# Impedance

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- Horowitz & Hill,
*The Art of Electronics, 2nd Ed.*, Ch. 1

## Contents |

## Impedance

The concept of impedance is basically an attempt to extend Ohm’s Law to devices that are not resistors. Ohm’s Law

relates voltage, *V*, and current, *I*, by a resistance, *R*. Impedance generalizes this by considering *V* and *I* as complex waveforms, and using the symbol *Z* to denote the (possibly complex, and possibly frequency-dependent) number that relates them. Just as a note, since *I* generally means current in electronics, the imaginary unit is generally denoted as *j* by electrical engineers. We’ll use that notation here as well.

### Capacitive Impedance

In discussing capacitors, we used capacitance *C* to relate current to the time-derivative of voltage:

Knowing from Fourier analysis that we can decompose any waveform into a sum of sin/cos functions, it makes sense to examine how this equation works for a sinusoid waveform like

In this case we get:

In that last line, where we’ve substituted *V* back in, we have an equation that looks very much like Ohms law, but instead of *R*, we have *V* = *I**Z*, with *Z*, for capacitors, given by:

Hence, Ohm’s law can be extended to capacitors if we generalize resistance to a complex impedance, and take the imaginary part of that impedance to be a frequency-dependent quantity. Apart from changing the phase of an incoming wave by , a capacitor can pass a high-frequency wave with very little attenuation, but can completely block a DC voltage with an infinite impedance.

### Inductive Impedance

Inductors, on the other hand, have a voltage that is dependent on the time derivative of current:

This time, let’s take the current to be a sinusoid given by:

In this case we get:

Again, the last line looks like Ohm’s Law, if we take the impedance of an inductor to be:

So an inductor is like a resistor that changes the phase of the incoming wave by , and resists higher frequencies more strongly than lower frequencies (and behaves just as a wire to a DC voltage).

## Mixed Circuits

There’s actually not a lot to say here. You can pretend that impedances are just complex resistors. They add when wired in series; they add reciprocally when wired in parallel. You’ll see that the rules for adding capacitors in parallel falls out naturally using the expressing for the impedance of a capacitor. Oh, and as we’ll see when discussing RC and LC filters, you can use the frequency-dependence of the impedances of capacitors and inductors to construct filters that purposely attenuate waveforms based on their constituent frequencies.

Finally, the idea of the Thévenin equivalent circuit (involving a voltage source and an equivalent series resistor) generalizes naturally to impedances. Now instead of an equivalent series resistor, we instead have an equivalent (and possibly frequency-dependent) series impedance:

*The Thévenin equivalent circuit, using impedances*