Alternating current
The Foundation Course on Physics
Professor Vojtěch Mornstein
Glencoe pp. approx. 680-693
Alternating current
• What is electric current? What is direct and alternating current?
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Alternating current
Alternating current (AC) is used for powering
various electrical devices. It is also efficient to be
used for long-distance delivery of electrical
energy (when using high voltage lines). To
describe alternating currents or voltages, we can
use similar quantities as used in the description
of a simple harmonic motion:
The cycle is one complete waveform.
Period T [s] is the time duration of one cycle.
Frequency [Hz = s-1] is the number of cycles per
second.
Alternating current
Amplitude is the maximum value of positive or negative
current (or voltage). Instantaneous values of
alternating current and voltage are:
It = Ipsin(2pft + f) [A] and
Ut = Upsin(2pft + f) [V],
where It and Ut are the instantaneous values of current or
voltage, Ip and Up are the amplitudes of current or
voltage (their peak values). f (phi) is the initial phase
angle which defines the values of I or U in time t = 0
Voltage is also denoted by the symbol of „V“ instead of „U“. There are some
advantages and disadvantages.
Alternating current
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A sinusoidal alternating voltage.
1 = peak voltage, also amplitude,
2 = peak-to-peak voltage,
3 = effective value of voltage (root mean square, rms)
– see next slide
4 = Period, T
Average value of alternating current or voltage is zero,
because their variation with time is sinusoidal.
Country Voltage Veff [V] Frequency [Hz]
Czechia 230 50
United Kingdom 230 50
United States 120 60
Japan 100 50, 60
Saudi Arabia 230 60
Tonga 240 50
Effective Voltage and Electrical Power
The effective voltage (current) is that value of direct voltage
(current) which would produce the same expenditure of
electrical energy in the circuit as the respective value of
direct voltage (current). The following equations hold
(under condition of sinusoidal time-course):
and
where Ip and Up are the amplitudes of current or voltage (their
peak values).
2
p
eff
I
I =
2
p
eff
U
U =
Effective Voltage and Electrical Power
Electrical power of alternating current:
P = UeffIeff =
[W = VA = watt = volt-ampere]
The Ohm’s law in its simple form is valid for AC
only when using effective values of voltage and
current.
R
U
RI
eff
eff
2
2
=
Reactance and Impedance
Reactance can be described as “resistance to the flow of electric charge in AC
circuits”. Impedance is the total reactance. In general, the reactance and
impedance Z depend on the frequency of the alternating current.
Reactance of a capacitor in an AC circuit:
[Ω],
where XC is the reactance of a capacitor of capacity C [F], f is the frequency
[Hz] of the alternating current, and w is angular frequency. The higher the
frequency of an alternating current, the smaller is the reactance of a
capacitor. In biological systems, membranes behave like capacitors!
• When uncharged, i.e. when the voltage across its plates equals zero, the
capacitor is charged by the greatest current.
• This means that the voltage is delayed with respect to the current with a
phase difference of +p/2, i.e. the amplitude of the current occurs by T/4
earlier than the amplitude of the voltage.
CfC
XC
p
1
2
1
==
Reactance and Impedance
Reactance of an inductor (or solenoid, coil) that is a part of an AC
circuit:
XL = wL = 2pfL [W]
where XL is the reactance of an inductor, and L is the self-inductance
[H] of the inductor. The higher the frequency of an alternating
current, the greater is the reactance of an inductor.
The alternating current passing through the winding of the solenoid
produces a magnetic field. This field induces a voltage in the
solenoid which has an opposite polarity to the voltage of the source
(see Lenz’s law).
This means that the current is delayed with respect to voltage with a
phase difference of –p/2, i.e. the amplitude of the current occurs by
T/4 later than the amplitude of voltage.
Impedance of a Circuit
It is often necessary to calculate the impedance of complex parts
of electric circuits. For example, the following formula can be
used for a resistor, capacitor and inductor (RCL) connected in
series (Fig.):
[W]
where R is the resistance. The current in such an electric circuit is
able to oscillate, which is very important for production and
transmission of electromagnetic waves (radiofrequencies). The
strongest oscillation (resonance) occurs when the impedance of
the circuit is minimal.
Fig. A resistor (R),
capacitor (C) and
inductor (L) connected in
series.
( )22
CL XXRZ −+=
Resonance in an RCL circuit
According to the last formula, the smallest Z can be
reached when
(XL – XC)2 = 0,
It means that
XL = XC.
(R does not depend on frequency, remains constant.)
This condition allows to find the resonance frequency:
( )
LC
f
LC
f
fC
fL
p
p
p
p
2
11
2
2
1
2
2
===
AC Transformer
This device is used for transformation of electric alternating voltages
from high to low values and vice versa. A transformer consists of
two solenoids with common core, i.e. primary (p, input) winding,
the secondary (s, output) winding, and the soft iron core – see Fig.
“Soft” means here that the iron is not permanently magnetised due
to the magnetic field produced by the primary winding.
Fig. The AC transformer. The iron core is grey-coloured.
During transformation, a part of the electric energy is lost, mainly by
conversion into heat. Therefore, ideal (efficiency = 100%) and
practical transformers can be distinguished.
AC transformer
The following formulas are used for some practical
calculations on transformers:
,
where V is the voltage across the primary (P) or secondary
(S) winding [V], I is the current through the winding [A], and
N is the number of turns in the winding. Thus, in the
transformer shown in the Fig., the voltage is reduced to one
half and current increased twice (np = 4, ns = 2).
P
S
S
P
S
P
I
I
N
N
V
V
==
AC transformer
The transformer efficiency is given by relationship between
the output and input power:
[%]
There are “copper” and “iron” energy losses in the
transformer. The first ones are due to Joule’s heat in copper
winding wires, the second are due to some magnetization and
demagnetization processes in the iron core.
100=
PP
SS
IV
IV

Transformer
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P
S
S
P
S
P
I
I
N
N
V
V
==
Transformer – real look
Windings are usually arranged
concentrically to minimize flux leakage
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216568
Transmission System
Power plant transformation transmission transformation consumption
Measuring instruments
• The construction of instruments for measuring electric
currents, voltages and other electric quantities are beyond
this lecture’s scope. However, it is absolutely necessary to
know that:
• Voltmeters are characterised by very high intrinsic
resistance. They must be connected in parallel to a
“resistor” or “appliance” of which the voltage difference is
to be measured. The measuring range of a voltmeter can
be inceased by “series resistors”.
• Ampermeters (ammeters) are characterised by very low
intrinsic resistance. They must be connected in series into a
circuit. The measuring range of an ammeter can be
increased by “shunts” – small resistors added in parallel.
A circuit with a voltmeter and
ammeter
Cell – source of electric
energy
A – ammeter, ampermeter –
very low impedance
Lamp – appliance, resistor
V – voltmeter – very high
impedance
February 2020