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An in-depth analysis of reactance and impedance in ac circuits. It covers the concepts of reactance for capacitors and inductors, the general solution for complex circuits, and the calculation of phase shifts. The document also discusses the power in reactive circuits and the use of complex quantities.
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Before we lo ok at some more examples using our technique of complex imp edance, let's lo ok at some related general concepts.
3.5.1 Reactance
First, just a rede ntion of what we already have learned. The term reactance is often used in place of imp edance for capacitors and inductors. Reviewing our de nitions of imp edances from Section 3.2 we de ne the reactance of a capacitor X~C to just b e equal to its imp edance:
However, an alternative but common useage is to de ne the reactances as real quantities. This is done simply by dropping the { from the de nitions ab ove. The various reactances present in a circuit can by combined to form a single quantity X , which is then equal to the imaginary part of the imp edance. So, for example a circuit with R, L, and C in series would have total imp edance
A circuit which is \reactive" is one for which X is non-negligible compared with R.
3.5.2 General Solution
As stated b efore, our technique involves solving for a single Fourier frequency comp onent such as V~ = V e{(!^ t+)^. You may wonder how our results generalize to other frequencies and to input waveforms other than pure sine waves. The answer in words is that we Fourier decomp ose the input and then use these decomp osition amplitudes to weight the output we found for a single frequency, Vout. We can formalize this within the context of the Fourier transform, whch will also allow us to see how our time-domain di erential equation b ecame transformed to an algebraic equation in frequency domain. Consider the example of the RC low-pass lter, or integrator, circuit of Fig. 7. We obtained the di erential equation given by Eq. 2. We wish to take the Fourier transform of this equation. De ne the Fourier transform of V (t) as