6.14:
Superposition Theorem for AC Circuits
The superposition principle finds application in the analysis of AC linear circuits with multiple independent sources.
Consider a circuit consisting of two resistors, an inductor, and two independent sinusoidal sources with different frequencies. The superposition theorem can be used to determine the voltage drop across the resistor.
According to the superposition theorem, the total output of a linear circuit with multiple inputs is equal to the sum of individual outputs from each input operating independently.
So, the circuit can be divided into multiple circuits, each with a single sinusoidal input.
Since impedance is frequency dependent, different frequency domain circuits exist for each frequency.
The voltage division rule and Ohm's law are applied to determine the output voltage of each circuit.
The obtained voltages are then expressed in the time domain.
The total output voltage is the sum of the voltages produced by each source individually.
When this voltage drop is plotted graphically, it results in a non-sinusoidal output, even though all the inputs are sinusoidal.
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.