### Thevenin Theorem

When performing network analysis, Thevenin's theorem is a very helpful tool. It allows for a non-varying portion of a circuit to be replaced with a simplified design, thus simplifying the analysis of the entire network. This equivalent circuit performs the same way as the original circuit would.

Thevenin's theorem states that any linear, two-terminal portion of a network can be replaced by a Thevenin equivalent circuit. A Thevenin equivalent circuit consists of a voltage source (V_{Th}) in series with a resistor (R_{Th}) where V_{Th} is the open-circuit voltage at terminals A-B and R_{Th} is the equivalent resistance at terminals A-B. This equivalent circuit can be seen in Figure 1 below. It is important to replace independent sources with their internal resistances when solving for R_{Th}, i.e. current sources are replaced with open circuits and voltage sources are replaced with short circuits.

*Figure 1*

#### Dependent Sources and Thevenin's Theorem

Thevenin's theorem can be applied when analyzing a circuit with dependent sources. In this case, all independent sources are turned off and the R_{Th} is calculated by applying a current source or voltage source at the open terminal. When using a voltage source, it can be assumed to be 1V for simple calculations. Using mesh analysis, find the current I_{o} at the output. When using a current source, it can be assumed to be 1A for easy calculations as well. Nodal analysis can be used to find the voltage at the a terminal. The equivalent resistance then becomes a simple Ohm's law calculation, seen in Equation 1. If R_{Th} takes a negative value, it means the circuit is supplying power to the terminals. Then to find V_{Th}, use mesh analysis with all independent/dependent sources included and solve for the open circuit voltage.

$$R_{Th}=\frac{1\text{ } V}{I_{o}} \; \; \text{[Equation 1]}$$

Now to apply this theory to an example problem.

*Figure 2*

Solving for R_{Th }first, redraw the circuit with the 12V source as a short circuit. Then excite the circuit using either a 1A current source or 1V voltage source at the a-b terminal. Using a current source at the a-b terminal produces the circuit in Figure 3 below.

*Figure 3*

Now to write the nodal analysis equations.

$$\frac{V_{1}}{10\text{ }\Omega}-1.5*I_{x}+I_{x}=0\; \; \text{[Equation 2]}$$

$$\frac{V_{o}-V_{1}}{6\text{ }\Omega}+\frac{V_{o}}{8\text{ }\Omega}=1\text{ }A\; \;\; \;\;\text{ [Equation 3]}$$

where,

$$I_{x}=\frac{V_{1}-V_{o}}{6\text{ }\Omega}\; \; \; \;\; \;\; \;\; \;\; \;\;\; \;\; \;\;\text{[Equation 4]}$$

Solving for V_{o }yields V_{o }= .888V or 888.8mV. Remember that R_{Th} = V_{o }/ I_{o}, so R_{Th} is equal to 888.8 mOhms.

Next, to find V_{Th}, replace the 12V source and remove the current source from the a-b terminal. Again, utilizing nodal analysis, the equations are as follows.

$$\frac{V_{1}}{14\text{ }\Omega}+\frac{V_{1}-12}{10\text{ }\Omega}-1.5*\frac{V_{1}}{14\text{ }\Omega}=0\; \; \text{ [Equation 5]}$$

$$\frac{V_{Th}}{8\text{ }\Omega}-\frac{V_{1}-V_{Th}}{6\text{ }\Omega}=0\; \; \; \;\; \;\; \;\; \;\; \;\;\; \;\; \;\;\; \;\; \;\;\text{[Equation 6]}$$

Solving for V_{1 }in Equation 6 yields V_{1 }= 1.75*V_{Th}. Plugging this into Equation 5 and solving for V_{Th} yields V_{Th} = 10.66V.

This theorem allows for the simplification of resistor and source configurations into to a one source and one resistor equivalent circuit. It is commonly used with varying loads, that way the load current and power dissipation can be calculated easily.

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