AC Circuits
Chapter 23
AC Circuits
Capacitive Reactance
 Phasor Diagrams
 Inductive Reactance
 RCL Circuits
 Resonance

Resistive Loads in AC Circuits
Ohm’s Law:
V
Vrms
I
I rms 
R
R
Vt  V0 sin 2ft 
Vt V0
I t   sin 2ft   I 0 sin 2ft 
R R
R is constant – does not depend on frequency
 No phase difference between V and I

Capacitive Reactance
At the moment a capacitor is connected to a voltage
source:
I0
V
+
-
Current is at its maximum
 Voltage across capacitor is zero

+
C
V=0
-
Capacitive Reactance
After a long time, the capacitor is charged:
V
+
-
+
C
V
-
Current is zero
 Voltage is at its maximum (= supply voltage)

Capacitive Reactance
Now, we reverse the polarity of the applied voltage:
I
V
+
+
-
+
C
Current is at its maximum (but reversed)
 Voltage hasn’t changed yet

V
-
Capacitive Reactance
Time passes; the capacitor becomes fully charged:
V
-
+
+
-
C
V
+
Current is zero
 Voltage has reversed to match the applied polarity

Capacitive Reactance
Apply an AC voltage source:
I(t) = I0sin(2ft+/2)
V(t) = V0sin(2ft)
+
C
an AC current is present in the circuit
 a 90° phase difference is found between the
voltage and the current

Capacitive Reactance
We want to find a relationship between the voltage
and the current that we can use like Ohm’s Law
for an AC circuit with a capacitive load:
Vrms  I rms  X C
We call XC the capacitive reactance, and calculate it
as:
1
XC 
2fC
units of capacitive reactance: ohms (W)
Capacitive Reactance
A particular example:
V0 = 50 V
f = 100 Hz
-
C = 750 mF
+
1
1
XC 

 2.12 W
-4
2fC 2  100 Hz  7.5  10 F
Capacitive Reactance
voltage vs. time
50
Vt  V0 sin 2ft 
40
30
20
voltage, V
10
0
-10
-20
-30
-40
-50
0
0.005
0.01
0.015
0.02
time, s
0.025
0.03
0.035
Capacitive Reactance
Vt  V0 sin2ft
Voltage and Current vs. Time


It  I0 sin 2ft  
2

50
40
30
voltage, current (V, A)
20
10
voltage
0
current
-10
-20
-30
-40
-50
0
0.005
0.01
0.015
0.02
time, s
0.025
0.03
0.035
Capacitive Reactance
Vt  V0 sin 2ft 
Voltage, Current, and Power vs Time


I t  I 0 sin  2 ft  
2

Pt  Vt I t
60
voltage, current, power (V, A, 10W)
40
20
voltage
current
0
power
-20
-40
-60
0
0.005
0.01
0.015
0.02
time, s
0.025
0.03
0.035
Capacitive Reactance





Power is zero each time either the voltage or current
is zero
Power is positive whenever V and I have the same
sign
Power is negative whenever V and I have opposite
signs
Power spends equal amounts of time being negative
and positive
Average power over time: zero
Capacitive Reactance
1
XC 
2fC

The larger the capacitance, the smaller the capacitive
reactance

As frequency increases, reactance decreases
XC  

DC: capacitor is an “open circuit” and

high frequency: capacitor is a “short circuit”
and X C  0
Phasor Diagrams
Consider a vector which rotates counterclockwise
with an angular speed   2f :
V
This vector is called
a “phasor.” It is a
visualization tool.
V0 sin(2ft)
V0
t=2ft
Phasor Diagrams
For a resistive load: the current is always
proportional to the voltage
V, I
V0 sin(2ft)
I0 sin(2ft)
I0 = V0 / R
V0
voltage phasor
t=2ft
current phasor
Phasor Diagrams
For a capacitive load: the current “leads” the voltage
by /2 (or 90°)
V, I
current phasor
V0 sin(2ft)
I0 sin(2ft + /2)
voltage phasor
t=2ft
Inductive Reactance
A coil or inductor also acts as a reactive load in an
AC circuit.
AC
Inductive Reactance
For a coil with a self-inductance L:
I
EMF   L
t
AC
Inductive Reactance
As the current increases through zero, its time rate
of change is a maximum – and so is the induced
EMF
I
EMF   L
t
AC
Inductive Reactance
As the current reaches its maximum value, its rate
of change decreases to zero – and so does the
induced EMF
I
EMF   L
t
AC
Inductive Reactance
The voltage “leads” the current in the inductor by
/2 (or 90°)
I
EMF   L
t
AC
Inductive Reactance
The inductive reactance is the Ohm’s Law constant
of proportionality:
Vrms  I rms X L
AC
X L  2fL
units of inductive reactance:
ohms (W)
Inductive Reactance
The voltage-current relationship in an inductive load
in an AC circuit can be represented by a phasor
diagram:
V, I
V0 sin(2ft)
voltage phasor
t=2ft
current phasor
I0 sin(2ft - /2)
Inductive Reactance
Mnemonic for remembering what leads what:
“ELI the ICEman”
EMF (voltage)
EMF (voltage)
inductor (L)
current (I)
capacitor (C)
current (I)
Inductive Reactance
X L  2fL

Larger inductance: larger reactance (more induced
EMF to oppose the applied AC voltage)

Higher frequency: larger impedance (higher
frequency means higher time rate of change of
current, which means more induced EMF to oppose
the applied AC voltage)
RCL Circuit
Here is an AC circuit containing series-connected resistive,
capacitive, and inductive loads:
R
C
AC
L
The voltages across the loads at any instant are different, but
a common current is present.
RCL Circuit
The current is in phase with
voltage in the resistor.
VL
I
The capacitor voltage trails the
current; the inductor voltage
leads it.
VC
We want to calculate the entire
applied voltage from the
generator.
VR
RCL Circuit
We will add the voltage phasors as vectors
(which is what they are.)
We start out by adding the reactive
voltages (across the capacitor and the
inductor).
This is easy because those phasors are
opposite in direction. The resultant’s
magnitude is the difference of the two,
and its direction is that of the larger
one.
VR
VL - V C
I
RCL Circuit
Now we use Pythagoras’ Theorem to add
the VL – VC phasor to the VR phasor.
V = VR2 + (VL - VC)2
VR
VL - V C
I
RCL Circuit
The current phasor is unaffected by our
addition of the voltage phasors.
It now makes an angle f with the overall
applied voltage phasor.
V = VR2 + (VL - VC)2
f
I
RCL Circuit
We can make Ohm’s Law substitutions for the voltages:
VC  IX C
VL  IX L
VR  IR
V  VR  VL  VC   I R  IX L  IX C 
2
2
2
V  I R  I X L  X C 
2
2
2
2
V  I R  X L  X C 
2
2
2
2
RCL Circuit
Our result:
V I R
2
2


 X L  XC
suggests an Ohm’s Law relationship for the combined loads in
the series RCL circuit:
V  IZ  I R   X L  X C 
2
2
Z  R2  X L  X C 
2
Z is called the impedance of the RCL circuit.
SI units: ohms (W)
RCL Circuit -- Power
If the load is purely resistive, the
2
average power dissipated is P  I rms R
V = VR2 + (VL - VC)2
We can use the phasor diagram to
relate R to Z trigonometrically:
VR I rms R R

  cos f
V
I rms Z Z
f
VL - VC
R  Z cos f
“power factor”
P  I rms R  I rms Z cos f
2
2
I
VR
RCL Circuit -- Resonance
Series-connected inductor and capacitor:
Capacitor is initially charged.
When connection is made,
+
discharge current I flows
through inductor. Induced EMF
opposes and limits discharge
current.
C
L
I
RCL Circuit -- Resonance
When capacitor is discharged,
current I slows and stops.
Decrease of magnetic flux in
inductor induces EMF that
opposes the decrease (and
continues the current I).
+
C
L
I
-
RCL Circuit -- Resonance
Induced current charges C with
the opposite polarity to its
original state. When the
capacitor is charged, the current I is stopped.
+
C
L
RCL Circuit -- Resonance
Now the capacitor begins to
discharge through the inductor
again – this time in the opposite
direction (new discharge current = -I).
+
Opposite induced EMF across
inductor again limits this new
discharge current.
The cycle continues.
C
-I
L
RCL Circuit -- Resonance
Energy is alternately stored in
the capacitor (in the form of
the electrical potential
energy of separated charges)
and in the inductor (in its
magnetic field). When the
magnetic field collapses, it
charges the capacitor; when
the capacitor discharges, it
builds the magnetic field in
the inductor.
C
L
RCL Circuit -- Resonance
This “LC oscillator” or “tuned tank circuit” oscillates at a natural
or resonant frequency of
f res 
1
2 LC
C
L
RCL Circuit -- Resonance
At the resonant frequency, how are the inductive and capacitive
reactances related?
f res 
XC 
1
2 LC
1

2f resC
1
LC

2C
C
2 LC
X L  2f res L  2L 
1
2 LC

The reactances are equal to each other.
L
L LC
LC


 XC
LC
C
LC
RCL Circuit -- Resonance
At the resonant frequency, when the inductive and capacitive
reactances are equal, what is the situation in the circuit?
VL
I
VC
VR
I
VR
RCL Circuit -- Resonance
At the resonant frequency, when the inductive and capacitive
reactances are equal, what is the impedance of the circuit?
Z  R2  X L  X C   R2  R
2
At resonance, the circuit’s impedance is simply equal to its
resistance, and its voltage and current are in phase.
If the resistance is small, the current may be quite large.