1
AC Circuits and Electromagnetic Waves
The first part of this chapter deals with circuits in which the currents and voltages vary
sinusoidally with time. The second part deals with the propagation of sinusoidally
varying electric and magnetic fields throu
gh space.
Alternating Current (AC) Circuits
We consider circuits consisting of combinations of resistors, capacitors, and inductors in
which the currents and voltages are sinusoidal. Assume the voltage source is given by
,
V
max
is the peak voltage and
f
is the frequency of oscillation.
Resistor circuit
If the AC voltage source is connected across a resistor, then the current also varies
sinusoidally and is in phase with the voltage.
.
v (t) =
V
max
sin(2
晴f
R
2
The power dissipated in the resistor is given by
Since
i
varies sinusoidally with time, then the power also varies with time. The average
power dissipated is
The average of sin
2
(2
ft) is ½. So,
We can write the average power as
,
where
.
I
rms
is called the ‘
r
oot

m
ean

s
quare’ current. It is obtained by first squaring the current,
then finding the average
(mean) of the square, and then taking the square root of this
average. It is the effective heating value
(or DC value)
of the time

varying current.
The
rms voltage is defined in a similar way
–
The rms current and rms voltage
are related by
Example
:
The line voltage in a house is nominally 120 volts rms. What is the maximum (peak)
voltage?
3
The
peak

to

peak
voltage is the difference between the maximum positive and maxim
um
negative voltages and is 340 V.
Capacitor circuit
If a sinusoidal voltage source is connected across a capacitor, then the
charge on the
capacitor and the current to the capacitor also vary
sinusoidally with time.
The voltage
and charge are related
by
v = q/C
, so the charge and voltage are in phase.
However,
the
charge lags the current by 90
o
, so
it is found that
the voltage lags the current by 90
o
.
The current

voltage relationship for a capacitor is given by
,
where
.
X
C
is called the
capacitive reactance
and has units of ohms. It is the quantity that limits
the current to the capacitor, similar to resistance. However, unlike resistance power
cannot be di
ssipated in a capacitor. This is because the current and voltage are 90
o
out of
phase.
This is analogous to pushing on a moving object. No work will be done by the
force if it is applied perpendicular to the displacement.
Note that the capacitive react
ance decreases with increasing frequency and capacitance.
v (t) =
V
max
sin(2
晴f
C
4
Example
:
A 0.02
F capacitor is connected to a 50

V rms AC voltage source which oscillates at 10
kHz. What is the rms current to the capacitor?
Inductor circuit
The v
oltage across an inductor depends on the time rate of change of the current and is
given by
This means that in an AC circuit, the voltage is sinusoidal and leads the current by 90
o
.
The cur
rent

voltage relationship for an inductor is given by
,
v (t
) =
V
max
sin(2
晴f
L
5
where
is called the
inductive reactance
.
As with the capacitor, no power can be dissipated in an ideal inductor (one with no
resistance). Note th
at the inductive reactance increases with increasing
f
and
L
.
Series LCR circuit
Consider an AC circuit containing a resistor, capacitor, and inductor in ser
ies. All have
the same current
(which is true
for any series circuit
)
. As previously describe
d, the
voltage across the resistor is in phase with the current, the voltage across the capacitor
lags the current by 90
o
, and the voltage across the inductor leads the current by 90
o
. This
means that the voltages across the inductor and capacitor are 180
o
out of phase. That is,
they subtract. The resulting voltage across the inductor and capacitor combination either
leads or lag
s
the voltage across the resistor, depending on whether
V
L
is greater than or
less than
V
C
.
Consequently, the rms vo
ltage across the inductor

capacitor combination is
Since the voltage across the LC combination is 90
o
out of phase with the voltage across
the resistor, then the total rms voltage must be obtained using the Pythagorean theorem
–
The
impedance
of the circuit is defined as
,
So we can write
v (t) =
V
max
sin(2
晴f
L
R
6
.
Z
has units of ohms and is a measure of the resistance of the circuit to the flow of current.
The
total current
and the
power dissipated in a series RLC circuit depend on the phase
shift between the total current and the total voltage. This phase shift depends on the ratio
of the out

of

phase to the in

phase voltage. Thus,
,
Or,
If
X
L
= X
C
, then the total phase shift is zero and we get maximum current and maximum
power dissipated in the resistor. (No power is dissipated in the inductor and the
capacitor.)
Resonance in a series LCR circuit
The total current in
the LCR circuit is given by
Since
X
L
and
X
C
depend on frequency, then
I
rms
depends on frequency. There is a
particular frequency for which
X
L
= X
C
, at which
I
rms
has its maximum value. At this
resonance frequency the voltages a
cross the inductor and capacitor exactly cancel, and all
the voltage drop is across the resistor. A plot of the current as a function of frequency
would look something like the following.
7
The resonance frequency,
f
0
, is give
n by
Or,
Example
:
Consider
an RLC circuit for which R = 1
0
, L = 0.2 mH,
C = 5
F
and the applied
voltage is V
rms
= 25 V?
What is the resonance frequency?
What would be th
e current in the circuit if
f
= 3 kHz?
What is the power dissipation in the circuit?
8
Transformers
:
A transformer consists of two coils which are closely coupled so that the flux generated
by one
coil (the primary) passes mostly through the other coil (the secondary).
The flux
coupling can be made nearly complete if the coils are wound around an easily
magnetizable core such as iron.
According to Faraday’s law, t
he
primary voltage is given by
and the secondary voltage is given by
Thus, we have
The secondary voltage can thus be larger or smaller than the primary voltage, depending
on the
turns ratio. If we assume that the power delivered into the primary is the same as
the power delivered to the load by the secondary,
then we find that
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That is, a transformer which steps up the voltag
e must step down the current, and vice
versa.
Example
:
A transformer has 20 primary turns and 100 secondary turns. If the primary voltage is
12
V, what is the secondary voltage?
If the load resistor on the secondary is 50
, t
hen
The current in the primary is then
Electromagnetic Waves
Electromagnetic waves consist
of oscillating electric and magnetic fields that propagate
through space at the speed of light. These wave
s can be produced by applying an
oscillating potential to an antenna. The
antenna could consist of a rod connected to
each side of an AC voltage source. The
voltage source would generate a sinusoidally
varying current and a sin
u
soidally varying
charge di
stribution in each rod. As a
consequence, the rods would generate
magnetic and electric
fields which would be
perpendicular to each other and would
radiate from the rods.
At distances
far away
from the antenna, the
configuration of the electromagnetic wa
ve
would look something lik
e that given in the
figure to the right
.
10
For a fixed frequency, both
E
and
B
vary sinusoidally in time and are in phase.
Both
E
and
B
are perpendicular to the direction of travel of the wave (
c
), and they are
perpendicular to
each other.
The relative orientations of
E
,
B
, and
c
are given by a right
hand rule (as, for example, y, z, and x for a Cartesian coordinate system).
It can be shown that the
speed
of the electromagnetic waves is given by
.
Usi
ng the accepted values of
0
= 4
x 10

7
T
m/A and
0
= 8.85 x 10

12
C
2
/(N
m
2
), this
equation gives
c = 3 x 10
8
m/s, which is the speed of light. This is to be expected
since
visible light is a
part of the electromagnetic spectrum.
It can also be shown t
hat the ratio of the magnitudes of E and B are always fixed and
given by
Electromagnetic waves carry energy. Specifically, the
intensity
of the wave, which is
power transmitted per unit area, is given by
Using the relationship between E and B and the expression for c, the intensity can also be
written as
Electromagnetic waves also carry momentum, even though they don’t have mass. If a
surface A completely absorbs an amount of
electromagnetic energy
U = IA
t
in a time
t
, the momentum absorbed by the surface is
(complete absorption)
If a surface completely reflects the radiation, then the momentum transferred to the
surface is
(complete reflection)
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Thus, when electromagnetic radiation strikes a surface it imparts a force to the surface
(since force is rate of change of momentum).
Example
:
The intensity of sunlight incident upon the earth is about 1,400 W/m
2
. What are
the
maximum values of the electric and magnetic fields associated with the radiation?
Solution:
How much solar power is incident upon the earth?
Solution:
The effective area of the earth seen by the sun is that of a circle wit
h the earth’s radius.
So,
If all the sun’s radiation is absorbed by the earth, then what is the force imparted by this
radiation on the earth?
Solution:
The power absorbed is the energy absorbed per second. This can be used t
o find the
momentum absorbed per second, which is the force.
Although this is a large force, it has an imperceptible effect due to the large mass of earth.
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