Superposition & Source Transformations

Superposition is a property of linear equations, not a special electrical effect. If a circuit contains linear elements and sources, any branch voltage or current is a linear combination of the independent source values. This makes source-by-source reasoning exact for voltages and currents.

Turning sources off #

To find the contribution of one independent source, set every other independent source to zero:

Dependent sources remain active because their values are tied to circuit variables. Internal source impedance remains as well; only the independent excitation is zeroed.

After solving each partial circuit, add signed responses:

$$x=\sum_k x_k.$$

Power does not superpose because $p=vi$ is bilinear and $i^2R$ is quadratic. Compute the final voltage and current first, then power.

Source transformations #

A Thevenin source $V_s$ in series with impedance $Z_s$ has the same terminal relation as a Norton source

$$I_s=\frac{V_s}{Z_s}$$

in parallel with $Z_s$. The transformation preserves external terminal voltage and current for every load.

It does not preserve internal branch current, source power, or stress. It is invalid if the impedance is not in series with the voltage source or not in parallel with the current source at the same two terminals.

A useful workflow #

Use source transformations to expose a reducible series/parallel structure; use superposition when individual source influence matters; use direct nodal or mesh analysis when decomposition multiplies the work.

Power does not superpose: the resistor power at the combined solution is not the sum of powers from the two partial solutions because $v^2/R$ is quadratic.