**Superposition:
Analyzing circuits with multiple sources.**

Caveat: Superposition only works for linear circuits. Circuits with R, L, C.

The current or voltage at any point in a circuit containing multiple sources (current and/or voltage) is the superposition (sum) of the currents or voltages imposed separately by each source.

**Procedure for analyzing a circuit using superposition.**

- Turn on the sources one at a time.
- For each source that is “off”,
replace it with its characteristic resistance. (Ideal voltage sources
*R*=0, a short circuit. Ideal current sources*R*=∞, an open circuit.) - Analyze for the desired quantity due to each source.
- The full solution for all sources active is the sum of the contributions due to each source individually.
- Carefully observe the signs of the individual voltages and currents. The polarity of the voltages and the direction of the currents may not be the same for all sources.

** Example:**
Find the Thevenin equivalent voltage,

Because the
network has 3 internal sources we will use superposition to solve the problem. Superposition
says that *V* is the sum of the voltages contributed from each of the 3
sources. *V* = *V*_{1} + *V*_{2} + *V*_{3}.
We will determine the voltage contribution from each voltage separately by
turning on only one source at a time. Sources that are “off” are replaced with
their characteristic resistance.

**Find
V_{TH} and R_{S} using superposition.**

**Solving for V_{1}
due to I_{S1}. **

The terminal voltage is the
voltage drop across *R*_{3}. We also note that *R*_{1}
and *R*_{2}+*R*_{3} form a current divider. To find *V*_{1}
we need *i*_{3} which we can find from the current divider
equation.

**Solving for V_{2}
due to I_{S2}. **

We see that
circuit reduces to almost the same problem as for *I*_{S1}, except
that the direction of the current is opposite. We have chosen to keep the same
direction of *i*_{3} to remind us that we need a negative sign. The
lower terminal will obviously be more positive that the upper terminal. We must
be very careful of the signs!

**Solving for V_{3}
due to V_{S3}. **

Again, the terminal voltage is
the voltage drop across *R*_{3}. However we also recognize that *R*_{1}+*R*_{2}
and *R*_{3} form a voltage divider. To find *V*_{3}
we simply use the voltage divider equation.

The *V*_{TH}
of this network as seen at the terminals is the sum of each of these component
voltages. Be careful of the signs!!

**To solve for the source
resistance, R_{S}, **we simply turn off ALL of the sources and
solve for the equivalent resistance of the resulting resistor network.

This work by L.A. Bumm is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.