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DC Network Analysis

# Thevenin's Theorem

Thevenin's theorem is a method of circuit analysis that involves reducing a complex circuit to a simple equivalent circuit. Thevenin's theorem is frequently used to make circuit analysis simpler.

Thevenin's theorem is especially useful in analyzing circuits where only
one particular resistor in the circuit (called the **load**) is subject
to change. Example:

Thevenin's theorem states: *Any linear electrical circuit can be replaced
at terminals A and B by a simple series circuit consisting of an equivalent
voltage source* *V*_{th} *and an equivalent resistance*
*R*_{th}.

Note: A **linear circuit** is an electronic circuit in which the
electronic components' values (such as resistance, conductance, capacitance,
inductance, gain, etc.) do not change with the level of voltage or current
in the circuit.

The equivalent series circuit will provide the same current through a
load as would the original circuit. The equivalent voltage
*V*_{th} is the voltage "seen" between the two terminals (A and B)
being considered in the original network with the load resistance removed.
The equivalent resistance *R*_{th} is the resistance "seen"
between the terminals of the original network when all voltage sources of
the original circuit are replaced by a short circuit (wire) and all current
sources are replaced by an open circuit (break).

The figure below shows the application of Thevenin's theorem to a simple network.
When Thevenin's theorem is applied, the current through *R*_{L}
of the figure below (view A) is the same as the current in *R*_{L} of
the figure below (view D).

It must be pointed out that Thevenin's theorem establishes equivalency for
*R*_{L} only. The current, voltage, and power of *R*_{L}
in the figure above (view D) are therefore the same as in the figure above (view A).
The power supplied to the circuit of figure above (view D) by
*V*_{th} is not the same as the power supplied to the
circuit of figure above (view A) by *V*.

Thevenin's theorem has the effect of removing a portion of a circuit to make the remaining circuitry easier to analyze. Use of Thevenin's theorem is further demonstrated by the following examples.

**Example 1:**

Find the Thevenin-equivalent circuit of the circuit shown in the figure below.

Solution:

**Calculating the equivalent voltage V_{th}**

*V*

_{th}.

Use Kirchhoff's voltage law and Ohm's law to calculate the current *I*
through elements in series (see the figure above):

The voltage between the points A and B can be figured from the voltages
*V*_{2} and *V*_{R2}. This is our equivalent
voltage *V*_{th} in the equivalent circuit:

**Calculating the equivalent resistance R_{th}**

Replace the voltage sources *V*_{1} and *V*_{2}
with a wire:

*R*

_{th}.

The resistance between the points A and B is equal to *R*_{1} and
*R*_{2} in parallel: 0.8 Ω. This is our equivalent
resistance *R*_{th} in the equivalent circuit:

**Example 2:**

Find the Thevenin-equivalent circuit of the circuit shown in the figure below.

Solution:

**Calculating the equivalent voltage V_{th}**

*V*

_{th}.

The current *I* through elements in series is equal to *I*_{1}.

The voltage between the points A and B can be figured from the voltages
*V*_{2} and *V*_{R2}. This is our equivalent voltage
*V*_{th} in the equivalent circuit:

**Calculating the equivalent resistance R_{th}**

Replace the current source *I*_{1} with a break and the voltage
source *V*_{2} with a wire:

*R*

_{th}.

The resistance between the points A and B is equal to *R*_{2}: 1 Ω.
This is our equivalent resistance *R*_{th} in the equivalent circuit: