# Impedance

### Reference Material

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

## Impedance

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

$V=IR, \,\!$

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 $\sqrt {-1}$ 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:

$\frac{dV}{dt}=\frac{I}{C}. \,\!$

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

$V=e^{j\omega t}=\cos {\omega t} + j\sin {\omega t}. \,\!$

In this case we get:

\begin{align} j\omega e^{j\omega t} & = \frac{I}{C}\\ j\omega V & = \frac{I}{C}\\ V & = I \frac1{j\omega C} \end{align}\,\!

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 = IZ, with Z, for capacitors, given by:

$Z_{c} = \frac1{j\omega C} \,\!$

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 $90^\circ$, 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:

$V=L\frac{dI}{dt}, \,\!$

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

$I=e^{j\omega t}=\cos {\omega t} + j\sin {\omega t}. \,\!$

In this case we get:

\begin{align} V & = Lj\omega e^{j\omega t}\\ V & = Lj\omega I \end{align}\,\!

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

$Z_ L=j\omega L \,\!$

So an inductor is like a resistor that changes the phase of the incoming wave by $90^\circ$, 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