Lab#2 - Department of Electrical & Computer Engineering

UNIVERSITY OF ALBERTA DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING EE 240 –Electrical Circuits I Laboratory Experiment #2

Fall 2011 Circuit Theorems

Objectives: The objectives of this laboratory experiment are to verify Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL), and to apply Thevenin's and superposition theorems to the analysis of electrical circuits. List of Equipment: a) A Digital Multi-Meter (DMM ): A DC (battery) powered measurement instrument equipped with a digital readout and equipped with a rotary switch, b) An Analog Volt Ohm Meter (AVOM): An AC (electric mains) powered electromechanical measurement instrument equipped with an analog readout (including a needle and a dial) using a moving coil movement, and equipped with rotary switches to select mode and range settings1, c) A DC Power Supply (DC Voltage Source), d) A Student Printed Circuit Board, and e) A decade-resistor box. I.

Introduction

The analysis of electrical circuits involves the determination of the voltages across and the currents through the constituent circuit elements. This analysis is based on a set of three basic laws, namely, Kirchhoff's voltage law (KVL), Kirchhoff's current law (KVL), and Ohm’s law. In practical situations, the analysis task may be simplified by using the Thevinin’s (or Norton’s) and superposition theorems. 1

For proper operation, this instrument must be used horizontally in order to balance the meter movement.

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Laboratory Experiment #2

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The experiments that follow will demonstrate the practical application of the above laws and theorems to the analysis of electrical circuits. The purposes of this laboratory are threefold: a) To verify the validity of KVL and KCL laws, b) To apply the superposition and Thevenin’s theorems to the analysis of electrical circuits and, and c) To confirm the validity of Thevenin’s and superposition theorems. II.

Theoretical Background

II. A

Kirchhoff's voltage law

Kirchhoff's voltage law states that the algebraic sum of the voltage drops across circuit elements around any closed loop in an electrical circuit at any instant of time is zero. II. B

Kirchhoff's current law

Kirchhoff's current law states that the algebraic sum of the currents flowing into (or out of) a node in an electrical circuit at any instant of time is zero. II. C

Thevenin’s theorem

Thevenin's theorem states that any circuit containing voltage and/or current sources and resistive elements can, when viewed from outside world (through a load), be replaced by the series combination of an ideal voltage source VTH and a resistor RTH . The Thevenin’s equivalent voltage source VTH happens to be the same as the open-circuit voltage VOC at the load terminals (i.e.VTH = VOC ). The Thevenin’s equivalent resistance RTH is the ratio of the open-circuit voltage VOC to the short-circuit current I SC at the load terminals. Alternatively, RTH can be obtained as the resistance looking back into the circuit from the load terminals with all of the voltage and/or current sources reduced to zero (or replaced by their internal resistances if these resistances are finite). II. D

Superposition theorem

The Superposition theorem states that the voltage across or the current through any circuit element in a linear electrical circuit consisting of a number of voltage and/or current sources acting together can be determined as the algebraic sum of the voltages across and currents through each element resulting from each source acting alone. III.

Laboratory Experiment Procedures This Laboratory Experiment consists of three parts numbered as parts III.A-III.C. Make sure that in the course of the laboratory experiment, the results obtained for each part are recorded in the corresponding tables included within the report section. To ensure an effective use of the laboratory time, the steps requiring calculations based on measured values or the steps which require written responses to

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Laboratory Experiment #2

questions can be completed at the completion of all measurements. Such steps appear in italics font in the Laboratory précis and the corresponding cells in the tables have a gray background. III.A Kichhoff’s voltage and current Laws Purpose: To verify the validity of KVL and KCL laws by using the electrical circuit shown in Figure 1 in the following. Experimental Procedure Step 1.

By using the DMM meter on the “Ω” range, measure the (actual) resistance of the resistors labeled as 1k 0 , 2k 2 , 2k 7 , and 4k 7 the student PCB board. Record the measured values in Table III.1. Table III.1

on

Measured Resistances

Nominal values

1k 0

2k 2

2k 7

4k 7

Measured values Step 2.

By using patch cords, connect suitable circuit elements to obtain the electrical circuit shown in Figure 1. I1

+ V1 -

I3

+ V3 R3 2k2

R1 2k7 I2

+ VS

Loop 1

-

20V

R2 1k

+ V2 -

IL= I3

Loop 2

RL 4k7

+ VL

-

Figure 1 Step 3.

Adjust the DC voltage source to generate a voltage of VS = 20V ± 0.1V by using the DMM meter V DC range. Record the measured value of VS in Table III.2. Table III.2

VS Step 4.

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Voltage Source Measurement

V DC

Measure and record the currents I1 , I 2 , and I 3 (= I L ) flowing through the designated circuit elements by using the DMM meter set to measure

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“I” using the rotary switch and record the results in Table III.3 . Please make sure to take the direction of the current flows into account. Table III.3

Current

Kirchhoff’s Current Law Verification

Measured Currents

Calculated Currents

% Error

I1 I2 I3 Algebraic sum of currents Step 5.

By using the DMM meter set to measure voltages, measure and record the voltage drops across resistors R1 , R2 , R3 and RL in Table III.4. Please make sure to include the voltage drop magnitudes (absolute values) in column 2, and their observed signs (relative to the polarities shown in the circuit diagram) in columns 3 and 4.

Step 6. Verify the validity of KCL law: Sum the currents entered in column 2 in Table III.3 and record the result in the last row of Table III.3. Verify the validity of KVL laws around both loops: Sum the voltage drops around each loop (taking into account the measured voltage drop magnitudes in column 2 and the observed voltage drop signs in column 3 and 4) and record the result in the last row of Table III.4. Table III.4

Voltage V1 V2 V3 VL VS Algebraic sum of voltages

Step 7. EE 240

Kirchoff’s Voltage Law Verification

Measured Magnitude

Observed Sign, Loop 1

Observed Sign, Loop 2

Calculated Voltages

% Error

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Redraw the circuit in Figure 1 in the space provided in the following, Write down the loop equations by using the measured values of the resistors and the measured voltage of the voltage source, Solve these equations for the loop currents, Use the results to determine the currents I1 , I 2 , and I 3 in terms of the loop currents. Enter the resulting currents in the third column of Table III.3. Determine the % Error between the measured and the calculated currents in accordance with % Error

Calculated I1 3 Measured I1 Calculated I1 3

3

100

and enter the results in the last column in Table III.3. Calculate the voltage drops across the resistors R1−3,L in accordance with Calculated V1

3, L

Calculated I1

3, L

Measured R1

3, L

Calculate the % Error between the measured and the calculated values of the voltage drops and enter them in the last column in Table III.4.

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III.B The Thevenin’s Theorem Purpose: To verify Thevenin’s theorem for the circuits in Figures 1 above and Figure 2 in the following. Experimental Procedure Verification of Thevenin’s theorem for the circuit in Figure 1

Step 1.

By using the DMM meter, measure the voltage across and the current through the load resistance RL in the circuit in Figure 1, and record the results in Table III.5.

Step 2.

Remove the load resistor RL from the circuit2.

Step 3.

Measure the open-circuit voltage VOC across the terminals where the load resistor RL was once connected by using the DMM meter on the V DC range, and record the result in Table III.5.

Step 4.

With the DMM meter connected across the same terminals as in Step 3, measure the short-circuit current I SC flowing through the

2

This is in preparation to derive the Thevenin’s equivalent circuit as seen by the load resistor RL .

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terminals where the load resistor RL was once connected, and record the result in Table III.5. Table III.5

Thevenin’s Equivalent Circuit Measurements for Figure 1

VL IL VTH = VOC I SC

Step 5.

The first method of determining the Thevenin’s equivalent resistance RTH is as follows: Method 1: •

Replace the 20V DC voltage source by a short-circuit, and

•

By using the DMM meter set on “Ω” range, measure the resistance across the terminals where the load resistance was once connected, and record the result in Table III.6. This measurement gives the value of the Thevenin’s resistance RTH .

Table III.6

Directly Measured RTH

RTH

Step 6.

The second method of determining the Thevenin’s equivalent resistance RTH is as follows: Method 2:

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•

Connect a decade resistor across the terminals where the load resistance was once connected,

•

Set the DMM meter to V DC range and connect it across the decade resistor,

•

Adjust the decade resistor until the measured voltage drop across this resistor becomes VOC / 2 ,

•

Read the value of the indicated resistance on the decade resistor. Record this value in Table III.7. This reading gives the value of the Thevenin’s resistance RTH .

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Laboratory Experiment #2 Decade Box Measured RTH

Table III.7

RTH

Step 7.

Obtain the theoretical Thevenin’s resistance by using the theoretical relationship RTH =

VOC I SC

Record this value in Table III.8. Step 8.

By using the Thevenin’s voltage source VTH , and by using the Thevenin’s resistance RTH calculated in Step 7, calculate the current IL =

VTH RTH + RL

through the load resistance RL (by using the measured value of RL ). Record this result also in Table III.8 below. Table III.8

Theoretical Calculations

RTH = VOC / I SC IL

Step 9.

Compare the RTH in Step 7 to the two RTH ’s in Step 5 and Step 6, and compare the I L in Step 8 to the I L in Step 4 in Experiment III.A (as recorded in Table III.3). Should the corresponding values be noticeably different, identify and correct any errors (in measurements or calculations).

Step 10. Build up the Thevenin’s equivalent circuit as a series connection of a voltage source VTH (by using the “B output” of the voltage source) and a resistor RTH (by using decade resistor box) as shown in Figure 2:

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Laboratory Experiment #2 RTH

+ VTH -

Figure 2 Make sure, To set the “B output” of the DC voltage source to the voltage VTH obtained in Step 3. To set the decade resistor to the RTH value obtained in Step 6. To Measure VOC , I SC , and RTH in the circuit in Figure 2. This circuit can replace the circuit in Figure 1. Use the Method 1 above to measure RTH and record the result in Table III.9. Step 11. Load the Thevenin’s equivalent circuit in Figure 2 by connecting the (nominally) 4k 7Ω load resistor RL across its two accessible terminals. Step 12. Measure the voltage across and the current through the resistor RL and record the result in Table III.9. Table III.9

Measurements for the Thevenin’s Equivalent Circuit

VTH = VOC I SC RTH VL IL

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Step 13. Compare the results in Table III.9 to the corresponding results in Step 1 and comment, in the space below, whether they are in close agreement.

Verification of Thevenin’s theorem for the circuit in Figure 3

Step 14. Modify the circuit in Figure 1 to obtain the circuit shown in Figure 33, where the voltage sources VS1 and VS 2 are to be set up in the next step. I1

+ V1 -

I3

R1 2k7 + VS1

+ VS2

-

20V

R3 2k2

I2 R2 1k

+ V3 -

IL= I3

+ V2

RL 4k7

-

-

-

Figure 3 Step 15. Set the “A Output” of the DC voltage source to generate a voltage VS1 of 20V DC , and its “B Output” to generate a voltage VS 2 of 10V DC . Record the measured values of VS1 and VS 2 in Table III.10. Table III.10 Measured Voltage Source Voltages

VS 1

VS 2 VDC

VDC

Step 16. By using the DMM meter, measure and record the voltage across the current through the load resistance RL in Table III.11.

3

This circuit consists of a second DC voltage source VS 2 connected in series with the resistor R2 .

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+ VL

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Laboratory Experiment #2 Table III.11

Load Measurements for Figure 3

Measured I L Measured VL Step 17. Disconnect the load resistor RL from the circuit in Figure 3 in preparation to derive Thevenin’s equivalent circuit across the load terminals. Step 18. Measure the open-circuit voltage VOC across the terminals where the load resistor RL was connected by using the DMM meter on the V DC range, and record the result in Table III.12. Step 19. By connecting the DMM meter across the same terminals measure the short-circuit current I SC , and record the result in Table III.12. Step 20. By using the first method of determining the Thevenin’s equivalent resistance, measure RTH and record the result in Table III.12. Step 21. Compare the measured RTH in Step 20 to the corresponding result obtained by using the theoretical relationship RTH =

VOC I SC

Record this value in Table III.12. Step 22. By using the Thevenin’s voltage source VTH , and by using the Thevenin’s resistance RTH calculate the current IL =

VTH RTH + RL

through the load resistance RL (using the measured value of RL ). Record the result in Table III.12. Table III.12 Thevenin’s Equivalent Circuit Measurements for Figure 3

VTH = VOC I SC Measured RTH RTH = VOC / I SC IL

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Step 23. In the space provided below, sketch the Thevenin’s equivalent of the circuit in Figure 1 loaded with the resistance RL , and by using the measured values for VTH and RTH , calculate the values of VOC , VL, and I L .

Step 24. Calculate the % Error between the measured values for VL and I L , and the corresponding calculated values in Step 23.

Step 25. In the space provided below, sketch the Thevenin’s equivalent of the circuit in Figure 3 loaded with the resistance RL , and by using the measured values for VTH and RTH , calculate the values of VOC , VL, and I L .

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Step 26. Calculate the % Error between the measured values for VL and I L , and the corresponding calculated values in Step 25.

Step 27. Do you expect RTH ’s to be the same for the circuits in Figure 1 and Figure3? Why?

Proceed to the next experiment without disconnecting the circuit in Figure 3. III.C The Superposition theorem Purpose: To verify the superposition theorem by using the circuit in Figure 3. EE 240

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Laboratory Experiment #2 Experimental Procedure

Step 1.

Determine the effect of the voltage source VS 1 acting alone By setting voltage source VS 2 to zero (i.e. by replacing a short-circuit in its place) in the circuit in Figure 2, and by having the voltage source VS 1 active. By measuring and recording, in Table III.13, the voltage drop across and the current flow through the load resistor RL .

Step 2.

Determine the effect of the voltage source VS 2 acting alone By setting voltage source VS 1 to zero (i.e. by replacing a short-circuit in its place) in the circuit in Figure 2, and by having the voltage source VS 2 active. By measuring and recording, in Table III.13, the voltage drop across and the current flow through the load resistor RL .

Step 3.

Add the measured voltages and currents through the load resistor RL and record them in Table III.13.

Step 4.

Enter the measured voltages and currents through the load resistor RL from Table III.11 in the last line of Table III.13. Table III.13 Superposition Theorem:

Voltage Source Acting Step 1:

VS 1

Step 2:

VS 2

Step 3:

Total

Measured

Measured

VL

IL

Measured Values from Table III.11 Step 5.

Verify the validity of the superposition theorem for the voltage across and the current through the load resistor RL by comparing the sum of the voltages and currents in Steps 1 and 2 with those measured in Step 16 in Section III.B.

Step 6.

Please disconnect your equipment, tidy your bench and have the instructor initial your data sheet before leaving the laboratory.

Step 7.

Does the experiment III.C verify the principle of superposition? Please explain!

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Step 8.

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What assumption regarding the internal resistance of a voltage source is made when a voltage sources is replaced by a short- circuit (i.e. what is the internal resistance of an ideal voltage source)?

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Laboratory Experiment #2

UNIVERSITY OF ALBERTA DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING EE 240 –Electrical Circuits I Laboratory Experiment #2

Fall 2011 Circuit Theorems

Laboratory Report Section Number: D1 D2 D3 D4 D5 D6 D7 D8 Station Number: 1 7 13 19

2 8 14 20

3 9 15 21

4 10 16 22

5 11 17 23

6 12 18 23

Team Members: Student Names

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ID Numbers

Grade