Inductance (and why it matters)
As you probably know, a guitar pickup is a coil of wire whose magnetic flux changes with the movement of the nearby guitar strings. A voltage is created across the coil that's proportional to the flux change, and that voltage is what we jack into our amplifier. The same phenomenon that makes this work is also responsible for the workings of transformers and inductors.
Not familiar with inductors? Along with resistors and capacitors, inductors are one of the three basic types of passive components. You don't see them very often in modern electronics, because just about everything they do can be done cheaper, smaller and quieter with active circuits. But once upon a time, they were essential to most EQ and filter circuits. Physically, an inductor is made from a coil of wire. Its effectiveness can be increased by putting a magnet in the middle. So not only is a guitar pickup an inductor, it has much greater inductance than any component you'll find in a parts catalog these days.
An inductor can be thought of as the opposite of a capacitor – it resists a current that is changing, while a capacitor resists a current that is constant. So when conducting an audio signal, an inductor's impedance is greater at higher frequencies, and a capacitor's impedance is less at higher frequencies. (A resistor's is the same regardless of frequency.)
Knowing that, let's look at the circuit in a guitar. It's nearly identical to one of the most common passive filter circuits, the LC resonant low-pass filter.
The capacitance in the guitar circuit comes mostly from the guitar cable. A little of it is provided by the capacitance in the pickup windings. We're going to figure the cable connects to some high-impedance input (your amp, or a pedal), so the only thing loading our circuit (the "R" in the diagram) is the resistance to ground through the volume and "tone" pot(s), which is also quite high.
A resonant low-pass filter has a flat response at lower frequencies, cuts highs sharply above a certain frequency, and has a peak centered around that frequency. That peak at the upper end of the guitar's spectrum is what gives pickups their "bite". The frequency and height of the peak can be predicted by the electrical parameters of the pickup coil, the cable and the guitar's volume and tone pots.
This is born out by our experience with common guitar pickups. A typical vintage Strat pickup has an inductance of around 2 henries and a winding capacitance in the neighborhood of 100pF. Hooked up with a cable that has a capacitance to ground of 500pF, we get a peak at 4594 Hz. A P‑90 is apt to have an inductance of about 6 henries, yielding a peak at 2653 Hz (with that same cable). If you've spent any time with these pickups, and have mucked around with EQ frequencies much, I think you'll agree that sounds about right.
This points out the significance of the cable you use to connect your guitar to the rest of your rig. Some players shell out for fancy cables that have a low capacitance, thinking that will transfer the signal more faithfully. They're missing the point. The cable tunes the pickup's resonance, and a better strategy would be to find one that sounds good to you with your guitar, measure its capacitance, and always shop for quality cables that match it.
Users of wireless systems that perceive a weakening of their tone are likely to blame the transmitter, when it may just be the elimination of the cable that's at fault.
This analysis also makes you want to give some thought to whether having "true bypass" in all your pedals is really such a good idea. If you connect your guitar to some "true bypass" pedals at the front of the stage, and then run a 20ft cable from there to your amp, you're showing the guitar pickups a drastically different capacitance when all the pedals are bypassed than when one of them is on. And chances are, it's the opposite of what you'd want – you're lowering the resonant frequency when you go for a pure, clean sound, and raising it when you go for some grit or processing. You might solve this by using some sort of buffer unit as the last effect on your pedal board. But if there's an electronic-switching pedal on there somewhere there's no need. That non-true-bypass pedal is buffering the signal whether it's in use or not.
The low-pass filter model is key to understanding what's really going on when you turn down the guitar's volume pot and hear the sound get duller. The volume pot sits in between the pickup and the cable, so when you turn it down you introduce series resistance and spoil the resonance. That's what kills the "sparkle". A lot of people think it's because the resistance of the turned-down pot and the cable capacitance form a first-order low-pass filter. They do, but most of what you're hearing is the resonance being killed by the separation of the cable capacitance from the pickup. The popular practice of adding a cap from the high terminal to the wiper can sort of compensate for the loss of highs, but it really doesn't preserve the peak. (I don't like complicated solutions any more than the next guy, but if you really wanted to solve this, you could probably design a pot that would add capacitance between the pickup and ground as the pot was turned down.)
Here's one for pedal steelers to ponder: I've wondered if the wonderful "oowww" some players get on held notes might be due to the use of a passive volume pedal. If you connect the steel to the pedal with a short cord, and run a long cable from there to the amp, the capacitance the pickup is seeing will increase as the pedal is floored, creating sort of a reverse wah. The inductance of a PSG pickup is massive (20 henries!), so this could be quite noticable. I'd be curious what steelers who've used both passive and active volume pedals think about this theory.
Ever wonder why using your bridge pickup and neck pickup together gives you more "shimmer" and less "honk" than the bridge pickup alone? It's because having the two pickups in parallel cuts the inductance in half. That makes the resonant frequency higher than what it is for a single pickup.
How much higher? The calculation we made of the Strat and P‑90 resonant frequencies was done with the formula for the frequency of a LC network:
Wait. What... math? It's OK. There are tools to save slackers like us from having to remember how to do algebra. If you do want to play along at home, a calculator with a square root key and a 1/X key will help a lot. And you need to know that C is given in farads. 500pF is .000,000,000,5 farads. So having both pickups on increases the frequency by a factor of the square root of 2 (1.4), right? No. You forgot that when they're both on, the pickups (and their winding capacitances) are loading each other. Please stop. This is making my head hurt.
What's way more entertaining anyway is using this knowledge to simulate pickups and guitar mods in a SPICE-modelling program. That's software that let's you diagram a circuit and test it, seeing gain, frequency response, distortion... right on the screen. Pretty cool. I use Electronics Workbench, Tina TI, and LT Spice. The later two are free. (Thank you, chip makers!) Here's how I model a 2-pickup Gibson in Tina:
All of this might leave you wondering why nobody told you about pickup inductance before. Wouldn't it be nice if manufacturers published the inductance of pickups? Or offered cables in various capacitances, just as strings are offered in different gauges? Maybe one of these days they will.
But if we keep obsessing about stuff that doesn't matter, like paper-in-oil capacitors and "50s style" wiring mods, they'll probably keep selling us snake oil. ∎