Superposition

What and why?

How do you eat an elephant? One bite at a time! So follows the superposition theorem: in a linear system, the value at any part of a circuit is equal to the sum of the responses at that same part due to any and all sources acting on it.

Um....what?

As confusing as it is in English, the theorem is much simpler in practice. Essentially, by "turning off" any voltage or current sources in a circuit, we can determine the value at some chosen node, $A$, by summing the values of $A$ when each voltage or current source is "on" alone.

Let's walk through an example, lifted from All About Circuits:

Since this circuit has 2 voltage sources, $B_1$ and $B_2$, we will have to calculate 2 sets of values. We will begin by "turning off" $B_2$, and replacing it with a short circuit:

Using nodal analysis at the topmost node, we obtain the following values:

We then repeat our analysis while "turning off" $B_1$ and replacing it with a short circuit:

And again perform nodal analysis at the topmost node, giving a second set of values:

Now, the tricky part. During the superposition of our two sets of values, we must take care to sum them according to the appropriate direction and polarity:

Which gives us our final result: