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Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

  • 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 for AC Circuits Examples

For better understanding, let us review the examples below:
1. Use the superposition theorem to find Io in the circuit in Figure.
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Let
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) ..... (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 for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

If we let Z be the parallel combination of –j2 and 8 + j10, then
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

and current Io is
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

or
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)  ........ (1.2) 
To get I“o, consider the circuit in Figure.(2b). For mesh 1,
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) .......  (1.3) 
For mesh 2,
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE).......(1.4) 
For mesh 3,
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE).......(1.5) 
From Equations.(1.4) and (1.5),
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Expressing I1 in terms of I2 gives
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)..... (1.6) 
Substituting Equations.(1.5) and (1.6) into Equation.(1.3), we get
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
or 
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Current I“is obtained as
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)..........(1.7) 
From Equations.(1.2) and (1.7), we write
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

2. Find vo in the circuit of Figure. using the superposition theorem.

Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

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 for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)....... (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 a 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. By voltage division,
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) ..... (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 for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

The equivalent circuit is now as shown in Figure. Let
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Figure. The solution of Figure.: (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 for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
In the time domain,
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) ..... (2.3) 
To obtain v3, we set the voltage sources to zero and transform what is left to the frequency domain.
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
The equivalent circuit is in Figure. Let 
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
By current division,
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
In the time domain,
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)  .... (2.4)
Substituting Equations.(2.2) to (2.4) into Equation.(2.1), we have 
Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

The document Superposition Theorem for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Network Theory (Electric Circuits).
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FAQs on Superposition Theorem for AC Circuits - Network Theory (Electric Circuits) - Electrical Engineering (EE)

1. What is the Superposition Theorem for AC circuits?
Ans. The Superposition Theorem is a powerful technique used in AC circuit analysis. It states that in a linear circuit with multiple voltage or current sources, the total voltage or current across any component is the sum of the individual voltages or currents caused by each source acting independently.
2. How does the Superposition Theorem work in AC circuits?
Ans. To apply the Superposition Theorem in AC circuits, we analyze the circuit separately for each source. For each analysis, we consider one source at a time and replace all other sources with their internal impedances. Then, we sum up the voltages or currents obtained from each analysis to find the total voltage or current in the circuit.
3. Can the Superposition Theorem be used for non-linear AC circuits?
Ans. No, the Superposition Theorem is only applicable to linear circuits. Non-linear components, such as diodes or transistors, do not satisfy the linearity requirement and cannot be analyzed using the Superposition Theorem.
4. What are the limitations of the Superposition Theorem in AC circuits?
Ans. The Superposition Theorem has certain limitations. It assumes that the circuit is linear, time-invariant, and has only independent sources. It cannot be used for circuits with dependent sources, sources that change with time, or circuits with non-linear components. Additionally, practical limitations such as component tolerances and manufacturing variations may affect the accuracy of the analysis.
5. Are there any practical applications of the Superposition Theorem in AC circuits?
Ans. Yes, the Superposition Theorem finds practical applications in AC circuit analysis. It allows us to simplify complex circuits by analyzing them in parts and then combining the results. This technique is particularly useful when dealing with circuits containing multiple sources, such as power distribution networks, where the individual contributions of each source need to be determined for accurate analysis and design.
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