This version (29 Jul 2020 19:55) was *approved* by dmercer.The Previously approved version (11 Sep 2019 20:20) is available.

The objective of this Lab activity is to verify Thévenin's theorem by obtaining the Thévenin equivalent voltage (V_{TH}) and Thévenin equivalent resistance (R_{TH}) for the given circuit. Verify the Maximum Power Transfer Theorem.

As in all the ALM labs we use the following terminology when referring to the connections to the M1000 connector and configuring the hardware. The green shaded rectangles indicate connections to the M1000 analog I/O connector. The analog I/O channel pins are referred to as CA and CB. When configured to force voltage / measure current -V is added as in CA-V or when configured to force current / measure voltage -I is added as in CA-I. When a channel is configured in the high impedance mode to only measure voltage -H is added as CA-H.

Scope traces are similarly referred to by channel and voltage / current. Such as CA-V , CB-V for the voltage waveforms and CA-I , CB-I for the current waveforms.

Thévenin's Theorem is a process by which a complex circuit is reduced to an equivalent circuit consisting of a single voltage source (V_{TH}) in series with a single resistance (R_{TH}) and a load resistance (R_{L}). After creating the Thévenin Equivalent Circuit, the load voltage V_{L} or the load current I_{L} may be easily determined.

One of the principal uses of Thévenin's theorem is to replace a large portion of a circuit, often a more complicated and uninteresting part, by a simple equivalent. The new simpler circuit enables rapid calculations of the voltage, current, and power which the more complicated original circuit is able to deliver to a load. It also helps to choose the optimal value of the load (resistance) for maximum power transfer.

Figure 1.

Figure 2: Thévenin Equivalent Circuit of Figure 1

2. The Maximum Power Transfer Theorem states that an independent voltage source in series with a resistance R_{S} or an independent current source in parallel with a resistance R_{S}, delivers a maximum power to the load resistance R_{L} when R_{L} = R_{S}.

In terms of a Thévenin Equivalent Circuit, maximum power is delivered to the load resistance R_{L} when R_{L} is equal to the Thévenin equivalent resistance R_{TH} of the circuit.

Figure 3: Maximum Power Transfer

ADALM1000 hardware module

Various Resistors (100 Ω, 330 Ω, 470 Ω, 1 KΩ and 1.5 KΩ)

1. Verifying the Thévenin's theorem:

a) Construct the circuit of figure 1 using the following component values:

R_{1} = 330 Ω

R_{2} = 470 Ω

R_{3} = 470 Ω

R_{4} = 330 Ω

R_{5} = 1 KΩ

R_{L} = 1.5 KΩ

V_{S} = +5V

b) Accurately measure the voltage V_{L} across the load resistance using ALM1000 Volt Meter Tool. Use the Volt Meter Tool by connecting channel CA to the + node of V_{L} and connect channel CB to the - node. V_{L} will be the difference between CA volts and CB volts. This value will later be compared to the one you will find using Thevenin Equivalent.

c) Find V_{TH}: Remove the load resistance R_{L} and measure the open circuit voltage V_{OC} across the terminals. Use the Volt Meter Tool by connecting channel CA to the + node of V_{OC} and connect channel CB to the - node. V_{OC} will be the difference between CA volts and CB volts. This is equal to V_{TH}. See figure 4.

Figure 4: Measuring the Thevenin Voltage

d) Find R_{TH}: Remove the source voltage V_{S} and construct the circuit as shown in figure 5. Use the ALM1000 Ohmmeter Tool to measure the resistance looking into the opening where R_{L} was. This gives R_{TH}. Make sure there is no power applied to the circuit before measuring with the Ohmmeter and the ground connection has been moved as shown.

Figure 5: Measuring the Thevenin Resistance R_{TH}.

e) Obtaining V_{TH} and R_{TH}, construct the circuit of figure 2. Create the value of R_{TH} using a series and or parallel combination of resistors from your parts kit. Using the Meter - Source Tool connect channel CA for the V_{TH} source and set the value to what you measured for VTH in step c).

Figure 6: Thevenin Equivalent Construction

f) With R_{L} set to the 1.5 KΩ used in step b) measure the V_{L} for the equivalent circuit and compare it to the V_{L} obtained in step b). This verifies the Thévenin theorem.

g) Optional: Repeat steps 1 b) to 1 f) for R_{L} = 2.2 KΩ

2. Verifying the Maximum Power Transfer theorem:

a) Construct the circuit as in figure 7 using the following values:

V_{S} = +5 V

R_{1} = R_{2} = 470 Ω

R_{3} = 1 KΩ

R_{L} = combinations of 1 KΩ and 100 Ω resistors ( figure 8 )

Figure 7: Circuit for Maximum Power Theorem

b) Use the Volt Meter Tool by connecting channel CA to the + node of V_{L} and connect channel CB to the - node across R_{L}. V_{L} will be the difference between CA volts and CB volts.

c) To find the value of R_{L} for which maximum power is transferred, vary the load resistances by constructing series / parallel combinations of 1 KΩ and 100 Ω for R_{L} between 500 Ω to 1400 Ω in 100 Ω steps as shown in figure 8. For each value of R_{L} write down V_{L}.

Figure 8. R_{L} configurations

d) Calculate the power for each load resistor value using P_{L}=V_{L}^{2}/R_{L}. Then, interpolate between your measurements to calculate the load resistor value corresponding to the maximum power (P_{L}-max). This value should be equal to R_{TH} of circuit in figure 7 with respect to load terminals.

1. Calculate the percentage error difference between the load voltages obtained for circuits of figure 1 and figure 2.

2. Using Voltage Division for circuit of figure 2, calculate V_{L}. Compare it to the measured values. Explain any differences.

3. Calculate the maximum power transmitted to the load R_{L} obtained for the circuit of figure 3.

**For Further Reading:**

DC Voltmeter Quick Start Guide (volt-meter-tool-1.2.exe)

DC Ohmmeter Quick Start Guide (ohm-meter-vdiv-1.2.exe)

DC Meter-Source Quick Start Guide (dc-meter-source-tool-1.3.exe)

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university/courses/alm1k/circuits1/alm-cir-4.txt · Last modified: 29 Jul 2020 19:55 by dmercer