# Chapter 7 AC Analysis

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2011
National Technology and Science Press.
or reproduced in any form or by any means without written permission of
the publisher.
Contents
1
2
3
4
5
Introduction
1.1 Resources . . . . . . . . . . . . .
1.2 Goals for Student Deliverables
1.3 Student Deliverables Checklist
1.4 Acknowledgements . . . . . . .
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7
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11
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Resistive Circuits
2.1 Kirchhoff’s Laws (2-3) . . . . . . .
2.2 Equivalent Resistance (2-4) . . . . .
2.3 Current and Voltage Dividers (2-4)
2.4 Wye-Delta Transformation (2-5) . .
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Analysis Techniques
3.1 Node-Voltage Method (3-1) . . . . . . . . . . . . . . . . . .
3.2 Mesh-Current Method (3-2) . . . . . . . . . . . . . . . . . .
3.3 Superposition (3-4) . . . . . . . . . . . . . . . . . . . . . . .
3.4 Thévenin Equivalents, Maximum Power Transfer (3-5, 3-6)
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Operational Amplifiers
4.1 Ideal Op-Amp Model (4-3) . . .
4.2 Noninverting Amplifier (4-3) .
4.3 Summing Amplifier (4-5) . . . .
4.4 Signal Processing Circuits (4-8)
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RC and RL First-Order Circuits
5.1 Capacitors (5-2) . . . . . . . . .
5.2 Inductors (5-3) . . . . . . . . . .
5.3 Response of the RC Circuit (5-4)
5.4 Response of the RL Circuit (5-5)
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4
CONTENTS
6
7
8
RLC Circuits
6.1 Initial and Final Conditions (6-1) . . . . . . . . . . .
6.2 Natural Response of the Series RLC Circuit (6-3) . .
6.3 General Solution for Any Second-Order Circuit (6-6)
6.4 Two-Capacitor Second-Order Circuit (6-6) . . . . . .
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84
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AC Analysis
7.1 Impedance Transformations (7-5) . . . . .
7.2 Equivalent Circuits (7-6) . . . . . . . . . .
7.3 Phase-Shift Circuits (7-8) . . . . . . . . . .
7.4 Phasor-Domain Analysis Techniques (7-9)
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91
. 91
. 97
. 100
. 102
AC Power
8.1 Periodic Waveforms (8-1) .
8.2 Average Power (8-2) . . .
8.3 Complex Power (8-3) . . .
8.4 The Power Factor (8-4) . .
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105
105
111
114
117
Frequency Response of Circuits and Filters
9.1 Scaling (9-2) . . . . . . . . . . . . . . . .
9.2 Bode Plots (9-3) . . . . . . . . . . . . . .
9.3 Filter Order (9-5) . . . . . . . . . . . . .
9.4 Cascaded Active Filters (9-7) . . . . . . .
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121
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130
10 Laplace Transform Analysis Techniques
10.1 s-Domain Circuit Analysis (10-7) . . . . . . . .
10.2 Step Response (10-8) . . . . . . . . . . . . . . .
10.3 Transfer Function and Impulse Response (10-8)
10.4 Convolution Integral (10-9) . . . . . . . . . . .
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133
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147
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157
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11 Fourier Analysis Techniques
11.1 Fourier Series Representation (11-2) . . . . . .
11.2 Circuit Applications (11-3) . . . . . . . . . . . .
11.3 Fourier Transform (11-5) . . . . . . . . . . . . .
11.4 Circuit Analysis with Fourier Transform (11-8)
A Parts List
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159
B LM317 Voltage and Current Sources
163
B.1 Variable Voltage Source . . . . . . . . . . . . . . . . . . . . . . 164
B.2 Current Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
CONTENTS
5
C TL072 Operational Amplifier
169
173
E Transient Response Measurement Techniques
177
E.1 Time Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
F Sinusoid Measurement Techniques
179
F.1 Amplitude and Phase Measurements . . . . . . . . . . . . . . 179
183
6
CONTENTS
Chapter 1
Introduction
This supplement to Circuits by Ulaby and Maharbiz contains 40 additional
end-of-chapter problems designed for three-way solution: analytical, simulation, and measurement. After solving the problem analytically the student continues by solving the same problem with NI Multisim and then
once again with NI myDAQ computer-based instrumentation and circuit
components. By iterating on each dimension of the problem until all three
agree students “triangulate on the truth” and develop confidence in their
analytical and laboratory skills.
Each problem requests at least one common numerical value for comparison among the three methods. The percent difference between simulated and analytical results as well as measured-to-analytical results indicates the degree to which the student has achieved a correct solution.
Normally simulation and analytical results agree to within a percentage
point, and measurements often agree with analytical results to within five
percent.
The problems are organized as four per chapter for Chapters 2 through
of the textbook in parentheses. Each problem contains the problem statement and sufficient detail to guide the student through the simulation and
physical measurement steps. Short video tutorials are linked to each problem to provide detailed guidance on Multisim techniques and ELVISmx
computer-based instruments for the myDAQ.
This document is fully hyperlinked for section and figure references,
document is the most efficient way to access all links, and clicking a video
hyperlink automatically launches the video in a browser. Within the PDF,
8
CHAPTER 1. INTRODUCTION
use ALT+leftarrow to navigate back to a starting point.
1.1
Resources
• Appendix A details the parts list required to implement all of the circuits and includes links to component distributors.
• Appendix B describes how to implement a variable voltage source
and two styles of current sources with the LM317 adjustable voltage
regulator. Many of the circuits require a DC voltage other than the
standard ±15V and 5V power supplies offered by the NI myDAQ.
The adjustable voltage source pictured in Figure B.3 on page 165 should
be constructed at the beginning of the term and left in place for subsequent circuits.
• Appendix C describes the Texas Instruments TL072 dual operational
amplifier used in many of the circuits. The op amp is frequently used
as a voltage follower to strengthen the 2 mA current drive of the myDAQ analog outputs. Appendix D describes the Intersil DG413 quad
analog switch used in many of the transient response problems.
• Appendix E details a laboratory technique to measure time constants
while Appendix F explains how to measure amplitude and phase
shift for sinusoidal signals.
• Appendix G lists all of the available video links.
1.2
Goals for Student Deliverables
Students should document their work in sufficient detail so that it could
be replicated by others. Present your work on the “Analysis” section as
you would on a standard problem set. Be sure to include a “Given” section
with your own drawing of the circuit diagram, a “Find” section that lists
the requested results for the problem, a detailed solution process, and a
clearly-identified end result. Do all of this work on engineering green paper
or in a lab book or as otherwise required by your instructor.
The “Simulation” section presents your work to set up the circuit simulation in NI Multisim and the simulation results you used to obtain meaningful information. Create a word processing document that contains an
organized set of screenshots with highlights and annotations as well as text
1.2. GOALS FOR STUDENT DELIVERABLES
and dialog box setup parameters for information not already visible on the
schematic – circle parameters that you entered or changed away from default values. Also include simulation results, again circling control settings
that you changed and highlighting regions where you obtained information. Figure 1.1 illustrates a screenshot from NI Multisim properly highlighted to indicate control settings that were adjusted away from default
values as well as regions on the screen where measurements were obtained.
Interpret the simulation results by writing them in standard form including
units, and write any additional calculations that were necessary to reach an
end result for simulation.
Figure 1.1: NI Multisim screenshot showing proper markings to indicate
control settings adjusted away from default values as well as regions where
measurement was obtained.
NOTE: Screen shots in Microsoft Word 2010 can be easily captured and
highlighted as follows:
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10
CHAPTER 1. INTRODUCTION
1. Select “Insert” tab and then “Screenshot,”
2. Choose the desired window or select “Screen Clipping” to define an
arbitrary region,
3. Select “Shapes,” and
4. Place circles or boxes to highlight important values.
The “Measurement” section presents your work to set up the physical
circuit and NI ELVISmx signal generators and measurement instruments.
guidelines for the “Simulation” section. Your instructor may require a
student ID when you work on the problem outside of scheduled class time.
Also include a schematic diagram showing all myDAQ connections.
Finally, the “Summary” section compares the requested numerical results from each of the three methods. Tabulate three results for each requested numerical quantity (analytical, simulation, and measurement) and
tabulate two percentage differences for each requested numerical quantity:
• Simulation-to-Analytical: [(XS − XA )/XA ] × 100%
• Measurement-to-Analytical: [(XM − XA )/XA ] × 100%
1.3. STUDENT DELIVERABLES CHECKLIST
1.3
Student Deliverables Checklist
1. Engineering paper or lab book – submit directly to instructor:
(a) Analysis
i. “Given / Find” section including original circuit
ii. Detailed solution
iii. End result clearly identified
(b) Simulation – interpreted results from simulation screen shots
(c) Measurement
i. Circuit schematic with myDAQ connections
ii. Interpreted results
(d) Results comparison table
2. Word processor document – submit electronically to instructor:
(a) Simulation screen shots
i. Circuit schematic
ii. Dialog box parameters with circles around entered or modified control values
iii. Simulation results marked up to highlight key results
(b) Photo of circuit on breadboard and myDAQ connections (if required)
(c) Measurement screen shots
i. ELVISmx signal generator instruments with circles around
entered or modified values
ii. ELVISmx measurement instruments marked up to highlight
key results and circles around entered or modified control
values
11
12
CHAPTER 1. INTRODUCTION
1.4
Acknowledgements
I gratefully acknowledge contributions from the following individuals:
• Tom Robbins (NTS Press) for his editorial support throughout this
project,
• Erik Luther (National Instruments) for his enthusiastic support of the
NI myDAQ product for engineering education,
• David Salvia (Penn State University) for his helpful suggestions regarding the design of this project, and
• Rose-Hulman students in Electrical Systems ES203 (Spring 2011) who
offered much helpful feedback on their experience with selected problems.
Ed Doering
Department of Electrical and Computer Engineering
Rose-Hulman Institute of Technology
Terre Haute, IN 47803
[email protected]
Chapter 2
Resistive Circuits
2.1
Kirchhoff’s Laws (2-3)
Determine currents I1 to I3 and the voltage V1 in the circuit of Figure 2.1
with component values ISRC = 1.8 mA, VSRC = 9.0 V, R1 = 2.2 kΩ, R2 =
3.3 kΩ, and R3 = 1.0 kΩ.
Figure 2.1: Circuit for Problem 2.1
14
CHAPTER 2. RESISTIVE CIRCUITS
NI Multisim Measurements
Enter the circuit of Figure 2.1 on the preceding page into NI Multisim and
measure the currents I1 to I3 and the voltage V1 .
• Place components from the “Virtual Components” palette
• Place a Simulate → Instruments → Measurement Probe for each current
• Place a Simulate → Instruments → Multimeter to measure the voltage
V1
• Use interactive simulation Simulate → Run
NI Multisim video tutorials:
• Find commonly-used circuit components:
• Measure DC current with a measurement probe:
• Measure DC voltage with a voltmeter:
NI myDAQ Measurements
Build the circuit of Figure 2.1 on the previous page. Use the myDAQ DMM
(digital multimeter) to measure the currents I1 to I3 and the voltage V1 .
• Implement the voltage source VSRC according to the circuit diagram
of Figure B.2 on page 164.
• Measure VSRC with the myDAQ DMM voltmeter and adjust the potentiometer to set the voltage as close to 9.0 volts as possible.
• Implement the current source ISRC according to the circuit diagram
of Figure B.4 on page 166. Use a 680 Ω resistor for the adjustment
resistor R.
• Measure ISRC with the myDAQ DMM ammeter and confirm that the
current is close to 1.8 mA. If you desire more precision, use a 1.0 kΩ
potentiometer for R3 and adjust it accordingly.
2.1. KIRCHHOFF’S LAWS (2-3)
NI myDAQ video tutorials:
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
Further Exploration with NI myDAQ
Resistor R3 shares the same current as the current source. Study the effect
of this resistor on the rest of the circuit.
1. Replace R3 with a 1.0 kΩ potentiometer.
2. Measure current I1 and record the range of currents you observe as
you adjust the potentiometer over its full range.
3. Repeat for currents I2 and I3 and voltage V1 .
4. Which of the four measured values appears to be independent of the
value of R3 ?
5. Which of the four measured values appears to depend on the value
of R3 ?
6. Propose an explanation for your observations.
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CHAPTER 2. RESISTIVE CIRCUITS
2.2
Equivalent Resistance (2-4)
Find the equivalent resistance between the following terminal pairs under
the stated conditions:
1. A-B with the other terminals unconnected,
2. A-D with the other terminals unconnected,
3. B-C with a wire connecting terminals A and D, and
4. A-D with a wire connecting terminals B and C.
Figure 2.2: Circuit for Problem 2.2
Use these component values:
• R1 = 10 kΩ
• R2 = 33 kΩ
• R3 = 15 kΩ
• R4 = 47 kΩ
• R5 = 22 kΩ
2.2. EQUIVALENT RESISTANCE (2-4)
NI Multisim Measurements
Enter the circuit of Figure 2.2 on the facing page into NI Multisim and use
the multimeter to measure each of the four resistances under the stated
conditions.
• Place components from the “Virtual Components” palette.
• Place a Simulate → Instruments → Multimeter and choose the ohmmeter setting (“Ω” button).
• Place a ground symbol and attach it to one of the multimeter terminals.
NI Multisim video tutorials:
• Find commonly-used circuit components:
• Measure resistance with an ohmmeter:
NI myDAQ Measurements
Build the circuit of Figure 2.2 on the preceding page. Use the myDAQ
DMM (digital multimeter) as an ohmmeter to measure each of the four
resistances under the stated conditions.
• Measure and record the resistance of each resistor individually; do
this before you connect the resistors together.
• Place the resistors to match the resistor orientations shown in Figure 2.2 on the facing page.
• The circuit need not connect to the myDAQ analog ground AGND
terminal; only Multisim requires the ground connection.
NI myDAQ video tutorials:
• DMM ohmmeter:
http://decibel.ni.com/content/docs/DOC-12938
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CHAPTER 2. RESISTIVE CIRCUITS
Further Exploration with NI myDAQ
Ohm’s Law states that a resistor creates a proportional relationship between its voltage and current v = iR where the resistance R is the proportionality factor. Setting the resistor voltage v to a known value and
measuring the resulting current with an ammeter provides another way
to measure resistance. Apply this method to measure each of the four resistances and compare with your previous results.
1. Apply the NI myDAQ 5-volt source to the terminals A and D. Use
the 5V and DGND (digital ground) terminals, with 5V connected to
terminal A and DGND to terminal D.
2. Use the DMM voltmeter to measure the voltage v as it appears at the
resistor network, and then record this value. Expect the voltage to be
slightly less than 5.0 volts, and also expect that it will vary somewhat
from one circuit connection to the next.
3. Use the DMM ammeter to measure the current i flowing into terminal
A; record this value, too.
4. Calculate the effective resistance R of the resistor network from your
two measurements, and then compare this value to your other measurements.
5. Repeat for the remaining three resistance measurements.
2.3. CURRENT AND VOLTAGE DIVIDERS (2-4)
2.3
Current and Voltage Dividers (2-4)
Apply the concepts of voltage dividers, current dividers, and equivalent
resistance to find the currents I1 to I3 and the voltages V1 to V3 .
Figure 2.3: Circuit for Problem 2.3
Use these component values:
• VSRC = 12 V
• R1 = 1.0 kΩ, R2 = 10 kΩ, R3 = 1.5 kΩ, R4 = 2.2 kΩ, R5 = 4.7 kΩ, and
R6 = 3.3 kΩ
NI Multisim Measurements
Enter the circuit of Figure 2.3 into NI Multisim. Use measurement probes
to measure each current; use voltmeter indicators to measure each voltage.
• Place components from the “Virtual Components” palette.
• Place a ground symbol and attach it to the negative terminal of the
voltage source.
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CHAPTER 2. RESISTIVE CIRCUITS
• Place a Simulate → Instruments → Measurement Probe for each current.
• Place a voltmeter indicator to display each voltage (see video tutorial
for details).
NI Multisim video tutorials:
• Find commonly-used circuit components:
• Measure DC current with a measurement probe:
• Measure DC voltage with a voltmeter indicator:
NI myDAQ Measurements
Build the circuit of Figure 2.3 on the preceding page. Use the myDAQ
DMM (digital multimeter) as a voltmeter to measure each of the three voltages; use the DMM as an ammeter to measure each of the three currents.
• Measure and record the resistance of each resistor individually; do
this before you connect the resistors together.
• Place the resistors to match the resistor orientations shown in Figure 2.3 on the previous page.
• Implement the voltage source VSRC according to the circuit diagram
of Figure B.2 on page 164.
• Measure VSRC with the myDAQ DMM voltmeter and adjust the potentiometer to set the voltage as close to 12.0 volts as possible.
NI myDAQ video tutorials:
• DMM ohmmeter:
http://decibel.ni.com/content/docs/DOC-12938
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
2.4. WYE-DELTA TRANSFORMATION (2-5)
2.4
Wye-Delta Transformation (2-5)
1. Find the currents I1 and I2 .
2. Determine the power delivered by each of the two voltage sources.
Figure 2.4: Circuit for Problem 2.4
Use these component values:
• V1 = 15 V and V2 = 15 V
• R1 = 3.3 kΩ, R2 = 1.5 kΩ, R3 = 4.7 kΩ, R4 = 5.6 kΩ, R5 = 1.0 kΩ,
and R6 = 2.2 kΩ
NI Multisim Measurements
Enter the circuit of Figure 2.4 into NI Multisim. Use measurement probes
to measure each current, and use the wattmeter to measure the power associated with each voltage source.
• Place components from the “Virtual Components” palette.
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CHAPTER 2. RESISTIVE CIRCUITS
• Place a ground symbol and attach it to the negative terminal of the
voltage source.
• Place a Simulate → Instruments → Measurement Probe for each current.
• Place a Simulate → Instruments → Wattmeter for each voltage source,
taking care to wire the wattmeters according to the passive sign convention.
NI Multisim video tutorials:
• Find commonly-used circuit components:
• Measure DC current with a measurement probe:
• Measure DC power with a wattmeter:
NI myDAQ Measurements
Build the circuit of Figure 2.4 on the preceding page. Use the myDAQ
DMM (digital multimeter) as an ammeter to measure each of the two currents; use the DMM as a voltmeter to measure each of the two voltage
source values.
• Measure and record the resistance of each resistor individually; do
this before you connect the resistors together.
• Place the resistors to match the resistor orientations shown in Figure 2.4 on the previous page.
• Use the myDAQ -15V power supply connection for the left voltage
source and the +15V power supply connection for the right voltage
source; connect AGND (Analog Ground) to the node identified by the
ground symbol.
• Measure the actual values of V1 and V2 when connected to the circuit;
expect them to be slightly less than 15 volts.
• Remember to insert the DMM ammeter in series between the voltage
source and the resistor.
2.4. WYE-DELTA TRANSFORMATION (2-5)
• Determine power as the product of measured voltage and measured
current.
NI myDAQ video tutorials:
• DMM ohmmeter:
http://decibel.ni.com/content/docs/DOC-12938
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
23
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CHAPTER 2. RESISTIVE CIRCUITS
Chapter 3
Analysis Techniques
3.1
Node-Voltage Method (3-1)
Apply the node-voltage method to determine the node voltages V1 to V4 for
the circuit of Figure 3.1 on the following page. From these results determine
which resistor dissipates the most power and which resistor dissipates the
least power, and report these two values of power.
Use these component values:
• Isrc1 = 3.79 mA and Isrc2 = 1.84 mA
• Vsrc = 4.00 V
• R1 = 3.3 kΩ, R2 = 2.2 kΩ, R3 = 1.0 kΩ, and R4 = 4.7 kΩ
NI Multisim Measurements
Enter the circuit of Figure 3.1 on the next page into NI Multisim. Use DC
operating point analysis to determine the four node voltages and the power
dissipated by each resistor.
• Display the net names and rename them to match the four node voltages V1 to V4 ; use only the numbers for the net names.
• Set up a Simulate → Analyses → DC Operating Point analysis to display the four node voltages and the power associated with each resistor.
26
CHAPTER 3. ANALYSIS TECHNIQUES
Figure 3.1: Circuit for Problem 3.1
NI Multisim video tutorials:
• Display and change net names:
• Find node voltages with DC Operating Point analysis:
• Find resistor power with DC Operating Point analysis:
NI myDAQ Measurements
Build the circuit of Figure 3.1. Use the myDAQ DMM (digital multimeter)
as a voltmeter to measure each of the four node voltages.
• Implement the voltage source VSRC according to the circuit diagram
of Figure B.2 on page 164.
3.1. NODE-VOLTAGE METHOD (3-1)
• Measure VSRC with the myDAQ DMM voltmeter and adjust the potentiometer to set the voltage as close to 4.00 volts as possible. Record
the actual voltage you measured.
• Implement the current source Isrc1 according to the circuit diagram
of Figure B.5 on page 167. Use a 330 Ω resistor for the adjustment
resistor R.
• Measure Isrc1 with the myDAQ DMM ammeter and confirm that the
current is close to 3.79 mA. Record the actual current you measured.
• Implement the current source Isrc2 according to the circuit diagram
of Figure B.4 on page 166. Use a 680 Ω resistor for the adjustment
resistor R.
• Measure Isrc2 with the myDAQ DMM ammeter and confirm that the
current is close to 1.84 mA. Record the actual current you measured.
NI myDAQ video tutorials:
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• Measure node voltage:
http://decibel.ni.com/content/docs/DOC-12947
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
Further Exploration with NI myDAQ
As you are by now aware, the analytical solution and the simulation results
always agree very well, largely because you can enter exact component
values into the simulator. However, the physical circuit component values
do not match the nominal values exactly: the 5%-tolerance resistors can
vary ±5% from the nominal value represented by the color-coded bands,
and the “1250/R mA” formula for the LM317-based current source is an
approximation.
Explore what happens when you recalculate the analytical solution using measured component values.
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CHAPTER 3. ANALYSIS TECHNIQUES
1. Recalculate the four node voltages using the nominal resistor values
and the measured values for Isrc1 , Isrc2 , and VSRC . Create a data table to compare these values with your analytical results in terms of
difference and relative difference.
2. Measure and record the five resistances R1 to R4 .
3. Recalculate the four node voltages using the measured resistor values
and the measured values for Isrc1 , Isrc2 , and VSRC . Create a data table to compare these values with your analytical results in terms of
difference and relative difference.
4. Summarize your results: What level of agreement did you achieve
between the analytical solution and the physical measurements?
3.2. MESH-CURRENT METHOD (3-2)
3.2
Mesh-Current Method (3-2)
Apply the mesh-current method to determine the mesh currents I1 to I4 .
From these results determine the voltage across the current source V1 .
Figure 3.2: Circuit for Problem 3.2
Use these component values:
• ISRC = 12.5 mA
• VSRC = 15 V
• R1 = 5.6 kΩ, R2 = 2.2 kΩ, R3 = 3.3 kΩ, and R4 = 4.7 kΩ
NI Multisim Measurements
Enter the circuit of Figure 3.2 into NI Multisim. Use “Measurement Probes”
and interactive simulation to measure the four mesh currents. Use the multimeter or a measurement probe to display the voltage across the current
source.
• Place the measurement probe on wires that carry only a single mesh
current; remember that many of the resistors carry two mesh currents.
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CHAPTER 3. ANALYSIS TECHNIQUES
NI Multisim video tutorials:
• Measure DC mesh current with a measurement probe:
• Measure DC node voltage with a measurement probe:
• Measure DC voltage with a voltmeter:
NI myDAQ Measurements
Build the circuit of Figure 3.2 on the previous page. Use the myDAQ DMM
(digital multimeter) as an ammeter to measure each of the four mesh currents; use the DMM voltmeter to measure the voltage across the current
source.
• Place the resistors to match the resistor orientations shown in Figure 3.2 on the preceding page. Use 1-inch jumper wires to establish
the top connections between the resistors to facilitate measurement of
the mesh currents.
• Implement the voltage source VSRC with the NI myDAQ -15V power
supply.
• Implement the current source ISRC according to the circuit diagram
of Figure B.4 on page 166. Use a 100 Ω resistor for the adjustment
resistor R.
• Measure ISRC with the myDAQ DMM ammeter and confirm that the
current is close to 12.5 mA. If you desire more precision, use a 1.0 kΩ
NI myDAQ video tutorials:
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
3.2. MESH-CURRENT METHOD (3-2)
Further Exploration with NI myDAQ
Some types of digital-to-analog converters require binary-weighted currents that can be selectively summed together. With a slight modification to
your existing circuit topology you can redesign it to produce mesh currents
that meet your own specifications such as those required by the digital-toanalog converter.
1. Consider the modified circuit of Figure 3.3. Apply mesh-current analysis to write a set of equations in terms of the indicated currents and
resistor values.
2. Choose resistor values that will establish the binary-weighted current
values I2 = ISRC /2, I3 = ISRC /4, I4 = ISRC /8, and I5 = ISRC /16,
and that will limit the current source voltage V1 to 5 volts or less.
3. Note: The standard parts list of Appendix A on page 159 includes
resistors that are close to the calculated values you need.
4. Build the circuit and measure all five mesh currents I1 to I5 .
5. Measure the current source voltage V1 .
6. Evaluate your results to determine how well the circuit produces the
desired binary-weighted currents.
Figure 3.3: Modified circuit for Problem 3.2 to produce binary-weighted
currents.
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CHAPTER 3. ANALYSIS TECHNIQUES
3.3
Superposition (3-4)
1. Apply the superposition method to determine the current IA and the
voltage VB , i.e., find the current IA1 due to the current source I1 acting
alone, the current IA2 due to the voltage source V2 acting alone, and
the current IA3 due to the voltage source V3 acting alone, and then
evaluate the sum IA = IA1 + IA3 + IA3 . Make use of current dividers
and voltage dividers as much as possible.
2. Use the superposition method to determine the voltage VB .
3. Apply the node-voltage method to find IA and VB , and then compare
these results to those of the superposition method.
Figure 3.4: Circuit for Problem 3.3
Use these component values:
• I1 = 1.84 mA
• V2 = 3.0 V and V3 = 4.9 V
• R1 = 1.0 kΩ, R2 = 2.2 kΩ, and R3 = 4.7 kΩ
3.3. SUPERPOSITION (3-4)
NI Multisim Measurements
Enter the circuit of Figure 3.4 on the facing page into NI Multisim. Use
interactive analysis and the voltmeter and ammeter indicators.
• Place the AMMETER_V and VOLTMETER_H components to display the
current IA and the voltage VB .
• Set the active source to its intended value, and then set all of the other
sources to zero. After stopping the simulator, press Ctrl+Z (“undo”)
two times to return the sources to their original values.
• Repeat to determine the responses due to each source acting alone.
NI Multisim video tutorials:
• Measure DC voltage with a voltmeter indicator:
• Measure DC current with an ammeter indicator:
NI myDAQ Measurements
1. Build the circuit of Figure 3.4 on the preceding page. Use the myDAQ
DMM to measure IA and VB when all sources are active.
2. Measure IA1 to IA3 by activating only one source at a time. Disable
the other sources by disconnecting the current source (replace it by
an open circuit) and disconnecting the voltage source and replacing
it with a jumper wire (short circuit).
3. Repeat for VB1 to VB3 .
4. Add your results together for each source acting alone, and then compare this result to your original measurement when all sources were
active.
• Implement the voltage source V2 according to the circuit diagram of
Figure B.2 on page 164.
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CHAPTER 3. ANALYSIS TECHNIQUES
• Measure V2 with the myDAQ DMM voltmeter and adjust the potentiometer to set the voltage as close to 3.00 volts as possible. Record
the actual voltage you measured.
• Implement the voltage source V3 with the NI myDAQ 5V power supply. Connect the myDAQ digital ground DGND terminal to the analog ground AGND at your breadboard.
• Measure V3 with the myDAQ DMM voltmeter when the circuit is connected. Expect to find this value slightly less than 5.0 volts. Record
the actual voltage you measured.
• Implement the current source I1 according to the circuit diagram of
Figure B.4. Use a 680 Ω resistor for the adjustment resistor R.
• Measure I1 with the myDAQ DMM ammeter and confirm that the
current is close to 1.84 mA. Record the actual current you measured.
NI myDAQ video tutorials:
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
3.4. THÉVENIN EQUIVALENTS, MAXIMUM POWER TRANSFER (3-5, 3-6)
3.4
Thévenin Equivalents, Maximum Power Transfer (3-5, 3-6)
1. Find the Thévenin equivalent of the circuit of Figure 3.5 at terminals
(a,b) as seen by the load resistance RL .
2. Determine the open-circuit voltage VOC that appears at terminals (a,b).
3. Determine the short-circuit current ISC that flows through a wire connecting terminals (a,b) together.
4. Determine the maximum power PLmax that could be delivered by this
circuit.
Figure 3.5: Circuit for Problem 3.4
Use these component values:
• VSRC = 10 V
• R1 = 680 Ω, R2 = 3.3 kΩ, R3 = 4.7 kΩ, and R4 = 1.0 kΩ
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CHAPTER 3. ANALYSIS TECHNIQUES
NI Multisim Measurements
1. Enter the circuit of Figure 3.5 on the preceding page into NI Multisim.
Connect a resistor RL as a load between terminals (a, b).
2. Use interactive analysis and measurement probes to determine the
open-circuit voltage.
3. Use interactive analysis and measurement probes to determine the
short-circuit current.
4. Run a parameter sweep to plot the load resistance power as a function of load resistance connected between terminals (a, b). Use a plot
cursor to determine the value of maximum power.
These tips provide more detail about the Multisim techniques for this
problem:
• Place a measurement probe on terminal b to display the load current.
• Place a measurement probe on terminal a referenced to the probe you
placed on terminal b to display the voltage across the load.
• Set the load resistance to a small yet finite value such as 0.1 Ω. Run
the interactive simulator to determine the short-circuit current.
• Set the load resistance to a large yet finite value such as 100 MΩ; enter
this value as 100MEG rather than 100M because ”m” means ”milli”
regardless of case. Run the simulator to determine the open-circuit
voltage.
• Set up a Simulate → Analyses → Parameter Sweep to plot P(RL)
over the range 1 Ω to 10 kΩ. Choose a linear plot type, select “DC Operating Point” for “Analysis to Sweep,” and plot 100 evenly-spaced
points to create a smooth curve.
• Use the plot cursors to find the maximum value of the load power.
Compare this value to the maximum power you calculated analytically.
3.4. THÉVENIN EQUIVALENTS, MAXIMUM POWER TRANSFER (3-5, 3-6)
NI Multisim video tutorials:
• Measure DC current with a measurement probe:
• Measure DC voltage with a referenced measurement probe:
• Use a Parameter Sweep analysis to plot resistor power as a function
of resistance:
• Find the maximum value of trace in Grapher View:
NI myDAQ Measurements
1. Build the circuit of Figure 3.5 on page 35. Calculate the Thévenin
equivalent circuit from the measurements taken in the next two parts.
2. Recall that the DMM voltmeter has very high resistance and thus appears as an open circuit. Connect the voltmeter between terminals
(a, b) to measure the open-circuit voltage.
3. Also recall that the DMM ammeter has very low resistance and thus
appears as a short circuit. Connect the ammeter between terminals
(a, b) to measure the short-circuit current.
4. Connect the variable load circuit shown in Figure 3.6 on the following
channels to measure the overall load voltage and the voltage that appears across the shunt resistor; this latter voltage is proportional to
the load current. Run the LabVIEW VI “VIPR.vi” (described below)
to display the load’s voltage, current, power, and resistance. First
sweep the potentiometer throughout its full range to get a sense of
the overall behavior, and then collect and tabulate at least 10 measurements of load power and load resistance; adjust the potentiometer to take measurements in 1 mW steps. Also record the maximum
LabVIEW “VIPR.vi” details:
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CHAPTER 3. ANALYSIS TECHNIQUES
Figure 3.6: Variable load with potentiometer (variable resistor) Rvar and
shunt resistor Rsh . The total load resistance is Rvar +Rsh . NI myDAQ Analog Input 0 (AI0) monitors the overall load voltage between terminals A-B
and Analog Input 1 (AI1) monitors the voltage across the shunt resistor; the
load current is the shunt resistor voltage divided by Rsh .
• The LabVIEW VI “VIPR.vi” measures the overall load voltage on analog input channel 0 (AI0+ and AI0-) and the shunt resistor voltage on
resistance for best accuracy. “VIPR.vi” calculates the load current as
the voltage on AI1 divided by the entered shunt resistance value, the
resistance as the load voltage divided by the current.
• The measured current value can become somewhat noisy, and “VIPR.vi”
applies a noise filter to improve your ability to read the display. The
noise filter calculates the average value of all of the measurements
accumulated since the last time the measured voltage changed by at
3.4. THÉVENIN EQUIVALENTS, MAXIMUM POWER TRANSFER (3-5, 3-6)
least 0.01 volts. Disable the noise filter, if desired.
• “VIPR.vi” is linked at the bottom of http://decibel.ni.com/
content/docs/DOC-16389. Download this source file and doubleclick it to open in LabVIEW; click the “Run” button to start the VI.
NI myDAQ video tutorials:
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
Further Exploration with NI myDAQ
Try this simple yet effective technique to directly measure Thévenin resistance:
1. Measure the open-circuit voltage at terminals (a, b),
2. Connect a variable resistor as the load (10 kΩ potentiometer works
well for this circuit),
3. Monitor the load voltage and adjust the potentiometer until the voltage is exactly one half of the open-circuit voltage,
4. Disconnect the potentiometer from the circuit, and
5. Measure the potentiometer resistance with an ohmmeter; this value
is the Thévenin resistance.
Apply this method to the circuit of this problem and compare your results to your other measurements of Thévenin resistance.
Explain why this method works. H INT: Consider a Thévenin equivalent circuit connected to a load resistor and recall what you know about
voltage dividers.
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CHAPTER 3. ANALYSIS TECHNIQUES
Chapter 4
Operational Amplifiers
4.1
Ideal Op-Amp Model (4-3)
1. Determine a general expression for vout in terms of the resistor values
and iS for the circuit of Figure 4.1 on the next page.
2. Find vout for these specific component values: R1 = 3.3 kΩ, R2 =
4.7 kΩ, R3 = 1.0 kΩ, and iS = 1.84 mA.
3. Determine the range of R2 for which −11 ≤ vout ≤ +11 volts.
NI Multisim Measurements
1. Enter the circuit of Figure 4.1 on the following page into NI Multisim.
Use the virtual three-terminal op amp model.
2. Measure vout for the given set of component values.
3. Plot vout as a function of R2 with a Simulate → Analyses → Parameter Sweep analysis of the “DC Operating Point” type. Increase the
number of points as needed to ensure an adequate number of measurements to characterized the op amp output voltage in the region
of the specified voltage limits.
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CHAPTER 4. OPERATIONAL AMPLIFIERS
Figure 4.1: Circuit for Problem 4.1
NI Multisim video tutorials:
• Use a Parameter Sweep analysis to plot resistor power as a function
of resistance:
• Measure DC node voltage with a measurement probe:
NI myDAQ Measurements
1. Build the circuit of Figure 4.1 with the given component values. Implement the current source Isrc1 according to the circuit diagram of
Figure B.4 on page 166. Use a 680 Ω resistor for the adjustment resistor R. When complete, measure and record the source current iS with
the DMM ammeter and confirm that it is close to 1.84 mA.
2. Measure the value of the vout with the DMM voltmeter.
4.1. IDEAL OP-AMP MODEL (4-3)
3. Replace R2 with a 10 kΩ potentiometer. Monitor vout and adjust the
potentiometer until the voltage reaches the specified limit. Disconnect the potentiometer from the circuit and then measure its resistance with the DMM ohmmeter.
• Use the Texas Instruments TL072 op amp described in Appendix C.
Follow the pinout diagram of Figure C.1 on page 170 for either of the
two available op amps in the package. You may also use an equivalent dual-supply op amp.
• Power the op amp with myDAQ +15V to VCC+ and -15V to VCC− .
Use AGND for the circuit ground.
NI myDAQ video tutorials:
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
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CHAPTER 4. OPERATIONAL AMPLIFIERS
4.2
Noninverting Amplifier (4-3)
The circuit in Figure 4.2 uses a potentiometer whose total resistance is R1 .
The movable stylus on terminal 2 creates two variable resistors: βR1 between terminals 1–2 and (1 − β)R1 between terminals 2–3. The movable
stylus varies β over the range 0 ≤ β ≤ 1.
1. Obtain an expression for G = vo /vs in terms of β.
2. Calculate the amplifier gain for β = 0.0, β = 0.5, and β = 1.0 with
component values R1 = 10 kΩ and R2 = 1.5 kΩ.
3. Let vs be a 100-Hz sinusoidal signal with a 1-volt peak value. Plot vo
and vs to scale for β = 0.0, β = 0.5, and β = 1.0.
Figure 4.2: Circuit for Problem 4.2
NI Multisim Measurements
1. Enter the circuit of Figure 4.2 into NI Multisim. Use these specific
components: OPAMP 3T VIRTUAL, AC VOLTAGE, and virtual linear
4.2. NONINVERTING AMPLIFIER (4-3)
potentiometer; see the video tutorial below to learn how to search for
parts by name. Set the AC voltage source frequency to 100 Hz.
2. Observe vs and vo with the oscilloscope. Vary the potentiometer value
over its full range of 0 to 100%, and then use the oscilloscope cursors
to measure the circuit gain for β = 0.0, β = 0.5, and β = 1.0.
3. Print screen shots of the oscilloscope for β = 0.0, β = 0.5, and β =
1.0. Use the same Channel A and Channel B vertical scale (volts per
division) for all three screen shots.
• Use the Simulate → Instruments → Oscilloscope with vs on Channel A and vo on Channel B. Run an interactive simulation until several
cycles of oscillation appear on the oscilloscope display.
• Use the cursors to measure the peak values of the input and output
signals, and then calculate the amplifier gain as the output value divided by the input value.
NI Multisim video tutorials:
• Find components by name:
• AC (sinusoidal) voltage source:
• Basic operation of the two-channel oscilloscope:
• Waveform cursor measurements with the two-channel oscilloscope:
NI myDAQ Measurements
1. Build the circuit of Figure 4.2 on the facing page with the given component values. Use the following myDAQ signal connections:
• AO0 (Analog Output 0) for vs ,
• AI0 (Analog Input 0) to display vs ; connect AI0+ to the source
voltage and AI0- to ground, and
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CHAPTER 4. OPERATIONAL AMPLIFIERS
• AI1 (Analog Input 1) to display vo ; connect AI1+ to the output
voltage and AI1- to ground.
Create the 100-Hz sinusoidal waveform with the NI ELVISmx Function Generator.
2. Observe vs and vo with the NI ELVISmx Oscilloscope. Vary the potentiometer value over its full range of 0 to 100%, and then use the
oscilloscope cursors to measure the circuit gain for β = 0.0, β = 0.5,
and β = 1.0.
3. Print screen shots of the oscilloscope for β = 0.0, β = 0.5, and β =
1.0. Use the same Channel 0 and Channel 1 vertical scale (volts per
division) for all three screen shots.
• Use the Texas Instruments TL072 op amp described in Appendix C.
Follow the pinout diagram of Figure C.1 on page 170 for either of the
two available op amps in the package. You may also use an equivalent dual-supply op amp.
• Power the op amp with myDAQ +15V to VCC+ and -15V to VCC− .
Use AGND for the circuit ground.
• The potentiometer terminals of Figure 4.2 on page 44 follow the standard pinout used by single-turn trim potentiometers. If your potentiometer does not label the pins, the stylus pin (Pin 2) is normally
placed between Pins 1 and 3.
NI myDAQ video tutorials:
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
Further Exploration with NI myDAQ
Signal amplifiers apply a gain G ≥ 1 to increase the amplitude of weak
signals, thereby making the signal information easier to use elsewhere in
the system. Use a voltage divider circuit (known as an attenuator in this
4.2. NONINVERTING AMPLIFIER (4-3)
context) to create a “weak” signal from a portable audio player or computer
audio output and then investigate how well the amplifier you built in this
problem can restore the original signal strength.
1. Add the two-resistor voltage divider circuit to your amplifier as shown
in Figure 4.3 on the next page.
2. Connect one plug of the 3.2 mm stereo cable supplied with your myDAQ kit to your audio player. Connect the other plug to the attenuator input vm as shown in Figure 4.3 on the following page to apply
the left channel of the stereo audio signal to the attenuator input.
3. Play some music and observe the signal vs with the oscilloscope. Confirm that signal is indeed “weak” – its amplitude should be well under one volt peak.
4. Observe the amplifier output signal vo with the oscilloscope. Confirm
that you can adjust the circuit gain to strengthen the music signal’s
amplitude.
5. Connect your earphones to the circuit output and listen as you vary
the circuit gain.
Do NOT disturb your circuit connections while you are wearing earphones. Accidently shorting together circuit connections can produce
a very loud and unexpected noise. Alternatively, use a speaker to listen to the amplifier output or hold the earphones some distance from
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CHAPTER 4. OPERATIONAL AMPLIFIERS
Figure 4.3: Circuit for Problem 4.2 with voltage-divider attenuator and audio signal connections. The music signal is vm , the attenuated signal to be
amplified is vs , and the amplified signal is vo .
4.3. SUMMING AMPLIFIER (4-5)
4.3
Summing Amplifier (4-5)
1. Design an op amp summing circuit that performs the operation vo =
−(2.14v1 + 1.00v2 + 0.47v3). Use not more than four standard-value
resistors with values between 10 kΩ and 100 kΩ. Refer to the resistor
parts list in Appendix A on page 159.
2. Draw the output waveform vo for the input waveforms v1 and v2
shown in Figure 4.4 and v3 = 4.7 volts.
3. State the minimum and maximum values of vo .
Figure 4.4: Input waveforms for Problem 4.3
NI Multisim Measurements
1. Enter the op amp summing circuit that you designed earlier. Use the
following components and instruments:
• Virtual three-terminal op amp model OPAMP 3T VIRTUAL
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CHAPTER 4. OPERATIONAL AMPLIFIERS
• Piecewise-linear voltage source PIECEWISE LINEAR VOLTAGE
for v1
• Pulse voltage source PULSE VOLTAGE for v2
• Four-channel oscilloscope
2. Plot vo and the three inputs v1 to v3 with the four-channel oscilloscope.
3. Use the oscilloscope display cursors to identify the minimum and
maximum values of vo .
Additional Multisim tips for this problem:
• Specify the PWL voltage source waveform by entering endpoints of
straight lines as time-voltage pairs. The triangle waveform of Figure 4.4 on the previous page requires only three entries to specify a
complete period. Select “Repeat data during simulation” to create a
periodic triangle waveform.
• Three fields need to be adjusted for the pulse voltage source to make
it match the required square waveform shape: “Initial Value,” “Pulse
Width,” and “Period.”
NI Multisim video tutorials:
• Basic operation of the four-channel oscilloscope:
• Piecewise linear (PWL) voltage source:
• Pulse voltage source:
• Find components by name:
NI myDAQ Measurements
1. Build the op amp summing circuit that you designed earlier. Use the
following myDAQ signal connections:
• AO0 (Analog Output 0) for v1 ,
4.3. SUMMING AMPLIFIER (4-5)
• AO1 (Analog Output 1) for v2 ,
• AI0 (Analog Input 0) to display either v1 or v2 ; connect AI0+ to
the input voltage of interest and AI0- to ground,
• AI1 (Analog Input 1) to display vo ; connect AI1+ to the output
voltage and AI1- to ground,
Create the triangle and square waveforms with the NI ELVISmx Arbitrary Waveform Generator; use 50 kS/s as the sampling rate.
I MPORTANT: The two waveform files must of the same length (10 ms).
2. Plot vo and v1 with the NI ELVISmx Oscilloscope. Repeat with v2 .
3. Use the oscilloscope display cursors to identify the minimum and
maximum values of vo .
• Use the Texas Instruments TL072 op amp described in Appendix C
on page 169. Follow the pinout diagram of Figure C.1 on page 170 for
either of the two available op amps in the package. You may also use
an equivalent dual-supply op amp.
• Power the op amp with myDAQ +15V to VCC+ and -15V to VCC− .
Use AGND for the circuit ground.
• Implement the constant voltage source v3 according to the circuit diagram of Figure B.2 on page 164. Adjust the potentiometer until the
measured voltage is as close to 4.70 volts as possible.
NI myDAQ video tutorials:
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
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CHAPTER 4. OPERATIONAL AMPLIFIERS
Further Exploration with NI myDAQ
Investigate the effect of the gain constants for waveform inputs v1 and v2 .
You can quickly and easily vary a resistor value by placing another resistor
in parallel with it, thereby reducing the effective resistance. Place a 10 kΩ in
parallel with the source resistor associated with waveform v1 and observe
the impact on the output voltage waveform. Plot the new output waveform, summarize the difference from the original waveform, and explain
why reducing the resistor value causes this change in appearance.
Repeat the experiment with the source resistor associated with waveform v2 .
4.4. SIGNAL PROCESSING CIRCUITS (4-8)
4.4
Signal Processing Circuits (4-8)
1. Design a two-stage signal processor to serve as a “distortion box”
for an electric guitar. The first-stage amplifier applies a variable gain
magnitude in the range 13.3 to 23.3 while the second-stage amplifier
attenuates the signal by 13.3, i.e., the second-stage amplifier has a
fixed gain of 1/13.3. Note that when the first-stage amplifier gain is
13.3 the overall distortion box gain is unity. The distortion effect relies
on intentionally driving the first-stage amplifier into saturation (also
called “clipping”) when its gain is higher than 13.3.
Use a 10 kΩ potentiometer and standard-value resistors in the range
1.0 kΩ to 100 kΩ; see the resistor parts list in Appendix A on page 159.
You may combine two standard-value resistors in series to achieve
the required amplifier gains.
2. Derive a general formula for percent clipping of a unit-amplitude sinusoidal test signal; this is the percent of time during one period in
which the signal is clipped. The formula includes the peak sinusoidal
voltage VP that would appear at the output of the first-stage amplifier with saturation ignored and the actual maximum value VS due to
saturation.
3. Apply your general formula to calculate percent clipping of a 1-volt
peak amplitude sinusoidal signal for the potentiometer dial in three
positions: fully counter-clockwise (no distortion), midscale (moderate distortion), and fully clockwise (maximum distortion). Assume
the op amp outputs saturate at ±13.5 volts.
4. Apply a 1-volt peak amplitude sinusoidal signal with 100-Hz frequency to the distortion box input and plot its output for the potentiometer dial in the same three positions as above. State the maximum
and minimum values of the distortion box output.
NI Multisim Measurements
1. Enter your design for the distortion box into NI Multisim. Use the virtual five-terminal op amp model for both stages. Connect the power
supply terminals ±13.5 volts. Apply the AC (sinusoidal) voltage source
as the signal input; configure the source for 1-volt peak amplitude
and 100 Hz frequency.
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2. Observe the distortion box input and output signals with the oscilloscope and vary the potentiometer value over its full range of 0 to
100%, and then use the oscilloscope cursors to measure the percent
clipping for the potentiometer settings 0%, 50%, and 100%.
3. Print screenshots of the oscilloscope display to show the distortion
box input and output signals for the three potentiometer settings in
the previous step.
4. Measure the maximum and minimum values of the distortion box
output.
• Use these specific components: OPAMP 5T VIRTUAL, AC VOLTAGE,
and virtual linear potentiometer.
• Remember that the five-terminal op amp symbol when initially placed
has its positive power supply connection on top; applying a vertical
flip to the symbol also flips the positive power supply connection to
the bottom.
• Place the “CMOS Supply (VDD)” as the op amp positive power supply connection and “CMOS Supply (VSS)” as the negative power supply connection.
• Use the basic two-channel oscilloscope with the distortion box signal
input on Channel A and its output on Channel B. Run an interactive simulation until several cycles of oscillation appear on the oscilloscope display.
• Take cursor measurements to determine the time duration of clipping
for a half-cycle of the sinusoidal signal. Divide this time by the duration of the entire half-cycle and multiply by 100%.
4.4. SIGNAL PROCESSING CIRCUITS (4-8)
NI Multisim video tutorials:
• Basic operation of the two-channel oscilloscope:
• Waveform cursor measurements with the two-channel oscilloscope:
• AC (sinusoidal) voltage source:
• VDD and VSS power supply voltages:
NI myDAQ Measurements
1. Build your distortion box circuit and use the following myDAQ signal connections:
• AI0 (Analog Input 0) to display the input signal; connect AI0+ to
the source voltage and AI0- to ground,
• AI1 (Analog Input 1) to display the output signal; connect AI1+
to the output voltage and AI1- to ground,
Create the 100-Hz sinusoidal waveform with the NI ELVISmx Function Generator.
2. Observe the distortion box input and output signals with the NI ELVISmx
Oscilloscope and vary the potentiometer over its full range. Use the
oscilloscope cursors to measure the percent clipping with the potentiometer dial in three positions: fully counter-clockwise (no distortion), midscale (moderate distortion), and fully clockwise (maximum
distortion).
3. Print screenshots of the oscilloscope display to show the distortion
box input and output signals for the three potentiometer settings in
the previous step.
4. Measure the maximum and minimum values of the distortion box
output.
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CHAPTER 4. OPERATIONAL AMPLIFIERS
• Use the Texas Instruments TL072 op amp described in Appendix C
on page 169. Follow the pinout diagram of Figure C.1 on page 170 for
the two available op amps in the package. You may also use a pair of
equivalent dual-supply op amp.
• Power the op amp with myDAQ +15V to VCC+ and -15V to VCC− .
Use AGND for the circuit ground.
• The TL072 op amp and similar devices saturate at approximately 1.5
volts under the supply voltage, consequently the actual saturation
levels are about ±13.5 volts. If you use a different type of op amp with
rail-to-rail outputs then you should expect to see the output saturation
levels match the measured values of the myDAQ 15-volt dual power
supply.
• Should you need to observe the output of the first-stage amplifier
with the oscilloscope for troubleshooting purposes, you must consider the ±10 volt input range limitation of the myDAQ analog input
channels. This range limit effectively makes you blind to any signal
activity between 10 volts and the myDAQ power supply of 15 volts.
• Take cursor measurements to determine the time duration of clipping
for a half-cycle of the sinusoidal signal. Divide this time by the duration of the entire half-cycle and multiply by 100%.
NI myDAQ video tutorials:
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
Chapter 5
RC and RL First-Order Circuits
5.1
Capacitors (5-2)
The voltage v(t) across a 10-µF capacitor is given by the waveform shown
in Figure 5.1.
1. Determine the equation for the capacitor current i(t) and plot it over
the time 0 to 50 ms.
2. Calculate the values of capacitor current at times 0, 25, and 30 ms.
Figure 5.1: Voltage waveform for Problem 5.1
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
NI Multisim Measurements
Oscilloscopes display a time-varying voltage as a function of time. The current through a component such as the capacitor in this problem can also be
displayed on an oscilloscope with a small-valued “shunt resistor” placed
in series with the component. The shunt resistor produces a proportional
voltage according to Ohm’s Law v = Ri where the resistance R serves as
the proportionality constant. A trade-off exists here: a small shunt resistance minimizes disruption to the surrounding circuit, but a large shunt
resistance maximizes the available signal to the oscilloscope.
1. Enter a circuit that contains the following components:
• 10-µF capacitor and 10-Ω shunt resistor connected in series
• ABM (Analog Behavioral Modeling) voltage source (ABM VOLTAGE)
connected across the capacitor-resistor combination; set up the
voltage value to match the waveform of Figure 5.1 on the previous page.
• Two-channel oscilloscope showing the capacitor voltage on Channel A and the shunt resistor voltage on Channel B.
Run interactive simulation, adjusting the oscilloscope settings to display the capacitor voltage and current with each waveform filling a
reasonable amount of the available display.
2. Use the oscilloscope display cursors to measure the capacitor current
at times 0, 25, and 30 ms. Divide the cursor measurement by the shunt
resistor value.
Additional Multisim tips for this problem:
• Build the ABM voltage source “Voltage Value” string by combining
the following functions:
– u(TIME) – Step function u(t)
– uramp(TIME) – Ramp function r(t)
– exp(TIME) – Exponential function et
For example, the string 800*(uramp(TIME) - uramp(TIME-0.01))
implements the first 20 milliseconds of the capacitor voltage waveform.
5.1. CAPACITORS (5-2)
• Place a DC voltage source someplace on the schematic sheet to enable
interactive simulation. Do not connect the DC source to the capacitor
circuit itself.
NI Multisim video tutorials:
• Basic operation of the two-channel oscilloscope:
• Waveform cursor measurements with the two-channel oscilloscope:
• ABM (Analog Behavioral Model) voltage source:
• Distinguish oscilloscope traces by color:
NI myDAQ Measurements
The myDAQ analog outputs AO0 and AO1 cannot source more than 2 mA
and still maintain the expected voltage output. Use an op amp voltage
follower (see Ulaby Section 4-7) to create a “strengthened” copy of the myDAQ analog output.
1. Construct a circuit similar to the Multisim circuit you created earlier,
i.e., place a 10-ohm shunt resistor in series with the capacitor, and connect the capacitor-resistor combination between the voltage follower
output and ground.
IMPORTANT: Electrolytic capacitors are polarized; ensure that the
positive-labeled terminal connects to the op amp output. Alternatively, if the capacitor is marked with a negative-labeled terminal,
connect this terminal to the shunt resistor.
Establish the following myDAQ connections:
• AO0 (Analog Output 0) to the voltage follower input,
AI0+ to the positive capacitor terminal and AI0- to the negative
capacitor terminal,
• AI1 (Analog Input 1) to display the shunt resistor voltage; connect AI1- to ground.
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
Create the capacitor voltage waveform of Figure 5.1 on page 57 with
the NI ELVISmx Arbitrary Waveform Generator; use 50 kS/s as the
sampling rate.
Adjust the NI ELVISmx Oscilloscope settings to display the capacitor
voltage and current with each waveform filling a reasonable amount
of the available display. Use a combination of edge triggering on
Channel 0 and the “Horizontal Position” control to place the upper
left corner of the voltage waveform at time 10 ms.
2. Use the oscilloscope display cursors to measure the capacitor current
at times 0, 25, and 30 ms. Divide the cursor measurement by the
shunt resistor value. Improve your measurement accuracy by using
the measured shunt resistance obtained by the myDAQ DMM ohmmeter.
• Use the Texas Instruments TL072 op amp described in Appendix C
on page 169 for the voltage follower. Follow the pinout diagram of
Figure C.1 on page 170 for either of the two available op amps in the
package. You may also use an equivalent dual-supply op amp.
• Power the op amp with myDAQ +15V to VCC+ and -15V to VCC− .
Use AGND for the circuit ground.
• Use Cursor 1 to take measurements on the shunt resistor voltage.
Leave Cursor 2 at time zero to make the “dT” indicator show time
directly.
• The shunt resistor voltage signal is relatively low amplitude. Expect
the waveform to jump vertically somewhat. Simply click the oscilloscope “Stop” button to freeze the display.
• Create three waveform segments with the ARB “Waveform Editor,”
two of length 10 ms and the third of length 30 ms. Choose the “Expression” option for each segment and enter expressions that include
the time variable t as needed. Set the “X Range From” value of the
30 ms segment to 0.03 seconds and set “Cycles” to 1. You can then enter the exponential function exp() as shown in Figure 5.1 on page 57.
For convenience, create the complete waveform with a unit amplitude and then adjust the “Gain” value of the Arbitrary Waveform
Generator to 8.
5.1. CAPACITORS (5-2)
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
• Increase current drive of analog output (AO) channels with an
op amp voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
5.2
Inductors (5-3)
The voltage v(t) across a 33-mH inductor is given by the sinusoidal pulse
waveform shown in Figure 5.2.
1. Determine the equation for the inductor current i(t) and plot it over
the time 0 to 0.4 ms. Assume zero initial inductor current.
2. Determine the time at which the inductor current reaches its maximum value.
3. Calculate the total range of inductor current, i.e., the maximum value
minus the minimum value.
Figure 5.2: Voltage waveform for Problem 5.2
NI Multisim Measurements
Oscilloscopes display a time-varying voltage as a function of time. The current through a component such as the inductor in this problem can also be
displayed on an oscilloscope with a small-valued “shunt resistor” placed
5.2. INDUCTORS (5-3)
in series with the component. The shunt resistor produces a proportional
voltage according to Ohm’s Law v = Ri where the resistance R serves as
the proportionality constant. A trade-off exists here: a small shunt resistance minimizes disruption to the surrounding circuit, but a large shunt
resistance maximizes the available signal to the oscilloscope.
1. Enter a circuit that contains the following components:
• 33-mH inductor and 10-Ω shunt resistor connected in series
• Two AC voltage sources (AC VOLTAGE) connected in series and
also connected across the inductor-resistor combination. Flip the
orientation of one of the sources so that the series combination
forms the difference of the two voltages. Set one voltage for a delay of 0.1 ms and the other for a delay of 0.3 ms; in this way only
a single cycle of the sinusoid appears across the pair of sources
as in the waveform of Figure 5.2 on the preceding page.
• Two-channel oscilloscope showing the inductor voltage on Channel A and the shunt resistor voltage on Channel B.
Run interactive simulation, adjusting the oscilloscope settings to display the inductor voltage and current with each waveform filling a
reasonable amount of the available display.
2. Use the oscilloscope display cursors to measure the time at which the
inductor current reaches its maximum value.
3. Use the oscilloscope cursors to determine the total range of inductor
current, i.e., the maximum value minus the minimum value.
NI Multisim video tutorials:
• Basic operation of the two-channel oscilloscope:
• Waveform cursor measurements with the two-channel oscilloscope:
• AC (sinusoidal) voltage source:
• Distinguish oscilloscope traces by color:
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
NI myDAQ Measurements
The myDAQ analog outputs AO0 and AO1 cannot source more than 2 mA
and still maintain the expected voltage output. Use an op amp voltage
follower (see Ulaby Section 4-7) to create a “strengthened” copy of the myDAQ analog output.
1. Construct a circuit similar to the Multisim circuit you created earlier,
i.e., place a 10-ohm shunt resistor in series with the inductor, and connect the inductor-resistor combination between the voltage follower
output and ground.
Establish the following myDAQ connections:
• AO0 (Analog Output 0) to the voltage follower input,
AI0+ to inductor terminal connected to the op amp output and
connect AI0- to the other inductor terminal,
• AI1 (Analog Input 1) to display the shunt resistor voltage; connect AI1- to ground.
Create the inductor voltage waveform of Figure 5.2 on page 62 with
the NI ELVISmx Arbitrary Waveform Generator; use 200 kS/s as the
sampling rate.
Adjust the NI ELVISmx Oscilloscope settings to display the inductor
voltage and current with each waveform filling a reasonable amount
of the available display. Use a combination of edge triggering on
Channel 0 and the “Horizontal Position” control to center the inductor voltage pulse.
2. Use the oscilloscope display cursors to measure the time at which
the inductor current reaches its maximum value; use the same time
reference as Figure 5.2 on page 62.
3. Use the cursors to determine the total range of inductor current, i.e.,
the maximum value minus the minimum value. Divide the cursor
measurement by the shunt resistor value. Improve your measurement accuracy by using the measured shunt resistance obtained from
the myDAQ DMM ohmmeter.
5.2. INDUCTORS (5-3)
• Use the Texas Instruments TL072 op amp described in Appendix C
on page 169 for the voltage follower. Follow the pinout diagram of
Figure C.1 on page 170 for either of the two available op amps in the
package. You may also use an equivalent dual-supply op amp.
• Power the op amp with myDAQ +15V to VCC+ and -15V to VCC− .
Use AGND for the circuit ground.
• Create three waveform segments with the ARB “Waveform Editor,”
two of length 0.1 ms on either end and the middle segment of length
0.3 ms. Choose the “Library” option for the middle segment and experiment with parameters until you obtain a single cycle with unit
amplitude. For convenience, create the complete waveform with a
unit amplitude and then set the “Gain” value of the Arbitrary Waveform Generator to 9.
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
• Increase current drive of analog output (AO) channels with an
op amp voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
5.3
Response of the RC Circuit (5-4)
Figure 5.3 shows a resistor-capacitor circuit with a pair of switches and
Figure 5.4 on the next page shows the switch opening-closing behavior as
a function of time. The initial capacitor value is −9 volts.
1. Determine the equation that describes v(t) over the time range 0 to
50 ms.
2. Plot v(t) over the time range 0 to 50 ms.
3. Determine the values of v(t) at the times 5, 15, 25, 35, and 45 ms.
Use these component values:
• R1 = 10 kΩ, R2 = 3.3 kΩ, and R3 = 2.2 kΩ
• C = 1.0 µF
• V1 = 9 V and V2 = −15 V
Figure 5.3: Circuit for Problem 5.3
NI Multisim Measurements
1. Enter the circuit of Figure 5.3 using the following components:
• VOLTAGE CONTROLLED SWITCH
5.3. RESPONSE OF THE RC CIRCUIT (5-4)
Figure 5.4: Switch positions for Problem 5.3
• ABM VOLTAGE (Analog Behavioral Modeling) voltage source; use
step functions (u(TIME)) to create the switch control waveforms
of Figure 5.4.
• Capacitor with initial value of −9 volts.
2. Name the net that connects the two switches to the capacitor. Set up
a Simulate → Analyses → Transient analysis with the end time set
to 0.05 seconds and with “Initial Conditions” set to “User-defined.”
Select the “Output” tab and add the capacitor voltage to the list of
analysis variables. Run the simulator to plot v(t).
3. Use the oscilloscope cursor to measure the values of v(t) at the times
5, 15, 25, 35, and 45 ms.
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
NI Multisim video tutorials:
• Find the maximum value of trace in Grapher View:
• Voltage-controlled switch:
• ABM (Analog Behavioral Model) voltage source:
NI myDAQ Measurements
1. Construct the circuit of Figure 5.3 on page 66 using the following components and NI ELVISmx instruments:
• Two normally-open Switches 1 and 4 contained in the Intersil
DG413 quad analog switch described in Appendix D on page 173.
Refer to the pinout diagram of Figure D.1 on page 174 and connect power according to the photograph of Figure D.2 on page 175.
• 9.0 volt source created with the LM317 variable voltage circuit
of Figure B.2 on page 164.
• 1.0 µF electrolytic capacitor. Connect the negative terminal of
the capacitor to ground.
• AO0 (Analog Output 0) to the switch control input of Switch 1.
• AO1 (Analog Output 1) to the switch control input of Switch 4.
• AI0 (Analog Input 0) to display the switch control voltage for
Switch 1; connect AI0+ to the switch control input and connect
AI0- to ground.
• AI1 (Analog Input 1) to display the capacitor voltage v(t); connect AI1+ to the positive side of the electrolytic capacitor and
connect AI1- to ground.
• Arbitrary Waveform Generator to create the switch control waveforms of Figure 5.4 on the previous page.
• Oscilloscope to view the Switch 1 control waveform and the capacitor voltage v(t). Adjust the Oscilloscope settings to display
the voltage v(t) so that the waveform fills a reasonable amount
of the available display. Use a combination of edge triggering
and the “Horizontal Position” control. You may find it helpful
5.3. RESPONSE OF THE RC CIRCUIT (5-4)
to set the “Acquisition Mode” to “Run Once” and then click the
“Run” button repeatedly until you capture a good trace.
2. Use the oscilloscope cursor to measure the values of v(t) at the times
5, 5, 25, 35, and 45 ms.
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
• Digital Writer (DigOut):
http://decibel.ni.com/content/docs/DOC-12945
Further Exploration with NI myDAQ
The circuit of Figure 5.3 on page 66 permits the capacitor to be charged to a
desired voltage by closing one of the switches that connects the capacitor to
a source. After charging, opening both switches should in principle allow
the capacitor to maintain its “charge” (or stored energy) indefinitely. However, the physical capacitor contains a nonideal dielectric material between
its plates that allows a slow trickle of current that eventually depletes the
stored energy. The nonideal dielectric can be modeled as a resistor in parallel with the capacitor plates.
Devise a method to estimate the value of the equivalent resistance that
connects the capacitor plates. Consider the half-life measurement technique of Figure E.1 on page 178 to measure the time constant. Connect the
switch control inputs to DIO0 and DIO1 and use the NI ELVISmx Digital
Writer to manually operate the switches. Use NI ELVISmx Oscilloscope to
display the capacitor voltage, taking special care to enable only one active
oscilloscope channel. Enabling both oscilloscope channels greatly reduces
the effective input resistance of the myDAQ analog inputs due to the rapid
switching between these channels to a common analog-to-digital converter.
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
5.4
Response of the RL Circuit (5-5)
The circuit of Figure 5.5 on the facing page demonstrates how an inductor can produce a high-voltage pulse across a load resistance Rload that is
considerably higher than the circuit’s power supply Vbatt , a 1.5-volt “AA”
battery. High-voltage pulses drive photo flash bulbs, strobe lights, and cardiac defibrillators, as examples.
Rs models the finite resistance of an electronic analog switch and Rw
models the finite winding resistance of the inductor.
1. Determine the load voltage v after the switch has been closed for a
long time.
2. Determine the equation that describes v(t) after the switch opens at
time t = 0.
3. Determine the magnitude of the peak value of v(t). How many times
larger is this value compared to the battery voltage Vbatt ?
4. State the value of the circuit time constant τ with the switch open.
Plot v(t) over the time range −τ ≤ t ≤ 5τ .
Use these component values:
• Rs = 16 Ω, Rw = 90 Ω, and Rload = 680 Ω
• L = 33 mH
• Vbatt = 1.5 V
NI Multisim Measurements
1. Enter the circuit of Figure 5.5 on the next page. Use the interactive
switch SPST (single pole, single throw) and a measurement probe to
determine v with the switch closed for a long time.
2. Connect the oscilloscope to monitor the voltage v(t). Run interactive
simulation, adjusting the oscilloscope settings to make the waveform
fill a reasonable amount of the available display in both the vertical and horizontal directions. Use edge triggering and the “Normal”
triggering mode to capture the transient when the switch opens. You
may wish to decrease the time step size of interactive simulation to
achieve higher resolution; see the tutorial video linked at the end of
this section.
5.4. RESPONSE OF THE RL CIRCUIT (5-5)
Figure 5.5: Circuit for Problem 5.4
3. Use the oscilloscope cursor to measure the magnitude of the peak
value of v(t).
4. Measure the time constant using the half-life technique described in
Figure E.1 on page 177.
NI Multisim video tutorials:
• Basic operation of the two-channel oscilloscope:
• Stabilize the oscilloscope display with edge triggering:
• Waveform cursor measurements with the two-channel oscilloscope:
• Find components by name:
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
NI myDAQ Measurements
1. Construct the circuit of Figure 5.5 on the previous page using the
normally-closed Switch 2 contained in the Intersil DG413 quad analog switch described in Appendix D on page 173. Refer to the pinout
diagram of Figure D.1 on page 174 and connect power according to
the photograph of Figure D.2 on page 175. Do not place actual resistors for Rs and Rw because these simply model the finite resistance
of the analog switch and inductor winding resistance. Create the
1.5 volt source with the LM317 variable voltage circuit of Figure B.2
on page 164 and connect it to the DG413 “Source (Input)” terminal;
connect the “Drain (Output)” terminal to the inductor.
Establish the following myDAQ signal connections to the DG413:
• DIO0 (Digital Input/Output 0) to the “Logic Control” (switch
control) input for Switch 2,
• AI0 (Analog Input 0) to display the switch control voltage; connect AI0+ to the switch control input and connect AI0- to ground,
• AI1 (Analog Input 1) to display the voltage v(t); connect AI1- to
ground.
Use the NI ELVISmx Digital Writer (“DigOut” on the NI ELVISmx Instrument Launcher) to operate DIO0 as an output. Toggle the button
for Line 0 to operate the analog switch. Use the NI ELVISmx DMM
voltmeter to measure v when the switch is closed.
2. Change the switch control voltage to AO0, Analog Output 0. Create
the switch control voltage with the NI ELVISmx Function Generator.
Choose the squarewave shape and adjust the amplitude and offset
to make the squarewave swing between 0 and 5 volts. Observe this
waveform on the oscilloscope to confirm your correct setup before
you connect it to the analog switch.
Adjust the NI ELVISmx Oscilloscope settings to display the voltage
v(t) so that the waveform fills a reasonable amount of the available
display. Use a combination of edge triggering and the “Horizontal
Position” control. You may find it helpful to set the “Acquisition
Mode” to “Run Once” and then click the “Run” button repeatedly until you capture a good trace. Alternatively, try increasing the squarewave frequency to keep the oscilloscope from timing out; a squarewave frequency of about (5τ )−1 Hz allows the voltage transient to
reach its final value before the switch closes again.
5.4. RESPONSE OF THE RL CIRCUIT (5-5)
3. Use the oscilloscope display cursors to measure the magnitude of the
peak value of v(t).
4. Measure the time constant using the half-life technique described in
Figure E.1 on page 177.
NI myDAQ video tutorials:
• Digital Writer (DigOut):
http://decibel.ni.com/content/docs/DOC-12945
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
Further Exploration with NI myDAQ
The switch model resistance Rs and the inductor winding resistance Rw
values used in the circuit of Figure 5.5 on page 71 were based on measurements taken with actual equipment, but may not necessarily match the
Measure the on-resistance of your analog switch and also measure the
using your measurements. Report the degree to which you see closer agreement between your theoretical and measured values for the time constant
τ.
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CHAPTER 5. RC AND RL FIRST-ORDER CIRCUITS
Chapter 6
RLC Circuits
6.1
Initial and Final Conditions (6-1)
The SPST switch in the circuit of Figure 6.1 on the following page opens at
t = 0 after it had been closed for a long time. Draw the circuit configurations that appropriately represent the state of the circuit at t = 0− , t = 0,
and t = ∞ and use them to determine:
1. vC (0), iC (0), and vC (∞), and
2. iL (0), vL (0), and iL (∞).
Use these component values:
• R1 = 680 Ω, R2 = 100 Ω, and R3 = 100 Ω
• Rsw = 10 Ω and Rw = 10 Ω
• L = 3.3 mH
• C = 0.1 µF
• Vs = 4.7 V
NI Multisim Measurements
Enter the circuit of Figure 6.1 on the next page using the SPST switch for
interactive simulation. Select Simulate → Interactive Simulation Settings
and set “Maximum time step (TMAX)” to 1e-006 to obtain the needed
resolution for this circuit.
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CHAPTER 6. RLC CIRCUITS
Figure 6.1: Circuit for Problem 6.1
1. Connect the two-channel oscilloscope to plot vC (t) and the voltage
across resistor R3 ; divide by the value of R3 to obtain the capacitor
current iC (t). Start interactive simulation and adjust the oscilloscope
settings to clearly show the two waveforms when the switch opens;
operate the switch with the space bar. Choose two different colors
for the oscilloscope traces to make them easy to identify. Take cursor measurements to determine vC (0) and iC (0) just before the switch
opens. Take another cursor measurement to determine vC (∞). Remember that “t = ∞” means the circuit has settled to it new steadystate value.
2. Reconnect the two-channel oscilloscope to plot vL (t) and the voltage
6.1. INITIAL AND FINAL CONDITIONS (6-1)
across resistor R2 ; divide by the value of R2 to obtain the inductor
current iL (t). Repeat the techniques from the previous step to display
the two waveforms. Take cursor measurements to determine iL (0)
and vL (0) just before the switch opens. Take another cursor measurement to determine iL (∞).
NI Multisim video tutorials:
• Basic operation of the two-channel oscilloscope:
• Waveform cursor measurements with the two-channel oscilloscope:
• Distinguish oscilloscope traces by color:
NI myDAQ Measurements
1. Construct the circuit of Figure 6.1 on the facing page using the following components and NI ELVISmx instruments (do not place resistors
Rsw and Rw because they simply model the finite resistance of the
analog switch and the inductor):
• Normally-closed Switch 3 contained in the Intersil DG413 quad
analog switch described in Appendix D on page 173. Refer to the
pinout diagram of Figure D.1 on page 174 and connect power
according to the photograph of Figure D.2 on page 175.
• 4.7 volt source created with 5V and DGND. The loading effect of
• AO0 (Analog Output 0) to the switch control input of Switch 2.
• AI0 (Analog Input 0) to display the voltage across the currentsensing resistor R2 for inductor current or R3 for capacitor current.
• AI1 (Analog Input 1) to display the capacitor voltage vC (t) or
inductor voltage vL (t).
• Function Generator to create the switch control waveform: choose
“Squarewave,” set the peak-to-peak amplitude to 5 V, the offset
to 2.5 V, and the frequency to 1 kHz.
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CHAPTER 6. RLC CIRCUITS
• Oscilloscope to view the current and voltage waveforms. Adjust
the Oscilloscope settings to display the voltage waveforms filling a reasonable amount of the available display. Use a combination of edge triggering and the “Horizontal Position” control.
You may find it helpful to set the “Acquisition Mode” to “Run
Once” and then click the “Run” button repeatedly until you capture a good trace.
2. Establish oscilloscope connections to display vC (t) and iC (t). Take
cursor measurements to determine vC (0) and iC (0) just before the
switch opens, and also measure vC (∞).
3. Modify the connections to display vL (t) and iL (t). Take cursor measurements to determine iL (0) and vL (0) just before the switch opens,
and also measure iL (∞).
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
6.2. NATURAL RESPONSE OF THE SERIES RLC CIRCUIT (6-3)
6.2
Natural Response of the Series RLC Circuit (6-3)
The SPST switch in the circuit of Figure 6.2 on the following page opens at
t = 0 after it had been closed for a long time.
1. Determine vC (t) for t ≥ 0.
2. Plot vC (t) over the time range 0 ≤ t ≤ 1 ms with a plotting tool such
as MathScript or MATLAB.
3. Determine the following numerical values; use either the equation
vC (t) or take cursor measurements from the plot you created in the
previous step:
• Initial voltage vC (0),
• Minimum value of vC ,
• Maximum value of vC ,
• Damped oscillation frequency fd = ωd /2π in Hz, and
• Damping coefficient α.
Use these component values:
• R1 = 220 Ω and R2 = 330 Ω
• L = 33 mH and C = 0.01 µF
• Vsrc = 3.0 V
NI LabVIEW video tutorials:
• Plot two functions of time:
• Take cursor measurements on a plot:
NI Multisim Measurements
1. Enter the circuit of Figure 6.2 on the next page using the SPST switch
for interactive simulation. Select Simulate → Interactive Simulation
Settings and set “Maximum time step (TMAX)” to 1e-007 to obtain
the needed resolution for this circuit.
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CHAPTER 6. RLC CIRCUITS
Figure 6.2: Circuit for Problem 6.2
2. Connect the two-channel oscilloscope to plot vC (t) over the time range
0 ≤ t ≤ 1 ms.
3. Determine the following numerical values with cursor measurements:
•
•
•
•
•
Initial voltage vC (0) just before the switch opens,
Minimum value of vC ,
Maximum value of vC ,
Damped oscillation frequency fd in Hz, and
Damping coefficient α.
• Measure the damped oscillation frequency by using the cursors to
measure the time between an integer number of oscillation cycles;
zero crossings are the easiest to identify. Determine the measured
oscillation period T and then take the reciprocal of this value for the
oscillation frequency.
• Measure the damping coefficient with the following procedure:
1. Place a cursor at the first peak value after the transient begins;
choose either a positive peak or a negative peak,
2. Place a second cursor at a peak value several cycles after the first
peak; choose the same type of peak (positive or negative) as you
did in the previous step,
6.2. NATURAL RESPONSE OF THE SERIES RLC CIRCUIT (6-3)
3. Record the voltage of the first peak as V1 ,
4. Record the voltage of the second peak as V2 ,
5. Measure the time difference between the two cursors and record
its value as T12 , and
6. Calculate α = ln(V1 /V2 )/T12 .
NI Multisim video tutorials:
• Basic operation of the two-channel oscilloscope:
• Waveform cursor measurements with the two-channel oscilloscope:
• Find the maximum value of trace in Grapher View:
NI myDAQ Measurements
1. Construct the circuit of Figure 6.2 on the facing page using the following components and NI ELVISmx instruments (do not place resistors
Rsw and Rw because they simply model the finite resistance of the
analog switch and the inductor):
• Normally-open Switch 1 contained in the Intersil DG413 quad
analog switch described in Appendix D on page 173. Refer to
the pinout diagram of Figure D.1 on page 174 and connect power
according to the photograph of Figure D.2 on page 175.
• 3.0 volt source created with the LM317 variable voltage circuit
of Figure B.2 on page 164.
• AO0 (Analog Output 0) to the switch control input of Switch 1.
• AI0 (Analog Input 0) to display the switch control voltage for
Switch 1; connect AI0+ to the switch control input and connect
AI0- to ground.
• AI1 (Analog Input 1) to display the capacitor voltage vC (t).
• Function Generator to create the switch control waveform: choose
“Squarewave,” set the peak-to-peak amplitude to 5 V and the
offset to 2.5 V.
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CHAPTER 6. RLC CIRCUITS
• Oscilloscope to view the switch control and capacitor voltage
waveforms.
2. Display the switch control waveform and vC (t) over the range 0 ≤
t ≤ 1 ms. Adjust the oscilloscope settings to display the voltage
waveforms filling a reasonable amount of the available display. Use
a combination of edge triggering and the “Horizontal Position” control. You may find it helpful to set the “Acquisition Mode” to “Run
Once” and then click the “Run” button repeatedly until you capture
a good trace. Choose a function generator frequency that allows the
natural response to occupy most of the display.
3. Determine the following numerical values with cursor measurements:
• Initial voltage vC (0) just before the switch opens,
• Minimum value of vC ,
• Maximum value of vC ,
• Damped oscillation frequency fd in Hz, and
• Damping coefficient α.
NOTE: Do not expect close agreement with your earlier analytical
and simulation results. Refer to the “Further Exploration” section below to learn why and the steps you can take to achieve closer agreement.
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
Further Exploration with NI myDAQ
The finite wire resistance of the physical inductor significantly contributes
to the total resistance of the series-connected loop. In fact, you should have
observed an unusually large mismatch between the physical circuit measurements and the analytical as well as simulated results. Try reducing the
330-ohm resistor value to account for the inductor resistance and obtain
6.2. NATURAL RESPONSE OF THE SERIES RLC CIRCUIT (6-3)
closer agreement between the physical circuit and the mathematical models.
Measure and record the resistance of your inductor with the DMM ohmmeter. How does this value compare on a percentage basis with the 330ohm resistor?
Next, connect a 10K potentiometer in parallel with the 330-ohm resistor.
Measure the resistance of this combination in series with the inductor; remember to disconnect the other circuit elements. Adjust the potentiometer
until the total measured resistance is 330 ohms. Reconnect your original
series RLC circuit including the potentiometer.
Repeat your earlier cursor measurements for initial voltage, minimum
and maximum values, damped oscillation frequency, and damping coefficient. Discuss the degree of improved match between the physical circuit
and its mathematical model.
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CHAPTER 6. RLC CIRCUITS
6.3
General Solution for Any Second-Order Circuit (6-6)
1. Develop a differential equation for vC (t) in the circuit of Figure 6.3.
Solve it to determine vC (t) for t ≥ 0. The component values are VS =
8 volts, RS = 680 Ω, C = 1.0 µF, L = 33 mH, and RW = 90 Ω.
2. Plot vC (t) from 0 to 5 ms using a tool such as MathScript or MATLAB.
Include hardcopy of the script used to create the plot.
3. Determine the following values for vC (t):
(a) Maximum value,
(b) Final value, and
(c) Damped oscillation frequency fd = ωd /2π.
Figure 6.3: Circuit for Problem 6.3
6.3. GENERAL SOLUTION FOR ANY SECOND-ORDER CIRCUIT (6-6)
NI LabVIEW video tutorials:
• Plot two functions of time:
• Take cursor measurements on a plot:
NI Multisim Measurements
1. Enter the circuit of Figure 6.3 on the facing page using the same component values listed in the problem statement. Implement the switch
with a VOLTAGE CONTROLLED SWITCH operated by a PULSE VOLTAGE
source configured to open the switch at time 1 ms; this delay makes
the initial transition easier to see.
2. Plot vC (t) from 0 to 5 ms with a Simulate → Analyses → Transient
analysis.
3. Use the Grapher View cursors to measure the following values for
vC (t):
(a) Maximum value,
(b) Final value, and
(c) Damped oscillation frequency fd = ωd /2π.
NI Multisim video tutorials:
• Pulse voltage source:
• Voltage-controlled switch:
• Plot time-domain circuit response with Transient Analysis:
NI myDAQ Measurements
1. Construct the circuit of Figure 6.3 on the preceding page using the
following components and NI ELVISmx instruments (do not place the
resistor RW as this simply models the finite wire resistance of the inductor):
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CHAPTER 6. RLC CIRCUITS
• Normally-open Switch 1 contained in the Intersil DG413 quad
analog switch described in Appendix D on page 173. Refer to
the pinout diagram of Figure D.1 on page 174 and connect power
according to the photograph of Figure D.2 on page 175.
• 8.0 volt source created with the LM317 variable voltage circuit
of Figure B.2 on page 164.
• 1.0 µF electrolytic capacitor. IMPORTANT: Observe proper polarity of the capacitor by connecting the negative terminal of the
capacitor to ground.
• AO0 (Analog Output 0) to the switch control input of Switch 1.
• AI0 (Analog Input 0) to display the switch control voltage for
Switch 1; connect AI0+ to the switch control input and connect
AI0- to ground.
• AI1 (Analog Input 1) to display the capacitor voltage vC (t).
• Function Generator to create the switch control waveform: choose
“Squarewave,” set the peak-to-peak amplitude to 5 V and the
offset to 2.5 V.
• Oscilloscope to view the switch control and capacitor voltage
waveforms.
2. Display vC (t) from 0 to 5 ms.
3. Use the oscilloscope cursor to measure the following values for vC (t):
(a) Maximum value,
(b) Final value, and
(c) Damped oscillation frequency fd = ωd /2π.
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
6.4. TWO-CAPACITOR SECOND-ORDER CIRCUIT (6-6)
6.4
Two-Capacitor Second-Order Circuit (6-6)
1. Develop a differential equation for each node voltage v1 (t) and v2 (t)
in the circuit of Figure 6.4. Solve each equation to determine v1 (t) and
v2 (t) for t ≥ 0. The component values are VS = 9 volts, R = 10 kΩ and
C = 0.1 µF.
2. Plot v1 (t) and v2 (t) on the same graph from 0 to 10 ms using a tool
such as MathScript or MATLAB. Include hardcopy of the script used
to create the plot.
3. Determine the time at which each node voltage reaches 50% of its
final value.
4. Discuss the difference in behavior of the two waveforms just after
time t = 0 and propose an explanation for this difference.
Figure 6.4: Circuit for Problem 6.4
NI LabVIEW video tutorials:
• Plot two functions of time:
• Take cursor measurements on a plot:
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CHAPTER 6. RLC CIRCUITS
NI Multisim Measurements
1. Enter the circuit of Figure 6.4 on the previous page using the same
component values listed in the problem statement. Implement the
voltage source as a step function using either a PULSE VOLTAGE source
or an ABM VOLTAGE source configured to place the step change at
1 ms; this delay makes the initial transition easier to see.
2. Plot v1 (t) and v2 (t) from 0 to 10 ms with a Simulate → Analyses →
Transient analysis.
3. Use the Grapher View cursors to determine the time at which each
node voltage reaches 50% of its final value.
NI Multisim video tutorials:
• Pulse voltage source:
• ABM (Analog Behavioral Model) voltage source:
• Plot time-domain circuit response with Transient Analysis:
NI myDAQ Measurements
1. Construct the circuit of Figure 6.4 on the preceding page using the
same component values listed in the problem statement. Implement
the voltage source using the NI ELVISmx Function Generator set to
squarewave mode. Adjust the function generator settings to match
the step-change voltage VS u(t); choose a frequency that is low enough
to allow the NI ELVISmx Oscilloscope to display the 10 ms range
without interruption.
2. Display both node voltages v1 (t) and v2 (t) at the same time from 0 to
10 ms.
3. Use the oscilloscope cursors to determine the time at which each node
voltage reaches 50% of its final value.
6.4. TWO-CAPACITOR SECOND-ORDER CIRCUIT (6-6)
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
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CHAPTER 6. RLC CIRCUITS
Chapter 7
AC Analysis
7.1
Impedance Transformations (7-5)
Determine the equivalent impedance Z looking into terminals A-B for the
circuit of Figure 7.1 at the following frequencies: 100 Hz, 500 Hz, 1000 Hz,
and 2000 Hz. Report your results in polar form.
Use these component values:
• R1 = 100 Ω and R2 = 90 Ω
• C = 1.0 µF and L = 33 mH
Figure 7.1: Circuit for Problem 7.1
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CHAPTER 7. AC ANALYSIS
NI Multisim Measurements
Figure 7.2 describes a laboratory technique to measure impedance of the
circuit at terminals A-B. The sinusoidal voltage source excites the circuit
with a known voltage v(t) and causes the current i(t). The impedance of
the circuit at a given operating frequency is Z = V/I where V and I are the
equivalent phasor representations of the time-domain voltage and current.
The magnitude of V is the same as the amplitude of v(t). Similarly, the
magnitude of I is the same as the amplitude of i(t). The sinusoidal voltage
source amplitude can easily be measured with the oscilloscope, but what
Figure 7.2: Circuit for Problem 7.1
Observe that the current i(t) entering Terminal A also flows through
resistor R1 . The voltage v1 (t) that appears across this resistor is directly
proportional to the current i(t) due to Ohm’s Law, consequently the current
can be indirectly measured as i(t) = v1 (t)/R1 .
The procedure to measure impedance therefore requires the following
steps:
1. Apply a sinusoidal voltage at the desired frequency to establish v(t);
typically a unit-amplitude voltage is convenient,
7.1. IMPEDANCE TRANSFORMATIONS (7-5)
2. Measure the voltage magnitude (also called amplitude and peak value)
VM of v(t),
3. Measure the voltage magnitude across the resistor R1 and divide by
the measured resistance R1 to determine the current magnitude IM ,
4. Calculate the impedance magnitude as VM /IM , and
5. Calculate the impedance phase by measuring the time difference tD
between the two sinusoids; convert this time difference to degrees
by multiplying by the sinusoidal frequency in Hz and then multiply
by 360 degrees. If the current waveform is delayed compared to the
voltage waveform then the phase sign is positive; if advanced, then
the phase sign is negative.
Use the above procedure to measure the impedance of the circuit at
frequencies 100 Hz, 500 Hz, 1000 Hz, and 2000 Hz. Activate the circuit
with the Simulate → Instruments → Function Generator and observe the
two voltage signals with the Simulate → Instruments → Oscilloscope .
• Choose different colors for the traces to make the voltage and current
traces easy to identify.
• Increase the time resolution of the simulation to improve the accuracy of your measurements, especially at the two higher frequencies:
select Simulate → Interactive Simulation Settings and enter 1e-006
seconds for Maximumum timestep (TMAX).
NI Multisim video tutorials:
• Basic operation of the two-channel oscilloscope:
• Distinguish oscilloscope traces by color:
• Waveform cursor measurements with the two-channel oscilloscope:
• Function generator:
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CHAPTER 7. AC ANALYSIS
NI myDAQ Measurements
The same impedance measurement technique described in the previous
Multisim section works well for the physical circuit, too. Activate the circuit with the NI ELVISmx Function Generator on AO0, and place an op amp
voltage follower between AO0 and the circuit; the voltage follower is necessary to boost the current drive of the analog output beyond its limit of
2 mA. Do not insert a physical resistor for R2 , because this resistor simply
models the finite wire resistance of the physical inductor.
Use the oscilloscope to display the voltage v(t) on AI1 and the resistor voltage v1 (t) on AI0. Measure the circuit impedance at the frequencies
100 Hz, 500 Hz, 1000 Hz, and 2000 Hz.
• Refer to Appendix C on page 169 for details on the TI TL072 dual
op amp device.
• The function generator sets it amplitude in terms of “peak-to-peak”
voltage; this is twice the amplitude of the sinusoid. Adjust this voltage to yield a unit-amplitude sinusoid.
• The oscilloscope “Display Measurements” panel under the waveform
display measures the peak-to-peak voltage of the waveforms. Set the
timebase to display at least two cycles to ensure that the measurement
is accurate. Display even more cycles to improve the accuracy and
stability of the measurement.
• You may also use the cursors to measure the amplitude (peak value).
• Use the oscilloscope cursors to measure the time difference between
zero crossings of the sinusoids. For this measurement reduce the
timebase value to maximize your ability to accurately measure the
time shift of the two sinusoids. Refer to the Multisim section to learn
how to properly determine the sign of the impedance phase.
7.1. IMPEDANCE TRANSFORMATIONS (7-5)
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
• Increase current drive of analog output (AO) channels with an
op amp voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
Further Exploration with NI myDAQ
The NI ELVISmx Bode Analyzer instrument provides a quick and effective
way to study circuit behavior over a range of frequencies, all within a single measurement step. The Bode Analyzer applies a sinusoidal signal (also
called a tone pulse) as the circuit stimulus on AO0, measures the actual circuit stimulus on AI0, and measures the circuit response on AI1. The Bode
Analyzer applies a series of tone pulses from low to high frequency and
then plots the circuit response – “gain” and “phase” – as a function of frequency. This is an automated form of the method you used earlier in this
problem.
The “Gain” display plots the ratio of the circuit response to the stimulus.
With the myDAQ connections described in the previous section the “Gain”
plot therefore displays the magnitude of the circuit impedance because the
current v(t) serves as the “stimulus” and the voltage proportional to i(t)
serves as the “response.” Simply multiply the “Gain” plot by the value of
R1 to see the impedance in ohms.
Set the following values:
• “Start Frequency” = 10 Hz,
• “Stop Frequency” = 10 kHz,
• “Peak Amplitude” = 1, and
• “Mapping” = Linear.
Try running the Bode Analyzer with its default step size of 5, and then
increase the step size until the plot is reasonably smooth and captures the
interesting features of the impedance curve.
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CHAPTER 7. AC ANALYSIS
Use the cursors to read specific values at the frequencies specified earlier, and compare to your previous measurements.
The impedance plot as a function of frequency offers a “birds eye” view
of the circuit behavior for a wide range of frequencies. For what frequency
range does the circuit appear inductive? Over what range does it appear
capacitive? What do you observe about the phase when the impedance
reaches its maximum value?
7.2. EQUIVALENT CIRCUITS (7-6)
7.2
Equivalent Circuits (7-6)
1. Determine the Thévenin equivalent circuit at the terminal pair A-B
for the circuit in Figure 7.3 using the open-circuit/short-circuit method.
Show the Thévenin impedance as a resistor in series with a single reactive element (capacitor or inductor) and determine the values of all
components in the equivalent circuit. The sinusoidal source VS is 3.0
volts amplitude and 500 Hz frequency.
2. Repeat with the source frequency increased to 1100 Hz.
3. Does the circuit seem to “change its personality” with different source
Use these component values:
• R1 = 90 Ω and R2 = 100 Ω
• C = 1.0 µF and L = 33 mH
Figure 7.3: Circuit for Problem 7.2
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CHAPTER 7. AC ANALYSIS
NI Multisim Measurements
1. Enter the circuit of Figure 7.3 on the previous page with an AC VOLTAGE
source. Set its “AC Analysis Magnitude” to 3 volts and leave the “AC
Analysis Phase” at its default zero value. Run a Simulate → Analyses
→ Single Frequency AC Analysis to measure the open-circuit voltage
magnitude and phase. Next, connect a 0.1 Ω resistor across the terminal pair A-B to approximate a short circuit and then measure the
magnitude and phase of the short-circuit current through this small
resistor. Calculate the Thévenin voltage and Thévenin impedance
from these measurements.
2. Repeat to obtain the open-circuit voltage and short-circuit current at
1100 Hz.
• Name the net connected to Terminal A to make it easier to find in the
AC Analysis “Outputs” tab.
• Choose “Magnitude/Phase” for the “Complex number format” option.
NI Multisim video tutorials:
• Measure phasor voltage with a Single Frequency AC Analysis:
• Display and change net names:
NI myDAQ Measurements
1. Construct the circuit of Figure 7.3 on the preceding page; do not include resistor R1 as this device simply models the finite winding resistance of the physical inductor. Build an op amp voltage follower
to strengthen the current drive of AO0 and use the voltage follower
output as VS . Create the sinusoidal voltage with the NI ELVISmx
Function Generator. Measure the open-circuit voltage magnitude and
phase, taking the source voltage VS as the phase reference. Connect a 10 Ω resistor across the terminal pair A-B to approximate a
short circuit. Measure the voltage across this resistor and divide by
its measured resistance to obtain the short-circuit current magnitude
7.2. EQUIVALENT CIRCUITS (7-6)
and phase. Calculate the Thévenin voltage and Thévenin impedance
from these measurements.
2. Repeat to obtain the open-circuit voltage and short-circuit current at
1100 Hz.
• The function generator sets it amplitude in terms of “peak-to-peak”
voltage; this is twice the amplitude of the sinusoid. Adjust this voltage to yield a sinusoid with a 3-volt amplitude.
• Use the NI ELVISmx Oscilloscope to display VS on AI0 and the voltage at Terminal A on AI1.
• The oscilloscope “Display Measurements” panel under the waveform
display measures the peak-to-peak voltage of the waveforms. Set the
timebase to display at least two cycles to ensure that the measurement
is accurate. Display even more cycles to improve the accuracy and
stability of the measurement.
• You may also use the cursors to measure the amplitude (peak value).
• Refer to Appendix F on page 179 to learn how to measure amplitude
and phase of a sinusoidal waveform.
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
• Increase current drive of analog output (AO) channels with an
op amp voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
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CHAPTER 7. AC ANALYSIS
7.3
Phase-Shift Circuits (7-8)
Figure 7.4 shows a phase-shift circuit based on op amps.
1. Write the general expression for the magnitude of Vout with frequency
taken as a variable. H INT: View the circuit as the cascade of two standard op amp circuits.
2. Write the general expression for the phase of Vout with frequency
taken as a variable.
3. Set C to 0.1 µF and set all resistors to 1.0 kΩ. Determine the frequency
in hertz at which Vout and Vin share the same magnitude. What is
the phase shift at this frequency?
Figure 7.4: Circuit for Problem 7.3
NI Multisim Measurements
Enter the circuit of Figure 7.4 with the component values listed in Part 3
of the problem statement. Use the AC voltage source for Vin with unit
amplitude and frequency set to the value you calculated in Part 3. Use
interactive analysis and the oscilloscope to measure the the magnitude and
phase of Vout .
• Use the AC VOLTAGE source. Set the “Voltage (Pk)” to 1 and “Frequency (F)” to the value you found in Part 3.
7.3. PHASE-SHIFT CIRCUITS (7-8)
• IMPORTANT: As a practical matter, connect a 100 kΩ resistor in parallel with the capacitor to prevent the output of the first-stage circuit
from saturating.
• Connect Simulate → Instruments → Oscilloscope so that you can
view Vin and Vout at the same time.
• Use oscilloscope cursors to measure magnitude and phase with the
technique described in Appendix F on page 179.
NI Multisim video tutorials:
• Basic operation of the two-channel oscilloscope:
• Waveform cursor measurements with the two-channel oscilloscope:
NI myDAQ Measurements
Construct the circuit of Figure 7.4 on the preceding page with the component values listed in Part 3 of the problem statement. Use the NI ELVISmx
Function Generator to create the sinusoidal voltage source on AO0; set its
frequency to the value you calculated in Part 3. Use the NI ELVISmx Oscilloscope to measure the node voltages Vin and Vout .
• The function generator amplitude control uses “peak-to-peak” units.
The amplitude of a sinusoid is one-half of its peak-to-peak value.
• IMPORTANT: As a practical matter, connect a 100 kΩ resistor in parallel with the capacitor to prevent the output of the first-stage circuit
from saturating.
• Use oscilloscope cursors to measure magnitude a phase with the technique described in Appendix F on page 179.
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
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CHAPTER 7. AC ANALYSIS
7.4
Phasor-Domain Analysis Techniques (7-9)
The circuit of Figure 7.5 operates at 1 kHz. Determine the node voltages
VA and VB .
Use these component values:
• R1 = 4.7 kΩ, R2 = 3.3 kΩ and R3 = 2.2 kΩ
• C1 = 0.047 µF and C2 = 0.1 µF
• V1 = 96 0◦ V and V2 = 36 − 90◦ V
Figure 7.5: Circuit for Problem 7.4
NI Multisim Measurements
Enter the circuit of Figure 7.5 and run an AC analysis to measure the voltage
magnitude and phase of VA and VB .
More specifically:
• Use the AC VOLTAGE source. Set the “AC Analysis Magnitude” and
“AC Analysis Phase” values according to the specified voltage source
values.
• Set up an Simulate → Analyses → AC Analysis to sweep the frequency over a range that includes 1 kHz; use a “Linear” sweep type
with “Vertical Scale” set to “Linear.”
7.4. PHASOR-DOMAIN ANALYSIS TECHNIQUES (7-9)
• Use the “Grapher View” cursors to measure the voltage magnitude
and phase at 1 kHz.
NI Multisim video tutorials:
• Measure frequency response with AC Analysis:
• Find the maximum value of trace in Grapher View:
NI myDAQ Measurements
Construct the circuit of Figure 7.5 on the preceding page. Use the NI ELVISmx
Arbitrary Waveform Generator to create the sinusoidal voltage sources; use
AO0 to create V1 and AO1 to create V2 . Use the NI ELVISmx Oscilloscope
to measure the node voltages VA and VB . Take V1 as the reference voltage
when measuring phase.
• Set the Arbitrary Waveform Generator sampling rate to its maximum
value of 200 kHz.
• Create a single period of a unit-amplitude 1 kHz sinusoid for each
output channel. Adjust the phase of the AO1 channel to match the
phase of V2 .
• Set the “Gain” controls of the Arbitrary Waveform Generator to match
the magnitudes of the specified voltage sources.
• Refer to Appendix F to learn how to use the oscilloscope to measure
magnitude and phase.
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
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CHAPTER 7. AC ANALYSIS
Chapter 8
AC Power
8.1
Periodic Waveforms (8-1)
Merriam-Webster defines a rectifier as a “device for converting alternating
current into direct current.” A half-wave rectifier does so by setting the negative portion of the waveform to zero. A full-wave rectifier negates the negative portion of the waveform, effectively reflecting it to its positive equivalent.
1. Plot the half-wave rectifier output and full-wave rectifier output for
each of the three standard waveforms shown in Figure 8.1 on page 109.
2. Determine the general expressions for the (a) average value and (b)
rms value for each of the six rectified waveforms.
3. Evaluate your expressions for average and rms values for Vm = 10 volts
and T = 10 ms.
NI Multisim Measurements
1. Create the half-wave rectified and full-wave rectified versions of the
three standard waveforms of Figure 8.1 on page 109 using the following approach:
• Create the unrectified original waveform with the Function Generator,
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CHAPTER 8. AC POWER
• Connect a wire to the + terminal of the function generator and
double-click to terminate the wire in “empty space” without connecting to any other terminal. Display the net names to determine the number of this net. Connect the Common terminal to
ground.
• Place the ABM VOLTAGE source, ground its negative terminal,
and enter the function positive(v(n)) (where n is the net
number of the wire connected to the function generator output)
to make the ABM source create the half-wave rectified version
of the function generator waveform,
• Place another ABM source (also with its negative terminal grounded)
with the absolute value function abs(v(n)) to create the fullwave rectified waveform, and
• Place an SPDT switch (single-pole double-throw) to conveniently
connect one ABM source or the other to the oscilloscope to display the rectified waveform.
2. Place a Measurement Probe to display the average value and rms
value of the rectified waveform. Determine the average and rms values of each of the rectified waveforms.
• The “Analog Behavior Modeling” (ABM) voltage source “senses” the
voltage of net n with the function v(n). The connection between the
function generator and the ABM source does not appear visually.
• The Multisim User Manual describes a wide variety of functions that
can be used by the ABM source. Visit ni.com/manuals and enter
“Multisim User Manual” into the search box. See the “Mathematical
Expressions” section of Chapter 7.
• Set up the Simulate → Instruments → Measurement Probe to display
V(dc) for average value and V(rms) for rms value.
8.1. PERIODIC WAVEFORMS (8-1)
NI Multisim video tutorials:
• Display and change net names:
• Measure RMS and average value with a measurement probe:
• Set the digits of precision of a measurement probe:
• ABM (Analog Behavioral Model) voltage source:
NI myDAQ Measurements
1. Create the half-wave rectified and full-wave rectified versions of the
three standard waveforms of Figure 8.1 on page 109 with the arbitrary
waveform generator on AO0; use 200 kS/s for the sampling rate. Display this signal on the oscilloscope using AI0. Extract the average
value of the waveform using the RC circuit shown in Figure 8.2 on
page 110 and display the output of this circuit on AI1.
2. Read the average value of the rectified waveform using either the cursor value or the “RMS” indicator below the waveform display for AI1.
Read the “RMS” indicator for AI0 to measure the rms value of the
rectified waveform. Display at least two periods of the waveform to
ensure accuracy of the “RMS” indicator.
• Set both channels of the oscilloscope to the same volts per division.
In this way the RC circuit output (average value) overlays the rectified waveform and you can visually see how the average value tends
toward the effective center of the waveform.
• IMPORTANT: Be sure to observe proper polarity for the electrolytic
capacitor. The capacitor may mark the negative lead rather than the
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NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
8.1. PERIODIC WAVEFORMS (8-1)
Figure 8.1: Waveforms for Problem 8.1
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Figure 8.2: RC circuit to extract average value of a waveform.
8.2. AVERAGE POWER (8-2)
8.2
Average Power (8-2)
The circuit shown in Figure 8.3 operates in sinusoidal steady state at 1500 Hz.
The voltage source amplitude is 3 volts. Find the average power delivered
by the voltage source.
Use these component values:
• R = 100 Ω,
• C = 1.0 µF, and L = 3.3 mH
Figure 8.3: Circuit for Problem 8.2
NI Multisim Measurements
Enter the circuit of Figure 8.3 with an AC VOLTAGE source, setting its peak
value and frequency according to the values specified in the problem statement. Place a Simulate → Instruments → Wattmeter instrument to measure
the voltage source power. The wattmeter displays average power in sinusoidal steady state.
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NI Multisim video tutorials:
• Measure average power and power factor with a wattmeter:
NI myDAQ Measurements
Build the circuit of Figure 8.3 on the preceding page. Activate the circuit
with the NI ELVISmx Function Generator on AO0, and place an op amp
voltage follower between AO0 and the remaining circuit; the voltage follower is necessary to boost the current drive of the analog output beyond
its limit of 2 mA.
Use the NI ELVISmx Oscilloscope to display the voltage source on AI0;
this voltage serves as the reference sinusoid for phase measurement. Display the voltage across the resistor on AI1, and realize that this voltage is
proportional to the voltage source current.
Calculate average power from three measurements: voltage source magnitude, current magnitude, and the phase difference between the source
voltage and current.
• The oscilloscope numerical display shows the RMS voltage of the
waveforms. Use the appropriate equation for average power based
on these RMS voltages.
• The oscilloscope numerical display also shows the peak-to-peak voltage of the waveforms. One half of this value is the amplitude of a
sinusoidal waveform. If you choose this measurement, use the appropriate equation for average power based on amplitudes.
• Refer to Figure F on page 179 to learn how to measure amplitude and
phase of a sinusoidal waveform.
NI myDAQ video tutorials:
• Increase current drive of analog output (AO) channels with an
op amp voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
Further Exploration with NI myDAQ
The power delivered by the voltage source depends on it frequency:
8.2. AVERAGE POWER (8-2)
1. Monitor the voltage source current on the oscilloscope and adjust the
voltage source frequency to maximize the current amplitude, thereby
maximizing the average source power delivered to the circuit. Record
the voltage source frequency fmax at which the source average power
is maximized, and then calculate the average power delivered by the
source at this frequency.
2. Calculate the impedance of the inductor and capacitor at fmax . Explain how these impedance values relate to maximizing the average
power delivered by the source.
3. Adjust the voltage source frequency in a range above and below fmax
and observe the phase difference between the source voltage and current. Discuss your observations.
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8.3
Complex Power (8-3)
The circuit shown in Figure 8.4 operates in sinusoidal steady state at 1000 Hz.
The voltage source amplitude is 2.5 volts.
1. Find the complex power in rectangular format for each of the four
circuit elements: SSRC , SR , SL , and SC .
2. Demonstrate conservation of complex power with these four values.
Use these component values:
• R = 100 Ω, C = 1.0 µF, and L = 3.3 mH
Figure 8.4: Circuit for Problem 8.3
NI Multisim Measurements
Enter the circuit of Figure 8.4 with an AC VOLTAGE source. Set the “AC
Analysis Magnitude” parameter to match the amplitude specified in the
problem statement.
8.3. COMPLEX POWER (8-3)
1. Run a Simulate → Analyses → Single Frequency AC Analysis to determine the complex powers SSRC , SR , SL , and SC in rectangular format.
2. Demonstrate conservation of complex power with these four values.
• The single-frequency AC analysis calls element power “P” (which
suggests average power only) but in fact calculates complex power
“S.”
NI Multisim video tutorials:
• Measure phasor voltage with a Single Frequency AC Analysis:
NI myDAQ Measurements
Build the circuit of Figure 8.4 on the facing page. Activate the circuit with
the NI ELVISmx Function Generator on AO0, and place an op amp voltage
follower between AO0 and the remaining circuit; the voltage follower is
necessary to boost the current drive of the analog output beyond its limit
of 2 mA.
Use the NI ELVISmx Oscilloscope to display the device voltage on AI0.
Display the voltage across resistor R on AI1, and realize that this voltage is
proportional to the current through every device in the circuit.
1. Measure the complex power for each of the four circuit elements using the following procedure:
• Read the RMS voltage Vrms from the oscilloscope numerical display,
• Read the RMS current Irms from the oscilloscope numerical display (divide the measured voltage by the resistance R),
• Measure the time shift between the voltage and current sinusoids with the oscilloscope cursors,
• Convert time shift to phase angle φZ = φv − φi (see Appendix F
on page 179 for details),
• Note whether the current waveform lags or leads the voltage; φZ
is positive when current lags and negative when current leads,
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• Express complex power as S = Vrms Irms 6 φZ , and
• Convert S to rectangular form.
2. Demonstrate conservation of complex power with these four values.
• Observe passive sign convention for each circuit element when measuring voltage and current: positive current enters the positive polarity terminal.
• Refer to Figure F on page 179 to learn how to measure amplitude and
phase of a sinusoidal waveform.
NI myDAQ video tutorials:
• Increase current drive of analog output (AO) channels with an
op amp voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
8.4. THE POWER FACTOR (8-4)
8.4
The Power Factor (8-4)
The circuit shown in Figure 8.5 on the next page is a “scale model” of two
industrial electric motors and a heating unit connected to a manufacturing plant power distribution network. The resistor/inductor combinations
R1 -L1 and R2 -L2 model the winding resistance and magnetic fields of the
motors. Resistor R3 models the heater coils. C represents the power factor
compensation equipment – essentially a capacitor bank with high power
capacity.
1. Determine the power factor of the uncompensated load, and draw its
power triangle to scale.
2. Determine the value of the compensation capacitor C required to improve the load power factor to 0.90 lagging.
3. Available power factor compensation capacitors include 0.1 µF, 1.0 µF,
and 10 µF; the cost of compensation equipment increases with capacitance. Choose the least expensive compensation capacitor closest to C
and then determine the power factor and power triangle (also drawn
to scale) of the compensated load.
Use these component values:
• R1 = 10 Ω, R2 = 100 Ω, and R3 = 100 Ω
• L = 3.3 mH and L = 33 mH
• VSRC with 1 volt amplitude and 2500 Hz frequency (actual industrial
motors operate at hundreds of volts and 50 Hz to 60 Hz frequency)
NI Multisim Measurements
Enter the circuit of Figure 8.5 on the following page with an AC VOLTAGE
source, setting its peak value and frequency according to the values specified in the problem statement. Place a SPST switch to conveniently engage or disengage the power factor compensation capacitor. Connect a
wattmeter to measure the power factor of the combined load and capacitor.
IMPORTANT: Reduce the interactive simulation maximum time step
TMAX to 1e-006 seconds; find this parameter at Simulate → Interactive
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Figure 8.5: Circuit for Problem 8.4
Simulation Settings. The default time step does not produce sufficient sample points per period at the operating frequency in this problem to allow
the wattmeter to produce an accurate power factor measurement.
1. Measure the power factor of the uncompensated load (switch open).
2. Close the switch to engage the compensation capacitor.
3. Measure the power factor of the compensated load.
NI Multisim video tutorials:
• Measure average power and power factor with a wattmeter:
NI myDAQ Measurements
Build the circuit of Figure 8.5. Do not place resistors R1 and R2 , as these
simply model the finite winding resistances of the two inductors.
Activate the circuit with the NI ELVISmx Function Generator on AO0,
and place an op amp voltage follower between AO0 and the remaining
8.4. THE POWER FACTOR (8-4)
circuit; the voltage follower boosts the current drive of the analog output
beyond its limit of 2 mA.
Use the NI ELVISmx Oscilloscope to display the load voltage on AI0.
Place a 10 Ω shunt resistor between the op amp output and the remaining
circuit; display the voltage across this shunt resistor on AI1, and recognize
that its voltage is proportional to the load current.
1. Measure the power factor of the uncompensated load.
2. Place your selected compensation capacitor into the circuit.
3. Measure the power factor of the compensated load.
• Place the myDAQ AI1 connections with proper polarity to measure
positive current flowing into the load.
• Recall that the power factor is cos(φZ ) where φZ = φv − φi , the difference between the load’s voltage phase and current phase.
• Refer to Appendix F on page 179 to learn how to measure the phase
of a sinusoidal waveform.
NI myDAQ video tutorials:
• Increase current drive of analog output (AO) channels with an
op amp voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
Further Exploration with NI myDAQ
Long-haul electrical energy distribution networks rely on transformers to
boost the voltage to hundreds of thousands of volts while simultaneously
reducing the current to very low levels – remember that power is the product of voltage and current, hence a given power can be transferred with low
voltage and high current or vice versa. Reducing the current reduces the
resistive losses in the transmission wires and improves efficiency.
Experience how power factor compensation impacts the amount of current that the utility must supply to the customer’s equipment:
1. Measure and record the RMS voltage and current as displayed on the
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2. Repeat these measurements for the compensated load.
3. Discuss which values remain similar and which values change significantly. Comment on the value of power factor compensation as far
as energy transmission efficiency is concerned.
Chapter 9
Frequency Response of Circuits and
Filters
9.1
Scaling (9-2)
Figure 9.1 on the next page shows a prototype bandreject filter with center
frequency ω0 = 1 rad/s. The prototype component values are R = 1 Ω,
L = 1.817 H, and C = 0.5505 F.
1. Apply magnitude and frequency scaling to the bandreject filter so
that R0 = 100Ω and L0 = 33 mH. Draw the finished circuit diagram.
2. Determine the center frequency in hertz of the scaled bandreject filter.
NI Multisim Measurements
1. Enter the circuit of Figure 9.1 on the following page using the scaled
component values calculated earlier. Drive the filter input with an
AC VOLTAGE source with “AC Analysis Magnitude” set to 1 V.
2. Plot the frequency response of the filter over the range 100 Hz to
10 kHz with Simulate → Analyses → AC Analysis. Change “Vertical Scale” to “Linear” and increase “Number of points per decade”
as needed to plot a smooth curve. Use a cursor to identify the filter’s
center frequency.
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Figure 9.1: Circuit for Problem 9.1
NI Multisim video tutorials:
• Measure frequency response with AC Analysis:
NI myDAQ Measurements
1. Build the circuit of Figure 9.1 using the scaled component values calculated earlier. Drive the filter input with AO0 strengthened by an
op amp voltage follower. Monitor the filter input with AI0 and the
filter output with AI1.
2. Plot the frequency response of the filter over the range 100 Hz to
10 kHz with the ELVISmx Bode Analyzer. Change “Mapping” to
“Linear” and increase “Steps” as needed to plot a smooth curve. Use
a cursor to identify the filter’s center frequency.
9.1. SCALING (9-2)
NI myDAQ video tutorials:
• Bode Analyzer:
http://decibel.ni.com/content/docs/DOC-12943
• Increase current drive of analog output (AO) channels with an
op amp voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
Further Exploration with NI myDAQ
NI Multisim provides a way to compare simulated results and physical
measurement results from NI myDAQ on the same ELVISmx instrument.
Study the video tutorial below to learn how to simultaneously display simulated and measured frequency response on the ELVISmx Bode Analyzer,
and then do the following:
1. Plot the frequency response of the simulated and physical circuit of
Figure 9.1 on the facing page using the scaled component values calculated earlier. Compare the two plots and discuss their similarities
and differences.
2. The simulated circuit is a model of the physical circuit and may not
capture every phenomenon of the real circuit. Recall that an inductor
is formed by hundreds of turns of very fine (small diameter) wire,
consequently the small yet finite resistance per unit length adds up to
form a significant resistance. Measure the resistance of your inductor
with the myDAQ DMM and then place a resistor with this value in
series with the ideal inductor in your Multisim circuit. Re-run the
simulator. Compare the two plots and discuss the performance of the
improved circuit model.
NI Multisim video tutorials:
• Combine Multisim simulation and myDAQ measurements in the
same instrument – Bode Analyzer:
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9.2
Bode Plots (9-3)
1. Determine the voltage transfer function H(ω) of the filter circuit shown
in Figure 9.2. Write your finished result in standard form suitable for
creating a Bode plot.
2. Substitute ω = 2πf to express the voltage transfer function in terms
of oscillation frequency f in hertz.
3. Generate Bode magnitude and phase plots for H(f ) using oscillation
frequency f as the independent variable. Use the following component values: R1 = 3.3 kΩ, R2 = 10 kΩ, C1 = 0.01 µF, and C2 = 0.1 µF.
4. Determine the following filter circuit properties by inspecting the Bode
plot:
(a) Low-frequency asymptotes for magnitude and phase
(b) High-frequency asymptotes for magnitude and phase
(c) Corner frequencies (this filter circuit has two such frequencies)
Figure 9.2: Circuit for Problem 9.2
9.2. BODE PLOTS (9-3)
NI Multisim Measurements
1. Enter the filter circuit of Figure 9.2 on the facing page. Drive the filter
input with an AC VOLTAGE source with “AC Analysis Magnitude” set
to 1 V. Use the three-terminal virtual op amp model OPAMP 3T VIRTUAL.
2. Plot the frequency response of the filter over the range 10 Hz to 100 kHz
with Simulate → Analyses → AC Analysis. Set “Vertical Scale” to
“Decibel” and “Sweep Type” to “Decade” to create a standard Bode
plot presentation of frequency response. Increase “Number of points
per decade” as needed to plot a smooth curve.
3. Determine the following filter circuit properties by inspecting the frequency response plot with cursors:
(a) Low-frequency asymptotes for magnitude and phase
(b) High-frequency asymptotes for magnitude and phase
(c) Corner frequencies; look for a change of 3 dB in magnitude from
an asymptote
NI Multisim video tutorials:
• Measure frequency response with AC Analysis:
NI myDAQ Measurements
1. Build the filter circuit of Figure 9.2 on the preceding page. Drive the
filter input with AO0. Monitor the filter input with AI0 and the filter
output with AI1.
2. Plot the frequency response of the filter over the range 10 Hz to 10 kHz
with the ELVISmx Bode Analyzer; note that this frequency range omits
the last decade compared to your analytical and simulation work. Increase “Steps” as needed to plot a smooth curve. IMPORTANT: Set
“Peak Amplitude” to 1 volt.
3. Determine the following filter circuit properties by inspecting the frequency response plot with cursors:
(a) Low-frequency asymptotes for magnitude and phase
(b) High-frequency asymptotes for magnitude and phase
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(c) Corner frequencies; look for a change of 3 dB in magnitude from
an asymptote
• The low- and high-frequency phase asymptotes of any filter are always integer multiples of 90◦ . If the phase plot does not seem to flatten
out enough, simply estimate the trend to the closest integer multiple
of 90◦ .
NI myDAQ video tutorials:
• Bode Analyzer:
http://decibel.ni.com/content/docs/DOC-12943
Further Exploration with NI myDAQ
Use the technique described in the video tutorial below to simultaneously
display the simulated and measured frequency response with the ELVISmx
Bode Analyzer. You may need to set “Op-Amp Signal Polarity” to “Inverted” to make the measured phase response overlay the simulated response.
Discuss the level of agreement between the two plots and explain any
discrepancies you observe.
NI Multisim video tutorials:
• Combine Multisim simulation and myDAQ measurements in the
same instrument – Bode Analyzer:
9.3. FILTER ORDER (9-5)
9.3
Filter Order (9-5)
The filter circuit shown in Figure 9.3 uses the component values R = 1.0 kΩ
and C = 1.0 µF.
1. Obtain an expression for H(ω) = Vo /Vi in standard form.
2. Substitute ω = 2πf to express H(ω) in terms of oscillation frequency
f in hertz.
3. Generate spectral plots for the magnitude and phase of H(f ).
4. Determine the cutoff frequency fc .
Figure 9.3: Circuit for Problem 9.3
NI Multisim Measurements
1. Enter the circuit of Figure 9.3. Drive the filter input with an AC VOLTAGE
source with “AC Analysis Magnitude” set to 1 V.
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2. Plot the frequency response of the filter over the range 1 Hz to 10 kHz
with Simulate → Analyses → AC Analysis. Set “Vertical Scale” to
“Decibel” and “Sweep Type” to “Decade” to create a standard Bode
plot presentation of frequency response. Increase “Number of points
per decade” as needed to plot a smooth curve.
3. Use cursors to measure the cutoff frequency fc (look for a change of
3 dB in magnitude).
NI Multisim video tutorials:
• Measure frequency response with AC Analysis:
NI myDAQ Measurements
1. Build the circuit of Figure 9.3 on the preceding page. Drive the filter
input with AO0. Monitor the filter input with AI0 and the filter output
with AI1.
2. Plot the frequency response of the filter over the range 1 Hz to 10 kHz
with the ELVISmx Bode Analyzer. Increase “Steps” as needed to plot
a smooth curve.
3. Use cursors to measure the cutoff frequency fc (look for a change of
3 dB in magnitude).
NI myDAQ video tutorials:
• Bode Analyzer:
http://decibel.ni.com/content/docs/DOC-12943
Further Exploration with NI myDAQ
Listen to the output of the Bode Analyzer to develop a more intuitive feel
for its operation. Connect AO0 AGND to your headphones similar to the
connection pictured in Figure 4.3 on page 48; connect the filter output to
both the right and left channels for best listening (the middle ring of the
audio plug carries the right channel signal). Comment on your listening
experience.
9.3. FILTER ORDER (9-5)
Do NOT disturb your circuit connections while you are wearing earphones.
Accidently shorting together circuit connections can produce a very loud
and unexpected noise. Alternatively, use a speaker to listen to the amplifier output or hold the earphones some distance from your ears.
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9.4
A telephone line provides sufficient bandwidth (3 kHz) for intelligible voice
conversations, but human hearing has a much higher bandwidth, typically
20 Hz to 20,000 Hz.
1. Design an active bandpass filter to mimic the bandwidth of a telephone line subject to the following constraints:
(a) Cascade a first-order active lowpass filter and a first-order active
highpass filter,
(b) Set the corner frequencies to 300 Hz and 3.0 kHz,
(c) Set the passband gain to 0 dB,
(d) Choose resistors in the range 1.0 kΩ to 100 kΩ, and
(e) Use a total of four fixed-value resistors and two fixed-value capacitors selected from the parts listed in Appendix A on page 159.
Draw the schematic diagram of your finished design.
2. Predict the performance of your finished design by calculating the
following values:
(a) Low-frequency passband corner in hertz,
(b) High-frequency passband corner in hertz, and
(c) Passband gain in decibels.
NI Multisim Measurements
2. Plot the frequency response of the filter over the audio frequency
range 20 Hz to 20 kHz with Simulate → Analyses → AC Analysis.
Set “Vertical Scale” to “Decibel” and “Sweep Type” to “Decade” to
create a standard Bode plot presentation of frequency response.
3. Evaluate the performance of your design by measuring the following
values:
(a) Low-frequency passband corner in hertz (move cursor to −3.0 dB
(b) High-frequency passband corner in hertz, and
(c) Passband gain in decibels.
NI Multisim video tutorials:
• Measure frequency response with AC Analysis:
• Set cursor to a specific value:
NI myDAQ Measurements
1. Build your bandpass filter. Drive the filter input with AO0. Monitor
the filter input with AI0 and the filter output with AI1.
2. Plot the frequency response of the filter over the audio frequency
range 20 Hz to 20 kHz with the ELVISmx Bode Analyzer. Increase
“Steps” as needed to plot a smooth curve.
3. Evaluate the performance of your design by measuring the following
values with cursors:
(a) Low-frequency passband corner in hertz (move cursor to -3.0 dB
(b) High-frequency passband corner in hertz, and
(c) Passband gain in decibels.
• You can obtain more accuracy on the low- and high-frequency corners by sweeping the frequency over a narrow range that brackets
the frequency of interest and substantially increasing the number of
sweep steps.
NI myDAQ video tutorials:
• Bode Analyzer:
http://decibel.ni.com/content/docs/DOC-12943
Further Exploration with NI myDAQ
Listen to the audible effects of your telephone line emulator:
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1. Remove the AO0 connection,
2. Connect your audio player’s output to your filter’s input using the
3.5 mm stereo audio cable included with your NI myDAQ product;
use test leads to connect the left channel similar to the connection
pictured in Figure 4.3 on page 48.
3. Connect your headphones to the filter output; connect the filter output to both the right and left channels for best listening (the middle
ring of the audio plug carries the right channel signal),
4. Play music through your filter, and
5. Compare the sound of the music “with filtering” and “without filtering” by temporarily shorting the high-pass filter capacitor and disconnecting the low-pass filter capacitor. Comment on your listening
experience.
Do NOT disturb your circuit connections while you are wearing earphones. Accidently shorting together circuit connections can produce
a very loud and unexpected noise. Alternatively, use a speaker to listen to the amplifier output or hold the earphones some distance from
For more sophistication, use the DG413 analog switch to conveniently
engage or disengage the telephone line emulator under control of one of
the myDAQ DIO digital outputs. Use one normally-open switch and one
normally-closed switch for single-line control of the two switches necessary
to engage/disengage the filter.
Chapter 10
Laplace Transform Analysis
Techniques
10.1 s-Domain Circuit Analysis (10-7)
1. Determine v(t) of the circuit shown in Figure 10.1 on the following
page for t ≥ 0, given that the switch is opened at t = 0 after having
been closed for a long time. Use the following component values:
Vsrc = 8 V, R1 = 470 Ω, R2 = 100 Ω, Rw = 90 Ω, C = 1.0 µF, and
L = 33 mH.
2. Plot v(t) from 0 to 5 ms using a tool such as MathScript or MATLAB.
Include hardcopy of the script used to create the plot.
3. Determine the following values for v(t):
(a) Initial value v(0),
(b) Final value of v(t),
(c) Minimum value of v(t), and
(d) Time to reach the minimum value of v(t).
NI LabVIEW video tutorials:
• Plot two functions of time:
• Take cursor measurements on a plot:
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CHAPTER 10. LAPLACE TRANSFORM ANALYSIS TECHNIQUES
Figure 10.1: Circuit for Problem 10.1
NI Multisim Measurements
1. Enter the circuit of Figure 10.1 using the same component values
listed in the problem statement. Implement the switch with a
VOLTAGE CONTROLLED SWITCH operated by a PULSE VOLTAGE source
configured to open the switch at time 1 ms; this delay makes the initial transition easier to see.
2. Plot v(t) from 0 to 5 ms with a Simulate → Analyses → Transient
analysis.
3. Use the Grapher View cursors to measure the following values for
v(t):
(a) Initial value v(0),
(b) Final value of v(t),
(c) Minimum value of v(t), and
(d) Time to reach the minimum value of v(t).
• Remember that Multisim voltages are all node voltages, i.e., a voltage
with respect to ground. The voltage v(t) in this problem exists be-
10.1. S-DOMAIN CIRCUIT ANALYSIS (10-7)
tween two nodes, however. Name the nets on either side of R2 (or display their default net numbers), click “Add expression” in the “Output” tab of the transient analysis setup panel, and enter an expression
of the form “v(pos)-v(neg)” where pos and neg denote the two
net names that connect to R2 with positive and negative polarity; the
expression forms the mathematical difference between the two node
voltages.
NI Multisim video tutorials:
• Pulse voltage source:
• Voltage-controlled switch:
• Plot time-domain circuit response with Transient Analysis:
• Display and change net names:
NI myDAQ Measurements
1. Construct the circuit of Figure 10.1 on the facing page using the following components and NI ELVISmx instruments:
• One normally-closed Switch 2 contained in the Intersil DG413
quad analog switch described in Appendix D on page 173.
• 8.0 volt source created with the LM317 variable voltage circuit
of Figure B.2 on page 164.
• 1.0 µF electrolytic capacitor. IMPORTANT: Observe proper polarity of the capacitor by connecting the negative terminal of the
capacitor to ground.
• AIO0 (Analog Output 0) to the “Logic Control” (switch control)
input for Switch 2.
• AI0 (Analog Input 0) to display the switch control voltage for
Switch 2; connect AI0+ to the switch control input and connect
AI0- to ground.
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CHAPTER 10. LAPLACE TRANSFORM ANALYSIS TECHNIQUES
• Function Generator to create the switch control waveform; set
the frequency to 100 Hz and adjust the amplitude and offset to
place the control waveform between 0 and 5 volts.
• Oscilloscope to view the Switch 2 control waveform and the
voltage vt (t).
2. Display v(t) from 0 to 5 ms.
3. Use the oscilloscope cursor to measure the following values for v(t):
(a)
(b)
(c)
(d)
Initial value v(0),
Final value of v(t),
Minimum value of v(t), and
Time to reach the minimum value of v(t).
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
Further Exploration with NI myDAQ
Switching circuits such as the one in this problem generally demand high
current from the power supply for brief periods of time. These high-current
pulses can cause spikes on the supply line that could disrupt proper operation of other connected devices such as digital microcontrollers. Connecting a capacitor between the power supply rail and ground provides a local
supply of temporary current for the switching circuit to stabilize the power
supply rail for other devices.
1. Observe the VSRC rail created by the LM317 on the oscilloscope; it
should still be set to 8.0 V. Estimate the magnitude of the voltage spike
and express its value as a percentage of 8.0 V.
2. Continue to observe the power supply rail as you connect a 10 µF capacitor between the VSRC rail and ground; place the capacitor in close
proximity to the switching circuit and remember to observe its polarity. Discuss the improvement in the stability of the power supply
rail.
10.2. STEP RESPONSE (10-8)
10.2
Step Response (10-8)
1. Determine the transfer function H(s) = Vo (s)/Vs (s) of the circuit
shown in Figure 10.2 on the next page. Write the transfer function in
simplified standard form with symbolic values.
2. Determine the output response vo (t) to the input vs (t) = 4u(t) by
working in the Laplace domain. Assume the capacitor is initially discharged.
3. Plot vs (t) and vo (t) on the same graph from 0 to 5 ms using a tool such
as MathScript or MATLAB for R = 5.6 kΩ and C = 0.1 µF. Include
hardcopy of the script used to create the plot.
4. Determine the following values for vo (t):
(a) Initial value vo (0+ ),
(b) Time to reach 50% of the initial value, and
(c) Final value.
NI Multisim Measurements
1. Enter the circuit of Figure 10.2 on the following page using the same
component values listed in the problem statement. Drive the circuit
input with a PULSE VOLTAGE source configured to produce vs (t) =
4u(t). Delay the pulse by 1 ms to make the initial step transition visible.
2. Plot vs (t) and vo (t) on the same graph from 0 to 5 ms with a Simulate
→ Analyses → Transient analysis.
3. Use the Grapher View cursors to measure the following values for
vo (t):
(a) Initial value vo (0+ ),
(b) Time to reach 50% of the initial value, and
(c) Final value.
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CHAPTER 10. LAPLACE TRANSFORM ANALYSIS TECHNIQUES
Figure 10.2: Circuit for Problem 10.2
NI Multisim video tutorials:
• Pulse voltage source:
• Plot time-domain circuit response with Transient Analysis:
NI myDAQ Measurements
1. Build the circuit of Figure 10.2 using the same component values
listed in the problem statement. Drive the circuit input with AO0 and
use the ELVISmx Function Generator to produce a zero-to-four volt
step transition with a period of 10 ms. Monitor the input voltage vs (t)
with AI0 and the output voltage vo (t) with AI1.
2. Display vs (t) and vo (t) on the ELVISmx Oscilloscope.
3. Use the oscilloscope cursors to measure the following:
10.2. STEP RESPONSE (10-8)
(a) Initial value vo (0+ ),
(b) Time to reach 50% of the initial value, and
(c) Final value.
NI myDAQ video tutorials:
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
Further Exploration with NI myDAQ
The circuit in this problem represents one implementation of an all-pass
filter. Set up the ELVISmx Bode Analyzer to measure the frequency response of the circuit; the necessary myDAQ connections should already be
in place. Set up the Bode analyzer controls as follows:
• Start frequency = 10 Hz
• Stop frequency = 10 kHz
• Steps = 10 per decade
• Mapping = linear
After running the frequency sweep set the “Gain” axis range to a minimum
of zero and a maximum of 2; double-click the numerical values at the top
and bottom of the axis display to set these values.
Study the response and then discuss the following questions:
1. Why is the circuit called an “all-pass” filter?
2. What is the general behavior of the phase response? More specifically,
what are the maximum and minimum values of phase shift?
3. Use the cursor to measure the frequency at the midpoint between the
maximum and minimum phase shift values. Compare this frequency
to the critical frequencies in the transfer function H(s) you derived
in the analytical section. H INT: Remember to account for angular
frequency versus oscillation frequency.
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CHAPTER 10. LAPLACE TRANSFORM ANALYSIS TECHNIQUES
10.3
Transfer Function and Impulse Response (10-8)
Figure 10.3 shows a passive highpass filter circuit. After you determine its
behavior (transfer function), design an input waveform to achieve a specified output waveform shape; do the first three parts of this problem symbolically to maintain generality:
1. Determine the transfer function H(s) = Y(s)/X(s) of the circuit.
2. Determine and sketch the impulse response h(t) of the circuit.
3. Create an input waveform x(t) that will cause the output waveform
y(t) to be a unit step function u(t). H INT: Work first in terms of H(s),
Y(s), and X(s).
4. Plot x(t) and y(t) on the same graph from 0 to 100 ms using a tool such
as MathScript or MATLAB for R = 10 kΩ and C = 1.0 µF. Include
hardcopy of the script used to create the plot.
5. Evaluate x(t) and y(t) at t = 50 ms.
Figure 10.3: Circuit for Problem 10.3
10.3. TRANSFER FUNCTION AND IMPULSE RESPONSE (10-8)
NI Multisim Measurements
1. Enter the circuit of Figure 10.3 on the preceding page using the same
component values listed in the analytical section. Drive the circuit
input with an ABM VOLTAGE source.
2. Plot x(t) and y(t) on the same graph from 0 to 100 ms with a Simulate
→ Analyses → Transient analysis.
3. Use the Grapher View cursors to measure the voltages x(t) and y(t)
at t = 50 ms.
Additional Multisim tips for this problem:
• Build the ABM voltage source “Voltage Value” string by combining
one or more of the following functions:
– TIME – Time function t
– u(TIME) – Unit step function u(t)
– uramp(TIME) – Unit ramp function r(t)
– Standard math operators: +, -, *, and /
NI Multisim video tutorials:
• ABM (Analog Behavioral Model) voltage source:
• Plot time-domain circuit response with Transient Analysis:
NI myDAQ Measurements
1. Build the circuit of Figure 10.3 on the facing page using the same component values listed in the analytical section. Drive the circuit input
with AO0 and use the ELVISmx Arbitrary Function generator to produce the waveform x(t) with a period of 100 ms. Set the voltage to
zero before the end of the period to allow the circuit sufficient time to
2. Display x(t) and y(t) on the ELVISmx Oscilloscope.
3. Use the oscilloscope cursors to measure the voltages x(t) and y(t) at
t = 50 ms.
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CHAPTER 10. LAPLACE TRANSFORM ANALYSIS TECHNIQUES
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
Further Exploration with NI myDAQ
The impulse function δ(t) is physically unrealizable as an input voltage
waveform, however, it can be approximated by a large-valued short-duration
rectangular pulse. Use the ELVISmx Arbitrary Waveform Generator to create a 10-volt pulse with very short duration compared to the 100 ms period.
View this input waveform and the circuit’s output (an approximation to the
impulse response) on the oscilloscope. Discuss the similarity and difference
between what you observe and what you calculated earlier as the circuit’s
h(t).
10.4. CONVOLUTION INTEGRAL (10-9)
10.4
Convolution Integral (10-9)
Figure 10.4 shows a passive lowpass filter circuit and its input voltage
waveform x(t).
1. Determine the impulse response h(t) of the circuit; work this and the
next part with symbolic values.
2. Determine the output of the circuit y(t) = x(t) ∗ h(t) by integrating
the convolution analytically, i.e., use Method 2 of Example 10-15 in
your text. Carry out this step using symbolic values.
3. Plot x(t) and y(t) on the same graph from 0 to 100 ms using a tool
such as MathScript or MATLAB for the following values: R = 10 kΩ,
C = 1.0 µF, A = 5 volts, and t0 = 50 ms. Include hardcopy of the
script used to create the plot.
4. Evaluate y(t) at the following times: 25 ms, 50 ms, and 60 ms.
Figure 10.4: Circuit and input voltage waveform for Problem 10.4
NI Multisim Measurements
1. Enter the circuit of Figure 10.4 using the same component values
listed in the problem statement. Drive the circuit input with an ABM VOLTAGE
source configured to produce the waveform x(t) with A = 5 volts
and t0 = 50 ms. Alternatively, use a PIECEWISE LINEAR VOLTAGE
source.
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CHAPTER 10. LAPLACE TRANSFORM ANALYSIS TECHNIQUES
2. Plot x(t) and y(t) on the same graph from 0 to 100 ms with a Simulate
→ Analyses → Transient analysis. Adjust the maximum time step
setting to plot at least 100 time points.
3. Use the Grapher View cursors to measure the output voltage y(t) at
the times 25, 50, and 60 ms.
Additional Multisim tips for this problem:
• Build the ABM voltage source “Voltage Value” string by combining
the following functions:
– TIME – Time function t
– u(TIME) – Step function u(t)
– Standard math operators: +, -, *, and /
NI Multisim video tutorials:
• ABM (Analog Behavioral Model) voltage source:
• Piecewise linear (PWL) voltage source:
• Plot time-domain circuit response with Transient Analysis:
NI myDAQ Measurements
1. Build the circuit of Figure 10.4 on the previous page using the same
component values listed in the problem statement. Drive the circuit
input with AO0 and use the ELVISmx Arbitrary Waveform Generator
to produce the waveform shown in the same figure with A = 5 volts,
and t0 = 50 ms. Set the period of the waveform to 100 ms.
2. Display x(t) and y(t) on the ELVISmx Oscilloscope.
3. Use the oscilloscope cursors to measure the output voltage y(t) at the
times 25, 50, and 60 ms.
10.4. CONVOLUTION INTEGRAL (10-9)
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
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CHAPTER 10. LAPLACE TRANSFORM ANALYSIS TECHNIQUES
Chapter 11
Fourier Analysis Techniques
11.1
Fourier Series Representation (11-2)
Consider the voltage waveform v(t) shown in Figure 11.1 on the following
page.
1. Determine if the waveform has dc, even, or odd symmetry.
2. Obtain its cosine/sine Fourier series representation.
3. Convert the representation to amplitude format and plot the amplitude line spectrum for n = 0 to 5 using A = 10 volts and T = 4 ms.
NI Multisim Measurements
1. Create the voltage waveform v(t) of Figure 11.1 on the next page
with a PIECEWISE LINEAR VOLTAGE source. Use the same amplitude and period as in the problem statement.
2. Plot and tabulate the amplitude line spectrum of v(t) with a Simulate
→ Analyses → Fourier Analysis :
(a) Set the “Frequency Resolution (fundamental frequency)” parameter to match the fundamental frequency f0 of the voltage waveform v(t).
(b) Leave the remaining parameters at their default settings.
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CHAPTER 11. FOURIER ANALYSIS TECHNIQUES
Figure 11.1: Voltage waveform for Problem 11.1
NI Multisim video tutorials:
• Find commonly-used circuit components:
• Piecewise linear (PWL) voltage source:
NI myDAQ Measurements
1. Connect myDAQ Analog Output 0 to Analog Input 0, i.e., AO0 to
AI0+ and AGND to AI0-.
2. Create the voltage waveform v(t) with the ELVISmx Arbitrary Waveform Generator using the same amplitude and period as in the problem statement. Set the sampling frequency to 200 kS/s.
3. Plot the power spectrum of v(t) on the ELVISmx Dynamic Signal Analyzer (DSA). Carefully adjust the panel controls to match the following settings:
• Input Settings:
(a) Source Channel = AI0
(b) Voltage Range = +/-10V
• FFT Settings:
11.1. FOURIER SERIES REPRESENTATION (11-2)
(a) Frequency Span = 10000
(b) Resolution (lines) = 400
(c) Window = None
• Averaging:
(a) Mode = RMS
(b) Weighting = Exponential
(c) Number of Averages = 5
• Frequency Display:
(a) Units = Linear
(b) Mode = Peak
• Scale Settings:
(a) Scale = Auto
• Cursor Settings:
(a) Cursors On = enabled
(b) Cursor Select = C1
4. Measure the amplitude spectrum for n = 0 to 5 using Cursor 1; take
the square root of the displayed cursor value “dVpkˆ2” to obtain the
voltage amplitude. IMPORTANT: Position Cursor 2 between a pair of
spectral lines to set its measured value to zero; the value displayed
as dVpkˆ2 is the difference between the two cursors and you want
Cursor 2 to serve as the zero reference.
• Use the cursor position buttons to make fine adjustments in the vicinity of a spectral line; these are the pair of gray diamonds at the bottom
center of the DSA.
• Double-click the upper limit value on the horizontal frequency axis
and select a lower value to zoom in on the lower-frequency spectral
lines. Do not change the “Frequency Span” value for this purpose
because this changes the measurement itself.
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
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CHAPTER 11. FOURIER ANALYSIS TECHNIQUES
Further Exploration with NI myDAQ
The ELVISmx Digital Signal Analyzer (DSA) represents a sophisticated instrument that performs a wide variety of frequency-domain measurements.
Experiment with the settings and discuss your findings:
• Averaging:
1. Choose Mode = Peak Hold and note that the “Restart” button
lower down becomes active. Also try Mode = None.
2. Number of Averages: Try different values including 0.
• Frequency Display:
1. Choose Units = dB; what advantage do you see in a logarithmic
display compared to a linear display?
The “FFT Settings” control the Fast Fourier Transform computation that
serves as the heart of the DSA. These critical settings must be carefully selected to obtain correct amplitude spectrum measurements of periodic signals. First learn how the DSA takes a measurement and then experiment
with the settings in a moment.
The DSA repetitively captures a snapshot of the input signal with duration “Resolution (lines)” (R) divided by “Frequency Span” (fspan ); this
time-domain record appears below the frequency display. Take a moment
to calculate this time duration from your current DSA FFT settings and confirm that the value does indeed match the upper limit of the time-domain
plot.
When measuring a periodic signal the captured time-domain signal
must contain an integer multiple of periods, consequently R/fspan divided
by the signal period T must be an integer N . Since the periodic signal frequency f0 is 1/T , the frequency span may be readily calculated as
Rf0
,
(11.1)
N
where fspan is the frequency span in Hz, R is the resolution in “lines” (sample points), f0 is the fundamental frequency of the periodic input signal in
Hz, and N is the number of periods captured. N = 10 cycles provides a
good starting point for most measurements.
Now, return the DSA settings to match those of your earlier work in the
NI myDAQ section of this problem. Calculate the value of N . Also calculate the values of N that result from choosing the other available values for
fspan =
11.1. FOURIER SERIES REPRESENTATION (11-2)
resolution R (the DSA offers a total of five resolutions). Change the DSA
FFT resolution to each of the other available values, and note the effects on
the frequency spectrum display and on the time-domain display. In particular, note the degree to which the amplitude line spectral values change.
Return the resolution to R = 400 lines. Calculate the frequency span
fspan for N = 10.5, i.e., for a time-domain record that contains ten periods
with a half-period tacked onto the end. Enter this value into the DSA and
note the degree to which the amplitude line spectral values change.
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CHAPTER 11. FOURIER ANALYSIS TECHNIQUES
11.2
Circuit Applications (11-3)
The sawtooth voltage waveform vs (t) shown in Figure 11.2 with A = 5 V
and T = 2 ms serves as the input to the circuit of Figure 10.2 on page 138.
1. Determine the Fourier series representation of vo (t).
2. Plot vo (t) and vs (t) with MathScript or MATLAB as follows:
(a) Time 0 ≤ t ≤ 5 ms,
(b) Sum of nmax = 100 terms, and
(c) Circuit components R = 5.6 kΩ and C = 0.1 µF.
Use sufficient time resolution to display Gibbs phenomenon ringing.
3. Measure the maximum value of vo (t) from the plot, and also measure
the first time at which the maximum value occurs after t = 0.
Figure 11.2: Voltage waveform for Problem 11.2
NI LabVIEW video tutorials:
• Plot two functions of time:
• Take cursor measurements on a plot:
11.2. CIRCUIT APPLICATIONS (11-3)
NI Multisim Measurements
1. Enter the circuit of Figure 10.2 on page 138 using the same component
values listed in the problem statement. Drive the circuit input with
a TRIANGULAR VOLTAGE source configured with the amplitude and
period specified in the problem statement.
2. Plot vo (t) and vs (t) using Simulate → Analyses → Transient analysis. Extend the plot time to 7 ms to allow the output to reach AC
steady-state. Increase the minimum number of time points as needed
to produce a smooth plot.
3. Measure the maximum value of vo (t) with the Grapher View cursors,
and also measure the first time at which the maximum value occurs
after t = 0; ignore the transient start-up behavior during the first
period.
NI Multisim video tutorials:
• Find commonly-used circuit components:
• Plot time-domain circuit response with Transient Analysis:
NI myDAQ Measurements
1. Build the circuit of Figure 10.2 on page 138 using the same component
values listed in the problem statement. Drive the circuit input with
AO0 and use the ELVISmx Arbitrary Waveform Generator to produce
the sawtooth waveform of Figure 11.2 on the facing page with the
amplitude and period specified in the problem statement. Monitor
the input voltage vs (t) with AI0 and the output voltage vo (t) with AI1.
2. Display vs (t) and vo (t) on the ELVISmx Oscilloscope.
3. Use the oscilloscope cursors to measure the maximum value of vo (t)
and the first time at which the maximum value occurs after t = 0.
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
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CHAPTER 11. FOURIER ANALYSIS TECHNIQUES
11.3
Fourier Transform (11-5)
1. Determine the Fourier transform of the rectangular pulse f (t) shown
in Figure 11.3.
2. Plot the amplitude spectrum |F(ω)| with MathScript or MATLAB as
follows:
(a) Frequency 0 ≤ f ≤ 4000 Hz (remember to convert angular frequency ω to oscillation frequency f ),
(b) A = 10, and
(c) τ = 1, 2, and 4 ms (create three distinct plots).
3. Determine the frequency at which the first null occurs in each of the
three plots.
4. Discuss the relationship between the rectangular pulse width and the
width of the main lobe of the amplitude spectrum.
Figure 11.3: Rectangular pulse waveform for Problem 11.3
11.3. FOURIER TRANSFORM (11-5)
NI Multisim Measurements
1. Create a circuit with a PULSE VOLTAGE source. Set the “Period” to
100 ms; set the remaining parameters as needed to create the pulse
shown in Figure 11.3 on the facing page with A = 10. Note that the
pulse must shift right to begin at t = 0; this shift does not affect the
amplitude spectrum.
2. Plot the amplitude line spectrum of f (t) with a Simulate → Analyses
→ Fourier Analysis for τ = 1, 2, and 4 ms (create three plots). Set the
following parameter values:
• “Frequency Resolution (fundamental frequency)” = 10 Hz,
• “Number of harmonics” = 400,
• “Display” = “Graph,” and
• “Vertical scale” = “Linear.”
3. Determine the frequency at which the first null occurs in each of the
three plots.
NI Multisim video tutorials:
• Find commonly-used circuit components:
NI myDAQ Measurements
1. Set up your myDAQ and ELVISmx instruments as follows:
(a) Connect myDAQ Analog Output 0 to Analog Input 0, i.e., AO0
to AI0+ and AGND to AI0-.
(b) Create the rectangular pulse f (t) with the ELVISmx Function
Generator in squarewave mode. Set the frequency to 10 Hz.
Adjust the amplitude and DC offset controls to match the pulse
waveform shown in Figure 11.3 on the preceding page with A =
10. Control the pulse width with the “Duty Cycle” control.
2. Plot the power spectrum of f (t) on the ELVISmx Dynamic Signal Analyzer (DSA) for τ = 1, 2, and 4 ms (create three plots). Adjust the
panel controls to match the following settings:
• FFT Settings:
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CHAPTER 11. FOURIER ANALYSIS TECHNIQUES
(a) Frequency Span = 4000
(b) Resolution (lines) = 400
(c) Window = None
• Averaging:
(a) Mode = None
• Frequency Display:
(a) Units = Linear
(b) Mode = Peak
• Scale Settings:
(a) Scale = Auto
3. Determine the frequency at which the first null occurs in each of the
three plots.
Further Exploration with NI myDAQ
1. Frequency spectrum plots normally possess much higher dynamic range
than their corresponding time-domain plots. Review the DSA plots
you created for the widest rectangular plot (τ = 4 ms), especially the
side lobe amplitudes beyond the first null frequency. Note how their
values appear quite small compared to the amplitude of the main
lobe. Now set the “Frequency Display” units to “dB” (decibels); you
can stabilize the display by setting the “Scale Settings” from “Auto”
to “Manual.” Discuss the merits of a logarithmic display scale compared to a linear display scale.
2. To further experience the advantages of a logarithmic display, repeat
the experiment with a single sinusoidal component. Set the function
generator to sinusoidal mode at 500 Hz and remove the DC offset.
Display the spectrum with “Linear” units and then with “dB” units.
Stabilize and improve the measurement by setting the “Averaging”
mode to “RMS.” Discuss your observations.
11.4. CIRCUIT ANALYSIS WITH FOURIER TRANSFORM (11-8)
11.4
Circuit Analysis with Fourier Transform (11-8)
The circuit of Figure 11.4 is excited by the double-pulse waveform shown
in the same figure.
1. Derive the expression for vo (t) using Fourier analysis.
2. Plot vs (t) and vo (t) on the same graph over the time span 0 ≤ t ≤ 5 ms
with MathScript or MATLAB for the following values: A = 5 volt,
T = 1 ms, R = 5.6 kΩ, and C = 0.1 µF.
3. Determine the value of vo (t) at times t = 2 ms and t = 3 ms.
Figure 11.4: Double-pulse waveform and circuit for Problem 11.4
NI Multisim Measurements
1. Enter the circuit of Figure 11.4 using an ABM VOLTAGE source and the
component values specified in the problem statement. Build the ABM
voltage source “Voltage Value” string with multiple step functions
u(TIME). Add, subtract, and delay the step functions as needed to
create the double-pulse waveform of Figure 11.4 with the parameters
specified in the problem statement.
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CHAPTER 11. FOURIER ANALYSIS TECHNIQUES
2. Set up a Simulate → Analyses → Transient to plot vs (t) and vo (t) over
the time span 0 ≤ t ≤ 5 ms. Choose “Set to zero” for the “Initial Conditions” parameter to properly model a discharged capacitor before
the first pulse occurs. Increase the minimum time step as necessary
to obtain a smooth plot at discontinuities.
3. Use the Grapher View cursors to measure the value of vo (t) at times
t = 2 ms and t = 3 ms.
NI Multisim video tutorials:
• Plot time-domain circuit response with Transient Analysis:
• ABM (Analog Behavioral Model) voltage source:
NI myDAQ Measurements
1. Build the circuit of Figure 11.4 on the previous page using the component values specified in the problem statement. Drive the circuit
with the ELVISmx Arbitrary Waveform Generator on AO0. Create the
double-pulse waveform of Figure 11.4 on the preceding page with the
parameters specified in the problem statement. Monitor vs (t) on AI0
and vo (t) on AI1.
2. Display vs (t) and vo (t) on the ELVISmx Oscilloscope. Adjust the settings for a time span of 5 ms and an appropriate scale (volts per division) to clearly see the output trace behavior. Adjust the triggering
level and horizontal position to place the leading edge of the first
pulse at the far left of the display. You will likely need to use the
“Run Once” acquisition mode to obtain a stable display; click “Run”
repeatedly until you obtain a satisfactory display.
3. Use the oscilloscope cursors to measure the value of vo (t) at times
t = 2 ms and t = 3 ms.
NI myDAQ video tutorials:
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
Appendix A
Parts List
Resistors
The following resistors are standard-value 5% tolerance 1/4 watt carbon
film devices. All listed resistors are available in resistor kits from Digi-Key,
Resistor Kit Description
365 pcs, 5 ea of 1.0Ω to 1.0MΩ
540 pcs, 30 values, 10Ω to 10MΩ
500 pcs, 64 values, 1.0Ω to 10MΩ
Supplier
Digi-Key
Jameco
Part #
RS125-ND
103166
271-312
See Resistor Color Codes at http://www.allaboutcircuits.com/
vol_5/chpt_2/1.html to learn how to read the color bands on carbon
film resistors.
Qty
1
2
1
1
1
Value (Ω)
10
100
330
470
680
Color Code
Brown - Black - Black
Brown - Black - Brown
Orange - Orange - Brown
Orange - Orange - Brown
Blue - Gray - Brown
160
APPENDIX A. PARTS LIST
Qty
4
1
1
2
1
4
2
1
1
1
2
1
Value (kΩ)
1.0
1.5
2.2
3.3
4.7
5.6
10
15
22
33
47
100
Color Code
Brown - Black - Red
Brown - Green - Red
Red - Red - Red
Orange - Orange - Red
Yellow - Violet - Red
Green - Blue - Red
Brown - Black - Orange
Brown - Green - Orange
Red - Red - Orange
Orange - Orange - Orange
Yellow - Violet - Orange
Brown - Black - Yellow
Potentiometers
The following potentiometers (variable resistors) are 3/8-inch square singleturn trimming style devices with 1/2-watt power rating.
Qty
1
1
2
Description
100Ω trimpot (Bourns 3386P-1-501LF)
1K trimpot (Bourns 3386P-1-103LF)
10K trimpot (Bourns 3386P-1-103LF)
Capacitors
Qty
1
1
2
1
1
Value (µF)
0.047
0.01
0.1
1
10
Type
Ceramic
Ceramic
Ceramic
Electrolytic
Electrolytic
Digi-Key #
3386P-101LF-ND
3386P-102LF-ND
3386P-103LF-ND
161
Inductors
Qty
1
1
Description
3.3-mH inductor (Murata Power Solutions 22R335C)
33-mH inductor (Murata Power Solutions 22R336C)
Digi-Key #
811-1295-ND
811-1294-ND
Active Devices and Integrated Circuits
N OTE: Texas Instruments offers free samples. Go to http://www.ti.com
and click “Sample & Buy” to get started.
Qty
3
1
1
Description
LM317L voltage regulator, 100mA (Texas Instruments)
TL072CP dual op amp (Texas Instruments)
Digi-Key #
296-17221-1-ND
296-1775-5-ND
DG413DJZ-ND
Circuit Specialists part number WB-102, http://www.circuitspecialists.
com/prod.itml/icOid/6885
Jumper Wire Kit
Circuit Specialists part number WK-1 (350 pieces, pre-formed, 22 AWG),
http://www.circuitspecialists.com/prod.itml/icOid/6920
Circuit Specialists part number MJW-70B (140 pieces, pre-formed, 22 AWG),
http://www.circuitspecialists.com/prod.itml/icOid/7590
Alligator clip style, cut in half with tinned ends.
Circuit Specialists part number M000F0003, http://www.circuitspecialists.
com/prod.itml/icOid/7682
162
APPENDIX A. PARTS LIST
Appendix B
LM317 Voltage and Current Sources
The Texas Instruments LM317 adjustable voltage regulator is a flexible device that when combined with suitable external resistors and the NI myDAQ power supply can serve as the basis for a fixed or variable voltage source and a fixed or variable current source. Figure B.1 shows the
LM317 package terminals as well as its schematic symbol. The LM317
sources current up to 1.5 amps, while the LM317L sources up to 100 mA.
See the datasheets available at http://www.ti.com; enter “lm317” in the
“Search by Part Number” field.
Figure B.1: LM317 adjustable voltage regulator: (a) package and terminals
for 1.5-amp device (TO-220 package), (b) package and terminals for 100mA device (TO-92 package), and (c) schematic symbol.
164
APPENDIX B. LM317 VOLTAGE AND CURRENT SOURCES
B.1
Variable Voltage Source
The circuit shown in Figure B.2 produces a variable voltage in the range
1.5 V to 13.5 V from the NI myDAQ +15V power supply. Figure B.3 on
the next page shows the recommended breadboard layout for this circuit.
Use bare-wire loops to facilitate easy connections with test leads to the NI
myDAQ ±15-volt dual power supply. The horizontal voltage “rails” follow
the top-to-bottom order of high to low voltage: +15 volts, variable voltage,
ground, and -15 volts. Build this circuit on the left edge of your breadboard
and leave it in place for all of your circuits projects.
Figure B.2: LM317 as a variable voltage source: (a) schematic diagram and
(b) equivalent circuit.
B.2
Current Source
The circuit shown in Figure B.4 on page 166 produces a current whose value
is approximately 1250/R mA. This circuit configuration “sources” current
B.2. CURRENT SOURCE
Figure B.3: Recommended breadboard layout for the LM317-based variable voltage source: (a) top view of breadboard showing the component
layout and voltage rail order, and (b) side view showing test lead connections between NI myDAQ and wire loops on the breadboard.
from the NI myDAQ +15V power supply and effectively operates as a current source with one terminal permanently attached to the NI myDAQ analog ground AGND.
The current source will operate as expected for circuits powered by the
NI myDAQ ±15 V dual power supply provided the following conditions
hold:
1. The requested current does not exceed the 30 mA current limit of the
NI myDAQ +15V power supply,
2. The voltage of the ungrounded current source terminal does not rise
higher than 13.5 V above ground, and
3. The current set resistor R does not exceed approximately 1.2 kΩ (the
minimum current ISRC is approximately 1 mA).
Figure B.5 on page 167 illustrates a similar current source that “sinks”
current to the NI myDAQ −15 V power supply. The current source will
operate as expected for circuits powered by the NI myDAQ ±15 V dual
power supply provided the following conditions hold:
1. The requested current does not exceed the 30 mA current limit of the
NI myDAQ −15 V power supply,
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166
APPENDIX B. LM317 VOLTAGE AND CURRENT SOURCES
Figure B.4: LM317 adjustable voltage regulator as a grounded current
source sourcing current from the NI myDAQ +15V power supply: (a)
schematic diagram, (b) equivalent circuit model, and (c) recommended layout with the standard breadboard layout of Figure B.3 on the preceding
page.
2. The voltage of the ungrounded current source terminal does not fall
lower than 13.5 V below ground, and
3. The current set resistor R does not exceed approximately 1.2 kΩ (the
minimum current ISRC is approximately 1 mA).
B.2. CURRENT SOURCE
Figure B.5: LM317 adjustable voltage regulator as a grounded current
source sinking current to the NI myDAQ -15V power supply: (a) schematic
diagram, (b) equivalent circuit model, and (c) recommended layout with
the standard breadboard layout of Figure B.3 on page 165.
167
168
APPENDIX B. LM317 VOLTAGE AND CURRENT SOURCES
Appendix C
TL072 Operational Amplifier
The Texas Instruments TL072 dual operational amplifier (“op amp”) provides two op amp devices in a single 8-pin package. For more details
see the datasheet available at http://www.ti.com; enter “tl072” in the
“Search by Part Number” field.
Figure C.1 on the following page shows the pinout diagram for the
TL072. Note the requirement for a dual power supply; the NI myDAQ
±15 V supply serves this purpose. Also note that the op amp device itself does not have a ground terminal. Instead the myDAQ AGND (analog
ground) establishes the ground reference.
Figure C.2 on page 171 shows the TL072 placed on the standard breadboard layout described in Figure B.3 on page 165, connected to power, and
NI Multisim provides a circuit model for the TL072: place the TL072ACP
device.
170
APPENDIX C. TL072 OPERATIONAL AMPLIFIER
Figure C.1: Texas Instruments TL072 dual op amp pinout diagram (top
view). The plastic package uses either a U-shaped cutout to indicate the
left side or an indented circle to indicate pin 1.
171
Figure C.2: Texas Instruments TL072 dual op amp placed on the standard
172
APPENDIX C. TL072 OPERATIONAL AMPLIFIER
Appendix D
The Intersil DG413 provides four analog switches; refer to Figure D.1 on the
next page for the pinout diagram. Each switch is bidirectional and operates
just like a physical SPDT switch but under the control of a digital (twolevel) control signal. Two of the switches are normally-open and the other
two switches are normally-closed. For more details see the datasheet available at http://www.intersil.com; enter “dg413” in the search box.
When connected and powered as shown in Figure D.2 on page 175 the
switches operate as expected over the full ±15 V range under the control
of the myDAQ digital outputs DIO0 to DIO7. The analog outputs AO0 and
AO1 may also serve as control signals provided they produce a voltage of
either 0 V (inactive switch state) or 5 V (active switch state).
NI Multisim provides a circuit model for the DG413: place the ADG413BN
device.
174
APPENDIX D. DG413 QUAD ANALOG SWITCH
Figure D.1: Intersil DG413 quad analog switch pinout diagram (top
view).Two of the switches are normally-open and the other two switches
are normally-closed. The switch positions indicate the normal (inactive)
state with the switch control voltage at low level. Switches 1 and 4 are
normally-open (NO) and Switches 2 and 3 are normally-closed (NC).
175
Figure D.2: Intersil DG413 quad analog switch placed on the standard
176
APPENDIX D. DG413 QUAD ANALOG SWITCH
Appendix E
Transient Response Measurement
Techniques
E.1
Time Constant
The general first-order circuit response takes the form
x(t) = x(∞) + [x(0) − x(∞)]e−t/τ , t ≥ 0,
(E.1)
where x(0) is the initial value at the start of the transient, x(∞) is the final
value, and τ is the time constant. Figure E.1 on the next page plots this
equation for the case of a final value lower than the initial value. This figure also shows the half-life time THL , defined as the time interval from the
onset of the transient at time t = 0 to the time at which x(t) reaches the
midpoint of the initial and final values. The half-life is easy to measure on
an oscilloscope from which the time constant follows as
τ=
THL
.
ln 2
(E.2)
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APPENDIX E. TRANSIENT RESPONSE MEASUREMENT TECHNIQUES
Figure E.1: General first-order circuit response showing initial value, final
value, and half-life time.
Appendix F
Sinusoid Measurement Techniques
F.1
Amplitude and Phase Measurements
Figure F.1 shows a pair of sinusoidal signals as displayed on an oscilloscope. The “Reference” sinusoid serves as the phase reference for the “Signal of Interest” and has a phase of zero degrees. Both sinusoids oscillate
at the same frequency f0 . Three measurements suffice to determine the
phasor representation of each sinusoid as follows:
Figure F.1: A pair of sinusoidal signals at the same frequency, one as the
reference and the other as the signal of interest. A and B indicate amplitude
measurements and C indicates the time shift measurement for phase.
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APPENDIX F. SINUSOID MEASUREMENT TECHNIQUES
1. The reference sinusoid is A6 0◦ ,
2. The phase shift θ in degrees is ±θ = C × f0 × 360◦ where C is the
absolute value of the time shift between the two sinusoids in seconds
and f0 is sinusoidal frequency in Hz; choose positive sign when the
signal of interest leads the reference (its zero crossing occurs before the
reference as pictured in Figure F.1 on the previous page) and negative
sign otherwise, and
3. The signal of interest is B 6 θ◦ .
Consider the NI ELVISmx Oscilloscope display of Figure F.2 on the facing page as an example of this measurement technique. Note that the “Volts
per Division” scales have been adjusted to make both sinusoids fill as much
of the screen as possible. The green trace on Channel 0 serves as the reference while the blue trace on Channel 1 is the signal of interest. The cursors
have been adjusted to measure the time shift between the two signals. Note
that the numerical display under the traces provides all necessary measurements:
1. The reference sinusoid amplitude is 1.696 volts divided by 2 (to convert from peak-to-peak) or 848 mV with a phase of zero degrees,
2. The phase shift is −65 µs × 2.502 kHz × 360◦ = −59◦ ; the phase is
negative because the signal of interest (blue trace) lags the reference,
i.e., the zero crossing occurs after the zero crossing of the reference,
and
3. The amplitude of the signal of interest is 427 mV divided by 2 or
214 mV.
From these measurements the phasor form of the reference is 8486 0◦ mV
and the phasor form of the signal of interest is 2146 − 59◦ mV.
F.1. AMPLITUDE AND PHASE MEASUREMENTS
Figure F.2: A pair of sinusoidal signals at the same frequency measured
by the NI ELVISmx Oscilloscope. After adjusting the cursors to measure
time shift the numerical display provides all required information in the
numerical display area.
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APPENDIX F. SINUSOID MEASUREMENT TECHNIQUES
Appendix G
NI LabVIEW MathScript Video Tutorials
• Plot two functions of time:
• Take cursor measurements on a plot:
NI Multisim & NI myDAQ Video Tutorials
• Compare simulated and physical DMM measurements:
NI Multisim Video Tutorials
Place components:
• Find commonly-used circuit components:
• Find components by name:
Sources:
• Function generator:
184
• AC (sinusoidal) voltage source:
• ABM (Analog Behavioral Model) voltage source:
• Pulse voltage source:
• Piecewise linear (PWL) voltage source:
• VDD and VSS power supply voltages:
Measure DC current:
• Measure DC current with a measurement probe:
• Measure DC mesh current with a measurement probe:
• Measure DC current with an ammeter indicator:
Measure DC voltage:
• Measure DC voltage with a voltmeter:
• Measure DC voltage with a referenced measurement probe:
• Measure DC voltage with a voltmeter indicator:
• Set the digits of precision of a measurement probe:
185
Measure DC node voltage:
• Measure DC node voltage with a measurement probe:
• Find node voltages with DC Operating Point analysis:
Measure DC power:
• Measure DC power with a wattmeter:
• Find resistor power with DC Operating Point analysis:
• Use a Parameter Sweep analysis to plot resistor power as a function
of resistance:
Measure resistance:
• Measure resistance with an ohmmeter:
Measure RMS and average value:
• Measure RMS and average value with a measurement probe:
Measure AC phasor voltage:
• Measure phasor voltage with a Single Frequency AC Analysis:
Measure frequency response:
• Measure frequency response with AC Analysis:
186
Measure AC power:
• Measure average power and power factor with a wattmeter:
Net names:
• Display and change net names:
Grapher View and oscilloscope cursor measurements:
• Find the maximum value of trace in Grapher View:
• Set cursor to a specific value:
Oscilloscope:
• Basic operation of the two-channel oscilloscope:
• Waveform cursor measurements with the two-channel oscilloscope:
• Distinguish oscilloscope traces by color:
• Stabilize the oscilloscope display with edge triggering:
• Basic operation of the four-channel oscilloscope:
Transient response:
• Plot time-domain circuit response with Transient Analysis:
• Voltage-controlled switch:
187
Combined Multisim / myDAQ measurements:
• Combine Multisim simulation and myDAQ measurements in the same
instrument – Bode Analyzer:
NI myDAQ Video Tutorials
See Electrical Circuits with NI myDAQ for more video tutorials and projects:
http://decibel.ni.com/content/docs/DOC-12654
NI ELVISmx Instruments for NI myDAQ:
• DMM ohmmeter:
http://decibel.ni.com/content/docs/DOC-12938
• DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12937
• DMM ammeter:
http://decibel.ni.com/content/docs/DOC-12939
• Function Generator (FGEN):
http://decibel.ni.com/content/docs/DOC-12940
• Arbitrary Waveform Generator (ARB):
http://decibel.ni.com/content/docs/DOC-12941
• Oscilloscope:
http://decibel.ni.com/content/docs/DOC-12942
• Bode Analyzer:
http://decibel.ni.com/content/docs/DOC-12943
http://decibel.ni.com/content/docs/DOC-12944
• Digital Writer (DigOut):
http://decibel.ni.com/content/docs/DOC-12945
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Measurement techniques:
• Measure current with a shunt resistor and DMM voltmeter:
http://decibel.ni.com/content/docs/DOC-12946
• Measure node voltage:
http://decibel.ni.com/content/docs/DOC-12947
• Increase current drive of analog output (AO) channels with an op amp
voltage follower:
http://decibel.ni.com/content/docs/DOC-12665
```