Superposition Theorem for AC Circuits

The superposition theorem states that the total output of a linear circuit with multiple inputs equals the sum of individual outputs from each input operating independently. Since impedance is frequency dependent, each frequency creates a separate frequency domain circuit. By analyzing each circuit individually using Ohm's law and voltage division, then summing the results, you can determine the total circuit response even when sources operate at different frequencies.

When sinusoidal sources operate at different frequencies, the circuit is not a true AC circuit and cannot be analyzed as a single sinusoid. Instead, superposition breaks the circuit into separate AC circuits, each analyzed at its own frequency. The individual sinusoidal outputs are then summed together, creating a composite waveform that is non-sinusoidal because it combines multiple frequency components.

When applying superposition, all sources except one are set to zero. A zero-volt voltage source becomes a short circuit, while a zero-ampere current source becomes an open circuit. These replacements eliminate inactive sources from the circuit, leaving only a single-frequency AC circuit that can be analyzed using standard phasor and impedance methods.

After deactivating all but one source, the resulting single-frequency circuit qualifies as an AC circuit and is analyzed using phasors and impedances. Phasors represent sinusoidal quantities in the frequency domain, while impedance accounts for frequency-dependent resistance from resistors, inductors, and capacitors. This approach simplifies calculations for each individual source before combining results.

Impedance varies with frequency, meaning each frequency component in a multi-source circuit experiences different impedance values. Superposition exploits this by decomposing the original circuit into separate circuits, each containing a single frequency. This frequency-dependent decomposition allows accurate analysis of complex circuits that would be difficult to solve as a single system.

After analyzing each single-frequency circuit using phasors and impedances, the resulting phasor voltages are converted back to time-domain sinusoidal expressions. These individual time-domain voltages are then summed to obtain the total output voltage of the original multi-source circuit, which may be non-sinusoidal depending on the frequency components involved.

Superposition relies on the principle that circuit outputs scale linearly with inputs. In linear circuits, doubling the input doubles the output, and combining inputs produces outputs that sum predictably. Nonlinear components like diodes or transistors violate this proportionality, making superposition invalid. AC circuits with resistors, inductors, and capacitors remain linear, preserving the validity of superposition analysis.