circuits lab exp 6 report
DESCRIPTION
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Tennessee State University College of Engineering, Technology, and Computer Science
Department of Electrical and Computer Engineering
ENGR 2001 CIRCUITS I LAB
Section 02
Lab Experiment #6
Thevenin’s Theorem, Norton’s Theorem, and Maximum Power Transfer
Vance Willis
Lab Partner: Tish Spalding
Instructor: Dr. Carlotta A. Berry
Lab Performed: October 13, 2005 Report Submitted: October 27, 2005
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ABSTRACT
The purpose of this experiment was to build a number of resistor circuits and take measurements to illustrate and verify Thevenin’s theorem for equivalent circuits and maximum power transfer. Thevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source (VTH) in series with a resistor (RTH). An analysis of the circuits by means of hand calculations, and simulations using the PSpice circuit simulation software was compared to experimental results. The maximum error noted was 4.70%. The maximum power theorem states that the maximum power transfer takes place when the load resistance is equal to the Thevenin resistance. This theory was also verified by experimentation, and analysis using the PSpice circuit simulation software. An inspection of the power vs. resistance graphs from both the experiment and the computer simulation show the peak power level to occur at the Thevenin resistance.
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TABLE OF CONTENTS
Abstract
I. Objective
II. Theory
III. Equipment
IV. Apparatus
V. Circuits
VI. Procedure
VII. Graphs
VIII. Results, Conclusions, and Recommendations
Appendix A Data Appendix B Formulas and Sample Calculations Appendix C References
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I. Objective: The purpose of this experiment was to build a number of resistor circuits and take measurements to illustrate and verify Thevenin’s theorem for equivalent circuits and maximum power transfer. Finally, the results were analyzed by an error analysis comparing the experimental results to the calculated results.
II. Theory:
Thevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTH in series with a resistor RTH, where VTH is the open-circuit voltage at the terminals, and RTH is the input or equivalent resistance at the terminals when the independent sources are turned off. In the case of an independent voltage source, it is turned off by replacing it with a short circuit. In the case of an independent current source, it is turned off by replacing it with an open circuit. Thevenin’s theorem is very important in circuit analysis, as it helps simplify a circuit. A large circuit may be replaced by a single independent voltage source and resistor.
inTH RR = (Thevenin resistance)
circuitopenTH VV −= (Thevenin voltage)
Similar to Thevenin’s theorem, Norton’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN is the open-circuit voltage at the terminals, and RN is the input or equivalent resistance at the terminals when the independent sources are turned off.
TH
THN R
VI = (Norton current)
THN RR = (Norton resistance) When analyzing a circuit, it is often useful to determine the load at which the circuit transfers the maximum power. The maximum power transfer theorem states that maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (RL=RTH).
TH
TH
RVp4
2
max = (Maximum power transfer)
5
III. Equipment: • Tektronix Digital Multimeter model # CDM250 • Tektronix Power Supply model # CPS250 • Resistors: 3.3 kΩ, 2.2 kΩ, and 1 kΩ • Potentiometer • Various resistors • Leads (2 pair) • Alligator Clips • Breadboard • PSpice software program
IV. Apparatus:
Experiment Parts 1, 2, 3, and 4: The apparatus used for this experiment consisted of a Tektronix digital multimeter (used in both voltmeter and ammeter modes), a Tektronix power supply, resistors (1 kΩ, 2.2 kΩ, and 3.3 kΩ), a potentiometer (parts 3 and 4 only), a breadboard, and alligator clips attached to the leads on the resistors. Figure 1 illustrates the apparatus configuration.
Figure 1 (Lab Apparatus used for Experiment Parts 1, 2, 3, and 4)
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V. Circuits Figure 2 is the circuit diagram for parts 1 and 2 of the experiment. Figures 3 and 4 are the circuit diagrams for parts 3 and 4 of the experiment, respectively.
b
a
Vance WillisENGR2001-0210/27/2005Lab #6 Report
V1 10Vdc R2 2.2k
R3
1k
R1
3.3k
Figure 2 (Circuit used for Experiment Parts 1 and 2)
b
a
Vance WillisENGR2001-0210/27/2005Lab #6 Report
V1 10Vdc R2 2.2k
R3
1k
R1
3.3k
RL
Figure 3 (Circuit used for Experiment Part 3)
7
b
a
Vance WillisENGR2001-0210/27/2005Lab #6 Report
Vth
Rth
RL
Figure 4 (Circuit used for Experiment Part 4)
8
VI. Procedure
Experiment Part 1: 1. Build the resistor circuit shown in Figure 2. 2. Connect the output of the power supply as shown and adjust the voltage to
10 V. 3. Connect a digital voltmeter across terminals a and b and record the voltage
measured (VTH). 4. Connect a digital ohmmeter across terminals a and b and record the
resistance measured (RTH). 5. Calculate the theoretical values for open-circuit voltage across terminals a
and b (VTH), and open-circuit resistance across terminals a and b (RTH) using hand calculations, and the PSpice circuit simulation program.
6. Perform an error analysis for the measured versus theoretical voltage and resistance values.
Experiment Part 2: 1. Build the resistor circuit shown in Figure 2. 2. Connect the output of the power supply as shown and adjust the voltage to
10 V. 3. Connect a digital ammeter across terminals a and b and record the current
measured (IN). 4. Calculate the theoretical value for short-circuit current across terminals a and
b (IN) using hand calculations, and the PSpice circuit simulation program. 5. Perform an error analysis for the measured versus theoretical current values.
Experiment Part 3: 1. Build the resistor circuit shown in Figure 3 (actual circuit). 2. Connect the output of the power supply as shown and adjust the voltage to
10 V. 3. Adjust the potentiometer to 1 kΩ. 4. Connect a digital voltmeter across the potentiometer and record the voltage
measured. 5. Connect a digital ammeter in series with the potentiometer and record the
current measured. 6. Using the measured values for voltage and current, calculate the power
dissipated in the potentiometer. 7. Calculate the theoretical values for voltage across the potentiometer, current
through the potentiometer, and power dissipated in the potentiometer using hand calculations, and the PSpice circuit simulation program.
8. Repeat steps 4, 5, 6, and 7 for each of the following potentiometer settings: 1.2 kΩ, 1.5 kΩ, 1.75 kΩ, 2 kΩ, 2.2 kΩ, 2.3 kΩ, 2.8 kΩ, and 3 kΩ.
9. Perform an error analysis for the measured versus theoretical voltage, current, and power values.
10. Create a scatter plot of the measured and calculated power vs. resistance.
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Experiment Part 4: 1. Build the resistor circuit shown in Figure 4 (Thevenin equivalent circuit). 2. Connect the output of the power supply as shown and adjust the voltage to
the calculated Thevenin voltage (VTH). 3. Combine various resistors as necessary to achieve the calculated Thevenin
resistance (RTH). 4. Adjust the potentiometer to 1 kΩ. 5. Connect a digital voltmeter across the potentiometer and record the voltage
measured. 6. Connect a digital ammeter in series with the potentiometer and record the
current measured. 7. Using the measured values for voltage and current, calculate the power
dissipated in the potentiometer. 8. Calculate the theoretical values for voltage across the potentiometer, current
through the potentiometer, and power dissipated in the potentiometer using hand calculations, and the PSpice circuit simulation program.
9. Repeat steps 5, 6, 7, and 8 for each of the following potentiometer settings: 1.2 kΩ, 1.5 kΩ, 1.75 kΩ, 2 kΩ, 2.2 kΩ, 2.3 kΩ, 2.8 kΩ, and 3 kΩ.
10. Perform an error analysis for the measured versus theoretical voltage, current, and power values.
11. Create a scatter plot of the measured and calculated power vs. resistance.
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VII. Graphs Figures 5 and 6 show scatter plots of the data accumulated during parts 3 and 4 of the experiment, respectively. The curves shown represent the power dissipated in the potentiometer as the load resistance was varied from 1 kΩ to 3 kΩ. Both the measured data and calculated theoretical data are shown.
Power Dissipated vs. Resistance(Experiment Part 3, Actual Circuit)
1.400
1.450
1.500
1.550
1.600
1.650
1.700
1.750
1.800
0.5 1 1.5 2 2.5 3 3.5
Resistance (kΩ)
Pow
er D
issi
pate
d (m
W)
CalculatedMeasured
Figure 5 (Scatter Plot of Power Dissipated vs. Resistance for Experiment Part 3)
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Power Dissipated vs. Resistance(Experiment Part 4, Thevenin Equivalent Circuit)
1.400
1.450
1.500
1.550
1.600
1.650
1.700
1.750
1.800
0.5 1 1.5 2 2.5 3 3.5
Resistance (kΩ)
Pow
er D
issi
pate
d (m
W)
CalculatedMeasured
Figure 6 (Scatter Plot of Power Dissipated vs. Resistance for Experiment Part 4)
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VIII. Results, Conclusions and Recommendations Experiment Part 1: For part 1 of the experiment (actual circuit), the open-circuit voltage and open-circuit resistance measurements closely agreed with theoretical results. The maximum error noted was 1.25%. Table 1 shows the calculated values for Thevenin voltage (VTH) and Thevenin resistance (RTH), as well as the error analysis. Figure 7 shows the results of a bias point analysis performed on the circuit using the PSpice circuit simulation software, which shows the 4 V potential (VTH) at terminals a and b. Figures 8 and 9 are enlarged views of the results of a DC sweep analysis from 0-1 ampere performed on the circuit using the PSpice circuit simulation software, which shows the y-intercept of the trace at 4 V (VTH), and the slope of the trace at 2320 Ω (RTH).
Table 1
(Calculated Data, Experimental Data, and Error Analysis for Experiment Part 1)
Calculated Measured Error Analysis
Open Circuit
Voltage VTH (V)
Open Circuit
Resistance RTH (Ω)
Open Circuit
Voltage VTH (V)
Open Circuit Resistance
RTH (Ω)
Open Circuit Voltage
VTH (%)
Open Circuit Resistance
RTH (%)
4 2320 3.95 2300 1.25% 0.86%
4.000V
I1
0Adc
0V
4.000V10.00VR3
1k
R1
3.3k
V1
10Vdc
R2
2.2k
Vance WillisENGR2001-0210/27/2005Lab #6 Report
Figure 7 (Results of PSpice Computer Simulation for Experiment Part 1)
(Bias Point Analysis to Verify VTH)
13
Figure 8
(Results of PSpice Computer Simulation for Experiment Part 1) (DC Sweep Analysis to Verify VTH and RTH)
(Lower Portion of Graph Enlarged)
14
Figure 9 (Results of PSpice Computer Simulation for Experiment Part 1)
(DC Sweep Analysis to Verify VTH and RTH) (Upper Portion of Graph Enlarged)
15
Experiment Part 2: For part 2 of the experiment (actual circuit), the current measurement agreed with the theoretical result. The error noted was 4.70%. Table 2 shows both the calculated and measured short-circuit currents (IN), and the error analysis. Figure 10 shows the results of a bias point analysis performed on the circuit using the PSpice circuit simulation software, which shows the 1.724 mA short-circuit current (IN). Figures 11 and 12 are enlarged views of the results of a DC sweep analysis from 0-1 volt performed on the circuit using the PSpice circuit simulation software, which shows the y-intercept of the trace at 1.724 mA (IN), and the slope of the trace being 1/2320, where 2320 Ω is the Thevenin resistance (RTH).
Table 2 (Calculated Data, Experimental Data, and Error Analysis for Experiment Part 2)
Calculated Measured Error Analysis
Short Circuit Current
IN (mA)
Short Circuit Current
IN (mA)
Short Circuit Current
IN (%)
1.724 1.643 4.70%
R1
3.3k
2.508mA
Vance WillisENGR2001-0210/27/2005Lab #6Report
R3
1k
1.724mA
R2
2.2k
783.7uA V20Vdc
1.724mA
V1
10Vdc
2.508mA
Figure 10 (Results of PSpice Computer Simulation for Experiment Part 2)
(Bias Point Analysis to Verify IN)
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Figure 11 (Results of PSpice Computer Simulation for Experiment Part 2)
(DC Sweep Analysis to Verify ITH and RTH) (Upper Portion of Graph Enlarged)
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Figure 12 (Results of PSpice Computer Simulation for Experiment Part 2)
(DC Sweep Analysis to Verify ITH and RTH) (Lower Portion of Graph Enlarged)
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Experiment Parts 3 and 4: For part 3 of the experiment (power dissipated in the actual circuit), the voltage and current measurements agreed with theoretical results. The maximum error noted was 4.33%. Table 3 shows a summary of the calculated voltages, currents and power, the experimental voltages, currents and power, and the percent error for each. For part 4 of the experiment (power dissipated in the Thevenin equivalent circuit), the voltage and current measurements also agreed with theoretical results. The maximum error noted was 3.06%. Table 4 shows a summary of the calculated voltages, currents and power, the experimental voltages, currents and power, and the percent error for each. Figure 13 shows the results of a DC sweep analysis from 0-3 kΩ performed on the circuit using the PSpice software. This is a graph of power dissipated in the load resistor vs. the load resistance, which agrees with calculated and experimental data (as shown in the graphs of Figures 5 and 6), and shows the maximum power being dissipated at 2320 Ω, which is the Thevenin resistance (RTH). In general, this experiment proves that the Thevenin equivalent circuit performs identical to the original circuit, and the maximum power transfer occurs at the Thevenin resistance. Some of the error detected in the experiment can be attributed to calibration errors in the instrumentation (multimeter), and resistor values not being exactly at nominal.
Table 3
(Calculated Data, Experimental Data, and Error Analysis for Experiment Part 3)
Meas. Calculated Measured
Calculated from
Measured Data
Error Analysis Pot.
Setting (kΩ)
Actual Res. (kΩ)
Current Thru Pot. (mA)
Voltage Across
Pot. (V)
Power Dissipated
in Pot. (mW)
Current Thru Pot. (mA)
Voltage Across
Pot. (V)
Power Dissipated
in Pot. (mW)
Current Thru Pot. (%)
Voltage Across
Pot. (%)
Power Dissipated
in Pot. (%)
1 1 1.205 1.205 1.452 1.166 1.191 1.389 3.22% 1.15% 4.33%
1.2 1.2 1.136 1.364 1.550 1.099 1.358 1.492 3.29% 0.41% 3.69%
1.5 1.5 1.047 1.571 1.645 1.014 1.564 1.586 3.16% 0.43% 3.57%
1.75 1.754 0.982 1.722 1.691 0.953 1.718 1.637 2.94% 0.24% 3.17%
2 2 0.926 1.852 1.715 0.899 1.852 1.665 2.91% 0.01% 2.90%
2.2 2.21 0.883 1.951 1.723 0.86 1.95 1.677 2.61% 0.07% 2.68%
2.3 2.29 0.868 1.987 1.724 0.845 1.984 1.676 2.61% 0.15% 2.76%
2.8 2.8 0.781 2.188 1.709 0.763 2.18 1.663 2.34% 0.34% 2.67%
3 3.01 0.750 2.259 1.695 0.733 2.25 1.649 2.33% 0.39% 2.71%
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Table 4 (Calculated Data, Experimental Data, and Error Analysis for Experiment Part 4)
Meas. Calculated Measured
Calculated from
Measured Data
Error Analysis
Pot. Setting
(kΩ) Actual Res. (kΩ)
Current Thru Pot. (mA)
Voltage Across
Pot. (V)
Power Dissipated
in Pot. (mW)
Current Thru Pot. (mA)
Voltage Across
Pot. (V)
Power Dissipated
in Pot. (mW)
Current Thru Pot. (mA)
Voltage Across
Pot. (V)
Power Dissipated
in Pot. (mW)
1 1 1.205 1.205 1.452 1.168 1.205 1.40744 3.06% 0.01% 3.04%
1.2 1.2 1.136 1.364 1.550 1.103 1.363 1.503389 2.94% 0.05% 2.98%
1.5 1.5 1.047 1.571 1.645 1.019 1.57 1.59983 2.69% 0.04% 2.73%
1.75 1.752 0.982 1.721 1.691 0.957 1.721 1.646997 2.58% 0.00% 2.58%
2 2 0.926 1.852 1.715 0.902 1.856 1.674112 2.58% 0.22% 2.37%
2.2 2.2 0.885 1.947 1.723 0.863 1.95 1.68285 2.48% 0.16% 2.33%
2.3 2.31 0.864 1.996 1.724 0.843 1.99 1.67757 2.42% 0.28% 2.70%
2.8 2.8 0.781 2.188 1.709 0.765 2.19 1.67535 2.08% 0.11% 1.97%
3 3 0.752 2.256 1.696 0.736 2.26 1.66336 2.11% 0.19% 1.92%
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Figure 13 (Results of PSpice Computer Simulation for Experiment Parts 3 and 4)
(DC Sweep Analysis to find Maximum Power Transfer)
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APPENDIX A
Data
Table 5 (Measured Data for Experiment Part 1)
Open Circuit Voltage
VTH (V)
Open Circuit Resistance
RTH (Ω)
3.95 2300
Table 6 (Measured Data for Experiment Part 2)
Short Circuit Current
IN (mA)
1.643
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Table 7 (Measured Data for Experiment Part 3)
Measured
Pot. Setting
(kΩ) Actual Res. (kΩ)
Current Thru Pot.
(mA)
Voltage Across Pot.
(V)
1 1 1.166 1.191
1.2 1.2 1.099 1.358
1.5 1.5 1.014 1.564
1.75 1.754 0.953 1.718
2 2 0.899 1.852
2.2 2.21 0.86 1.95
2.3 2.29 0.845 1.984
2.8 2.8 0.763 2.18
3 3.01 0.733 2.25
Table 8 (Measured Data for Experiment Part 4)
Measured
Pot. Setting
(kΩ) Actual Res. (kΩ)
Current Thru Pot.
(mA)
Voltage Across Pot.
(V)
1 1 1.168 1.205
1.2 1.2 1.103 1.363
1.5 1.5 1.019 1.57
1.75 1.752 0.957 1.721
2 2 0.902 1.856
2.2 2.2 0.863 1.95
2.3 2.31 0.843 1.99
2.8 2.8 0.765 2.19
3 3 0.736 2.26
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APPENDIX B
Formulas, Sample Calculations, and Error Analysis Formulas:
vip = (Power)
inTH RR = (Thevenin resistance)
circuitopenTH VV −= (Thevenin voltage)
TH
THN R
VI = (Norton current)
TH
TH
RVp4
2
max = (Maximum power transfer)
% error = 100*ltheoretica
measuredltheoretica − (Percent error)
Calculations:
022003300
10=+
− THTH VV 4=THV V (Calculation of Thevenin voltage using
nodal analysis)
232022003300
)2200)(3300(1000 =+
+== abTH RR Ω (Calculation of Thevenin resistance
equivalent resistance technique)
310724.12320
4 −×===TH
THN R
VI A (Calculation of Norton current)
3
22
max 10724.1)2320(4
)4(4
−×===TH
TH
RVp W (Calculated maximum power transfer)
% error = %86.0100*2320
23002320=
− (Percent error, Thevenin resistance)
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APPENDIX C
References
Alexander, Charles K. and Matthew Sadiku, Fundamentals of Electric Circuits, 2nd
Edition, McGraw Hill, 2004. Berry, Dr. Carlotta A. Circuits I Lab Study Guide for ENGR2001. Tennessee State
University