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DC/AC Fundamentals: A Systems
Approach
Thomas L. Floyd
David M. Buchla
RC Circuits
Chapter 10
Ch.10 Summary
Sinusoidal Response of RC Circuits
When resistance and capacitance are connected in series,
the phase angle between the applied voltage and total current
is between 0 and 90, depending on the values of resistance
and reactance.
VR
VS
VS
VR leads VS
VC
VC lags VS
R
C
VS
I
VS
I leads VS
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Impedance of Series RC Circuits
In a series RC circuit, the total impedance is the
phasor sum of R and XC.
R is plotted along the positive x-axis.
X 
  tan 1 C 
 R 
XC is plotted along the negative y-axis.
Z is the diagonal
R
It is convenient to
reposition the phasors
so they form an
impedance triangle.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd

XC
R

Z
Z
XC
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Ch.10 Summary
Impedance of Series RC Circuits
Sketch the impedance triangle and show the
values for R = 1.2 kW and XC = 960 W.
Z  R 2  XC2  (1.2 kΩ2  (0.96 kΩ2  1.33 kΩ
X 
  tan 1 C 
 R 
1  0.96 kΩ 
 tan 

 1.2 kΩ 
 39
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
R = 1.2 kW

39 o
XC = 960 W
Z = 1.33 kW
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Ch.10 Summary
Series RC Circuit Analysis
Ohm’s law is applied to series RC circuits using Z, V, and I.
V  IZ
I
V
Z
Z
V
I
Because I is the same everywhere in a series circuit, you can
obtain the various component voltages by multiplying the
impedance of that component by the current, as the following
example demonstrates.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Series RC Circuit Analysis
Assume the current in the previous example is 10 mA. Sketch
the voltage phasor diagram. (The impedance triangle from the
previous example is shown for reference.)
The voltage phasor diagram can be found using Ohm’s law.
Multiply each impedance phasor by 10 mA (as shown below):
VR = 12 V
R = 1.2 kW

39 o
XC = 960 W
x 10 mA
=

39 o
VS = 13.3 V
VC = 9.6 V
Z = 1.33 kW
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Phase Angle vs. Frequency
Reactance phasors can only be drawn for a single
frequency because X is a function of frequency.
R
As frequency changes, the
impedance triangle for an RC
circuit changes as illustrated
here because XC decreases
with increasing f. This
determines the frequency
response of RC circuits.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
3
2
Increasing f
1
Z3
XC3
f3
XC2
f2
XC1
f1
Z2
Z1
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Ch.10 Summary
Application
A series RC circuit can be used to produce a phase lag by a
specific amount between an input voltage and an output by taking
the output across the capacitor. This circuit is a basic low-pass
filter, a circuit that passes low frequencies and rejects all others.
This filter passes low frequencies up to a frequency called the
cutoff frequency.
V

Vin
Vout
VR
f
f
(phase lag)
Vin
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Vout
(phase lag)
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Ch.10 Summary
Application
Reversing the components in the previous circuit produces a
circuit that is a basic lead network. This circuit is a basic highpass filter, a circuit that passes high frequencies and rejects
all others. This filter passes high frequencies down to a
frequency called the cutoff frequency.
V
Vout

Vin
R
Vin
(phase lead)
Vout
Vout
VC
Vin

(phase lead)
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Application
An application showing how a phase-shift network is
useful is the phase-shift oscillator, which uses a
combination of RC networks to produce a 180o phase
shift that is required for the oscillator to work.
Amplifier
Rf
Phase-shift network
C
C
R
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
C
R
R
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Ch.10 Summary
AC Response of Parallel RC Circuits
For parallel circuits, it is useful to introduce two new quantities
(susceptance and admittance) and to review conductance.
Conductance is the reciprocal of
resistance.
1
G
R
Capacitive susceptance is the
reciprocal of capacitive reactance.
1
BC 
XC
Admittance is the reciprocal of impedance.
1
Y 
Z
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
AC Response of Parallel RC Circuits
In a parallel RC circuit, the admittance phasor is the sum of the
conductance and capacitive susceptance phasors:
Y  G2  BC2
From the
diagram, the
phase angle
is:
B 
  tan 1 C 
G 
VS
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
G
BC
Y
BC

G
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Ch.10 Summary
AC Response of Parallel RC Circuits
Draw the admittance phasor diagram for the circuit.
The magnitudes of conductance, susceptance, and admittance
are:
1
1
1
BC 
 2(10 kHz)(.01 μF)  628 mS
G 
 1 mS
XC
R 1 kΩ
Y  G2  BC2  (1mS)2  (0.628 mS)2  1.18 mS
VS
R
1 kW
C
0.01 mF
f =10 kHz
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
BC
628 mS

Y
1.18 mS
G = 1 mS
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Ch.10 Summary
Analysis of Parallel RC Circuits
Ohm’s law can be applied to parallel RC circuits using Y, V,
and I.
I
V
Y
I  VY
I
Y
V
Because V is the same across all components in a
parallel circuit, you can obtain the current in a given
component by simply multiplying the admittance of the
component by the voltage, as illustrated in the
following example.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Analysis of Parallel RC Circuits
If the voltage in the previous example is 10 V, sketch the
current phasor diagram. The admittance diagram from the
previous example is shown below for reference.
The current phasor diagram can be found from
Ohm’s law. Multiply each admittance phasor by 10 V.
BC = 0.628 mS
Y=
1.18 mS
G = 1.0 mS
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
x 10 V
=
IC = 6.28 mA
IS =
11.8 mA
IR = 10 mA
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Ch.10 Summary
Phase Angle of Parallel RC Circuits
Notice that the formula for capacitive susceptance is
the reciprocal of capacitive reactance. Thus BC and IC
are directly proportional to f:
BC  2fC
As frequency increases, BC
and IC must also increase, so
the angle between IR and IS
must increase.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
IC
IS

IR
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Ch.10 Summary
Equivalent Series and Parallel RC
Circuits
For every parallel RC circuit there is an equivalent series RC
circuit at a given frequency. The equivalent resistance and
capacitive reactance are shown on the impedance triangle:
Req = Z cos 

Z
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
XC(eq) = Z sin
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Ch.10 Summary
Series-Parallel RC Circuits
Series-parallel RC circuits are combinations of both series and
parallel elements. These circuits can be solved by methods from
Z1
series and parallel circuits.
Z2
For example, the
components in the
green box are in series:
RR1
CC1
RR22
CC22
Z1  R12  XC21
The components in
the yellow box are in
parallel:
R2 X C 2
Z2 
R22  X C2 2
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
The total impedance can be found by
converting the parallel components to
an equivalent series combination,
then adding the result to R1 and XC1
to get the total reactance.
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Ch.10 Summary
Measuring Phase Angle
An oscilloscope is commonly used to measure phase angle in
reactive circuits. The easiest way to measure phase angle is to set
up the two signals to have the same apparent amplitude and
measure the period. An example of a Multisim simulation is shown,
but the technique is the same in lab.
Set up the oscilloscope so that two
waves appear to have the same
amplitude as shown.
Determine the period. For the wave
shown, the period is
 20 μs 
T  (8.0 div)
  160 μs
 div 
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Measuring Phase Angle (Cont’d)
Next, spread the waves out using the SEC/DIV control in order to
make an accurate measurement of the time difference between
the waves. In the case illustrated, the time difference is
 5 μs 
t  (4.9 div)
  24.5 μs
 div 
The phase shift is calculated from
 24.5 μs 
 Δt 
  360  55o
     360  
T
 160 μs 
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
The Power Triangle
As shown earlier, you can multiply the impedance phasors for
a series RC circuit by the current to obtain the voltage
phasors. The earlier example is shown below for review:
VR = 12 V
R = 1.2 kW

39 o
XC = 960 W
x 10 mA
=

39 o
VS = 13.3 V
VC = 9.6 V
Z = 1.33 kW
Multiplying each value in the left-hand triangle gives you the
corresponding value in the right-hand triangle.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
The Power Triangle (Cont’d)
Multiplying the voltage phasors by Irms (10 mA) gives the power
triangle values (because P = VI ). Apparent power is the product
of the magnitude of the current and magnitude of the voltage and is
plotted along the hypotenuse of the power triangle.
VR = 12 V
x 10 mA
=
Ptrue = 120 mW
Pr =
96 mVAR
VC =
9.6 V
VS = 13.3 V
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Pa = 133 mVA
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Ch.10 Summary
Power Factor
Power factor is the ratio of true power (in W) to apparent
power (in VA). Volt-amperes multiplied by the power
factor equals true power. Power factor can be determined
using:
PF  cos 
Power factor can vary from 0 (for a purely reactive
circuit) to 1 (for a purely resistive circuit).
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Apparent Power
Apparent power consists of two components; the true power
component, which does the work, and a reactive power
component, that is simply power shuttled back and forth
between source and load.
Ptrue (W)
Some components such as
transformers, motors, and
generators are rated in VA
rather than watts.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
Pr (VAR)
Pa (VA)
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Ch.10 Summary
RC Circuit Frequency Response
When a signal is applied to an RC circuit, and the output is taken
across the capacitor as shown, the circuit acts as a low-pass filter.
As the frequency increases, the output amplitude decreases.
Vin
10 V dc
0
Vout
1010
VV
rms
rms
10 V rms
10 V dc
100
WW
W
100
W
100
ƒƒ ƒ= =110kHz
20
kHz
kHz
111mmF
mFFF
V rms
VV8.46
rms
10 V dc 1.57
0.79
rms
0
V out (V)
9.98
9.98
Plotting the response:
8.46
8.46
9
8
7
6
5
4
3
1.57
1.57
0.79
0.79
2
1
0.1
0.1
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
11
10
10 20
100
f (kHz)
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Ch.10 Summary
RC Circuit Frequency Response
Reversing the components, and taking the output across the
resistor as shown, the circuit acts as a high-pass filter.
As the frequency increases, the output amplitude also increases.
Vin
10 V dc
0
Vout
10VVrms
rms
10
10 V rms
10 V dc
1m F
111
mmm
FFFHz
ƒ = 100
ƒ ƒ= =1 10
kHzkHz
9.87V rms
5.32
VVrms
0.63
rms
100 W
W
100
100WW
100
0 V dc
Vout (V)
9.87
9.87
Plotting the response:
5.32
5.32
10
9
8
7
6
5
4
3
0.63
0.63
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
2
1
0
0.01
0.1
0.1
11
10
10
f (kHz)
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Ch.10 Summary
Key Terms
Impedance
The total opposition to sinusoidal current
expressed in ohms.
Phase angle The angle between the source voltage and
the total current in a reactive circuit.
Capacitive The ability of a capacitor to permit current;
susceptance the reciprocal of capacitive reactance,
(BC) measured in siemens (S).
Admittance (Y) A measure of the ability of a reactive circuit to
permit current; the reciprocal of impedance,
measured in siemens (S).
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Key Terms
Power factor
The relationship between volt-amperes and
true power or watts. Volt-amperes multiplied
by the power factor equals true power.
Frequency
response
In electric circuits, the variation of the output
voltage (or current) over a specified range of
frequencies.
Cutoff
frequency
The frequency at which the output voltage of
a filter is 70.7% of the maximum output
voltage.
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
1. If you know what the impedance phasor diagram
looks like in a series RC circuit, you can find the
voltage phasor diagram by
a. multiplying each phasor by the current
b. multiplying each phasor by the source
voltage
c. dividing each phasor by the source voltage
d. dividing each phasor by the current
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
2. A series RC circuit is driven with a sine wave. If
the output voltage is taken across the resistor, the
output will
a. be in phase with the input.
b. lead the input voltage.
c. lag the input voltage.
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
3. A series RC circuit is driven with a sine wave. If you
measure 7.07 V across the capacitor and 7.07 V
across the resistor, the voltage across both
components is
a. 0 V
b. 5 V
c. 10 V
d. 14.1 V
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
4. If you increase the frequency in a series RC circuit,
a. the total impedance will increase
b. the reactance will not change
c. the phase angle will decrease
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
5. Admittance is the reciprocal of
a. reactance
b. resistance
c. conductance
d. impedance
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
6. Given the admittance phasor diagram of a parallel
RC circuit, you could obtain the current phasor
diagram by
a. multiplying each phasor by the voltage
b. multiplying each phasor by the total current
c. dividing each phasor by the voltage
d. dividing each phasor by the total current
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
7. If you increase the frequency in a parallel RC
circuit,
a. the total admittance will decrease
b. the total current will not change
c. the phase angle between IR and IS will
decrease
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
8. The magnitude of the admittance in a parallel RC
circuit will be larger if
a. the resistance is larger
b. the capacitance is larger
c. both a and b
d. none of the above
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
9. The maximum power factor occurs when the
phase angle is
a. 0o
b. 30o
c. 45o
d. 90o
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Quiz
10. When power is calculated from voltage and current
for an ac circuit, the voltage and current should be
expressed as
a. average values
b. rms values
c. peak values
d. peak-to-peak values
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Ch.10 Summary
Answers
1. a
6. a
2. b
7. d
3. c
8. d
4. c
9. a
5. d
10. b
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
© 2013 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved