Superposition

The principle of superposition supposes that for some vector $\vec{x}\in {\mathbb{R}}^n$, it is made up of a linear combination of multiple constituent vectors (due to vector addition). This holds for all linear functions, i.e., where $f(x+y)=f(x)+f(y)$.

$$\left[\begin{array}{c}{x}_0\\ x_1\\ ⋮\\ x_n\end{array}\right]=\left[\begin{array}{c}{x}_0\\ 0\\ ⋮\\ 0\end{array}\right]+\cdots +\left[\begin{array}{c}0\\ 0\\ ⋮\\ x_n\end{array}\right]$$

This result turns out to be pretty useful in analysis in physics, where multiple forces add up and can be expressed as a single vector, or with electric field lines.

Applications

• Linear circuits follow this principle, allowing us to exploit this in circuit analysis. See below.
• Vector fields follow this principle. This has broad applications in electromagnetism.

Circuits

Superposition supposes that in a linear circuit with multiple independent sources, we can calculate the current or voltage at any point in the network as the algebraic sum of all individual contributions of each source acting alone. If we don’t analyse the contributions of all sources, we may get a non-linear response.

What does this mean? We go source by source, by shorting voltage sources and opening current sources, and analysing each sub-circuit.

$$I_O=I_O^{{}^{\prime }}+I_O^{{}^{\prime \prime }}$$ $$v_o=v_o^{{}^{\prime }}+v_o^{{}^{\prime \prime }}$$

We can apply superposition using different combinations, instead of just one source at a time, i.e.:

$$\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}{x}_1\\ x_2\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ x_3\\ x_4\end{array}\right]=\text{etc...}$$

We try not to analyse circuits with dependent sources, but it’s possible. We can still analyse independent sources one-by-one, but we must keep dependent sources unchanged as they are given (i.e., neither removed nor forced on or off).

AC analysis

Unlike our other circuit analysis techniques, superposition works in the phasor domain with some modifications. Take two main cases: one is where we have all inputs with the same frequency $\omega$. In that case, we do essentially the same thing as in DC analysis.

Case two is where the sources don’t all have the same frequency and direct superposition doesn’t apply. We instead find the outputs each at a different frequency (i.e., if some sources were one frequency and some the other, we find the contributions for the given frequency). Then, we convert to the time domain and add up:

$$y(t)=y_1(t)+y_2(t)+\cdots +y_n(t)$$