# Analyzing AC Circuits using the Superposition Theorem

This won't be a "ground-up" explanation of the Superposition Theorem as it applies to DC circuits (which is usually how the Theorem is introduced). It is assumed that the reader is already familiar with the topic of using Superposition to solve a DC circuit and this tutorial will simply show how it applies to AC circuits.

### Brief Introduction on the Superposition Theorem:

First, consider a circuit with two or more independent sources. The Superposition principle states that the voltage across (or current through) an element in a linear circuit is the sum of the voltage across (or current through) that element due to each independent source acting alone. Superposition is based on the Linearity Property which states that the response to a sum of inputs is the sum of the response to each separate input. The Linearity Property is (not surprisingly) valid only for linear circuits where the output is linearly related (directly proportional) to its input.

### Steps to analyzing Circuits using the Superposition Theorem:

1. "Turn off" all independent sources except one source. Determine the output voltage/current due to the remaining source.
2. Repeat step #1 for each remaining independent source.
3. Find the total contribution by adding all of the individual contributions from each independent source.

### Application to AC circuits:

Since the Superposition Theorem applies to linear circuits, it can be used to analyze AC circuits in the same manner as with DC circuits. The theorem becomes critical if the AC circuit has sources that are operating at different frequencies.

# For circuits with sources operating at different frequencies:

Recall that impedance depends on frequency. Therefore, when using Superposition, we must construct a circuit in the frequency domain for each separate frequency. The total response is then obtained by adding the individual responses in the TIME-DOMAIN.

Adding individual responses in the frequency/phasor domain is wrong!!! Remember that the frequency factor (e^jwt) is implicit in sinusoidal analysis and this factor will change for every angular frequency (omega). The exception to this would be a situation where all of the sources are operating at the same frequency (in which case the individual responses could be summed in phasor form.)

Let's take a look at an example problem using the Superposition Theorem:

Continue on to Superposition example #1...