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:

${\begin{aligned}j\omega e^{j\omega t}&={\frac {I}{C}}\\j\omega V&={\frac {I}{C}}\\V&=I{\frac {1}{j\omega C}}\end{aligned}}\,\!$

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:

$Z_{c}={\frac {1}{j\omega C}}\,\!$

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:

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

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