Short Topical Videos
- Horowitz & Hill, The Art of Electronics, 2nd Ed., Ch. 1
The concept of impedance is basically an attempt to extend Ohm’s Law to devices that are not resistors. Ohm’s Law
relates voltage, , and current, , by a resistance, . Impedance generalizes this by considering and as complex waveforms, and using the symbol to denote the (possibly complex, and possibly frequency-dependent) number that relates them. Just as a note, since generally means current in electronics, the imaginary unit is generally denoted as by electrical engineers. We’ll use that notation here as well.
In discussing capacitors, we used capacitance 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 back in, we have an equation that looks very much like Ohms law, but instead of , we have , with , 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.
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).
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