Experiment 4: Thevenin’s Theorem and Maximum Power Transfer

Objectives:

- To prove Thevenin’s theorem experimentally.- To prove the maximum power transfer theory for DC and AC circuits.

Equipments:

1. DC power supply.

2. Function generator (Signal generator).

3. Digital multi-meters.

4. Resistors. 100 Ω, 470 Ω, 1 kΩ, 2.2 kΩ, 3.3 kΩ, 4.7 kΩ and 10 kΩ.

5. Variable resistor.

6. Variable capacitor.

7. Variable inductor: 17.5 mH.

Theoretical Background:

1. Thevenin’s Theorem:

Thevenin’s theorem allows us to transform a complicated circuit by replacing it with

simpler one, which is called Thevenin equivalent circuit. Thevenin’s theorem could

be applied to AC circuits in the same manner as has been applied to DC circuits. In

the case of DC circuit; Thevenin’s equivalent circuit consists of a single independent

voltage source (Eth) in series with a resistor (Rth). But in case of AC circuit;

Thevenin’s equivalent circuit consists of a single independent voltage source (Eth) in

series with impedance (Zth). The resulting voltage-current relationship at the load

resistance remains unchanged by this transformation. In this experiment we will

verify Thevenin’s theorem by constructing both the original complicated circuit and

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the simplified Thevenin equivalent circuit and measuring the voltage and current at

the load for both circuits.

To find the variable of interest (voltage or current); remove the related branch

(assume it connected between the two terminals a and b) from the circuit and calculate

(Eth) which is the voltage between the terminals between terminals a and b (in most

cases, this voltage is called the open circuit voltage).

Short out all the independent voltage sources and open all the independent current

sources (this can be done only theoretically, not in practice) and calculate Zth across

the terminals a and b. Draw the Thevenin’s equivalent and replace the branch that has

been removed and find using basic rules the desired variable (voltage, or current etc.).

(See Fig.1)

Fig. 1: Thevenin’s Theorem

2. Maximum power transfer

The power transferred from a supply source to a load is at its maximum when the

resistance of the load is equal to the internal resistance of the source. In the other

words: a resistive load will be consumptive maximum power from the supply when

the load resister is equal to the equivalent (Thevenin) network resister.

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According to the Thevenin equivalent circuit in Fig. 2, the condition for maximum

power transfer is:

RL=RTh (1)

And the maximum transferred power is:

Pmax ¿V Th

2

RTh (2)

R Th

0

R LV Th

Fig. 2: Thevenin equivalent circuit.

A graph of PL versus RL is shown in Fig.3. The maximum value of power which

occurs when RL=RTh is also shown in this figure.

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Fig. 3: The power delivered to the load vs. the load resistance.

In AC circuits; the max power will be transferred to a load when the load impedance

is the conjugate of the Thevenin’s impedance across the load terminals (ZL=ZTh¿ ). (see

Fig. 4).

Fig. 4: Maximum power transfer for ac circuit

Procedure:

Pre-Lab # 1: Perform the following exercise before coming to the lab to do this experiment. You will not be allowed to work on the experiment until the pre-lab has been completed.For the circuit shown in Fig. 5, consider R1= 470 Ω, R2= 1.0 kΩ, R3= 2.2 kΩ, R4=

3.3 k Ω, and Vs=12 V. Calculate Eth and (show your work here)

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Fig. 5. Circuit used in verification of Thevenin’s theorem

V Th= V oc = 8.16 ….………. V RTh= 1157.5…………..….. ΩI R 4 = ………0.00183………… A. (Current through R4 resistor)

Procedure 1: Thevenin’s Theorem

1. Construct the circuit shown in Fig. 5.

2. Measure the current through the resistor R4. I = ………… A.

Regarding the current I through R4 resistor, do your measurements agree with the

theoretical analysis?

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3. Remove R4 and measure the open circuit voltage Voc (between terminals a and b)

(refer to Fig. 6.) V Th= V oc = …………. V.

Regarding Voc, do your measurements agree with the theoretical analysis?

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.4. Replace R4 with a wire and measure the short circuit current I sc (refer to Fig. 7.)

I sc= ………. A.

5. calculate Rth such that RTh=V oc

I sc

RegardingRTh, do your measurements agree with the theoretical analysis?

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Fig. 6: Measuring V oc

Fig. 7: Measuring I sc

6. Using an ohmmeter, adjust the resistance of the potentiometer to the value of

RTh you calculated in step 5.

7. Using a voltmeter, adjust the terminal voltage of the power supply to the value of

V oc you calculated in step 2.

8. Construct Thevenin equivalent circuit as shown in Fig. 2.

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9. Measure the current through R4 resistor. I = ………… A. Regarding the current

through R4 resistor I, do your measurements in step 2 (original circuit) and step 7

(equivalent circuit) agree?

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Comments and discussion

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Procedure 2: Maximum power transfer for DC circuits.

Pre-lab 2: Find the expected (theoretical) value of RLthat satisfies the maximum

power transfer and find the value of the maximum power)

1. Connect the circuit shown in Fig. 8 with the values indicated in the figure. For the

load resistance you can use a variable resistor.

2. Vary the load resistance according to table 1. For each value, measure the voltage

at the load and calculate the power delivered to the load and complete the table.

R L

V Th =1 0 V

R Th

1 0 0 O h m

0

Fig. 8: Maximum power transfer for DC circuit.

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Table 1: Load capacitance values.

RL V L I L PL

10 Ω

30 Ω

50 Ω

70 Ω

90 Ω

100 Ω

120 Ω

150 Ω

170 Ω

200 Ω

Comments and discussion

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Procedure 3: Maximum power transfer for AC circuits.

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Pre-lab 3: Find the expected (theoretical) value of CLthat satisfies the maximum

power transfer and find the value of the maximum power)

1. Remove the circuit and connect the circuit shown in Fig. 9. R1=100 Ω, V1=5V,

1000 Hz, L1=17.5 mH.

2. Set the function generator to 5 V p−p, 1 kHz, sinusoidal wave.

3. Measure the internal resistance of the inductor and make R2=R1+R(inductor).

4. Vary the load capacitance according to table 2. For each value, measure the voltage

of the load resistance, and calculate the power delivered to the load.

PR2 ¿V R2

2

R2

5. Complete table 2.

C L

cvar

L 1

17.5m H

1 2

V 15 V a c0 V d c

R 1

100

V

0

R 2

W

Fig. 9: Maximum power transfer for AC circuit.

Table 2: Load capacitance values.

CL |XCL| V R 2 PR2

0.1 µF

0.25 µF

0.5 µF

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0.75 µF

0.9 µF

1.1 µF

1.3 µF

1.4 µF

1.45 µF

1.5 µF

1.7 µF

2 µF

3 µF

5 µF

7 µF

9 µF

10 µF

6. Draw the relationship between the power delivered at load and |XCL| using the

data from table 2.

Comments and discussion

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Questions:

1. Prove equations (1) and (2) using the circuit analysis.

2. Calculate the efficiency of the circuit given in Fig. 8 at the maximum power

transfer case (at RL=RTh) and compare it to the maximum power transfer. Explain

your result.

Efficiency=Pout

P¿=

PL

PSource

3. Find the value of RL that gives the maximum power transfer. Then, find the

transferred power to the load and the efficiency at that value.

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0

R 32k

R 5

1k

R 2

1k

R L

R 42kV 1

1 0 V d c

R 1

1k

Conclusions: ( Conclude the whole experiments and find he percentage error

between theoretical and practical values )

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Graphs: (Include your graphs if required)

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