Thevinin Equivalance

Thevinin's theorem states that any circuit can be modeled by a voltage source and a resistor in series. To verify this claim, we will model a circuit in pspice, then solve for the node voltage both analytically and numerically.

The circuit in question for this laboratory is 2 independent voltage sources, with 3 cable resistances, and 2 load resistances. The circuit diagram follows:

All of the components have known values, except for RL2, which is the load we will use to test this theory.
If we want to find the equivalent voltage across the two terminals of RL2, we can remove it from the circuit and perform nodal or mesh analysis until we obtain the voltage across RL1.

As RC3 is floating, (not part of a closed path to ground), it has potential on both sides of its leads equal to the potential across RL1. Thus we can neglect it for the rest of our calculations. Once the mathematics are performed, it is determined that the voltage across RL1 is approximately 8.64 Volts. In pspice we can simulate a break in the circuit by adding a resistor which has enormous resistance in comparison to the rest of the circuit.

It is evident that the analytical model and the numerical model agree on the predicted voltage.

The next step in the analysis of this circuit is to determine the Thevinin resistance. In the general case, a circuit's thevinin resistance can be found by removing all independent sources and replacing them by either short circuits or open circuits. If the independent source is a voltage source, it is replaced by a short, and if the independent source is a current source it is replaced by an open.

For the circuit in question, we arrive at the following model.

The resistance between terminals a and b can be found by combining RC1, RC2, and RL1 in parallel, then combining the result with RC3 in series.

The Rth or Thevinin resistance should be 69.946 Ohms.

Thus, our complicated circuit can be replaced with the following Thevinin Equivalent:

Now that we have a simple model of our circuit, we may want to know how the circuit will respond to changes in the load across a and b. For instance, if we place a variable resistor in between the terminals we can see how voltage and current vary in response to load.

It is evident that we will have different power output across RL2 depending on our load resistance.

Suppose for example we have a component which only functions if its terminal voltage is greater than 8V. If we wanted to find the minimum resistance that this load could offer, we could consult the above graph and find that the necessary resistance should be approximately .8k or 800 Ohms. If we wanted to solve this problem analytically, we could perform circuit analysis on the Thevinin equivalent circuit.

Kirchoff's Voltage Law says that the voltage across RL2 must be equal to the product of the current through RL2 and its resistance. The current through the entire circuit is equal to the quotient of 8.64 by 65.9+RL2. Using these two constraints, it is determined that the minimum resistance of RL2 must be approximately 824.3 Ohms. This is consistent with what we found on the graph above.

Now that we know the relevant parameters and constraints, we can begin to build our circuit and see if it matches the predicted values.

In order to simulate the Thevinin Resistance, we chose a resistor which was close to the calculated value. The load was simulated by a variable resistance box, and the power supply was a variable power supply.

The data collected on the Thevinin circuit was acurate to the predicted values by better than 2 percent.

The voltage between terminals a and b was measured with the load resistance set to 824, and again with the circuit open.

Then the original circuit was constructed and tested in the same manner.

The actual circuit deviated from the predicted values by less than a single percent. Its superior accuracy may be due to a fortuitous combination of resistor and power supply errors.

As a final analysis, it is also interesting to note when the power that is supplied to RL2 would be maximum. Using the maximum power transfer theorem, it is determined that .2785 watts can be delivered to that load if its resistance is about 67 Ohms.

To verify this, we chose values of RL2 which were half as much, equal to, and twice as much as the Thevinin Resistance. In each configuration we measured the voltage across the load and calculated the power output.

It is important to note that in the ideal power zone, the voltage across the load is nearly half of the Thevinin Voltage. Additionally, it is interesting to see the symmetry when the resistance is multiplied or divided by the same number. Finally, the predicted power output, 2.785 Watts, can be found in the table at the appropriate voltage, thereby completing our verification of Thevinin's Theorem.