Laboratory #4
Series and Parallel AC Circuits
I.
Objectives
1. Understand the phasor relationship between the voltage phasor of each branch
of the series AC circuit and the total voltage phasor.
2. Understand the phasor relationship between the current phasor of each branch
of the series AC circuit and the total current phasor.
3. Understand the phasor relationship between current phasor and voltage phasor
of each branch of a series parallel AC circuit.
II. Principle
RL Series Circuit
1.
i = I msinwt
IXL
R
V
Re
v
I = Im 0°
VR Ve = IRe
j
q
L
VL
Fig. 4.1. RL series circuit.
According to Kirchhoff’s voltage law (Let Re be small and negligible):
v = vR + vL = iR + L
d ( I m sin wt )
di
= RI m sin wt + L
dt
dt
= RI m sin wt + w LI m cos wt = I m ( R sin wt + wL cos wt )


R
wL
= I m  R 2 + w 2 L2 (
sin wt +
cos wt ) 
R 2 + w 2 L2
R 2 + w 2 L2


(4-1)
= I m  R 2 + w 2 L 2 sin(wt +  ) 


The phase angle difference θ between the current phasor and the voltage
phasor can be found:
 = tan −1
wL
(4-2)
R
Since the coil of an inductor has the equivalent series resistance R, the phase
angle between the voltage phasor and the current phasor is not 90, as shown in
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
51
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
the above vector diagram.
2.
RC Series Circuit
i = I msinwt
VR Ve = IRe

R
j
I = Im 0°
VC
Re
v
C
IXC
V
Fig. 4.2. RC series circuit.
3.
RC Parallel Circuit
I
iR
v = Vmsinwt
R
iC
C
IC
j

V = Vm 0°
IR
Fig. 4.3. RC parallel circuit.
According to Kirchhoff’s current law (Let Re be small and negligible):
i = iR + iC =
v
dV
+C
R
dt
(4-3)
Vm sin wt
d (Vm sin wt )
+C
R
dt
V sin wt
1
= m
+ wCVm cos wt = Vm ( sin wt + wC cos wt )
R
R
v = Vm sin wt =


1
 1

wC
R
= Vm  ( ) 2 + w 2C 2 (
sin wt +
cos wt ) 
1
1
 R

( ) 2 + w 2C 2
( ) 2 + w 2C 2


R
R
(4-4)
1
= Vm ( ( ) 2 + w 2C 2 sin(wt +  ))
R
The phase angle difference θ between the current phasor and the voltage phasor
can be found:
(4-5)
 = tan −1 wCR
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
52
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
4.
RL Parallel Circuit
IR

iR
v = Vmsinwt
j
V = Vm 0°
iL
R
IL
L
I
Fig. 4.4. RL parallel circuit.
5.
RL Series and Parallel Circuit
vR1
iL
iR2
R1
vT
IR2
iT
vR2
R2L
R
IL
R
VR2
L
IT
T
L
VT
VR1
Fig. 4.5. RL series and parallel circuit.
III. PSPICE simulation
Please use PSPICE to simulate iT、iR、iC、iL、vT、vR、vC、vL of all circuits in
this experiment. Time span is from 0 s to 50 ms. Please attach the circuit diagram,
the voltage waveform diagram, and the current waveform diagram to the pre-lab
report and mark the peak value of these waveforms.
IV. Components and instruments
1.
2.
Components
(1) Resistors: 50 Ω × 2
(2) Capacitor: 0.047 F × 1
(3) Inductor: 1 mH × 1
Instruments
(1) Digital wattmeter
(2) Oscilloscope
(3) Digital multimeter
(4) Function generator
V. Experimental methods and results
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
53
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
1. Using the digital wattmeter to measure RC and RL series circuits
(1) Connection diagram is as below:
Digital
Wattmeter
I+
I
V+
-
V
- 5 Vrms
10 kHz
K1
K2
R
50 Ω
R
50 Ω
C
0.047 μF
L
1 mH
Fig. 4.6. Digital wattmeter.
(2) Record:
RC series circuit (close K1)
vT (V)
iT (A)
PT (W)
RL series circuit (close K2)
pf
(leading or lagging)
vT (V)
iT (A) PT (W)
pf
(leading or lagging)
2. RC series circuit
(1) Connection diagram is as below:
a
it
50 Ω
b
5 Vrms v
T
10 kHz
0.047 μF
c
Fig. 4.7. RC series circuit.
(2) Use the oscilloscope to observe the voltage waveforms across R and C and
measure the phase difference:
a). The attenuation ratio of all differential probes are set to be X200.
b). In CH1, the positive terminal is connected to point a and the negative
terminal is connected to point b to measure the voltage waveform across
R, where the oscilloscope is set to be 40 V/div. and 5 ms/div.
c). In CH2, the positive terminal is connected to point b and the negative
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
54
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
terminal is connected to point c to measure the voltage waveform across
C, where the oscilloscope is set to be 40 V/div. and 5 ms/div.
d). In CH3, the positive terminal is connected to point a and the negative
terminal is connected to point c to measure the voltage waveform of the
RC series circuit, where the oscilloscope is set to be 40 V/div. and 5
ms/div.
e). Setting of measured value: Press Measure button first, then press the
first button on the right-hand side of the screen to enter the setting of the
measurement value. The first and second sources are selected as CH1
and CH2, respectively, and the auto-measurement is selected as the
f).
phase. Then press Exit to complete the measurement settings. The
measured value is the phase angle difference between R and C.
Press Save to save the waveform(s) in USB.
3. RL series circuit
(1) Connection diagram is as below:
it
a
50 Ω
b
1 mH
vT
c
Fig. 4.8. RL series circuit.
(2) Use the oscilloscope to observe the voltage waveforms across R and L, and
measure their phase angle difference:
a).
The attenuation ratio of all differential probes are set to be X200.
b).
In CH1, the positive terminal is connected to point a and the
negative terminal is connected to point b to measure the voltage
c).
d).
waveform across R, where the oscilloscope is set to be 40 V/div.
and 5 ms/div.
In CH2, the positive terminal is connected to point b and the
negative terminal is connected to point c to measure the voltage
waveform across C, where the oscilloscope is set to be 40 V/div.
and 5 ms/div.
In CH3, the positive terminal is connected to point a and the
negative terminal is connected to point c to measure the voltage
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
55
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
waveform across the RC series circuit, where the oscilloscope is
set to be 40 V/div. and 5 ms/div.
Setting of measured values: Press Measure button first, then press
the first button on the right-hand side of the screen to enter the
setting of the measurement value(s). The first and second sources
are selected as CH1 and CH2, respectively, and the automeasurement is selected as the phase. Then press Exit to complete
the measurement settings. The measured value is the phase angle
difference between R and L.
e).
4. RC Parallel Circuit
(1) Connection diagram is as below:
a
it
iC
iR
5 Vrms vT
10 kHz
50 Ω
0.047 μF
b
Fig. 4.9. RC parallel circuit.
(2) Record:
vT (V)
iT (A)
iR (A)
iC (A)
PT (W)
PR (W)
PC (W)
(3) Calculate the power factor and phase angle using the above measured
values:
pfR
(leading or lagging)
pfC
(leading or lagging)
pfT
(leading or lagging)
θR
(deg.)
θC
(deg.)
θT
(deg.)
5. RL Parallel Circuit
(1) Connection diagram is as below:
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
56
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
a
it
iR
iL
50 Ω
5V
v
10 kHz T
50 Ω
1 mH
b
Fig. 4.10. RL parallel circuit.
(2) Record:
vT (V)
iT (A)
iR (A)
iL (A)
PT (W)
PR (W)
PL (W)
(3) Calculate the power factor and phase angle using the above measured
values:
pfR
(leading or lagging)
pfL
(leading or lagging)
pfT
(leading or lagging)
θR
(deg.)
θL
(deg.)
θT
(deg.)
VI. Problems and conclusion
1.
2.
3.
4.
5.
6.
Compare the differences between the measured values and the simulated
value of PSPICE.
Compare the values recorded in 1, why is VT different from VR + VC in series
circuit?
In the series and parallel experiments, the total power is the sum of the branch
power?
Taking the current as a reference, draw the phasor diagram of each branch
voltage in 4 and find the sum, and compare it with the measured value.
Taking the voltage as a reference, draw the phasor diagram of each branch
voltage in 5 and find the sum, and compare it with the measured value.
Conclusion:
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
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王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
Laboratory #4 Report
Group:
Name:
I.
Student ID:
Experimental methods and results
1. Using the digital wattmeter to measure RC and RL series circuits
(1) Connection diagram is as below:
Digital
Wattmeter
I+
I
V+
-
V
- 5 Vrms
10 kHz
K1
K2
R
50 Ω
R
50 Ω
C
0.047 μF
L
1 mH
Fig. 4.6. Digital wattmeter.
(2) Record:
RC series circuit (close K1)
vT (V)
iT (A)
PT (W)
RL series circuit (close K2)
pf
(leading or lagging)
vT (V)
iT (A) PT (W)
pf
(leading or lagging)
2. RC series circuit
(1) Connection diagram is as below:
a
it
50 Ω
b
5 Vrms v
T
10 kHz
0.047 μF
c
Fig. 4.7. RC series circuit.
(2) Use the oscilloscope to observe the voltage waveforms across R and C and
measure the phase difference:
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
58
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
g). The attenuation ratio of all differential probes are set to be X200.
h). In CH1, the positive terminal is connected to point A and the negative
terminal is connected to point B to measure the voltage waveform across
R, where the oscilloscope is set to be 40 V/div. and 5 ms/div.
i). In CH2, the positive terminal is connected to point B and the negative
terminal is connected to point C to measure the voltage waveform across
C, where the oscilloscope is set to be 40 V/div. and 5 ms/div.
j). In CH3, the positive terminal is connected to point A and the negative
terminal is connected to point C to measure the voltage waveform of the
RC series circuit, where the oscilloscope is set to be 40 V/div. and 5
ms/div.
k). Setting of measured value: Press Measure button first, then press the
first button on the right-hand side of the screen to enter the setting of the
measurement value. The first and second sources are selected as CH1
and CH2, respectively, and the auto-measurement is selected as the
phase. Then press Exit to complete the measurement settings. The
measured value is the phase angle difference between R and C.
l). Press Save to save the waveform(s) in USB.
3. RL series circuit
(1) Connection diagram is as below:
it
a
50 Ω
b
1 mH
vT
c
Fig. 4.8. RL series circuit.
(2) Use the oscilloscope to observe the voltage waveforms across R and L, and
measure their phase angle difference:
a).
The attenuation ratio of all differential probes are set to be X200.
b).
In CH1, the positive terminal is connected to point A and the
negative terminal is connected to point B to measure the voltage
waveform across R, where the oscilloscope is set to be 40 V/div.
and 5 ms/div.
c).
In CH2, the positive terminal is connected to point B and the
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
59
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
negative terminal is connected to point C to measure the voltage
waveform across C, where the oscilloscope is set to be 40 V/div.
and 5 ms/div.
In CH3, the positive terminal is connected to point A and the
negative terminal is connected to point C to measure the voltage
waveform across the RC series circuit, where the oscilloscope is
set to be 40 V/div. and 5 ms/div.
Setting of measured values: Press Measure button first, then press
the first button on the right-hand side of the screen to enter the
setting of the measurement value(s). The first and second sources
d).
e).
are selected as CH1 and CH2, respectively, and the automeasurement is selected as the phase. Then press Exit to complete
the measurement settings. The measured value is the phase angle
difference between R and L.
4. RC Parallel Circuit
(1) Connection diagram is as below:
a
it
iC
iR
5 Vrms vT
10 kHz
50 Ω
0.047 μF
b
Fig. 4.9. RC parallel circuit.
(2) Record:
vT (V)
iT (A)
iR (A)
iC (A)
PT (W)
PR (W)
PC (W)
(3) Calculate the power factor and phase angle using the above measured
values:
pfR
(leading or lagging)
pfC
(leading or lagging)
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
pfT
(leading or lagging)
60
θR
(deg.)
θC
(deg.)
θT
(deg.)
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南
5. RL Parallel Circuit
(1) Connection diagram is as below:
a
it
iR
iL
50 Ω
5V v
T
10 kHz
50 Ω
1 mH
b
Fig. 4.10. RL parallel circuit.
(2) Record:
vT (V)
iT (A)
iR (A)
iL (A)
PT (W)
PR (W)
PL (W)
(3) Calculate the power factor and phase angle using the above measured
values:
pfR
pfL
pfT
θR
θL
θT
(leading or lagging)
(leading or lagging)
(leading or lagging)
(deg.)
(deg.)
(deg.)
II.
Problems and conclusion
1.
2.
3.
4.
5.
6.
Compare the differences between the measured values and the simulated
value of PSPICE.
Compare the values recorded in 1, why is VT different from VR + VC in series
circuit?
In the series and parallel experiments, the total power is the sum of the branch
power?
Taking the current as a reference, draw the phasor diagram of each branch
voltage in 4 and find the sum, and compare it with the measured value.
Taking the voltage as a reference, draw the phasor diagram of each branch
voltage in 5 and find the sum, and compare it with the measured value.
Conclusion:
Profs. Li Wang and Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan
61
王醴、郭泰豪教授, 國立成功大學電機系, 臺灣台南