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…
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.
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Q1: How does the superposition theorem apply to AC circuits with multiple frequency sources?
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.
Q2: Why does a circuit with multiple sinusoidal sources at different frequencies produce non-sinusoidal output?
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.
Q3: What happens to inactive sources when applying superposition to multi-source circuits?
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.
Q4: How are phasors and impedances used in superposition analysis?
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.
Q5: What is the relationship between frequency dependence and circuit decomposition in superposition?
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.
Q6: How do you convert individual frequency domain results back to the time domain?
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.
Q7: Why is superposition limited to linear circuits in AC analysis?
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.