RCRC Circuits Circuits
Chapter 10
Thomas L. Floyd
David M. Buchla
DC/AC Fundamentals: A Systems DC/AC Fundamentals: A Systems ApproachApproach
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.10 Summary
Sinusoidal Response of RC Circuits
R C
VS
VR leads VS
VSVR
VC lags VS
VS VC
I
I leads VS
VS
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
In a series RC circuit, the total impedance is the phasor sum of R and XC.
R is plotted along the positive x-axis.
XC is plotted along the negative y-axis.
It is convenient to reposition the phasors so they form an impedance triangle.
Z is the diagonal
Ch.10 Summary
Impedance of Series RC Circuits
R
XC1tan
R
Z
R
Z
XCXC
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Sketch the impedance triangle and show the values for R = 1.2 k and XC = 960 .
Ch.10 Summary
Impedance of Series RC Circuits
kΩ 1.33kΩ (0.96kΩ (1.2 2222 CXRZ
39
kΩ 1.2
kΩ 0.96tan
tan
1
1
R
XC
Z = 1.33 kXC = 960
R = 1.2 k
39o
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Ohm’s law is applied to series RC circuits using Z, V, and 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.
Ch.10 Summary
Series RC Circuit Analysis
I
V Z
Z
V I IZV
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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):
Ch.10 Summary
Series RC Circuit Analysis
x 10 mA =
Z = 1.33 kXC = 960
R = 1.2 k
39o 39o
VR = 12 V
VC = 9.6 VVS = 13.3 V
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Reactance phasors can only be drawn for a single frequency because X is a function of frequency.
Ch.10 Summary
Phase Angle vs. Frequency
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.
Z3
XC1
XC2
XC3
Z2
1
2
f
f
f
3
R Increasing f
3
2
1
Z1
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.10 Summary
Application
(phase lag)
(phase lag)
V
Vin Vout
VinVout
VR
ff
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Reversing the components in the previous circuit produces a circuit that is a basic lead network. This circuit is a basic high-pass filter, a circuit that passes high frequencies and rejects all others. This filter passes high frequencies down to a frequency called the cutoff frequency.
Ch.10 Summary
Application
(phase lead)
V
R(phase lead)
VinVout
Vout
VinVC
Vout
Vin
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.10 Summary
Application
Amplifier
R R R
C C CPhase-shift network
Rf
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
For parallel circuits, it is useful to introduce two new quantities (susceptance and admittance) and to review conductance.
Ch.10 Summary
AC Response of Parallel RC Circuits
Conductance is the reciprocal of resistance. R
G1
Capacitive susceptance is the reciprocal of capacitive reactance. C
C XB
1
Admittance is the reciprocal of impedance.Z
Y1
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
In a parallel RC circuit, the admittance phasor is the sum of the conductance and capacitive susceptance phasors:
From the diagram, the phase angle is:
Ch.10 Summary
AC Response of Parallel RC Circuits
22CBGY
VS G BCBC Y
G
G
BC1tan
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Draw the admittance phasor diagram for the circuit.
The magnitudes of conductance, susceptance, and admittance are:
Ch.10 Summary
AC Response of Parallel RC Circuits
mS 1kΩ 1
11
RG mS 628μF) kHz)(.01 (102
1
CC X
B
mS 1.18mS) (0.628mS) (1 2222 BGY C
VSR C
f =10 kHz
1 k 0.01 mFY
G = 1 mS
BC
628 mS1.18 mS
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Ohm’s law can be applied to parallel RC circuits using Y, V, and I.
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.
Ch.10 Summary
Analysis of Parallel RC Circuits
V
IY VYI
Y
IV
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.10 Summary
Analysis of Parallel RC Circuits
x 10 V=
x 10 V=Y =
1.18 mS
G = 1.0 mS
BC = 0.628 mS
IR = 10 mA
IC = 6.28 mA
IS = 11.8 mA
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Notice that the formula for capacitive susceptance is the reciprocal of capacitive reactance. Thus BC and IC are directly proportional to f:
Ch.10 Summary
Phase Angle of Parallel RC Circuits
fCBC 2
As frequency increases, BC and IC must also increase, so the angle between IR and IS must increase.
IC IS
IR
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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:
Ch.10 Summary
Equivalent Series and Parallel RC Circuits
Z
Req = Z cos
XC(eq) = Z sin
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Series-parallel RC circuits are combinations of both series and parallel elements. These circuits can be solved by methods from series and parallel circuits.
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.
The components in the yellow box are in parallel:
Ch.10 Summary
Series-Parallel RC Circuits
R C
R2 C2
Z1
R1 C1
R2 C2
Z2
21
211 CXRZ
22
22
222
C
C
XR
XRZ
For example, the components in the green box are in series:
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Measuring Phase Angle
μs 160div
μs 20div) (8.0
T
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
The phase shift is calculated from
55o
Ch.10 Summary
Measuring Phase Angle (Cont’d)
μs 24.5div
μs 5div) (4.9
t
360
μs 160
μs 24.5360
T
Δt
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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:
Ch.10 Summary
The Power Triangle
x 10 mA =
Z = 1.33 kXC = 960
R = 1.2 k
39o 39o
VR = 12 V
VC = 9.6 VVS = 13.3 V
Multiplying each value in the left-hand triangle gives you the corresponding value in the right-hand triangle.
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Multiplying the voltage phasors by Irms (10 mA) gives the power triangle values (because P = VI ). 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.
Ch.10 Summary
The Power Triangle (Cont’d)
VR = 12 V
VS = 13.3 V
VC = 9.6 V
x 10 mA=
x 10 mA=
x 10 mA=
Ptrue = 120 mW
Pa = 133 mVA
Pr = 96 mVAR
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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:
Power factor can vary from 0 (for a purely reactive circuit) to 1 (for a purely resistive circuit).
Ch.10 Summary
Power Factor
PF cos
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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.
Ch.10 Summary
Apparent Power
Some components such as transformers, motors, and generators are rated in VA rather than watts.
Ptrue (W)
Pa (VA)
Pr (VAR)
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
10 V dc
VoutVin
100
1 Fm10 V dc
0
10 V dc
0
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.
Plotting the response:
1ƒ = 1 kHz
8.46 V rms10 V rms 100
Fm1.57 V rms
10 V rms
1ƒ = 10 kHz100
Fm0.79 V rms
10 V rms
1ƒ = 20 kHz
100
Fm
Ch.10 Summary
RC Circuit Frequency Response
Vout (V)
9.98
8.46
1.570.79
0.1 1 10 20 100f (kHz)
9
8
7
6
5
4
3
2
1
Vout (V)
9.98
8.46
1.570.79
0.1 1 10 20 100f (kHz)
9
8
7
6
5
4
3
2
1
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
Vin
10 V dc
0
Vout
0 V dc10 V dc 100 1 Fm
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.
Plotting the response:
ƒ = 100 Hz
0.63 V rms10 V rms
100 1 Fm
ƒ = 1 kHz
5.32 V rms10 V rms
100 1 Fm
ƒ = 10 kHz
9.87 V rms10 V rms
100 1 Fm
Ch.10 Summary
RC Circuit Frequency Response
Vout (V)
f (kHz)
9.87
5.32
0.6300.01 0.1 1
10
9
8
7
6
5
4
3
2
1
10
Vout (V)
f (kHz)
9.87
5.32
0.63
00.01 0.1 1
10
9
8
7
6
5
4
3
2
1
10
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
The total opposition to sinusoidal current expressed in ohms.
The ability of a capacitor to permit current; the reciprocal of capacitive reactance, measured in siemens (S).
The angle between the source voltage and the total current in a reactive circuit.
A measure of the ability of a reactive circuit to permit current; the reciprocal of impedance, measured in siemens (S).
Ch.10 Summary
Key TermsImpedance
Phase angle
Capacitive susceptance
(BC)
Admittance (Y)
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
The frequency at which the output voltage of a filter is 70.7% of the maximum output voltage.
In electric circuits, the variation of the output voltage (or current) over a specified range of frequencies.
The relationship between volt-amperes and true power or watts. Volt-amperes multiplied by the power factor equals true power.
Ch.10 Summary
Key Terms
Power factor
Frequency response
Cutoff frequency
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
5. Admittance is the reciprocal of
a. reactance
b. resistance
c. conductance
d. impedance
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
9. The maximum power factor occurs when the phase angle is
a. 0o
b. 30o
c. 45o
d. 90o
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
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
Ch.10 Summary
Quiz
DC/AC Fundamentals: A Systems ApproachThomas L. Floyd
© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved
1. a
2. b
3. c
4. c
5. d
6. a
7. d
8. d
9. a
10. b
Ch.10 Summary
Answers
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