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# Solving Circuits with Impedances

Module by: Don Johnson. E-mail the author

Summary: Introducing the advantages of using impedance and how to use it.

As we unfold the impedance story, we'll see that the powerful use of impedances suggested by Steinmetz greatly simplifies solving circuits, alleviates us from solving differential equations, and suggests a general way of thinking about circuits. Because of the importance of this approach, let's go over how it works.

1. Even though it's not, pretend the source is a complex exponential. We do this because the impedance approach simplifies finding how input and output are related. If it were a voltage source having voltage v in =pt v in p t (a pulse), still let v in = V in ei2πft v in V in 2 f t . We'll learn how to "get the pulse back" later.
2. With a source equaling a complex exponential, all variables in a linear circuit will also be complex exponentials having the same frequency. The circuit's only remaining "mystery" is what each variable's complex amplitude might be. To find these, we consider the source to be a complex number ( V in V in here) and the elements to be impedances.
3. We can now solve using series and parallel combination rules how the complex amplitude of any variable relates to the sources complex amplitude.

A common error in using impedances is to keep the time-dependent part, the complex exponential, in the fray. The entire point of using impedances is to get rid of them in writing circuit equations and in the subsequent algebra. The complex exponentials are there implicitly (they're behind the scenes). Only after we find the result do we raise the curtain and put things back to together again. In short, when solving circuits using impedances, t t should not appear except for the beginning and end.

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