6.14 : Superposition Theorem for AC Circuits

Consider encountering a circuit in a steady state where all its inputs are sinusoidal, yet they do not all possess the same frequency. Such a circuit is not classified as an alternating current (AC) circuit, and consequently, its currents and voltages will not exhibit sinusoidal behavior. However, this circuit can be analyzed using the principle of superposition.

The principle of superposition stipulates that the output of a linear circuit with several concurrent inputs is equivalent to the cumulative outputs when each input operates independently. The inputs to the circuit are the voltages from the independent voltage sources and the currents from the independent current sources.

When all inputs except one are set to zero, the remaining inputs become 0-V voltage sources and 0-A current sources. Given that 0-V voltage sources equate to short circuits and 0-A current sources correspond to open circuits, the sources linked to the other inputs are replaced by open or short circuits. What remains is a steady-state circuit with a single sinusoidal input, which qualifies as an AC circuit and is analyzed using phasors and impedances.

Hence, the principle of superposition is employed to transform a circuit with multiple sinusoidal inputs at varying frequencies into several separate circuits, each with a singular sinusoidal input. Each of these AC circuits is then analyzed using phasors and impedances to determine its sinusoidal output. The aggregate of these sinusoidal outputs will coincide with the output of the initial circuit.