% are the angles made by a typical radius with the axes of the wires. This function must vanish on the surfaces of the wires and we have approximately (8-1) APn(~ cos vAO + BPn(~ cos t>) = 0, APn(- cos &) + BP J- cos fcs) = ft Hence n must be a root of Pn(- cos MPn(- cos fe) = [Pn(- cos 0) and in region (2) /Y„-function| should be included. Fig. 11,29. The cross-section of a conical antenna and boundary spheres. ANTENNA THEORY 471 Another method of approach is based, on considering two cones of unequal length with a generator at the common apex 0 (Fig. 11.30). The field in region (1) is similar to that in region (-1) of the preceding problem. The field in region (2) consists of waves corresponding to a different set of values for n since in this region the field must be finite for 8 = jr and consequently T(B) = APn(- cos 9). The values of n are then obtained from the equation PK(— cos $) =0. If tit-is small, the roots are approximately >i = m + 60/K, m - 0,1,2, ■ ■ Transmission of waves on a wire energized near one of its ends may also be studied by considering waves on a cone surmounted by a small sphere (Fig. 11.31) The electromotive force may be applied either between the cone and the spherical surface round the circumference AB or between the apices at 0, one apex belonging to the original cone and the other to a small cone leading to the surface of the sphere. Approximate solutions of these problems can be found without much difficulty and extended to wires of other shapes. These solutions will supplement the conclusions drawn from the solution of the principal problem of two equal cones of equal length. Fig. 11.30. The cross-section of 1 conical antenna and boundary spheres. B Fio. 11.31. The cross-section of a conical antenna and a small sphere at its apex. 11.9. Spherical Antennas Equation (2-3) is the general expression for the terminal admittance Yt of a biconical antenna consisting of two equal and oppositely directed cones. When the cone angle $ and the length / are sufficiently small, the total current 1(1} is so nearly equal to zero that the terminal admittance is determined substantially by the complementary current. But as the cone angle increases the total current makes ;in increasingly important contribution to'the admittance. When the cone angle ^ is nearly 90 degrees, the biconical antenna becomes a pair of nearly hemispherical conductors fed by a cone transmission line from the center (Fig. 11.32). In this case the complementary current 1(1) becomes small and the terminal admittance is determined largely by the total current 1(1). That the complementary current becomes relatively small can be seen if we observe that in the present case the cone line is approximately a disc line with variable separation between the " parallel planes." When the separation is small compared with the wavelength, all transmission modes except the principal are attenuated and the energy associated with these modes is concentrated in ■IV ELECTROMAGNETIC WAV ES Chap, II the region where (he conical feeders join l he sphere direction toward the center) is comparable to the the two hemispheres The capacitance representing this local storage of energy will be small compared with the external------' ■ ■ - The depth of i h is region (in rhr separation between the edges of capacitance between the hemispheres. Flo. 11.32. The cross-section of a Fin. 11.33. A spherical antenna fed spherical antenna or a double cone along an arbitrary circle of parallel, of large angle \p surmounted by large spherical caps. Thus we may assume that the approximate voltage distribution over the aperture of the cone transmission line in Fig. 11.32 or in the more general case shown in Fig. 11.33 is governed by the principal wave. Let V be the total transverse voltage at r = / = «, where a is the radius of the sphere; then Ee(a,9) = 0 except in the interval \p < 8 < \j/ + i?, where Ee(a,9) = IrrKa sin 0 Ka sin 9 ' (9-1) In accordance with equations (1-1) the meridian electric intensity for r > a mav be represented as follows y (9-2) In the theory of spherical harmonics it has been established that an arbitrary function /(8) (subject to certain limitations) can be expanded in the form AO) = Z anPi(cos 6), an = 2"+ ' f J(9)Pln(cos 6) unOdB. n = l 2»(» + 1) J|) Hence the coefficient A„ in the expansion for aEe(a,8) is 60(2;/+ 1) T*0J = 30(2» +l)[Pn(cos ý + Ů) - Pn(cos^)\ »(» + IJK ' ■ (9-3) ANTENNA THEORY 473 From (2) and (1 I) we obtain the remaining intensities rnf(r,B) = -F\Z ^^r~ Pl(coS 8), t * ti(n+l)Kn(iM iu0*E, m rE4» - ^/':—Pk(fxa 8). „=i nk„(ipa) The conduction current in the sphere, flowing in the direction of decreasing coordinate d, is 00 1(9) = E ln(0), h(8) = lirAJMJ/ sin 9 Pl(cos 9), (9-4) where Mn IS the radial admittance of the nth zonal wave at the surface of the sphere k«(i0a) Jn(Pa) - ifam Vt'n($a) r,l$'n(8i) + J§(0aj]' The conjugate complex power flow across the aperture of the cone is ~*f aEUa,8)I(9)d9 18O0x^2 " 2» + 1 - = V2 E : , n M„[P„(cos + + i» - P.(cos W2. K f*=i»(»+1) From this we obtain the average admittance at the aperture of the cone line Yav = 2\I>*/F2. This value may be taken as the approximate value of the terminal admittance Yt — Yav The exact value would be obtained if we used the exact expression for Ee(a,9) instead of the approximation (1). Let us compare this expression with the ratio of the total current I($) flowing across the edge of the upper hemisphere to the transverse voltage IMA 60* " In 4- 1 , - -~ = E , . n Mn sin * Pi (cos ^)[P„(cos # + tf) - Pn(cos m (9-5) V- K ,= i n{n + 1) When i^ + t?/2 = tt/2 this ratio represents one component of the terminal admittance as defined by (2-3). For other values of \p we should take the average value* [Ity) + Ity + &)]/lF' The characteristic impedance of the cone line is / Ů \b + ů\ K = 60 log I cot - tan —-— 1 60 log + sin (, ů\ *' si" I f + 2 I - sin - *The inequality of I(p) and I(p + t?) is due to secondary waves and affects the input impedance only indirectly through these waves. For the principal current 17-1 l'JwKCTHtJMA(;NK'l'K: WAVES ClUr. 11 When t? is small compared with ip + then wc have approximately 120 sin K = Hh'l sin + ^ sin + 0 In this case we also have, except when n is large, P„(coS pff| - P„(cos $ = #Pl |^cos + 01. Substituting these approximations in Yt = Yav and in (5), we obtain (9-6) (9-7) — = 7t sin ^ sin ^ + - j E + a^SfiW ^ [cos U + - J . The two expressions tend to become equal as approaches zero. For large values of ft the original terms in the expansions fbi Yt and I(^)/l-' must be retained since (6) will no longer be a good approximation. While the individual terms are small, the approximation (6) will lead to a divergent series if used for unrestricted values of ft. Thus we have the following approximate expression for the terminal admittance ^ 36007T * 2n + l #E »-!»(»+ 1) M4P„(cos $ + (?) - P„ (cos (9-8) When computing this expression the first few terms may be replaced by the corresponding terms in the series (7) for Yt. The terminal admittance tends to infinity as d approaches zero; this is as it should be since the capacitance between the two hemispheres will increase indefinitely as the distance between the edges diminishes. On the other hand, Fig. 11.4 shows that when n increases, GK rapidly approaches zero and only a few terms are needed to compute G{ quite accurately. If i? = (it — only the odd terms in (8) remain; thus 72007T £ 4th + 3 M / it- $\~ For small values of the real part may conveniently be computed from (7); thus (9-9) T, 4ot + 3 0a). 2^(m + l)(2m+l) The values of the associated Legendre functions and the A's are (9-10) 2(« + &! 1 ■ 3 ■ 5 ■ ■ ■ (2m + 1) 2-4-6 ■ 2w #8o4y1-- ANTENNA TIlEoUV 3()(4^4-3)n„,,^eos'r~ 2iuea log X/j as $ —> 0. The following direct method of proving this asymptotic property is also useful for computing Bt from (8). For large values of n we may use the asymptotic formulae P„(cos 8) Assuming that f, + &/2 = x/2, substituting in the imaginary part of (8), rejecting the first term, expanding the coefficient in a power series, and retaining only the principal term, we obtain * 1 -cos2m# ( I. B'm W fi: mm Kog § + As m increases the difference between the corresponding terms in the exact series for Bt and in the above approximate series approaches zero as V£wi% hence the series formed by subtracting one of these series from the other converges more rapidly than-the original series. An approximate formula for the conductance of a large spherical antenna can also be obtained by regarding the sphere as a 180-degree wedge transmission line; thus Mu ELECTROMAGNETIC WAVES" Chap, i I GtttiBa/\2<)1 -nv/Mtx. lIiíh approximation it fairly good even for moderate valuei of 8a. The outstanding feature of the spherical antenna which is shared by other " broad surface " radiators is the comparatively slow variation of lis impedance with frequency as contrasted with the rapid variation of the impedance of thin wires. 11.10. The Reciprocity Theorem The Reciprocity Theorem may be briefly stated as follows: the positions of an impedanceless generator and ammeter may be interchanged without affecting the ammeter reading. One type of proof is very similar to the proof of the Reciprocity Theorem for electric networks (see section 5.1). In this section we shall prove the theorem first for transmission lines and then for antennas. Consider a transmission line of length / and let E\(x) and E-2(x) be two distributions of applied series electromotive forces; then 5=-Z71 + £lWí I --YTU i 2 rrr 1 r- / \ Čil — =-ZI2 + E2(x), -£--YV* Multiplying the first equation by I2, the last by Vx and adding, we obtain d dx fFJ2) = -Zhl2 - YVXV2 + EA2. Integrating from x = 0 to x = /, and rearranging the terms, we ha.ve f Exh dx = f ZIJ2 dx + fl YVXV2 dx + FJ2 t/rj i/q «/n The first two terms on the right are obviously symmetric in the subscripts; the last is also symmetric since Fyl2 = Z(/)/,(/)/2(/) - Z(0)h{Q)I2(0), where 2(0) and Z(l) are the terminal impedances. Therefore JElI2 dx = f E2Ii dx. This is the general Reciprocity Theorem for transmission lines. In the special case when Ex(x) and E2(x) are concentrated in infinitely short intervals at x = £1 and x = £2> we have J*» /»fe+o Eidx = l1(i2) / E2(x)dx, «1-0 ^d-o ANTENNA THEORY •177 .a If the impressed electromotive forces are equal, then J2(£i) = Ii(%2)-In three dimensions we write curl Ei = —Mi - iwnHi, curl H\ — (g + iwe)Eu curl E2 = — M2 — iwy.H2, curl H2 = (g + iwt)E2. Multiplying scalarly the first equation by H2, the last by £1, and subtracting,* we have Ex - curl H2- H2- curl £, = Ml ' H2 + iu>Hx ■ H2 + (g + ft««)Ei • E2. The left side equals div (H2 X £1); multiplying by dv, integrating over a volume (v), replacing the volume integral of the divergence by the surface integral of the normal component, and rearranging the terms, we have JJfMx-Ihdv^- J j Jiic/xHx ■ H2 dv - f ff(g+ -E2dv + ff(H2X Et)n dS, where (S) is the boundary of (0). The first two terms on the right are symmetric in the subscripts. The last term is also symmetric since we are free to choose (0) as an infinite sphere, in which case EB = vHp, Ev = — rjHe, and consequently and j j (H2 X Et)«JS = -1, ff (He,iHe.2 + Hv,iHvA) dS, j j JMx-H2dv = J f' Jm2- Hxdv. (10-1) This equation holds even if g, ft, and e are functions of position. Consider now a special case of two antennas (Fig. 11.34) energized at points A and B-. Two voltages Fx (A) and F2(B) may be applied by means of two infinitely small magnetic current loops round points A and B, carrying the following magnetic currents KX(A) = -Fx{A)y K2(B) = -F2(B). Under these conditions (1) becomes Fx(A)I2(A) = F2{B)h{B). * Subtracting rather than adding since the electromagnetic equations are antisymmetric. 478 ELECTRl (MAGNETIC WAVES Chap, m ANTENNA TUB >RY The above is the most Irequenlly used reciprocity theorem. Another theorem is: the positions of a generator and a voltmeter, both of infinite impedance, may be interchanged without affecting the voltmeter reading. The more general form of this theorem for transmission lines is fl f,V2dx = t fA\dx, where fi(x) and /a(-v) are two current distributions applied in shunt with the line. Similarly in the three dimensional case we have ////a One corollary of the Reciprocity Theorem is: the directivity patterns of an antenna are the same whether the antenna is used as a transmitter or as a receiver. This follows at once from the definition of such patterns. The directive pattern of a transmitter is explored at an " infinite " distance by a tuned doublet dy^ so oriented that maximum power is received by the doublet. The directive pattern of a receiver is found by obtaining its re-Fie,. 11.34. Illustrating the Reci- Sponse to plane waves arriving from differ-piocity eoiem. g^jrectjonSj tne eleccrjc vectors of these waves being so oriented that maximum power is received by the antenna. 11.11. Receiving Antennas In the case of a receiving antenna the electromotive force is impressed at all points of the antenna and the complete theory of a receiving antenna depends on the solution of the most general transmitter problem. On the other hand, the solution for a transmitting antenna energized at the center is sufficient, in view of the Reciprocity Theorem, to enable us to find the performance of the same antenna when the electromotive force is removed and the internal impedance of the generator is used as a load. Thus consider an antenna and a tuned current element at a large distance from it (Fig. 11.35), and suppose first that the antenna is being used as a transmitter. Let p be the moment of the current distribution in the antenna when an electromotive force V is acting at A, p = fl(z) dz. The voltage V impressed on the element and hence the current in it |B Fig. 11.35. Illustrating the Reciprocity Theorem. Vv(B) - 2\r irip!e-il3r 2\rR l wliere K is the radiation resistance of the element. Now let the current element act as transmitter with the antenna as receiver, then the current in the element and consequently the electric intensity impressed on the antenna are V _ hVlc-^ h(B) = -, E- 2\rR By the Reciprocity Theorem the current at A in response to this field is equal to I\{B); expressing this in terms of E, we have h(A) = h(B) = pE ~V if /(a) dz Eh V Za + ZL' where h is the effective height of the antenna, and ZA, ZL are respectively the antenna and load impedances. Thus the current in the antenna when used as a receiver has been expressed in terms of the current distribution in the antenna when used as a transmitter. In applying the Reciprocity Theorem care should be taken not to change the impedances. INK IMPEDANCE CONCEPT CHAPTER XII The Impedance Concept 12.1. In Retrospect Most concepts grow with their use. Originally the word " number " meant what we now call " integer "; then in successive steps the meaning has been broadened to include rational fractions, irrational fractions, negative numbers, and finally complex numbers. The impedance concept is no exception and it has grown considerably in the half-century of its existence. The term " impedance " was proposed in 1886 by Oliver Heaviside for the voltage/current amplitude ratio in a circuit comprised of a resistor and an inductor.* In 1889 a further impetus to its use in the same sense was given by Oliver Lodge.f Three years later F. Bedell and A. Crehore proposed the term " impediment " for a similar ratio in a circuit including a capacitor as well as a resistor and an inductor. However, there was no real need for a separate term and the word " impedance " was soon used in a broader sense. It was not very long before it was used for any voltage/current ratio, expressed as a complex number with its absolute value equal to the amplitude ratio and its phase to the phase difference between the voltage and current. Tt is outside the scope of the present book to follow the historical development of the impedance concept and the above remarks have been made only to give a probable date of its inception. What concerns us primarily is the actual meaning of the concept and its uses. First let us consider stationary fields and the simpler concept resistance. The resistance of a conductor AB (Fig. 12.1) is defined as F/I, where V is the electromotive force along AB and I is the current in the conductor. The current in a B the conductor is the same across all cross-sections and Fig. 12.1. A wire be- the voltage between the terminals AB is independent tween two termm.i Q£ tjle patn along which the electric intensity is integrated. Thus our definition of the resistance requires no further qualification. The situation becomes quite different when an alternating voltage is applied between the terminals A and B. The voltage depends on the path joining the terminals and the current varies along the conductor. * The Electrician, July 23, 1886, p. 212. t Electrical Review, May 3, 1889. 480 If the distance between the terminals is small (Fig. 12.2), the voltage is nearly the same for paths going " more or less directly " from A to B and the current flowing out of A, let us say, is nearly equal to the current flowing into B. The input impedance is then defined as the voltage divided1 by the input current. In practice, the distance between the terminals is " small " if the voltage is nearly independent of the path from one terminal to the other. In theory we might be tempted to assume that the terminals are infinitely close; but unfortunately the capacitance, and hence the admittance, between terminals of finite size tends to infinity as the distance between them approaches zero. This is a purely theoretical situation; in practice the distance is merely small and the capacitance is also usually small. A way out of this difficulty is found by assuming that the terminals are tapered off to mere points (Fig. 12.3). If such conical terminals are coaxial, then the admittance between them, ignoring the rest of the circuit, is Fig. 12.2 whose are close together. Fig. 12.3. A wire with pointed terminals which can be brought infinitely close together without making the capacitance between the terminals infinite. rVV + h2 60X log 1 4- cos ý sin ý 60\ log h 4- -vV + h2 (1-1) where h and a are respectively the height and the maximum radius of each cone. If h = a, we have Wla , (1_2) Y = 7^ = /0.O84O - mhos. 60X log (1 4- V2) This admittance is very small unless the waves are very short; even if it is doubled or trebled by increasing h, it will have a negligible effect on the impedance of the circuit unless the latter is very large. If a is kept constant while h approaches zero, we have ^ = /0.0524 4; 60XA XA (1-3) this admittance is equal to the admittance of two parallel circular discs of radius a, separated by distance h. As h approaches zero its value increases indefinitely but, except for very short waves, h has to be very small indeed before the impedance of the entire circuit is noticeably affected. The conception of impedance does not depend on the existence of ter- 482 ELECTROMAGNETIC WAVES i in o THE IMPEDANCE CONCEPT ■ih;i initials. Thus in dealing with the transmission «>l waves on perleť 1:1 y eon ducting parallel wires (Fig. 12.4), the series impedance per unit length "I the transmission line is defined as the following ratio Vbc - Pad Z = / a ab. (Ill a B Fig. 12.4. Illustrating the impedance per unit length of a transmission line. In defining the transverse voltage Vnc the path of integration must hi D c restricted to the plane perpendicular to I he " * ' wires; it does not have to lead directly front one wire to the other and may deviate con sidcrably from a straight path as long as ii remains in the transverse plane; but it should not leave the plane. The transverse voltagt can be measured directly only when the separa tion between the wires is small and the im pedance of the leads connecting b and c with the terminals of the volt meter does not appreciably affect the measurement. The impedance per unit length of the parallel pair of wires as defim .1 above does not have quite the same meaning as the resistance per unil length of a wire, or the impedance per unit length of a long coil. In tilt latter cases there exist actual voltages between two points on the wire or between two turns of the coil, and the impedance is similar to the input impedance of a circuit. But in the transmission line consisting of pel fectly conducting wires, the voltage between a and b along the wire i zero; so is the voltage between c and d. Yet the line as a whole acts i if it had an impedance between a " terminal " a, d and another terminal b, c. The next step in extending the impedance concept was to include ratios of electric and magnetic intensities. This extension has served to unify many diverse problems concerning the reflection of electromagnetic waves so that each problem has become a special case of a general problem. Con sider for example a metal tube (or a coaxial pair) filled with two homogeneous dielectrics and let the boundary between the dielectrics be perpendicular to the axis of the tube (Fig. 12.5). For each transmission mode the field distribution in transverse -mff/m/i/Mulmm. planes is independent of the constants of the di- 12.5. Reflection In electric and the reflection and transmission co- ?.ave ?uides at a simP1' // / t I discontinuity, efficients depend solely on the ratio Kz jKz of the wave impedances in the direction of the tube. The same general formula applies when plane waves are incident normally or obliquely to the plain interface between two media, provided either E or H is parallel to tin W//Ě interface; it applies when cylindrical waves are incident normally on cylindrical bodies and when spherical waves are incident normally on spherical bodies. The impedance normal to a surface separating two media is defined as the ratio of the tangential electric intensity to the tangential magnetic intensity. The impedance normal to a thin film with a sheet of infinite impedance just behind it is equal to the surface impedance of the film itself, that is, the ratio of the electric intensity to the current density. In general the only practical means of approximating in effect a sheet of infinite impedance behind the film is to place a good conductor a quarter wavelength (or an odd number of quarter wavelengths) behind the film. The wavelength in question should be that in the direction normal to the film. Any wave guide may be terminated in a resistance film, normal to its axis, and designed to absorb all the energy carried by a wave traveling in one particular transmission mode; we need only make the surface resistance Rs of the film equal to the wave impedance of the incoming wave Ra = Kg (1-5) The surface resistance of a thin film of thickness t is substantially equal to its d-c resistance Rs — 1/gt. Since the conductivity of very thin films is not necessarily equal to the conductivity of the material of the film in bulk, Rs should be measured. For the dominant mode in air-filled coaxial pairs we have Ks — 120tt and the impedances are matched when Rs = 1207T. The d-c resistance of an annular film is R = ST log ~ : (1-6) where a and b are the radii of the boundaries of the film. If the radii of the coaxial pair are equal to a and b, then the matching condition becomes R = K, where K is the integrated characteristic impedance of the coaxial pair. In the case of circular wave guides the matching resistance films are circular discs. To determine the surface resistance of these discs the d-c resistance between a pair of concentric circular electrodes in contact with the discs may be measured. Since from (6) we'have Rs - 2ttR log b> 628833 IM i.i ECTROMAGNETIC WAVES < 'ii.ii-. ii where a and l> arc flic radii of die electrodes, I In- matching condition (5) may be expressed in terms of this measured resistance R - — Jog - . 2ir a For transverse magnetic and transverse electric waves this becomes respectively R = 2x log 2x^1 log ■ In the case of rectangular guides the matching films are rectangular. If the dimensions of the rectangle are a and b and the d-c resistance R i measured between the sides having the length a, then R = (b/a)R„ and the matching condition (5) becomes R = (b/a)Kz. We have seen that a sufficiently thick conductor can he regarded as an impedance sheet whose surface impedance is equal to the intrinsic impedance ij of the conductor. If the thickness / of the conductor is small compared with the radius of curvature, then the surface impedance is fj coth at. When an actual conductor is replaced by an impedance sheet, consisting of a sheet of finite impedance over a sheet of infinite impedance, many problems are simplified. No energy can flow across an infinite impedance and the media on either side of die impedance sheet may be treated as electrorriagnetically independent. Also it becomes unnecessary to consider the field in the conductor itself. 12.2. Wave Propagation between Two Impedance Sheets Consider two parallel impedance sheets CD and C'D' (Fig. 12.6) and let die surface impedance of each be Z. Then the field will be symmetric x C )il i) u 11 urn i n milium 11 n i niiuiiuim iiiiD' Fic. 12.6. Parallel impedance sheets. with respect to the plane AB midway between the sheets. If we confine our attention to fields in which the //-lines are parallel to the jy-axis, the non-vanishing field intensities will be Ex, £2, Hv. From the assumed symmetry about the central plane we conclude that the electric lines arc either perpendicular or parallel to this plane. In the former case a perfect i ill. IMPEDANCE ct INCEPT 485 conductor and in the latter a sheet of infinite impedance may be inserted at the plane, without disturbing the field distribution. When the jta-plano is a perfect conductor, then for progressive waves in the positive z-dircetion we have Hv — J cosh yx e~ -Vz r2 + T2 = - (2 I) Ex = J~-coshyxe Vs, Ez = /-r-sinlx yxe T:, toe toe where T and y are respectively the longitudinal and transverse propagation constants and / is the conduction current density in AB. The condition for natural waves is then -— tanh yb = — Z, toe (2-2) where b is the distance between AB and CD. Thus we have the following expressions for the propagation constants w * \ b2 where w is a root of the following equation 13 tanh w = —tiaSZ = ß\ ißbZ (2-3) (2-4) When Z = 0, we have to = mi. If n = 0, y = 0, Ez vanishes identically and we have the principal wave. If | BbZ \ is small compared with tj, then one root of (4) is small compared with unity and we have approximately w 2 = ifibZ r = „ ißZ j,. , , • _ toe (2-5) We have already seen that this formula can be obtained directly from the integral equations of electromagnetic induction by assuming that the longitudinal displacement current is zero. Actually the longitudinal electric intensity varies almost linearly with x and the magnetic intensity is approximately constant Ez = -ZJ Hv = J ( _ ißbZx2\ V 2, b2) ,— Yz Then the current density in the sheet CD is -/ 4- (ißbZ/2v)J; thus this density is slightly different from the current density in the sheet AB. ELECTRI (MAGNETIC WAVES' c'iiai. 12 THE IMPEDANCE O >NCEPT ■I «7 At the other exlreme when i lie right hand side of (4) is infinite we have W = i(w + })ir, and when the right-hand side is large hut finite, then w = i(n + For the principal wave n = 0 and w = /V/2 — irr)/28bZ. The longitudinal electric intensity is nearly sinusoidal, rising from zero at the plane AB to a maximum at the plane CD. The current in the sheet CD nearly vanishes and the dielectric between the planes serves as the " return circuit." In solving (4) for intermediate values of the right-hand side we shall consider only the case in which Z is a pure resistance Z = R, k = BbR bR 60X' the numerical coefficient in the last equation corresponding to free space between the planes. Since w is complex, we write w = u + iv. Substituting in (4) and separating the real and imaginary parts, we have u sinh u cos v — v cosh u sin v = k sinh u sin v, v sinh u cos v + u cosh u sin v = —k cosh u cos v. Eliminating k, we obtain u sinh 2« = v sin 2v. (2-6) Since u is essentially real, the left side of this equation is positive and the values of v are confined to the following intervals 7T 3x 0< v<-, * between two perfectly conducting phines (fig. 12.9) carrying equal currents in the same direction. At the plane of the film the magnetic intensity vanishes and we have in effect a surface of infinite impedance. The above equations will apply to this case if _ _ R = 2RBba. ential equations (4.12-18) appropriate _ _ to the actual problem. For each trans- ^ ]2g Resifitancc sheet betwcctl mission mode these equations can be perfectly conducting planes, converted into the conventional form of transmission line equations in which the .v-coordinate is suppressed. Eliminating Ez we have dz = -\M»-i^2)Hv> -67 = -1íj}ěEx Integrating each of these equations with respect to x from x = 0 to x = b, we have dV dz /itůfibp {v2 — u2)p 2uvp\ dl __ _ ioxa ^ \ a iwtab coeab / ' dz pb ' v = Jqe*Jx> P = jjjo Hvdx- I is the conduction current in the plane AB between y — 0 andy = a; the constant p is unity when Hy is uniform. When k is small, p is nearly unity, the second term in the expression for the distributed series impedance is negligible, the third term becomes R/a, and the equations reduce to the engineering form based on neglecting the longitudinal displacement currents. When k is comparable to unity, the expressions for the distributed series impedance and shunt admittance become complicated; as k becomes large, these expressions are again simplified. If the AB-p\nne is a surface of infinite impedance, the equation for the transverse propagation constant becomes -— coth ~yb — —Z. tost This equation has no solution in the vicinity of y = 0 and it defines transmission modes similar to the higher transmission modes in the preceding case 490 ELECT U< )MA(iNliTIC WAVES Chap. i:. 12.3. On Impedance and Reflection of Waves at Certain Irregidaritics in Wave Guides Let us consider a wave guide of rectangular cross-section (Fig. 12.10) bounded by two planes of zero impedance ,v = 0, x = a, and two planes of infinite impedance y = 0, y — b. Such a wave guide is an idealization of a pair of parallel conducting strips; the infinite impedance sheets serve to eliminate the edge effect, and thus simplify the mathematical problem. The transverse electric intensity in the plane z = 0 is, in general, an arbitrary function of x and. are of zero Impedance }'■ This function determines completely the field and the other two of }n the wave guide. We shall take a simple case as infinite impedance. an ex;lmp]e and assume fh.at Es = 0, Ey(x,y,0) = f(y). In this case Hy = Hz — 0. Then for an infinitely long guide the field intensities are of the following form Ev = Z En cos 22 e-Tn*} Tn = S - »=o b V e> o Fig. 12.10. Rectangular wave guide; two faces „ ftiry — lZ MnEn cos —r- e n=0 b (3-1) = — 7—7 Y, nMJin sin ^ e~v^. In the plane 2 = 0 we have do £» — 2- cos //a, = — T, MHEn cos n = 0 7i7ry Consequently the conjugate complex power flow across z = 0 is ** = & {MoEQEt % 11 M.nEnE*i). (3-2) (3-3) The coefficients £„ are obtained by expanding/(_y) in a cosine series of the form (2); thus E^-jj(y)dy = ~, En. =-J j(y) cos—y~dy, (3-1) where Wis the transverse voltage between the conducting strips. Thus the power flow across the plane z = 0 may be expressed in terms of the transverse voltage and the form of its distribution over the plane. The input impedance may now be defined so that = \YiVW. (3-5) ii ie tMPEDANX F 1 < »NCEPT •I') I Substituting from (;l) in (3) and using (5), we obtain Yl~-Vb + 12r,bkrnE0E$- (3~6) The first term is the characteristic admittance of the guide to the principal wave. If X > 2b, the second term is imaginary. As X increases indefinitely we have nir B lb b r„ «x and the reactive part of the input admittance approaches zero regardless of the form of the voltage distribution. Thus for " low frequencies " the input impedance oi the guide is nearly equal to its characteristic impedance to the principal wave. Let us consider a specific numerical example in which the electric intensity Ey in the plane z = 0 is zero except in the interval d < y < d 4- s, Fig. 12.11. Two wave guides joined together. where it is Vjs. This distribution approximates that at the mouth of a wave guide of height s, joining another wave guide of larger height b (Fig. 12.11). In this case we have . 2b r r =- s 0 nirs L mr(d -4- s) . met sin ■---— sin —- b b 1 = 4b . mrs tit (2d + s) —- sin —7- cos mrs 2b 2b When d + s/2 = b/2, we obtain E'2m ^2m+l = 0, E0 (-)»-*> sin ^. niTTS b Substituting in (6), we have Y,- = G; + iBi} G,: = M = aj-nb, and 2?W7r.^ M 2„2 zC IT S , (, 2mirs\ T2toot2 As the wavelength increases, this ratio becomes approximately 2ntKS b3 M -K~s2\,ti 1 — cos m 492 II IX "I'Kt MAGNETIC WAVES Chap. 13 By (3.7-53) wc have M "ft + + iwCi M If j = 0.01*, then \\ b J c-",loel-°-338 + ii(T)2 When / = 1000 and b = 0.02, 2b/\ = 1.3 X 10~7. Bi/M = 5.6 X 10~7. Even if/is raised to 106, the ratio of the susceptance to the characteristic admittance is still small. But when the frequency becomes so high that X is comparable to b, then the susceptance may become appreciable. Consider now a wave guide of uniform rectangular cross-section with a metal diaphragm or iris across it (Fig. 12.12). Again we assume that two faces of the guide are of zero impedance and the other two of infinite impedance. Let a wave in the dominant mode impinge on the iris. The re- Fig. 12.12. An iris in a rectangular wave guide. fleeted and transmitted waves will consist of higher order waves as well as the dominant one. Let the electric intensity of the incident wave in the plane of the iris (the xy plane), be Eh. The electric and magnetic intensities of the transmitted field are of the form (2). Since the total electric intensity tangential to the iris is continuous across the plane of the latter, the intensity of the reflected wave in this plane is E% = (£o - Eh) + Z En cos . From this we obtain the magnetic intensity of the reflected field HI = M0(E0 - £S) + £ MnEn cos ^ . Ti-i b The magnetic intensity is continuous over the aperture of the iris; there we have Ht + Hx = Hx, so that -M0E0 + M0(E0 -2$)+ E MnEn cos ~^ = - £ MnEn cos ^ 71=1 b n = \i b THE IMPEDANCE CONCEIT 493 Transposing the terms, we obtain Multiplying by ntty M0E0 = E MnEn cos »i=0 * mry E El cos n=o b (3-7) (3-8) and integrating over the iris,* we have MoEiElab = M0E0Etab + \ab £ MnEnE%, n=l Eh 1 _ . , 1 " Mn E„E* En 1> 2n = lMo EUEq This equation represents the reciprocal of the transmission coefficient across the iris for the dominant wave. Now the admittance (6) of either half of the wave guide as seen from the iris may be represented in the following form h ' M 2^iM0E0Et Consequently we have l/p = 1 + Y/M. Comparing with (7.13-10) we find that the transmission coefficient for the dominant wave is the same as if the iris acted as an admittance Ys = 2Y~ in shunt with a transmission line whose characteristic admittance is M. The impedances of the two faces of the iris are thus in parallel, as is evident by inspection. That the iris should act as an admittance in shunt with the guide could have been assumed to begin with, since the voltage of the dominant wave is continuous at the iris and the current is discontinuous. When X > 2b, then Y and Ys are pure reactances. But when X < 2b, Ys has in general a real component. While there is no loss of power at the iris, some power is carried beyond the iris and reflected back in other modes than the dominant; this represents an effective loss of power in the dominant wave. The theory of transverse irises in wave guides in which all four faces are perfect conductors is similar to the above. Thus if the edges of the iris are parallel to the £-vector of the dominant wave, then in the range a < X < 2a, the iris acts as an inductive reactance in shunt with the guide. If the £-vector is perpendicular to the edges, the effective reactance is negative and the iris acts as a capacitor. Similarly a transverse wire par- * This is permissible even though (7) is true only over the aperture since (8) vanishes outside the aperture. 494 ELECTUoma(;nktk' waves Ciiai'. 12 allel to the /'-'-vector acts as a shunt inductance. These phenomena are not peculiar to high frequencies. When a conventional inductance coil or a capacitor is inserted in shunt with a transmission line consisting of parallel wires and operated at low frequencies, the field of the dominant wave is disturbed and the local field in and around the inserted structure abstracts some energy from the dominant wave during one half cycle and returns it during the other half; the dominant wave therefore suffers reflection. 12.4. The Impedance Seen by a Transverse Wire in a Rectangular Wave Guide The impedance seen by a transverse wire in a rectangular wave guide (Fig. 12.13) is of interest because it approximates the impedance seen by a coaxial pair (Fig. 12.14) when the shorter side b is small compared with a quarter wavelength so that the current distribution in the wire is substantially uniform. There are at least three methods available for the calcula- Fie.. 12.13. I Fig. 12.14. tion of this impedance. We can express the free space field of the current I in the wire by a contour integral of the type given in problem 10.9 and add to it another field, expressed by a similar integral, so as to satisfy the boundary conditions at the surface of the guide. The contour integrals are then evaluated. Another method consists in considering the total field in the guide as due to superposition of the free space fields of the current in the wire and its images in the walls of the guide. A third method is based on the following considerations. If the radius of the wire is small, then the field at the surface of the wire is nearly equal to that produced by an infinitely thin current filament on the axis of the wire. The latter filament can be regarded as the limit of a current strip; and the current strip represents a known discontinuity in H, which can be expanded in a series of the form E#n sin nwx/a, appropriate to T£-waves. Then the complete field is determined and the impedance is obtained as the ratio — bEy/I, where Ev is the intensity on the surface of the wire. Thus if the guide extends to infinity in both directions, then by the second method we obtain for the real and imaginary parts of the impedance seen THE imit.i)ANCE roNt'F.n' 495 from a wire of radius r respectively 00 R = luWoOSr) + 21 M2t,pa) - E M2npa + 2Bd) - E J0(2npa + 8a - 2pd)]; X = -ivBi[No(fir) + 2 E N0(2nBa) 00 CO - E N0(2nBa + 2Bd) - E N0(2npa + pa - 2pd)]. Thus in the case of thin wires the resistance component is nearly independent of the radius and the reactance is a constant depending on a and d plus a logarithmic function of r. In fact we have X(r2) -X(n) = vIoB--X r2 A simple expression for R can be obtained either by the first or by the third method. Thus in the frequency range between the absolute cut-off and the next higher it will be found that R a \ 4«V As the frequency increases and passes the cut-off frequencies for the successive TE-waves other terms will be added and we shall have in general R = Ri + i?2 + i?3 H----, where n n2\2Y112 . , mrd is the resistance corresponding to the ?zth transmission mode. The above impedance is that looking into the wave guide extending to infinity in both directions. The two semi-infinite halves of the guide are in parallel and consequently the impedance looking into either half is 2Z. If now a < X < 2a, so that only the dominant wave carries power to any distance along the guide, and if we have a conducting piston at distance / from the axis of the wire, then the resistance component of the impedance seen by the wire becomes vb/ 4xA/t X2\"1/2 • a«/ Vll'C^TA1"^ 8in -, where X is the wavelength along the guide. The additional factor comes in because of reflection of the dominant wave by the piston. On the other 496 Kl.k(Tl« MAGNETIC waves Chap, 12 hand (In- reactance represents the local field round I lie wire and is suhstan tially unaffected l>y the piston unless it is quite close to the wire. The methods outlined above can be used equally well for calculating the mutual impedance between two parallel wires in the guide and hence for the solution of any problem involving a system of parallel wires. The above results can be generalized by assuming a sinusoidal distribution of current in the wire. Problems 1.1. Prove that if ^ is the angle between two directions (8\,; zir = A„ sin 0 -j- Az cos 0, vis = A„ cos 8 — Az sin 0; zip = yir sin 0 + ^8 cos 8, Az = zfr cos 8 — Ae sin 0. 1.3. Orthogonal curvilinear cylindrical coordinates («,£>,z) may be defined by-means of functions of a complex variable » + /'» = F(x + iy), x + iy = J{u + iv), z Prove that ds* = |/'(« + «0 I V«2 + **J + 1.4. Bicylindrical coordinates (u,ů,z) may be defined as follows: , .<, , a+ (x+iy) u + tfl «+»>= log-;——rr , * + ty = « tanh —-— , z = z. Show that = I log (a + x)* + y* (a - *)2 + .y2' sinh u ů = tan" M K3NETIC WAVES 1.5. Elliptic coordinates (ii,i'),z) may lie defined liy the following equations: x+ iy = I cosh (// H- id), 2=2, Show that x = I cosh u cos d, y — I smb. U sin rfji = 1/2(cosh In - cos 2#)( I, n=l»P pi cos»*>i= £ ( — )»-»» —-- TO=o ml (« - /"-mpm cos wco. If p < I, then p and / are interchanged in the first equation; the second equation remains unaltered. 2.6. Obtain the following identities 1 C ept 1 C e"' 2« J(o p2 2tt« J(C)i5K+l = /, />0; = nrt>0- 2.7. Let F(t) be a function which vanishes for / < 0, starts rising linearly at / ■= 0 and reaches unity in r seconds; subsequently it remains constant. Show that F(t) 1 r 1 - e-pT _ _- I -_ evt 2irhJ{C) p'2 dp. 5(H) ELECTROMAGNETIC WAVES Note that this Integral is the difference of two integrals, each representing a linear function of/; one function starts from / = 0 and the other from / m t. Fig. 1 2.8. Show that for the function defined by Fig. 2 1 r 1 — e~-»T — e-p(T+T> + e-pvr+M F\t) = y~T~ I---—T-- e* dp. Fig. 2 2.9. Show that the spectrum of a sinusoid of unit amplitude and finite duration F(l) = 0, f < 0; P cos (W + jp), 0 < / < T; = o, />r; JO) _ p-(p-ia1T J _ ď-(p+tu)r "I eiv H-- Note that where 1 p - g-fr* 4jr» |_ p — ÍCÚ ' p + ÍC0 Si(fi#) = r-.l •-- + —— ). 4xí \p — tia p 4- iu/ The spectrum Si is independent of the duration T of the sinusoid; it is the spectrum of the sinusoid starting at t = 0 and continuing indefinitely. The second term in S(p) is a similar sinusoid, starting at / = T with just the right phase to cancel the first sinusoid ever after. Thus, for the semi-infinite sinusoid beginning at / = 0, I'KOlil.l'.MS 501 have 4TriJ(c)\p — ioi p + ice/ 2ir/J(C.) p cos (p — o> sin tp p2+^ epi dp. V Š 2.10. For a circuit consisting of a resistor and an inductor in series Z{p) = R + /)/.. I f a steady electromotive force V is impressed at / = 0 and is discontinued at / = r, then /(/) *-(l- 0 < / < r, K. and ic If ^?t/L is large, then for / > r 7(7) = afOIWH r l Fig. 3 2.11. For a circuit consisting of a resistor and a capacitor in scries Z(p) = R+l/pC. Show that for 0 < t < r, T t and for t > t, 0—\AAAA r. The electric charge could be obtained directly by the contour integral method if, instead of the impedance Z(p), we used the impediment Z(p) = pR + 1/C. 2.12. For a circuit consisting of a resistor, a capacitor, and an inductor in series Z(p) = R + pL + 1/pC. Show that for 0 < / < T V ■ mm r. t lit) = hit) + R + 2piL R + 2p2L' where p\ and pi are the zeros of Zip), and for / > t Ht) = hit) - hit - t). ■vWWv- r l Fig. 5 502 ELECTRl (MAGNETIC WAVES 2.13. An electromotive force r§ =0, / < 0; = V cos (oil + 0; is impressed on a circuit consisting of R, L, and C in series. Show that yeHut+v) ye-i(wt+v) I'p\(p\ cos

1 /(/) = 22 (iw) + 22 (-ho) + (?t+ w2)(i? + 2plL) Fpz(p2 cos (p — oi sin p)(?Pjt + The sum of the first two terms is the real part of Fe^'+^/Ziiw); this is the steady state term and could have been obtained directly. Obtain the solution for the case R = 0 and oi = the natural frequency of the circuit. 2.14. In a transmission line described by (2.10—3) the impressed electromotive force E(x) is a progressive wave Ee~'^ix over a section 0 < x < / and is zero elsewhere. Show that 7T; I--**N»v} lirt J id) E(x) m no y + 0 EeTl, x < 0; Ee~Tx, x> I; F{x) = 2A"(r + iBi) 1 _ f! 2K-(-r + ^i) y£f-'fti £ I; Ee~Tx, Ee~l r2 + p? 2(-r + «p\) 2(Y + iBL) 0 < x < I. Prove these equations first by evaluating the contour integrals and second by the method of section 7.9. Note that y = —iBi is not a pole of the complete integrands and hence contributes nothing either when x < 0 or when x > /; in the region 0 < x < I, the integrand must be split into two terms, and y — — iB\ is a pole and contributes the " forced term." In the second method the interval (0,/) again differs from the remainder of the line in that when x is an interior point the definite integrals must be split at this point because the integrands are different on its two sides; but when x is an exterior point such splitting is unnecessary. Consider also the special case B\ = 0 when the applied electromotive force is uniform, equal to £ in the interval (0,1) and to zero outside this interval. PROBLEMS 2.15. Consider the function shown in big. 6. Show that F(x)-— f — (1 - f - e~3y° -\----)dy 2mJ(c) 7 50.1 In Ju- dy. 'CO 7(1 + ' ■ / The poles of the integrand are given by 7 = 0 and 1 + e-^ = 0; that is, (2m + 1)7t2 To = 0, ym = Hence, F(x) = 0 if x < 0; if x > 0, then , m = 0, ±1, ±2, 1 1 00 I 1 2 00 1 ^) = 2 + «S77/%^2+.„So2^W (2m + \)irx F(x) 1 a a X Fig. 6 2.16. Consider a function/(*) which vanishes identically outside the interval (0,/), so that -">* dx. (1) /(*)= f dee) S(y)er*dy, S(y) = 2rri JO Consider a new function F(x) which vanishes identically for x < 0 and is periodic of period / for x > 0, being equal to/(*) in the interval (0,/). Then, F(x) = f S(y) (1 + e->1 + hence, for x > 0, 5(14 ELECTROMAGNETIC WAVES lf/(#) ia ti real function, let ■ so that 2 r1 4W 2?/7t.v >V c°s ■—j— dx, Then S&S = *«o + £ (k cos + £B sill ^ . (5) r tfW^ " A^M ™" (S) defineS a Pcri0dlc i" ^ Interval (-«,« ) and not only .n(0,=c . Equation (2), on the other hand, define, a function which vanishes for x < 0 and is given by (2) for x > 0 The coefficients a b of the Fourier series can be obtained either from the conven-tional formulae (4) or from (1) and (3). Thus an - ibn m, j flf{x)s-f»* dx, Trl = fel 1 >/o / 2,17. .Consider the function defined by Fig. 7 in the interval (0,/). Prove that in this interval where /(*) = f + | («„ cos 2^f + bn sin |f| , *, = it fsin ^T^+i) _ s;n Wl ^ 2^ „„ 2™ / ,(\ tefeifi 2^ . ^ . 2«x/ ■ A — — cos —---cos t Fig. 7 Prove also that in this interval m. - t + vz ^in cos + | Cos^. I'KOltlT.MS 505 Likewise, prove thut in the same interval ,, , 4A ™ 1 . nits , >iir ( , , A , wtt.v /M = V?«sin17HmTV/ + 2)sm"T- 4.1. Show that the external inductance L, the capacitance C, and the conductance G per unit length of two coaxial cylinders whose radii are a and b,b>a, are b „ 2tt£ * 1 Its L = — log - , C 2-rr « log G = log : 4.2. Show that for two parallel wires whose axes are separated by distance /, large compared with the radii, we have approximately L = ^log^=, C T Vab , G log Vat log Va~b 4.3. Show that if the radii a and b of two metal spheres are small compared with the distance I between their centers, then the capacitance is approximately C = 4ire a b I 4.4. Consider two perfectly conducting concentric spheres and a homogeneous conducting medium between them. Along some radius imagine a filament, insulated from the rest of the medium, and let an impressed electromotive force sustain a steady current / from the outer sphere to the inner. Calculate the field between the spheres, the impressed electromotive force, and the conductance between the spheres. What happens if the medium between the spheres is a perfect dielectric? 4.5. Show that the total force exerted by an electric particle qi moving with velocity »i on a particle qs moving with velocity vn is „ _ ? iff art 2 Ajziis^ia X I'i) X S5 4ire?i2 4**fs Show then that the force between two electric current elements of moments lj± and 1Va is Vhh(ru X h) X k F = 4ir/n (An electric current element is a short filament carrying uniform current.) 5.1. Discuss free and forced oscillations in two inductively coupled simple series circuits. Consider in particular the case of two high Q circuits, tuned to the same frequency and show that in this case there exists a critical coefficient of coupling £ = JLis/VLnZ.22 = l/\^Q\Qi such that for k > £ the current in the secondary circuit passes through two maxima and one minimum while for k ^ £ there is only m mt III (MAGNETIC WAVES OIIC maximum. Show thai if k the relative hand width (in the .same sense a.s for simple tunetl circuits) is V2/QiQ2 if Qi and {)■• arc of the same order of magnitude, but it is approximately 1/Qi if Q\ S> Qi~^> I. 5.2. Thevenin's Theorem: At a pair of accessible terminals any linear network containing one or more impedanceless generators acts as a generator whose electromotive force equals the voltage appearing across the terminals when no load impedance is connected and whose internal impedance is the impedance measured between the terminals if all the generators arc short-circuited. Prove it. 6.1. Consider the wave produced by an electric current element situated at the origin along the z-axis. Let / be the radial current flowing outward through the hemisphere 9 < x/2 and VX>e. the transverse voltage along a meridian (or along any path in the surface r = const.) from the radius 9 = 0 to the radius 9 = tt. Show that V and / satisfy the following transmission equations dV dr c im 2 \ ~ + 7 — -t)l, dl ~dr 6.2. Consider the wave produced by a small electric current loop situated in the jry-plane at the origin. Let K be the radial magnetic current flowing outward through the hemisphere 8 < ir/2 and U the magnetomotive force along a meridian from the radius 0 = 0 to the radius 0 = tt. Show that K and U satisfy the following transmission equations dK — = —twfiirU, dU 'dr = -I — + ---,)K. 6.3. Consider two equally and oppositely charged conductors. The regions subtended by the conductors and bounded by electric lines are called tubes of flow; the regions bounded by equipotential surfaces are equipotential layers. Show that the tubes of flow are in parallel with each other and that the equipotential layers are in series. Hence show that the capacitance of the two conductors is C = where F(s,u,v) dS is the area of the normal cross-section of a typical elementary tube; s is the distance along the lines of flow, and u, v are the coordinates of a point on one of the conductors. The corresponding formula for the conductance is obtained if e is replaced by g. 6.4. Assume a conducting cylinder of radius fl, placed in a uniform electric field normal to the axis of the cylinder. Find the charge distribution on the cylinder. 6.5. Assume a conducting cylinder of radius a and a uniformly charged filament parallel to the cylinder at distance / from the axis. Find the field when the charge per unit length of the cylinder is equal and opposite to that on the filament. Consider two cases. (1) the filament is outside the cylinder, (2) the filament is inside the cylinder. i'kohit.MS 507 6.6. Assume a circular cylinder of radius a whose permeability is ai placed in a medium with permeability nt normally to a uniform magnetic field. Find the field inside lbe ev linder and the reflected field outside the cylinder. 0.7. Si ilve h.4 if the cylinder is replaced by a sphere. O.K. Solve 6.6 if the cylinder is replaced by (1) a sphere, (2) a spherical shell. 6.9. Solve problem 6.6 if the cylinder is replaced by a cylindrical shell. 7.1. Show that the admittance seen by the generator in Fig. 7.4 is Y.mj^lK cosh r{ 4- Zi sinh r&[K cosh T(/ ■ £) + Z2 sinh F(/- £)] and that the impedance seen by the generator in Fig. 7.5 is Z = -jj [K sinh r| + Zi cosh r|PC sinh V (/ f) + Za cosh T(/- £)]. 7.2. Consider a transmission line of length /, terminated at both ends into its characteristic impedance and let the impressed series voltage per unit length be Ee^1, where X is the distance from one end. Find the transverse voltage and longitudinal current V{x) J 2T iLrs- 1 2Y r-7 i T + y ■ 7" r r+7 ,-ru-z) 7.3. Let Zj be connected in shunt with a line at distance / from the input terminals and let the line be terminated in Zi at distance h beyond Z%. Find the input impedance by two methods: (1) using (7.6-2) or (7.6-6), (2) using (7.11-11). 7.4. An attenuator is a device which, when inserted in a transmission line, absorbs power without introducing reflections. Design a symmetric jT-type attenuator and calculate the attenuation ratio. 7.5. Design an attenuator, using series resistors only. 7.6. Discuss resonance in a non-dissipative transmission line shorted at one end and terminated into a capacitor at the other end. Show that if the terminal capacitance C\ is small compared with the total d-c capacitance CI of the line, then the longest resonant wavelength is 4(/+ A), where C\ = Cl\. 7.7. Treat die problem of section 7.S by another method. Starting with the impedances Zj, and Zu looking respectively to the left and to the right from the generator, determine (in the case shown in Fig. 7.4) 4- 0) and — 0) in terms of -/(£); then use (7.4-10) to obtain the voltage and current distribution. Treat similarly the case shown in Fig. 7.5. 8.1. Consider n parallel thin wires and let Em be the electric intensity impressed uniformly on the mth wire. Show that the currents in the wires may be obtained from Z~L Zmklk = Em, m = 1,2, • • ■ n, k where Zm,m is the sum of the internal and external impedances of the mth wire and Zmk = (l/2ir)MpKo(v!mh)) where 4a is the interaxial distance between the wires. .SOU EUJCTromagnktic wavi.cs I'ROW.kms 509 Dismiss I he special case <>C Iwn equal wires cnemi/.ed in parallel (/',', /','•.) and in push-pull (Ei = — E»<= J-ZÍ). Show that if the intcrnxial separation in the latter case is small, then E = |2Zť + (iwju/w) log //«]/, where Z< is the internal impedance per unit length of each wire. 8.2. Prove that in the case of three equal, perfectly conducting, equispaced, coplanar wires, energized in parallel (Ei = £2 = £3) the approximate ratio of the current in the middle wire to that in either of the other wires is 1 — log 2/Iog IIa, where / is the distance between adjacent wires and a is the radius. 8.3. Prove that if the interaxial distances between three equal, parallel wires are small and if E2 = Es = 0, then h + h — — h and a larger fraction of the total current flows in the wire nearest to the first wire. 8.4. Show that the inductance of a solenoid of radius a, coaxial with a perfectly conducting shield of radius b, is L = — |gj , where / is the length, S the area of the cross-section of the solenoid, and N is the total number of turns. Show that the circulating current in the shield is opposite to that in the solenoid and that the current ratio is a4/ (i2 — a2). 8.5. Obtain the exact expressions for the internal impedance of a conducting cylindrical shell: (1) with an external return, (2) with an internal return. Obtain the transfer impedance. 8.6. Taking into consideration only the principal wave discussed in section 8.14, obtain the expressions for the input impedance and the current in a large circular loop and in a rhombus fed at one of its vertices. 8.7. Consider an infinitely long electric current filament, carrying current 7, and a conducting cylinder of radius a whose generators are parallel to the filament. Find the current distribution and the power dissipated in the cylinder on the assumption that the distance I between the filament and the axis of the cylinder is small. In particular consider the high frequency case when the current is near the surface of the cylinder. 8.8. Discuss the dominant transverse magnetic wave in a rectangular wave guide. For this wave magnetic lines form a single set of loops (Fig. 6.20) and the longitudinal electric intensity has only one maximum. Obtain the expressions for the field, the longitudinal current, and the transverse voltage between the axis and the walls of the guide. Show that the cut-off wavelength is Xc = 2ab/Va2 + b2 and that the attenuation constant a = 231(8' + bz)/{qab(a2 + b2) Vl - v\ wllere v = x/\e. Obtain the integrated impedances _abKt Tt2abKs 4(a2 + b-) 4abK, 64(a2 + b2) ' 5rV+ b2)' where the wave impedance at a typical point is Ks = tj Vl — v2. 8.9. In problem 8.8 the longitudinal propagation constant vanishes when X = Xc and then the electric lines become parallel to the guide. Assuming two conducting planes normal to these lines, we obtain a parallelopipedal cavity with free oscillations in it. Show that the energy content is W = mbV2/%c, where c is the dimension of the cavity parallel to E and V is the maximum voltage amplitude. Show that the lota! power loss in the cavity and the Q are £R lr" Vab . W£ Al__**_ W "V L'2 2,\i* + a2)] ' Q 9tX[l + i&^ůr' If b = «, then Q = TtTjf/ft V2(a + 2c). 8.10. Discuss circular magnetic waves (that is waves with circular magnetic lines) in a circular tube. Find the field and show that the cut-off" frequencies are Xc ,m - 27r«/*m where km is a typical zero of/of*). For the lowest cut-off Xc = \3\d, d = 0.76X„, where d = 2a. Show that the attenuation constant is a = 9l/r)a Vl — v2 and that 1 K = *' 4ir' r,i Zwkjx® , K, = ijVl - v2. 48 Vl - v2 and Kw,t = 30 Vl - v2 if the tube is filled For the lowest mode Ky, 1 '■ with air. 8.11. Show that the cut-off frequencies of circular electric waves are given by Xc,m = 2ira/km, where km is a non-vanishing zero of /i(.v). For the lowest mode X„ = 0.820*/. Obtain the attenuation constant a = 9lv2/vaV\ - v2. 8.12. Discuss circular magnetic waves in perfect dielectric " wires." 9.1. Obtain the radiation intensity and the power radiated by a uniform current loop of any radius a: $ = \5ir(8ayi2fl(Ba s\n 0), Mt) dt. 0 9.2. Obtain the radiation intensity and the power radiated by the condenser antenna (by a pair of parallel circular plates, energized from the axis of the condenser so formed): °Vvi(^ sin 0), * = 960ir da r2l}a 9.3. Calculate the power radiated from an open end of a coaxial pair. W where V is the voltage across the open end. iL( s t v2 360\X2log£/a/ 9.4. Obtain the radiation intensity of a rhombic antenna in free space. Consider the case in which progressive waves are established in the antenna and assume that the amplitude is unaffected by radiation. 10.1. Prove the orthogonality of the T-functions corresponding to regions enclosed by perfectly conducting cylindrical surfaces; that is, show that J J t1t1 dS = 0, where (S) is the cross-section of the cylindrical guide and Th T2 are functions corre- 510 ELECTROMAGNETIC WAVES 10. sputiding tn tin- same boundary condition (V - 0 or OT/flu 0) hut to different values of x. 10.2. Consider a uniform plane wave impinging on a conducting wire of small radius a in such a way that 11 is normal to the wire. Let the angle $ between the wire and the direction of the plane wave be arbitrary. Find the scattered wave and the current in the wire. 10.3. Study the normal incidence of a uniform plane wave on a homogeneous cylinder with arbitrary electromagnetic properties and of arbitrary radius a. Consider both cases: (1) £ is parallel to the cylinder, (2) // is parallel to the cylinder. , 10.4. Consider the problem of reflection of uniform plane waves from a perfectly conducting sphere. Find the reflected field and study some special cases. 10.5. Consider a perfectly conducting sphere and a current element in the direction of some radius. Find the field. 10.6. Solve the preceding problem for a small electric current loop coaxial with some radius. 10.7. Obtain the field of a typical transverse current element in a metal tube of circular cross-section. 10.8. Consider a cylindrical cavity of radius a and height h and inside it a uniform electric current filament parallel to the axis of the cavity. Find the impedance seen by the filament. 9. Prove that K0 (C, where G = 60tt'zs2/K-\- and C = 2ta + 60 (s — a)/K'iVa, s being the distance between the axes of the wires and K the characteristic impedance. Show that the resonant wavelength is approximately X = 4/ + 4v0CK = 4/ + (8/tt) a log (s/a) + 240 (s - d)/K. 12.1. Find an approximate expression for the reactance of the iris shown in Fig. 12.11 to the dominant wave in the frequency range between the absolute cut-off and the next higher when all faces of the wave guide are conductors. Consider the case in which the Ti-lines are parallel to the edges of the iris and then the case in which they are normal. Show that in the first case the iris possesses an inductive reactance and in the second capacitive. Questions and Exercises 1. What is the capacitance of a sphere of radius 1 cm. in free space? 1.1 ppf, 2. What is the magnetic intensity inside a conducting wire of radius a carrying a uniform current 7? 7p/2xa2. 3. What is the internal magnetic energy per unit length of a wire carrying a uniformly distributed current? where Li = p/Stt. 4. What is the interna! inductance of a copper wire ? An iron wire whose relative permeability is 100? 0.05 ph, 5 ph per meter. ol if NTH iNS AND EXERCISES 511 5. What is the capacitance between parallel plates in air, one millimeter apart, if the radius of each plate is 10 cm ? 278 ppf (except for the edge effect). 6. What is the capacitance in Ex. 5 if mica is inserted between the metal plates? 7. What is the capacitance per meter of a coaxial pair, with air between the cylinders, for the diameter ratio e? What is the inductance? 55.6 ppf, 0.2 ph. 8. Estimate the external inductance of a circular loop of radius i>, made of wire of radius a. pb log b/a. 9. What is the approximate resistance of a 40 watt electric bulb? 10. What is the motional electromotive force developed in a rectangular wire loop rotating in a uniform magnetic field with the frequency u> radians per second? Assume that initially the plane of the loop is normal to the field and that S is the area of the loop. BSce sin to/. 11. What is the order of magnitude of the Q of coils employed in radio communication ? The Q of a coil is defined as the ratio u>L/R where R is the resistance of the coil. 12. The power factor of a capacitor is defined as the reciprocal of its Q — u>C/G, where G is the conductance of the capacitor. What is the order of magnitude of the power factor of capacitors employed in radio communication? 13. What is the resonant frequency and the characteristic impedance of a circuit in which L = 10 mh, C = 100 ppf? /= {\/2ir)\W = 159 kilocycles per second, K ■= 10,000 ohms. 14. Let the Q of the circuit in Ex. 13 be 200. What is the series impedance (1) at resonance, (2) at twice the resonant frequency, (3) at half the resonant frequency ? 50, 50 4- /15.000, 50 - /15,000 ohms. 15. In Ex. 14 what is the shunt impedance (the impedance measured across the coil or the capacitor? (1) at resonance, (2) at twice the resonant frequency, (3) at half the resonant frequency? 2 X 10°, 22 - «6667, 22 + «6667 ohms. 16. In the case of natural oscillations in the above circuit how long would it take for the amplitude to decrease by 1 neper? 400 microseconds. 17. In the case of natural oscillations what is the rate of decay in nepers (1) per second, (2) per radian, (3) per cycle? &/2Q, 1/2(9, ir/Q. 18. In a series resonant circuit, what is the approximate ratio of the current at resonance to that at twice (or half) the resonant frequency? 1.5(2. 19. In a parallel resonant circuit, what is the approximate ratio of the voltage across the capacitor at resonance to that at twice the resonant frequency ? 20. What is the radiation resistance of a wire one meter long, energized at the center, when X0 = 10 m ? 1.97 ohms. 21. What is the electric intensity at distance 100 km from a current element radiating 10 watts in free space? 300 microvolts per meter. 22. In Ex. 21 assume that the current element is at the ground surface (assumed to be a perfect conductor) and normal to it. What is the intensity? 424 microvolts per meter. 23. What is the electric intensity if the current element is replaced by a small current loop ? 24. Estimate the capacitance between the outside surfaces of a capacitor formed by two parallel circular discs of radius a, distance h apart (the external capacitance of the discs). Assume a S5, h. Roughly m lo« a/h. 512 ELECTROMAGNETIC VVAVT'.S 25. Whiit is the reflection coefficient in the case of two to one impedance mismatch ? =fc h 26. What is the amplitude of the reflection coefficient if the line is terminated into K(l + i) ? 0.45. 27. What is the resonant impedance of a dissipative quarter wave section of a transmission line short-circuited at the far end? 4K/a\. 28. Express the answer in Ex. 27 in terms of the series resistance R per unit length, assuming that there are no shunt losses. 8K'*/R\. 29. Express the shunt conductance by an equivalent series resistance. GK2. 30. Express the quarter wave resonant impedance in Ex. 27 in terms of the Q. (4MKQ. 31. What is the Brewster angle when the ground Q is unity? When Q is small? When Q is large? „ /_A_ IKŘ \h 32. What is the Q of an air-filled cylindrical resonator of radius 20 cm and height 5 cm, assuming copper walls ? 33. Show that for an air-filled cylindrical cavity Q = o/(a + h), where k = ttV}. 34. Calculate the longitudinal electric intensity in a coaxial pair having air as , j ■ i _ 60/ / b a\ the dielectric. = — I Za log - 4- Zb log - ) where Z„ and Zb are the internal A.\ p pf impedances of the cylinders. 35. How does the total longitudinal displacement current between coaxial cylinders compare with the conduction current in the inner cylinder? 36. If the diameter of the outer cylinder is fixed, what is the diameter ratio for ... , i a b which the attenuation is minimum r log - = 1 + - , - = 3.59. a b a 37. What are the conditions for maximum Q in a coaxial section when the diameter of the outer cylinder is small compared with the length of the section? 38. What is the approximate Q of a single circular turn of wire if the radius of the wire is a and that of the loop b\ Q = (8a log b/a)r)/(K. 39. What is the approximate Qof a doublet antenna assumed so short that radiation 3«X (21 \ 7} losses can be neglected ? q = -—z. \ log — — 1 it* . 2irt \ a / ui 40. What is the expression for the maximum received power Wr in terms of the ^power JVt radiated by the transmitter ? 1VT = (l/l6ir'i)gig2(K/r)2fFt. If the directivities are measured with respect to short doublets, then WT = Q.0l42gtg-iQi/r)2fPt. In terms of the effective areas of the receiver and transmitter, Wr/Wt = iSriSV/XV8. BIBLIOGRAPHICAL NOTE A complete bibliography on the Subject of wave theory would make a book of sizable proportions. Those who are interested in following the subject into periodical literature will find it advantageous to start with the following references which have been chosen mainly to call attention to the names of recent contributors in this field. Barrow, W. L., " Transmission of Electromagnetic Waves in Hollow Tubes of Metal," Proc. I.R.E., October, 1936. Brii.louin, L., " Theoretical Study of Dielectric Cables," Electrical Communication, April, 1938. Carson, J. R., " The Guided and Radiated Energy in Wire Transmission," AJ.E.E. JL, October, 1924. Carson, J. R., Mead, S. P., and Schelkunoff, S. A., "Hyper-frequency Wave Guides — Mathematical Theory," Bell System Tech. Jour., April, 1936. Carter, P. S., " Circuit Relation in Radiating Systems and Applications to Antenna Problems," Proc. I.R.E., June, 1932. Chu, L. J., and Stratton, J. A., " Forced Oscillations of a Prolate Spheroid," Jour. App. Phys., March, 1941. Condon, E. U., " Forced Oscillations in Cavity Resonators," Jour. App. Phys., February, 1941. Foster, R. M., " A Reactance Theorem," Bell System Tech. Jour., April, 1924; " Directive Diagrams of Antenna Arrays," Bell System Tech. Jour., April, 1926. Fry, T. C, " Plane Waves of Light — III," Jour. Opt. Soc. Amer. and Rev. Set. Instr., June, 1932. Gray, M. C, "Horizontally Polarized Electromagnetic Waves over a Spherical Earth," Phil. Mag., April, 1939. Hahn, W. C, " New Method of die Calculation of Cavity Resonators," Jour. App. Phys., January, 1941. EIallen, E., "Theoretical Investigations into the Transmitting and Receiving Qualities of Antennas," Nova Acta (Upsala), November, 1938. Hansen, W. W. and Richtmeyer, R. D., " On Resonators Suitable for Klystron Oscillators," Jour. App. Phys., March, 1939. Kino, L. V., " On the Radiation Field of a Perfectly Conducting Base Insulated Cylindrical Antenna over a perfectly Conducting Plane Earth, and the Calculation of Radiation Resistance and Reactance," Phil. Trans., Ser. A., November 2, 1937. Page, L. and Adams, N. I., 7 „ " The Electrical Oscillations of a Prolate Spheroid," Phys. Rev., May, 1938. Ramo, S., " Space Charge an,- ' teld Waves in an Electron Beam," Phys. Rev., August 1, 1939. 513 514 BIBLIOGRAPHICAL NOTU Suuthworth, G. C, " Certain Factors Affecting the Gain of Directive Antennas " I'm:. I.K.E., September, 1930. Schelku*o,,t, S. A., The Electromagnetic Theory of Coaxial Transmission Lines and Cylindrical Shields," Bell Syst. Tech. Jour., October 1934 vaw dbr Pol B, and Bremmer &, "The Propagation of Radio Waves over a bmitely Conducting Spherical Earth —III," Phil. Mag., June, 1938 SYMBOLS USED IN TEXT NOT INCLUDED IN TABLE I, PAGE 61 page page frequency in cycles per Oil attenuation constant = second 21 re(r) 23 i: imaginary unit 14 phase constant = im(r) 23 k: impedance ratio 212 V intrinsic impedance 81 p: oscillation constant 23 A wavelength 23 p\ with subscript): trans- V frequency ratio 317 mission coefficient 211 ? growth constant = refj>. 22 9 (with subscript): reflec- a intrinsic propagation tion coefficient 210 constant 81 v: wave velocity 23 X (with subscript): transfer A: magnetic vector poten- ratio 206 tial 128 angular velocity 11: susceptance = im(F) 27 — im(p) 21 C: Euler's constant 48 r propagation constant 23 F: electric vector potential 128 A Laplacian 12 G: conductance = re(T) 27 radiation intensity 333 Q: of a medium 83 complex power 31 Q: of a circuit 115 stream function 174 R: resistance = re(Z) 27 0: solid angle 161 W: power work or energy X: reactance = im(2) 27 9t : intrinsic resistance 82 * No page references are given for W and since their exact significance varies somewhat from one formula to another. In each case the symbol is clearly defined in the context. INDEX Absolute cut-off, 155 polarization, 93 value, 15 Absorption of plane wave by thin plate, 247 Admittance, 27 coefficients, 103 expansion in partial fractions, 121, 123 in shunt with rectangular guide, 493 uniform line, 212, 217 terms of complex power, 31 of parallel circular discs, 481 Ampere's law, 66, 101, 149 Amplitude, 15, 21 distribution function, 376, 383 Angle of incidence, 252 reflection, 253 refraction, 255, 257 total internal reflection, 256 Angular points, 143 Antenna theory, 441 ff Arrays, of radiators, 335ff broadside, 342, 345, 348, 352 continuous, 347, 348, 351 end-fire, 342, 345, 351 linear, 342 rectangular, 353 with assigned null directions, 350 binomial amplitude distribution, 349 nonuniform amplitude distribution, 349 uniform amplitude distribution, 342 Associated Lcgendre functions, 45, 53, 402, 474 Attenuation constant, 23, 82 expressed as power ratio, 196 for chain of transducers, 109 circular guide, 324, 390, 509 conductors, 89 deformed circular guide, 398 rectangular guide, 318, 321, 388, 389, 508 TE-waves in wave guide, 387 TM-waves in wave guide, 386 uniform line, 196, 197 unit of, 26 in cylindrical shields, 304 magnetostatic shields, 308 loss, 304 ratio, 314 Attenuators, 507 Average characteristic impedance ot antennas, 462 cylindrical wires, 293 power, 31 for plane waves in cylindrical guides, 385 spherical waves, 403, 405 uniform linev 191 Baffles in wave cuides, 392 Bedell, F.j 480 Bel, 25 Bending of wave guides, 324 Bessel functions, 45ff expansion in Fourier series, 300 Biconical antennas, 441 ff nonsymmetric, 470 Bilinear functions, 17, 199 Boundary conditions, 46 518 INDEX Boundary conditions, at interface between two media, 73, 88,157, 164, 172 for perfectly conducting cylinders, 383 Brewster angle, 253, 260, 512 Capacitance, 27, 64 coefficients, 165 of concentric spheres, 64, 436 cylindrical wire, 290 parallel plates, 100 sphere, 64 spherical caps on antennas, 465 thin circular wire, 182 two conductors, 506 spheres, 505 wedge, 181 sheet, 270 Capacitors, 27, 98, 99, 511 representing system of conductors, 166 Caucby-Riemann equations, 180 Cavity resonators, coaxial pair as, 280 conical, 288 cylindrical, 267, 437 parallelopipedal, 508 spherical, 294, 298 Characteristic admittance, 198 See Characteristic impedance constants of uniform line, 195 impedance, 81 of bent wave guide, 326, 328, 329 chain of transducers, 110 coaxial cylinders, 244, 276, 277 cone line, 287, 473 conical antenna, 450, 454 cylindrical wire, 290 inclined wires, 293 laminated medium, 315 plane wave between parallel .Strips, 243 Characteristic impedance, of series circuit, 115 tapered line, 222 uniform line, 193 velocity, 76, 82, 256 wavelength, 82 Circular aperture in absorbing screen, 356, 367 electric waves, 298, 390, 509 magnetic waves, 294, 441, 509 wave guides, 322, 389, 411, 483 Circularly polarized waves, 249 Circulating current sheet, 273 waves, 328, 409 Circulation, 7 Closed line, see Short-circuited line Coaxial cones, TEM waves on, 286 cylinders, characteristic impedance of, 244 distributed constants of, 505 half wavelength section of, 280 natural waves between, 390 TEM waves between, 275 TM waves between, 418 uniform cylindrical waves between, 269 Coils wound on magnetic cores, 310 Complementary current waves, 449, 450, 466, 470 Complex numbers, 14 point functions, 6 potential, 179ff power, 31 for transducer, 106 uniform cylindrical wave, 262 line, 191 plane wave, 245 flow, 79 Poynting vector, 80, 249, 250 space factor, 335 spectrum, 35 stream function, 179, 183 variables, 14 functions of, 179 Concentric spheres, 63, 90 waves between, 435 INDEX 519 Conductance, 27, 62 of large spherical antenna, 475 Conductivity, 60 Conductors, 60 intrinsic constants of, 83 in dielectric medium, 159 Cone of constant phase, 409 silence, 337 transmission line, 287, 472 Conical antenna, 441ff waves, 432 Conjugate complex power, 106 numbers, 16 Constant current generator, 204, 228 voltage generator, 204, 229 Contact forces, 70 Contour (C) in propagation constant plane, 41, 43 integrals for step and impulse functions, 34 lines and surfaces, 3 Convection current density, 68 Coordinate lines and surfaces, 9 systems, 8 bicylindrical, 497 curvilinear cylindrical, 497 cylindrical, 8 elliptic, 498 rectangular, 8 spherical, 8 spheroidal, 498 Cornu's spiral, 362, 363, 367 Coulomb's law, 64, 70 Crehore, A., 480 Critical frequency, 281 Crosstalk, 236, 279 Curl, definition, 7 differential expressions for, 11 Current distribution, in antennas, 449, 466 distant field of, 331 field of, 126ff, 139 in circular guide, 323 cylindrical cavity, 268 nonuniform line, 208 Current distribution, in rectangular guide, 321 uniform line, 200 sheets, 74, 76, 244 as shunt generators, 245 field of parallel, 245 transfer ratio, 108, 206, 223 Currents across a closed surface, 72 Curvature effect, 324, 391, 435 Cut-off frequency, 113 for bent wave guide, 329 circular guide, 151^323, 390 coaxial cylinders, 391 cylindrical guide, 384 dielectric wires, 427 rectangular guide, 317, 387 TE waves, 381 TM waves, 377, 387 triangular guide, 394 wavelength, 152, see Cut-off frequency Cycles per second, 21 Cylindrical antenna, input impedance of, 461, 463 cavity with coaxial plunger, 272 guided waves, 430 shields, 304 wave guides, 383ff waves" 24, 260, 312, 406, 410, 430 Decibel, 25 Dielectric constant, 63 loss, 440 plate, waves over, 428 wires, waves in, 425 Differential invariants, 12 Diffracted field, 266, 355 Diffraction by narrow slit, 274 by thin wire, 266, 315 Fraunhofer, 365 f Fresnel, 365ff Direct capacitance, 167 Direction components, 2 Directional derivatives, 4 Directive gain, 335, see Directivity pattern, 335 520 INDEX 521 Directivity, 335 of electric current clement, 336 loop, 338 horns, 362 end-fire array, 346 vertical antenna, 339 wave guide as radiator, 360 Disc transmission line, 261, 471 Discontinuities, continuous distribution of, 208 in wave guides, 482, 491 moving -surface, 75 Displacement, 64 current density, 67 density, 62, 63 Distributed constants, of medium,8l uniform line, 112 see Transmission equations mutual admittance, 235 impedance, 235 Divergence, definition, 6 differential expressions for, 10 Dominant transmission mode, 156 wave, 281 between coaxial conductor?! 420 impedance sheets, 488 in biconical antennas, 442 circular guide, 322, 390 rectangular guide, 316, 38?> 410, 508 Double layer of charge, 160, 180 mode transmission line, 250 Eccentricity effect, 285 Edge effect, 358 Effective area, 360, 365 Electric current density, 60, 68 on perfectly conducting sJieeh 157 element, directive properties o>: 336 distant field of, 133 field of, 129, 412ÍF moment of, 129 radiation from, 133, 334 Electric current filament, 75, 175, 264, 300 inside metal tube, 424 loop, 70, 98 directive properties of, 338 distant field of, 163 equivalent to magnetic double layer, 162 field of, 163 impedance of, 144 inductance of, 68, 293, 312 inside sphere, 298 radiation from, 147 sheet, 74, 158 strip, field of, 303 horns, 360fF intensity, 60 radiation vector, 332 Electrically oriented waves, 407 Electrolytic cell, 70 Electromagnetic constants of medium, 81, 84, 85 equations, see Maxwell's equa tions field in terms of two scalar wave functions, 382 radiation vectors, 333 induction, laws of, 66 quantities, table of, 61 Electromechanical impedance, 27 Electromotive force, 60 of self-induction, 68, 99 Electrostatic fields, 159 induction, 62 potential, 159 reciprocity theorem, 166 Elliptically polarized waves, 249, 378, 382, 407 End effects in antennas, 465 Energy content of cylindrical cavity, 268, 508 densities, 78 of system of parallel wires, 176, 177 stored in cylindrical resonators, 438 Energy, theorems, 77 for harmonii In Id , 79 nondissipative lines, 239 stationary lields, 168 uniform lines, 191 Equiamplitude planes, 88 Equiangular spirals, 22 Equiphase planes, 88, 256, 282 surfaces, 6, 24, 415, 432 Equipotential layers, 506 lines, 378, 382 ■surfaces, 169, 283 Equivalence theorem, 158 for stationary fields, 164 uniform lines, 217 Euler's constant, 48 Exponential functions, 18 integral, 56 oscillations, 19 waves, 39, 86, 282 Exponentially tapered line, 222 External capacitance, 263, 511 impedance, 98 of infinite plane strip, 266 thin wire, 263 inductance of a finite wire, 263 loop, 146, 511 Factorial, generalized, 47 Faraday, M., 63, 65 Faradav's law, 66, 98, 101, 149,150 Field slice, 76, 78 Filters, 112ff band pass, 114 high pass, 113, 151, 155, 223 low pass, 113 Force between two charges, 64, 70 moving charges, 505 on moving charge, 72 Forced oscillations, 29 in simple parallel circuit, 119 series circuit, 115 waves, 40 in dielectric wire, 428 metal tube, 423 Fourier integral, 32, 35 series, 32, 503, 504 Fraunhofer diffraction, 365 Free oscillations, 29 space constants, 82 Frequency, 21, 26 ratio, 317, 378, 381 Fresnel diffraction, 365ff integrals, 57, 361, 363 zones, 367 Fringing capacitance, 272 Functions of position, 3 Fundamental electromagnetic equations, 60fF Gain, 335 in logarithmic units, 336 terms of solid angle, 351 of broadside arrays, 348 electric current element, 337 loop, 339 end-fire arrays, 346 General radiation formula, 333 Generator impedances, 107 Generators, 68, 70 in phase, 141 push-pull, 140, 143, 189 of infinite impedance, 189, 190, 202, 217, 228, 478 zero impedance, 189, 190, 201, 217, 229 Giorgi, G., 60 Gradient, definition, 4 differential expressions for, 10 Green's function, 33 theorems, 12, 168, 176, 384 Ground conductivity, 469 effect, 337, 353 Growth constant, 22 Hankel functions, 49 Harmonic oscillations, 21 Plight frequency impedance of uniform line, 233 resistance of finite wire, 264 metallic conductor, 264 V' INDEX INDEX 523 1 ligh frequency resistance of parallel wires, 284 impedance field, 307 Q circuits, 117, 119, 120, 505 dielectrics, 318 Hobson, E. W., 53, 406 Hollow metal sphere, 294, 298 Huygens, C, 357 source, 354 distant field of, 366 Hybrid waves, 154 Hyperbolic functions, 499 Hypergeometric functions, 53 Images, method of, 169 of various elementary sources, 171 Imaginary part of complex number, 14 unit, 14 Impedance, 27 circle, 199 coefficients, 102, 313 concept, 26, 319, 480 coupled, 228 diagram for conical antenna, 457, 459 input, see Input impedance in series with line, 211, 217 terms of complex power, 31 matching, 136, 211, 219, 232 mismatch, 232, 305 normal to conducting plate, 246 interface, 89, 252 surface, 80, 483 thin film, 483 of a loop, 99, 146 ratio, 212, 258, 304, 320, 487 of two current filaments, 308 metals, 305 seen b>r transver.se wire in rectangular guide, 494 sheets, 74, 80, 431, 484ff Impedances, diagrammatic representation of, 28 Impediment, 27, 480 Impcdors, 26, 97 in parallel, 101 series, 101 Impressed currents, 68, 138, 189 electromotive force, 68, 189 field, 157 forces, 70 potential, 164 Impulse function, 31, 413 Incident wave, 252, 254, 431 Inclined wires, 292, 469 Index of refraction, 83 Indicial admittance, 34 Induced admittance, 229 current theory of shielding, 307 impedance, 228 Inductance, 27 coefficients of parallel wires, 177, 179 of cylindrical wire, 290 loop, 68, 293, 312 solenoid, 273, 508 thin circular wire, 183 two cylindrical wires, 284 Induction theorem, 158, 355, 357 for stationary fields, 164 uniform lines, 217 Inductively coupled circuits, 505 Inductor, 27, 98, 99 Infinite current sheet, 379, 382 impedance sheet, 74, 246, 483, 489 Infinitely long wire, 262, 417 Initial phase, 21 Input admittance, expansion in partial fractions, 121, 232 of coaxial cylinders, 420 conducting plate, 247 conical antennas, 454 infinite wire, 418 parallel wires, 423 series circuit, 116, 117 uniform line, 198, 232 wedge line, 274 impedance, 42, 107, 481 I npul Impi 'I......I, expansion in partial li.ictions, 121,232 of antenna inside sphere, 295 antennas of arbitrary shape, 459ff chain of transducers, 108 conical antennas, 447, 449, 454ff resonator, 289 cylindrical antennas, 461,463 infinite wire, 291 loop in spherical cavity, 299 nonuniform line, 206 parallel circuit, 119 strips, 490 terminated line, 228 uniform line, 197, 232 Insertion loss, 309 Instantaneous energy fluctuations, 118 Integral cosine and sine functions, 56 Integrated characteristic impedances, 319, 320, 324, 508, 509 Intensity ratios, 26 Internal capacitance of parallel discs, 263, 272 impedance, 98, 263 of conducting wire, 264, 277 inductance of finite wire, 263 Intrinsic impedance, 76, 81 propagation constant, 81 reactance, 82 resistance, 82 Inverse impedance, 198 points, 16 terminal impedance, 452, 453, 457 Iris in wave guide, 492 Irregularities in wave guides, 490 Isotropic medium, 62 Iterative structures, 108, 236, 312 Jahnke and Emde, 48 Kirchhoff's laws, 101, 102 Lagrange, J. L., de, 61 Laminated conducting cylinders, 279 shields, 309, 312ff Laplace transform, 35 Laplace's equation, 164, 173, 179, 283, 383 Laplacian, 12 Large distance, 132 Lcgendre functions, 54, 401, 402 of fractional order, 447ff Level lines and surfaces,' 3 Line integral, 7 source, uniform, 174 Linear differential equations, solution of, 19 Linearly polarized waves, 248, 378, 382 Lodge, O., 480 Logarithmic derivative, 18 of generalized factorial, 48 increment (or decrement), 22 spiral, 22 Lorentz, H. A., 128 Low frequency capacitance of parallel discs, 263 resistance of finite wire, 263 impedance field, 307 MKS system of units, 60 Magnetic charge, 69 current, 66 density, 66, 69 element, vector potential of, 131 filament, 75 sheet, 74, 158 strip, field of, 303 displacement, 65 current, 68 doublet, 70 field, 64 flux, 65 density, 65 intensity, 65 lines, 64 ','1 Ini) i. \ INDEX 525 Magnetic pole, 69 radiation vector, 332 shields, 305 Magnetically oriented waves, 407 Magnetomotive force, 66 Magnetostatic double layer, 162 potential, 159 of electric current loop, 162 shielding, 308 Major radiation lobe, 343, 347, 350 Matching of impedances, 136, 211, 219, 232 resistance films, 483 Mathieu equation, 394 Maxima and minima of superposed waves, 215 Maximum received power, 136, 218, 512 Maxwell, J. C, 66 Maxwell's equations, 70, 81 differential form of, 73 integral form of, 69 special forms of, 94 Mechanical impedance, 27 Modified Bessel functions, 50 Modular constant, 385 Modulus, 15 Moment of current distribution, 134 current element, electric, 129 magnetic, 162 Motional electromotive force, 68, 71 Multiple reflections, 224 transmission lines, 235 Mutual admittance, 105, see Mutual impedance electrostatic induction, coefficients of, 166 Mutual impedance, 102, 105 in terms of mutual power, 107 of antenna and sphere, 295, 299 current elements, 134 loops, 145, 146 terminated lines, 228 inductance of current loops, 146 power, 106, 137, 372 Mutual radiation resistance of current elements, 138 loops, 145, 33v Natural frequency, see Natural oscillations oscillation constants, see Natural oscillations oscillations, 29 in cylindrical cavity resonators, 267, 437 hollow metal sphere, 296, 299 simple series circuit, 118 slightly dissipative network, 122 uniform lines, 229, 232 propagation constants, see Natural waves waves, 40 between coaxial cylinders, 390 parallel impedance sheets, 485 in circular guides, 389 cylindrical guides, 383 dielectric plates, 429 multiple lines, 235 rectangular guides, 387 Neper, 25 Networks, lOOff Neumann number, 424 Nondissipative medium, constants of, 82 Nonharmonic current distribution, 138 Nonuniform lines, 205, 437 first order corrections for, 209 terminated section of, 237 wave functions for, 207 with constant r, 210, 238, 291, 460 Null directions, 343, 347 Oblique incidence, 251, 358 Ohm's law, 60 One-directional wave in nondissipative medium, 245 uniform line, 203, 218 Open end of wave guide, radiation from, 359 line, 194, 198, 200, 219, 231, 233, 237 Optimum thickness of conducting shell, 278 Orthogonality, 509 Oscillation constant, 23, see Natural oscillations mode, 156, 234 Output impedance, 107, 206, 460 Parallel circuit, 119, 120, 511 conducting strips, 243, 490 bending of, 324, 328 current filaments on cylinder, 302 cylinders, 283, 284 impedance sheets, 484 planes, surface charges on, 91 waves between, 261, 411 plates, 99 wires, constants of, 505 currents in, 507 energized in push-pull, 140,143, 422, 508 mutual impedance of, 372 potentials for, 141 power radiated by, 372, 373 transmission equations for, 149 waves on, 39,72, 283,320, 421 Partial differential equations, reduction of, 44 fractions, 121, 232 reflection and transmission coefficients, 224 Perfect conductor, 74 dielectric, 62, 74 magnetic conductor, 170 Perfectly conducting wires, 140, 142 Perforated cylindrical cavity, 269 Period of oscillation, 21 revolution, 21 Permeability, 65 Phase, 15,21,23 amplitude pattern, 252 constant, 23, 24, 26, 82 of bent wave guide, 326, 328 chain of transducers, 109 rectangular guide, 318 uniform line, 196 length, 452 slowness, 25 velocity, 23, 25 Plane conducting strip, 266' earth, waves over, 259, 431 of incidence, 251 waves, 24, 242, 410 Pohl, R. W., 60 Point charge in semi-infinite me dium, 172 functions, 3 generators, 41, 42, 190, 201, 202, 217 sources, 189, 201 Poisson's equation, 176 Polarization, 90 currents, 92, 157 Poles of impedance function, 30, 61 Potential coefficients, 165 distribution on perfectly conducting wires, 140 functions, 282,'286, 399 of line charges, 181ft magnetostatic double layer, 161 point charges, 169, 170 simple and double layers, 165 layer, 160 Power absorption, 218, 288, 320, 324, 439 dissipation, 62, 116, 321 factor, 511 flow in rectangular guide, 319 ratio for circular aperture, 357 ratios, 25 received by load resistance, 136 reflection and transmission coefficients, 212 Poynting vector, 78 526 index Primary constants nfiiiMlhini, HI,W uniform line, 195 Principal oscillation mode, 155 transmission mode in antenna region, 445, 447 voltage and current, 449 wave, 281, see Dominant wave Principle of conservation of charge, 143, 151* energy, 77 Progressive waves, 26 cylindrical, 300 in bent guides, 325 nonuniform lines, 206 uniform lines, 193 on a wire, 348 on coaxial cones, 287 Propagation constant, 23, 81 of chain of transducers, 109 coaxial cylinder, 276, 277 compound conductors, 280 iterative structure, 237 plane wave between parallel strips, 243 rectangular guides, 317 TE waves, 381 TEM waves, 281, 286 TM waves, 377 uniform line, 112, 192 waves between impedance sheets, 484 Proximity effect, 285 Push-pull, generators in, 140,143,189 Q definition, 22, 83 in terms of energy and power, 116 of conical resonators, 289 cylindrical cavity, 269,439, 512 doublet antenna, 512 parallel circuit, 119 rectangular cavity, 509 resonant section of coaxial pair, 280 series circuit, 115 spherical cavity, 296, 299 Q, of uniform line, 230 wire loop, 512 Quarter wave section of uniform line 219 Quasi-conductor, 84 Quasi-dielectric, 84 Quasi-static waves, 302 Radial admittance 272, see Radial impedance capacitance, 272 impedance,. 261 of biconical antennas, 442 coaxial cylinders, 269, 277 coil on magnetic core, 311 current filaments, 308 cylindrical waves, 273, 301, 304, 306 spherical waves, 401, 404 propagation constant, 407 Radians per second, 21 Radiated power, 333 effect of radius of wire on, 341 field of current element in terms of, 134 from antenna energized at ccn ter, 373 current element, 133 loop, 509 open end of coaxial pair, 509 perfectly conducting wire, 144 two current elements, 137 parallel wires, 372, 373 Radiation from current sheets, 35 1 open end of rectangular guide, 359 progressive current waves on a wire, 348 intensity, 333, 509 approximate forms of, 351, 352 pattern, 335 for broadside array, 348 conical antennas, 469 end-fire array, 347 index 527 Radiation, resist am c of current element, 134 loop, 147 perfectly conducting wire, 144 vertical antenna, 340 vectors, 332, 334 Reactance, 27, 30 Real part of complex number, 14 Receiving antennas, 478 Reciprocity theorem for antennas, 477 electric networks, 103 mutual inductance coefficients, 179 uniform lines, 202, 476 Rectangular aperture in absorbing screen, 355, 358, 367, 368 wave guides, 154, 316, 484, 490, 494, 508 bending of, 324, 328 Reflected field, 157, 433 potential, 164 waves, 218, 252, 294 Reflection, 156 at oblique incidence, 251 discontinuities in wave guides, 492 charts, 213ff coefficients, 232 as functions of the impedance ratio, 212 for dielectric plate, 429 iterative structures, 237 magnetostatic shields, 309 nonuniform lines, 226 power, 212 uniform line, 210 sections, 226 plane waves, 247, 253, 254, 257, 258 waves over plane earth, 433 formation of wave functions using, 227 from circular plate, 358 Reflection, in wave guides, 320, 4''0 loss, 304ff Refracted field, 157 waves, 255 Region enclosed by a curve, 185 Relative derivative, 18 dielectric constant, 83 directions of E, II, V, 242 permeability, 83 polarization, 91 width of resonance curve, 1.17, 506 Resistance, 27, 30, 62, 480 film as guide terminator, 483 between perfectly conducting planes, 489 normal to metal plate, 90 sheets, 246 Resistor, 27, 98 Resonance, 115 curves, 115, 117, 271 in cylindrical cavities, 269 slightly nonuniform lines, 237 spherical cavities, 292, 296, 299 uniform lines, 507 Resonant frequency, 115 impedance of cylindrical an ten nas, 464 lengths of antennas, 454, 456, 465 wavelength 237, see Resonance-Response of linear system to arbi trary force, 32, 36,37 unit impulse, 36, 37 voltage step, 36 Retarded potential, 128 Rhombic antenna, 461 Riemann space, 409 Root mean square radial electric current, 403 magnetic current, 405 Roots of JJk) = 0, 389 Jim = 0, 390 Scalar potential, 128, 375, 380 product, 2, 16 Schwarz transformation, 18411 ».>{ INDEX index 529 Secondary electromagnetic constants, 81 Sectorial waves, 153 Self-impedance of finite wire, 373 Self-reactance of antenna, 295 Semi-infinite line, 108 Series circuit, 29,37, 115, 501,511 distributed constants, 188, 195 generator, 189 impedance, 39 reactance for impedance matching, 221 Shielding, 303fF attenuation ratio, 314 effectiveness, 247 efficiency, 306 improvement, 305 ratio, 308, 310 Short-circuited line, 194, 197, 200, ■230, 233,237 Short-circuiting caps, 280 Shunt admittance, 39 distributed constants, 188, 195 generator, 189 susceptance for impedance matching, 220 Simple layer of charge, 160 transmission line, 156 Sinusoidal current distribution on wire, 369 functions, 31, 142 Skin depth, 90 effect, 265 Sliding wire between parallel wires, 71 Slightly noncircular wave gi::des,397 nonuniform lines, 209, 237 Small distance, 132 Solenoids, 99, 273, 508 Solid angle, 161, 350 Sommerfeld, A., 430 Space factor, 335ff Spectrum, 35, 500 Spherical antennas, 471 caps on ends of antennas, 450, 465 harmonics, 45, 53, 55, 401, 472 Spherical shields, 306, 309 waves, 24, 285, 410 Spheroidal antennas, 461 Stationary fields, 159, 179, 282 waves, 26, 294 cylindrical, 300 in bent wave guide, 327 uniform line, 195 Steady-state oscillations, 29 Step function, 31 Stieltjes integral, 34 Stream function, 174 for parallel filaments, 175, 176 Strength of double layer, 160, 162 impulse, 31 Successive approximations, 208, 231, 291 reflections, 224 Surface admittance, 80 charges, 91, 93 conductance, 80 divergence, 7 impedance, 80, 98, 277, 278, 296, 483 resistance, 80, 278, 483 self-impedance, 279 transfer impedance, 278 waves, 430 Susceptance, 27 of cone line, 475 System of conductors, 165 TE waves, see Transverse electric TEM waves, see Transverse electromagnetic TM waves, see Transverse magnetic Terminal admittance of conical antennas, 452,471 impedance of conical antennas, 450,452,459 impedance, reflection from, 216 Terminals, 480 Thevenin's theorem, 506 Time growth constant, 26 Toroidal cavity, 269 Transducers, 104, 201, 236 Transducers, chains of, 108, 236 Transfer admittance, 105 constant of chain of transducers, 109 impedance, 105, 237 ratios, 206, 308, 310, 313 Transformation of impedances, 219 Transient oscillations, 29 Translation formula for radiation vectors, 334 Transmission coefficients, 223, see Reflection coefficients for iris in wave guide, 493 equations for bent wave guides, 326 coaxial cylinders, 276 concentric spheres, 436 conical guides, 406 current element, 506 loop, 506 disc line, 261 parallel strips, 243 wires, 144, 188, 283 perfectly conducting wires, 143 rectangular guide, 321 TE waves, 155,381, 404 TEM waves, 282, 286 TM waves, 151, 378, 400 uniform lines, 188 plane waves, 242 waves between impedance sheets, 489 lines, 39,112,148,188,502,507 as transducers, 201 constants of, 39, 188, 195 dissipative, 192, 196, 200 nondissipative, 195, 198, 215 wave functions, for, 192 modes, 155, Z35, 411 in antenna region, 446 theory of shielding, 307 Transmitted field, 157 potential, 164 waves, 218, 255, 257 Transverse electric waves, 154 Transverse electric waves, average energy in, 385 between coaxial cylinders, 391 cylindrical, 300, 408 £-lines for, 395 in cylindrical cavities, 438 guides, 383, 484 rectangular tube, 154, 316, 388 plane, 380 spherical, 403 electromagnetic waves, 154, 242, 260, 320 between coaxial cylinders,275 on coaxial cones, 286 cylindrical wires, 290 parallel wires, 283 plane, 281 spherical, 285 magnetic waves, 154 average energy In, 385 between coaxial cylinders,390 cylindrical, 303, 408 ij-lines for, 396 in cylindrical cavities, 438 guides, 383, 484 rectangular guides, 508 plane, 375 spherical, 399 wire in wave guide, 494 Tubes of flow, 506 Two-dimensional fields, 299 stationary, 173 Two-wire shield, 307 Uniform cylindrical waves, 260 field slice, 76, 78 lines, see Transmission tines plane waves, 87, 242, 433 Unit complex number, 16 impulse, 32 contour integral for, 35 source, 32 step, 31 contour integral for, 34 Units in MKS system, 60 530 INDEX Universal resonance curves, 117 Vector, 1 components of, 2 point function, 6 potential, 12J5 nonharmonic, 138 of current distribution, 132,331 product, 3, 16 Vectors, addition and subtraction of, 1, 2 Vertical antennas, 339,461, 463 Voltage transfer ratio, 206, 223 Watson, G. N., 49 Wave equation, 86 functions, 23 guides, 148 circular, 150, 322 of miscellaneous cross-sections, 392 variable cross-section, 405 rectangular, 154, 316 triangular, 393 impedance, 38, 206, 244, 250, 282, 286, 482 for TE waves, 381 TM waves, 377 normals, 24 potentials, 128 velocity, 23 Wavefront, 75, 139 Wavelength, 23 Waves, 22 at interface between two media, 88 Waves, between coaxial cyiinď i 275 concentric spheres, 435 impedance sheets, 484 from arbitrary distributions of sources, 204 in conductors, 88 dielectric wires, 425 dielectrics, 87 transmission lines, 39, 188tE on coaxial conductors, 418 inclined wires, 292, 469 infinitely long wire, 417 parallel wires, 39, 188, 421 single wire, radiation from, 31H over dielectric plate, 428 plane earth, 431 Wedge, potential of line charge in side, 183, 185 transmission line, 274, 475 Wheeler, H. A., 285 Wires energized unsymmetrically, 470 normal to conducting disc, 510 of finite length, field of, 369,371 radius, constants of, 263 power radiated by, 341 with tapered terminals, 481 Zermeck, J., 430 Zero impedance sheet, 74, 246,485 Zeros of impedance function, 30, 37, 118, 297 Zonal harmonics, 451 waves, 442 'I I RLE< TR< (MAGNETIC WAVES ChaPi A If the guard plales arc removed, the electric lines near the edges of mil parallel strips will bulge out as shown in big. 4.1 and the magnetic line* will bend round to enclose one conductor or the other. Subsequent analysis will show that a wave of this modified type may exist at all frequencies and that the shape and distribution of the electric and magnetic lines are independent of the frequency, for such a wave the edge effect is small if b is small com pared to a since the energy is distributed largely bfl tween the plates. If b is small compared to a, the strips can be bent into cylinders to form coaxial conductors with nearlj equal radii (Fig. 8.3). Electric lines will run alone radii and magnetic lines will be coaxial circles between the conductors. There will be a slight " curvaturt effect " instead of the edge effect. The curvature effect is comparatively small; thus if the radii of the conductors are a and b (b > a), then by the parallel plane formula (using the average circumference for the approximate length of the magnetic lines), we have* Fig. 8.3. Coaxial cylinders of nearly equal radii. *) 1200 - a) ir(b + a) b + a If b = 2«, this gives K = 40 ohms; the exact value is 41.6. Since the voltages along various parts of a given radius are added while the magnetomotive force is the same for all magnetic lines, the characteristic impedance of a coaxial pair is the sum of the characteristic impedances of coaxial shells into which the space between the conductors might be subdivided. Thus if b — a is divided into n equal parts, the exact value of K may be expressed in the following form K = lim 120 (b -«) Jn-\ L(2h —1 + 1 )a+b (2n-3)a+3b + 1 .n-l)b_ a+(2n-l)b as n Taking again b — 2a and choosing n = 2, we obtain K = 41.1; this value differs from the exact value by about 1 per cent. Let us now return to waves in an unlimited medium. With transverse dimensions fading out of the picture, we fix our attention on the field intensities E and H, rather than on their integrated values, and define the ratio E/H as the wave impedance in the direction of wave propagation. A uniform plane wave can be generated by a plane current sheet of uniform * When a numerical value is ascribed to the intrinsic impedance, free space is usually assumed. WAVES, WAVE GUIDES AND KKSONATORS I 245 dt nsity. Consider such a sheet in the .yv plane and let its density be /„. Since the electric intensity is continuous at the sheet while the magnetic intensity is discontinuous, we have £,(+0) = £,(-0), Hv(+0) - H„(-0) = -/„ The current sheet acts as a shunt generator and sends out plane waves in both directions Et(z) = -hJ*~"> = -hJ^-a\ z>0, K{z) - -h]*e°\ H-(z) = A/V", z < 0. The complex power (per unit area) contributed to the field by the im- I iressed forces is If the medium is nondissipative, then the power carried by each wave per unit area in an equiphase plane is *+ = £E+(z)[ff+(z)]* = hJJZ, *" = -hE-(z)[H-(z)}* = hJJt. The sum is equal to the power contributed to the field. The total power carried by a uniform plane wave in an unlimited medium is infinite and the wave cannot possibly be started by an ordinary generator. The principal reason for considering such waves at all is their simplicity, combined with the fact that at great distances from any antenna and in a sufficiently limited region the wave is nearly plane. If the medium is nondissipative it is possible to send all the energy in one direction only. Consider two parallel equal current sheets (1) and (2), a quarter wavelength apart, and let the currents be in quadrature. If the current in the left-hand sheet (2) is 90 degrees ahead, then the right-hand wave generated by it will be in phase with the right-hand wave generated by the sheet (1); the two waves will reinforce each other. The left-hand wave from (1) will be 180 degrees out-of-phase with the left-hand wave from (2); the two waves will destroy each other to the left of the plane (2). The electric intensity of the wave produced by the sheet (1) will directly oppose the electric intensity of the second sheet and reduce the total intensity at that sheet to zero; hence the second sheet contributes no power and may be taken to be a perfect conductor. The electric intensities of the two waves reinforce each other at the sheet (1). Assuming that this sheet is in the plane z = 0, we have therefore E+(z) = -vj^!, H+{z) = -/iff*i z > 0. The power emitted by the sheet is twice that which would be emitted by an isolated sheet.