AC Circuits
–
page
1
of
12
RESISTORS, CAPACITORS AND INDUCTORS
IN AC CIRCUITS
INTRODUCTION AND
THEORY
Many electric circuits use batteries and involve direct current (dc). However, there are
considerably more circuits that operate with alternating current (ac),
when
the charge
flow
reverses direction periodically.
In an “ac” circuit, the
most common
generators serve the same
purpose as a battery s
erves in a dc circuit:
they give energy to the moving
electric charges
but they change the direction of magnetic forces periodically.
Since electrical outlets in a house provide alternating current, we all use ac circuits routinely.
If the voltage and the current
alternate sinusoidally with time
we can write:
€
v
v
(
t
)
V
m
⋅
sin(
ϖ
t
ϕ
)
€
i
i
(
t
)
I
m
⋅
sin(
ϖ
t
)
Where
v and i
represent the
instantaneous
voltage and current when we are considering
their variation with time explicitly.
V
m
and I
m
are the
amplitude
or
peak value
of the voltage and current
V =
V
m
/2
1/2
and I =
I
m
/2
1/2
without subscripts refer to the RMS values.
f
is the
ordinary frequency
and represents the number of complete oscillations per
second
ω = 2
π
f is the
angular frequency
.
φ is the
phase difference
between the voltage and current.
Resistors And Ohm's Law In AC Circuits
The
schematic
below is that
of
a
n
ac circuit formed by plugging a toaster into a wall socket.
The heating element
of
the toaster is essentially a thin wire of resistance R and becomes red
hot when the electrical energy is dissipated in it
According to Ohm’s Law, t
he
instantaneous
volta
ge v across a resistor is proportional to the
instantaneous
current
i flowing through it.
Figure 1: Resistor in an AC circuit
R
V
i
AC Circuits
–
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2
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12
The next
two
graph
s show
the voltage
across the resistor
and
the
current
flowing through the
resistor as a func
tion of time.
Figure 2: V and I: Sinusoidal variation in phase
Figure 3: Phasor Diagram
From the graph in Figure 2
the peak value of voltage
across a pure resistor
is reached at the
same time with the peak value of the current. Therefore, t
he current and voltage
are
said to
be in phase, and mathematically this is expressed as
In the Figure 3, the
radial vectors
also called
phasors
rotate with angular velocity ω
representing the current and the voltage across the resistance. The lengths of these phasors
represent the peak current I
m
and voltage V
m
. The y components
of these phasors
are
and they are equal to the y
–
values from
the graph in figure 2 at any time.
The
phasor
diagram
shows that the voltage and the current are in phase
.
Resistance, Reactance
And
Impedance
The ratio of
the
voltage to
the
current in a resistor is
called
its
resistance
.
As well, in a circuit
where the
current is proportional to
the
voltage
,
the circuit is
called
a
linear circuit
.
This is
happening when the circuit contains only resistors, capacitors and/or inductors.
Resistance
does not depend on frequency, and in
a
r
esistor the voltage and the current
are in phase.
However, circuits with only resistors
,
capacitors, and solenoids
are not very
useful in some
Instantaneous voltage
V
–
(blue
),
and current
i
–
(
red
)
V(t),
i
(t)
t
I
m
V
m
AC Circuits
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page
3
of
12
cases
.
If the circuit contains
also,
diodes
or
transistors,
the circuit is no longer
linear
.
In most practical cases,
the ratio of
the
voltage to
the
current depend
s
on
the
frequency and
in general there
is
a phase difference
between the voltage and the current. In this general
case,
the ratio of
the
voltage to
the
current
is called
I
mpedance
and i
t
is denoted with the
symbol Z
. Resistance is a spe
c
ial case of impedance. A very important
case is that in which
the voltage and
the
current are out of phase by 90°: this is an important case because when
this happens, no power is lost in t
he circuit and
the ratio of
the
voltage to
the
current is called
th
e
reactance
,
denoted with
the symbol
X
.
Reactance can be caused by capacitors or by
inductors.
Capacitors
In AC Circuits
Capacitors store electric charge. They are used with resistors in
timing
circui
ts
because it
takes time for a capacitor to fill with charge. They are
also
used to
smooth
varying DC
supplies by acting as a reservoir of charge. They are also used in filter circuits
because
capac
itors easily pass AC
sign
als but they block DC
signals.
The voltage on a capacitor
depends o
n the amount of charge stored
on its plates.
If we denote the instantaneous value
of the current with i(t):
The current flowing
off the positive
plate is
equal to
the current
flowing
into the negative plate
and by
definition is the rate at which
the
charge
Q
is being stored. From the definition of the
capacitance
as a function of the charge Q and the potential across the capacitor
,
it fol
lows that
But
the charge Q on the capacitor equals the integral of the current with respect to time.
In the indefinite integral above, the constant of integration was set to zero so that the average
charge on the capacitor would be zero (we are
starting with an uncharged capacitor).
AC Circuits
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12
Therefore, the voltage across the capacitor:
The last equation shows that
the current and
the
voltage are out of phase
by 90
0
.
The
capacitive reactance
X
C
is equal to:
and it can be defined
as the r
atio of the magnitude of the voltage to magnitude of the curr
ent
in a capacitor (and that is Ohm’s Law for the capacitor!)
Looking at the
difference of
phase, the voltage across the capacitor is 90°, or one quarter
of
period
, behind the
current.
The same phase difference
φ = 90°
is
reflected in the phasor
diagrams. Since t
he vertical component of
any
phasor arrow represents the instantaneous
value of its quantity
and the
phasors are rotating counter clockwise
t
he phasor representing
V
C
is
90°
behind
the
current
.
Figure 4: V and I: Sinusoidal variation for Capacitor
Figure 5: Capacitor Phasor Diagram
As we have seen before,
when the voltage and the current differ in phase by 90
0
, the
resistance is called
reactance
.
Another
important
difference between reactance and resistance
is that the reactance is
frequency dependent
and for a capacitor, it
decreases with frequency.
Instantaneous voltage
V
–
(blue
), and current
i
–
(
red
)
AC Circuits
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page
5
of
12
Inductors In AC Circuits
An inductor is usually a coil of wire.
The
resistance
of an ideal indu
ctor
is negligible, as is its
capacitance.
However, the voltage across an inductor is influenced
by changes of
its own
magnetic field
.
Faraday's la
w of electromagnetic induction states that the
current
i(t)
in the coil
sets up a magne
tic field, whose magne
tic flux
Φ
B
is proportional to the field strength
B
, which
in turn
is proportional to the current
.
Therefore, the
s
elf inductance of the coil, denoted L is defined as
:
However,
Faraday's law gives the emf
induced in a coil due to a change in the m
agnetic flux
According to Kirchhoff’s First Law the
emf is
a voltage rise; therefore,
the voltage drop v
L
across the inductor
should be:
v
L
(
t
)
−
e
L
d
Φ
B
d
t
d
d
t
L
⋅
i
(
t
)
L
⋅
d
d
t
I
m
⋅
s
i
n
(
ϖ
t
)
ϖ
L
⋅
I
m
⋅
c
o
s
(
ϖ
t
)
V
m
⋅
s
i
n
(
ϖ
t
π
2
)
As in the case of the capacitor, we
define the
inductive reactance
X
L
as the ratio
of the
magnitudes of the voltage and
the
current, and from the equation above we see that
X
L
= ωL.
It is worth noting
the analogy to Ohm's law: the voltage is proportional to the current, and the
peak voltage and currents are related by
V
m
= X
L
.I
m
AC Circuits
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Figure 6: V and I: Sinusoidal variation for Solenoid
Figure 7: Solenoid Phaso
r Diagram
From figures 6 and 7 it follows that the voltage and the current through the solenoid are in
phase
:
the voltage across the inductor
has its maximum when the current is changing most
rapidly, which is when the current is
passing through
zero. The
refore, the
voltage across the
ideal inductor is 90°
(or
)
ahead of the current, (i
.
e
.
it reaches its
maximum
one quarter
of
the cycle
before the current does).
The same conclusion is drawn from
the phasor diagram.
We should also note
that
for a coil
the
reactance is
frequency dependent
in the sense that it
increases with frequency
.
Summary:
Resistance, Reactance and Impedance
The following is a summary of the relationship between
voltage and current in linear
circuits:
The
impedance
is the general term
for the ratio of
the
voltage to
the
current.
Resistance
is the special case of impedance when φ = 0,
Reactance
the special case when φ =
±
90°.
Component
Resistor
Capacitor
Inductor
Difference of
Phase between
Voltage and
Current
Voltage and
Current are in
phase
Voltage lags
behind Current by
π
/2
Current
lags
behind Voltage
by
π
/2
Ohm’s Law
R
V
R
I
X
c
V
C
I
1
ϖ
C
X
L
V
L
I
ϖ
L
V(t),
i
(t)
t
AC Circuits
–
page
7
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R
esistor and
C
apacitor connected in Series
When we connect components together, Kirc
h
hoff's laws apply at any instant. So the voltage
v
s
(t) across
a
resistor and capacitor
in series is just
v
s
(t) = v
R
(t) + v
C
(t)
However
,
the addition is
more
complicated be
cause the two are not in phase:
they add to give
a new sinusoidal voltage, but the amplitude
V
S
is
less
than V
R
+ V
C
as it can be seen from
figur
e 11.
Figure 10
:
Resistor and Capacitor in Series
Figure 11
:
R

C Series
Phasor Diagram
However, using
Pythagoras' theorem
in figure 11 we have
:
V
S
2
V
R
2
V
C
2
Using Ohm’s Law and expressing the three v
oltages and substituting in the equation above,
we obtain the impedance Z
RCS
and the phase difference
ϕ
between voltage V
RCS
and the
current I:
I
⋅
Z
R
C
S
2
I
⋅
R
2
I
⋅
X
C
2
Z
R
C
S
R
2
1
ϖ
C
!
"
#
$
%
&
2
and
t
a
n
ϕ
−
V
C
V
R
−
X
C
R
−
1
ϖ
R
C
R
V
S
i
C
v
s
er
ie
s
=
v
R
+
v
C
b
u
t
V
s
er
ie
s
>
V
R
+
V
C
.
V
C
v
seri
es
=
v
R
+
v
C
but
V
seri
es
>
V
R
+
V
C
.
The
am
plit
ude
s
and
the
RM
S
volt
age
V
R
v
ser
i
es
=
v
R
+
v
C
but
V
seri
es
>
V
R
+
V
C
.
The
am
plit
ude
s
I
V
R
V
C
V
RCS
ϕ
AC Circuits
–
page
8
of
12
R
esistor and Inductor co
nnected in Series
Similarly, when
we connect
a solenoid in series with a resistor,
the instantaneous
voltage v
s
(t)
acros
s the
resistor and
inductor
in series
will be:
v
s
(t) = v
R
(t) + v
L
(t)
And again,
the amplitude
V
RLS
will be always less
than V
R
+ V
L
as
it can be seen from figure
13.
Figure 12
:
Resistor and Inductor in Series
Figure 13
:
R

L Series
Phasor Diagram
Applying again
Pythagoras' theorem
, from figure 13 we have
:
V
R
L
S
2
V
R
2
V
L
2
Using Ohm’
s Law to express the three voltages in the equation above, we obtain the
impedance Z
RLS
and the phase difference
ϕ
between voltage V
RLS
and the current I:
I
⋅
Z
R
C
S
2
I
⋅
R
2
I
⋅
X
L
2
Z
R
C
S
R
2
ϖ
L
2
and
t
a
n
ϕ
V
L
V
R
X
L
R
ϖ
L
R
R
V
S
i
L
v
s
eri
es
=
v
R
+
v
C
b
ut
V
s
eri
es
>
V
R
+
V
C
.
T
h
e
a
m
pli
V
L
v
seri
es
=
v
R
+
v
C
but
V
seri
es
>
V
R
+
V
C
.
The
am
plit
ude
s
and
the
RM
S
volt
age
s V
do
not
V
R
v
seri
es
=
v
R
+
v
C
but
V
seri
es
>
V
R
+
V
C
.
The
am
plit
ude
s
and
the
RM
I
V
R
V
L
V
RLS
ϕ
AC Circuits
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9
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12
R
esistor, C
apacitor and Inductor connected in Series
Now
we connect
a capacitor, and solenoid in series with a resistor,
the instantaneous
voltage
v
s
(t) acros
s the
resistor, capacitor
and
inductor
in series
will be:
V
RCLS
(t) = v
R
(t) + v
C
(t)
+
v
L
(t)
But, t
he amplit
ude
V
RCLS
will be always less
than V
R
+ V
C
+
V
L
because of the same reason
explained before.
Figure 14
:
Resistor, Capacitor and Inductor in Series
Figure 15
:
R

C

L Series
Phasor Diagram
Applying again
Pythagor
as' theorem
, from figure 15 we have
:
V
R
C
L
S
2
V
R
2
V
L
−
V
C
2
Using Ohm’s Law to express the four voltages in the equation above, we obtain the
impedance Z
RCLS
and the phase difference
ϕ
between voltage V
RCLS
and the current I:
I
⋅
Z
R
C
L
S
2
I
⋅
R
2
I
⋅
X
L
−
I
⋅
X
C
2
Z
R
C
S
R
2
ϖ
L
−
1
ϖ
C
"
#
$
%
&
'
2
and
t
a
n
ϕ
V
L
−
V
C
V
R
X
L
−
X
C
R
ϖ
L
−
1
ϖ
C
R
Since
the inductive and capacitive phasors are 180° out of phase, their reactances tend to
cancel
each other
.
This happens at
resonance
when X
L
= X
C
. At resonance
ϕ
= 0, the
impedance Z = R has
a minimum and the current through the circuit can reach very large
I
V
R
V
L
V
RLS
ϕ
R
V
S
i
V
C
v
ser
i
es
=
v
R
+
v
C
but
V
seri
es
>
V
R
+
V
C
.
The
am
plit
ude
s
and
the
RM
S
volt
age
s V
do
V
L
v
ser
i
es
=
v
R
+
v
C
but
V
seri
es
>
V
R
+
V
C
.
The
am
plit
V
C
v
seri
es
=
v
R
+
v
C
but
V
seri
es
>
V
R
+
V
C
.
The
am
plit
ude
AC Circuits
–
page
1
0
of
12
values that could be damaging to the circuit.
APPARATUS
RLC circuit board with resistors, capacitors and inductor
o
Resistors:
100
Ω, 1 W; 33 Ω, 5 W; 10 Ω, 10 W
o
Capacitors: 100 µF, 16 V and 330 µF, 16 V (capacitance
values may vary by ±20 %)
o
Inductor: 8.2 mH @ 1 kHz, 6.5 Ω maximum DC resistance,
0.8 A current rating RMS, 3/4” I.D. x 1

3/4” O.D.
Dual channel oscilloscope with sinuso
idal voltage signal generator incorporated.
BNC cables.
Banana plug patch cords.
PROCEDURE
In this experiment you will be applying a sinusoidal signal to different circuits and will analyze
the effect of the resistors, capacitors and inductors on the cu
rrent and the relative phase
between the voltage applied and the current.
Familiarize yourself with the apparatus to be used. The function generator and the
oscilloscope are a single unit. There should be a “TEE” connected to the output of the
function
generator portion of the unit, with one end of the “TEE” connected directly to CH1 of
the oscilloscope (to g
ive you the input signal) and t
he other end going
to the points of the
circuit where the source is going to be connected. The voltage collected acro
ss different
components of the circuit will be going to CH2
of the oscilloscope.
1.
Select the mode of the function generator to “sinusoidal” and then select a signal in the
range of 100KHz.
AC Circuits
–
page
11
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12
2.
Using the
“T” splitter to apply this signal to “Chanel 1”
of
the o
scilloscope and to the
“source” in the
circuit
in figure 10.
3.
Adjust
the amplitude of the
sinusoidal
signal
, and make sure that the ‘OFFSET’
knob is
pushed in.
Setup the time/div so that only one cycle appears on the oscilloscope.
4.
Collect the voltage acr
oss the resistor and the capacitor and apply it on “Chanel 2” of
the oscilloscope. Compare with what you see on CH1. Are CH1 and CH2 in phase for
both the resistor and the capacitor? Explain.
5.
Read the relative phase for each component from the oscilloscop
e by comparing the
position of the signal on CH1 relative to CH2.
6.
Measure the resistance of the resistor R with an ohm

meter and calculate the capacitive
reactance X
C
and the capacitance of the capacitor C.
Table # 1: Data for Resistor and Capacitor in
Series
7.
Repeat steps 2 through 6 for the circuits in
figure 12 and determine the relative phase,
the inductive reactance and the induction of the solenoid.
Table # 2: Data for Resistor and Inductor in Series
Component
ϕ
(
s)
ϕ
(rad)
tan
ϕ
R
(
Ω
)
f
(Khz)
ϖ
=
2
π
f
(
S

1
)
C
(F)
Capacitive
Reactance
X
C
(
Ω
)
Resistor
Capacitor
Component
ϕ
(
s)
ϕ
(rad)
tan
ϕ
R
(
Ω
)
Inductive
Reactance
X
L
(
Ω
)
f
(Khz)
ϖ
=
2
π
f
(
S

1
)
L
(H)
Resistor
Inductor
AC Circuits
–
page
12
of
12
8.
Repeat steps 2 through 6 for the circuit in figure 14 and determine the relative phase,
the overall reactance
X
L

X
C
and the impedance Z of the circuit.
Table # 3: Data for Resist
or, Capacitor and Inductor in Series
Component
ϕ
(
s)
ϕ
(rad)
tan
ϕ
R
(
Ω
)
Overall
Reactance
X
L

X
C
(
Ω
)
f
(Khz)
ϖ
=
2
π
f
(
S

1
)
Z
(
Ω
)
Resistor
Capacitor &
Inductor
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