# Superposition Theorem (ac)

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You will recall from Chapter 8 that the superposition theorem eliminated the need for solving simultaneous linear equations by considering the effects of each source independently. To consider the effects of each source, we had to remove the remaining sources. This was accomplished by setting voltage sources to zero (short-circuit representation) and current sources to zero (open-circuit representation). The current through, or voltage across, a portion of the network produced by each source was then added algebraically to find the total solution for the current or voltage.
The only variation in applying this method to ac networks with independent sources is that we will now be working with impedances and phasors instead of just resistors and real numbers.
The superposition theorem is not applicable to power effects in ac networks since we are still dealing with a nonlinear relationship. It can be applied to networks with sources of different frequencies only if the total response for each frequency is found independently and the results are expanded in a non-sinusoidal expression, as appearing in next chapters.
One of the most frequent applications of the superposition theorem is to electronic systems in which the dc and ac analyses are treated separately and the total solution is the sum of the two. It is an important application of the theorem because the impact of the reactive elements changes dramatically in response to the two types of independent sources. In addition, the dc analysis of an electronic system can often define important parameters for the ac analysis.
We will first consider networks with only independent sources to provide a close association with the analysis of Chapter 8.
Example 1: Using the superposition theorem, find the current I through the 4-Ω reactance ($X_{L_{2}}$ ) of [Fig .1].
Fig. 1: Example 1.
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For dependent sources in which the controlling variable is not determined by the network to which the superposition theorem is to be applied, the application of the theorem is basically the same as for independent sources. The solution obtained will simply be in terms of the controlling variables.
Example 2: Using the superposition theorem, determine the current $I_2$ for the network of [Fig. 5]. The quantities $\mu$ and h are constants.
Fig. 5: Example 2.
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