Again, a pickup is only an AC generator. The model is the generator (voltage), and the internal impedance which is the series resistance of the windings and the inductance of the windings as a coil inductor, in parallel'ish with the effective inter-winding capacitance.

You can, of course use a current model. If you recall the Thevenin and Norton conversion mechanisms (if you don't, first call them), a voltage source in series with an impedance is equivalent to a current source in parallel with the transformed impedance.

From a First Principles perspective, in electronics, current is the principle from which the others are derived. The basic unit is the electron and how many are passing thru in a second.

The DC resistance of a pickup coil, while a somewhat accurate indicator of the number of turns of wire in a pickup, and the relative output of a pickup, for

__exactly the same types of pickups only__, is

__only__ accurately meaningful as the measure of resistance of the wire. Since a few different wire gauges are used in pickup coils, with the somewhat random actual resistance per unit measure of wire for any given gauge (+/-20% or more - it's awful dang thin guys), there is significant variation of little meaning in a resistance measurement, compared to the variation of significant meaning as it relates to gauge and the actual number of turns (since the inductance of a coil is proportional to the square of the change in the turns ratio.....).

This, coupled with the frequency response variations of the different magnet formulations and the varying production strength of the actual magnetization of each magnet, the DC resistance of the coil is

__only__ the

__accurate indication__ of the

__DC resistance of the coil__. Based solely on a resistance reading, the DiMarzio HS series with resistance readings around 20+K should be scorchers but they are not. The Duncan Designed Scorcher, with a resistance reading around 20+K is.

The generator and its internal impedance:

A generator can develop its maximum output potential only when there is no load. When there is a load, a voltage divider is formed between the internal impedance of the generator and the external impedance of the load.

When the output of the generator is shorted, a maximum short circuit current flows that is a function of this internal load. In a sense of speaking (and a working model thereof) a DC battery (a collection of individual cells arraigned not unlike a battery of artillery - attributed to Benjamin Franklin) is really a device that generates a maximum potential (voltage) at all times, and only its internal resistance increases as a function of its declining charge. It appears to have no output voltage only since its internal impedance becomes infinite. A battery tester in effect creates said voltage divider and measures the voltage developed across only the external part thereof.

*definition*I use the term

__impedance__ in its full meaning since resistance is only one component (and only real) part thereof. Real only means that its effect,

__by itself__, can be represented in the real plane. For a full, accurate, and honest representation (which IS REQUIRED for any understanding approaching any degree of truth, accuracy, or meaningful insight), of impedance, the complex plane and/or phasor representation must be used.

Any attempt at understanding the actual effects in play within an AC circuit without a functional understanding of AC circuit theory will be no more accurate than describing the taste of various flavors of ice cream without actually opening the containers and tasting them.

While it takes an understanding of AC circuit analysis for all to fully "see" what is happening within a passive electric guitar, all having said understanding would make posting about it moot.

*/definition**definition*Unit

The identity unit for any given object type. In mathematics this is usually one (or 1). For instance, the hypotenuse of a unit square (one with each side of length 1) is the square root of 2. This is a clue, what is the length of said model hypotenuse of a unit tesseract, a four dimensional "qube"?

For a pickup model, in its dimensionless form, it has an output of 1, and an impedance of 1. I didn't say "one what", they're just all "1".

*/definition*For a DC-like analysis (I didn't say a DC only) one unit pickup coil is usually driving an impedance of 250K on both the volume and tone controls (we'll pretend that this is at a frequency where the impedance is the same as 250K) as well as the input to an amp. While we'd all like to believe that the input impedance of an amp is 1 Meg Ohms, it usually is not. A good value to use is 500K to make the math’s easy.

Our single coil pickup, in generally reality (not relativity), has an internal resistance around 5 to 8K Ohms. We'll pick 5K263.

So, the effective external load is 250K||250K||500K or 100K. This is being driven by a generator with 5 K of internal output impedance. The external voltage developed is 100K/(100K + 5K263 K) or 0.95 of the internal voltage developed (1).

Now, if we have two of these unit pickups in parallel, each developing exactly the same signal and voltage, we effectively are driving the same load with a generator having an internal impedance of 2K632 Ohms. This develops an output voltage of 100K/(100K + 2K263 K) or 0.974 of the internal voltage developed (1).

This isn't much higher, but the harmonic content from two pickups spaced apart is richer than from one alone, and most effective pickup amp and cable loads are really a good bit lower in impedance. The human ear hears frequency variation much better than amplitude variation. (This makes the whole tremo'Leo thing even more interesting since an effective variation of amplitude of 6% (one half-step in pitch) would have never been noticed. "Yo Sparky, what's that there shiny lever on yer geetar do? Oh, not much...")

The increase in the "width" of the sound is what we notice since, not only is the harmonic content fuller, the effective impedance of our combined "meta-pickup" is halved, with a resulting increase in the center of the response bandwidth of 1/2 octave.

The effective instantaneous output from two pickups in parallel is the average of their instantaneous output levels.

Having two pickup coils in series causes their outputs to add. It also causes the effective internal impedance to double. This is why 500K controls are better for such structures; the same current loading is effected. The added inductance results in the center of the response bandwidth being 1/2 octave lower, and, since most non-welding humbucking pickups are comprised of individual single coils that are generally "weak sisters" of their single coil counterparts, they need all the gentle loading that we can give them.

And again, this was all just DC-like cheating, things are really more complex..........

Ratios:

While the pickup spacing on a Strat is pleasing to the human eye, there is little other basis to said spacing. One might argue that the harmonic content along the string, solely as a function of sensing along a string for any given same plucked note, is based on the ratio of the distance from the bridge saddle. A coil placed nearer to the saddle than a traditional single coil, is likely between 1/2 and 2/3 closer. One would argue that the harmonic content is then 2:1 or 3:2 higher. The closer to the saddle, the higher the harmonic content, the lower the signal.

Piezo saddles sound the way that they do because they are sensing the strings at the, er, saddle. However, since they are sensing

*pressure variations* and not

*string excursion*, their output is higher than zero (but their internal impedance is most high).

This is why I preach selecting the bridge coils in a humbucker as a matter of the tone desired, and selecting the corresponding hum canceling coil in the neck humbucker only then as a matter of convenience. Of course, this is on a unit basis, slug coils sound differently than screw coils, even in a neck pickup (yada, yada, yada).

So, what you are doing has always made perfect sense since it's all just

__physics__ anyway.Good luck and happy routing. Remember, it's harder to put the wood back in......................