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Superposition Theorem for AC Circuit

Since ac circuits are linear, the superposition theorem applies to ac circuits the same way it applies to superposition theorem dc circuits.

Superposition Theorem AC Circuit

The theorem becomes important if the circuit has sources operating at different frequencies. In this case, since the impedances depend on frequency, we must have a different frequency domain circuit for each frequency.

The total response must be obtained by adding the individual responses in the time domain. It is incorrect to try to add the responses in the phasor or frequency domain. It is because the exponential factor ejωt is implicit in sinusoidal analysis, and that factor would change for every angular frequency ω. It would therefore not make sense to add responses at different frequencies in the phasor domain. Hence, when a circuit has sources operating at different frequencies, one must add the responses due to the individual frequencies in the time domain.

Superposition Theorem AC Circuit Examples

For better understanding, let us review the examples below :
1. Use the superposition theorem to find Io in the circuit in Figure.(1).
superposition theorem ac circuit
Figure 1
Solution :
Let
superposition theorem ac circuit
(1.1)
where I'o and I"o are due  to the voltage and current sources, respectively. To find I'o , consider the circuit in Figure.(2a).
superposition theorem ac circuit
Figure 2
If we let Z be the parallel combination of -j2 and 8 + j10, then
superposition theorem ac circuit
and current I'o is
superposition theorem ac circuit
or
superposition theorem ac circuit
(1.2)
To get I"o, consider the circuit in Figure.(2b). For mesh 1,
superposition theorem ac circuit
(1.3)
For mesh 2,
superposition theorem ac circuit
(1.4)
For mesh 3,
superposition theorem ac circuit
(1.5)
From Equations.(1.4) and (1.5),
superposition theorem ac circuit
Expressing I1 in terms of I2 gives
superposition theorem ac circuit
(1.6)
Substituting Equations.(1.5) and (1.6) into Equation.(1.3), we get
superposition theorem ac circuit
or
superposition theorem ac circuit
Current I"o is obtained as
superposition theorem ac circuit
(1.7)
From Equations.(1.2) and (1.7), we write
superposition theorem ac circuit

2. Find vo in the circuit of Figure.(3) using the superposition theorem.
superposition theorem ac circuit
Figure 3
Solution :
Since the circuit operates at three different frequencies (ω = 0 for the dc voltage source), one way to obtain a solution is to use superposition, which breaks the problem into single-frequency problems. So we let
superposition theorem ac circuit
(2.1)
where v1 is due to the 5 V dc voltage source, v2 is due to the 10 cos 2t V voltage source, and v3 is due to the 2 sin 5t A current source.

To find v1, we set to zero all sources except the 5 V dc source. We recall that at steady state, a capacitor is an open circuit to dc while an inductor is short circuit to dc. There is an alternative way of looking at this. Since ω = 0, jωL = 0, 1/jωC = ∞. Either way, the equivalent circuit is as shown in Figure.(4a). By voltage division,
superposition theorem ac circuit
(2.2)
To find v2, we set to zero both the 5 V source and the 2 sin 5t current source and transform the circuit to the frequency domain.
superposition theorem ac circuit
The equivalent circuit is now as shown in Figure.(4b).Let
superposition theorem ac circuit
superposition theorem ac circuit
Figure 4. Solution of Figure.(3) : (a) setting all sources to zero except the 5 V dc source, (b) setting all sources to zero except the ac voltage source, (c) setting all sources to zero except the ac current source.
By voltage division,
superposition theorem ac circuit
In the time domain,
superposition theorem ac circuit
(2.3)
To obtain v3, we set the voltage sources to zero and transform what is left to the frequency domain.
superposition theorem ac circuit
The equivalent circuit is in Figure.(4c). Let
superposition theorem ac circuit
By current division,
superposition theorem ac circuit
In the time domain,
superposition theorem ac circuit
(2.4)
Substituting Equations.(2.2) to (2.4) into Equation.(2.1), we have
superposition theorem ac circuit

Untuk Bahasa Indonesia baca Teorema Superposisi Rangkaian AC
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