Circuits Lab Exp 6 Report

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)

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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


V1 10Vdc R2 2.2k

R3

1k

R1

3.3k

RL

Figure 3 (Circuit used for Experiment Part 3)

7

b

a


Vth

Rth

RL

Figure 4 (Circuit used for Experiment Part 4)

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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 (%)


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


Figure 7 (Results of PSpice Computer Simulation for Experiment Part 1)

(Bias Point Analysis to Verify VTH)

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Figure 8

(Results of PSpice Computer Simulation for Experiment Part 1) (DC Sweep Analysis to Verify VTH and RTH)

(Lower Portion of Graph Enlarged)

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(DC Sweep Analysis to Verify VTH and RTH) (Upper Portion of Graph Enlarged)

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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)


IN (mA)


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


(Bias Point Analysis to Verify IN)

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(DC Sweep Analysis to Verify ITH and RTH) (Upper Portion of Graph Enlarged)

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(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)


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)


Voltage Across

Pot. (V)

Power Dissipated

in Pot. (mW)


Voltage Across

Pot. (V)

Power Dissipated

in Pot. (mW)


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)


RTH (Ω)

3.95 2300



IN (mA)

1.643

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Measured

Pot. Setting


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


Measured

Pot. Setting


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

Documents

Circuits Lab Exp 6 Report