What's going on with Series electronics is pretty unusual - there have only been a handful of other guitars that use this sort of noise reduction system (Fender Strat Elite and the original Paul Reed Smith Bass both from the 80s as far as I know).
A magnetic pickup is composed of a coil of wire sitting in a magnetic field. The pickup is positioned so the strings are in the magnetic field as well. When a string containing iron moves it disturbs the magnetic field of the pickup, which induces an electric signal in the coil. That tiny signal is the sound of your string vibrating which you can amplify.
The coil not only responds to magnetic disturbances from the strings, but will pick up electromagnetic interference (EMI) as well. This is where hum and buzz come from.
Single coil pickups are very sensitive to EMI. This led to the development of many kinds of dual-coil pickups to try to fix the problem. A clever guy figured out that with two matched pickups, you can get rid of the hum by connecting the second pickup backwards. The EMI created the same buzz in each pickup, but when you connect them out of phase, the buzz is cancelled.
That's great, but the sound from the strings is also out of phase, which means very little bass and a tinny sound. But an even more clever guy noticed that if you flipped the magnets in the pickup over AND wired the pickup backwards, then you could cancel the EM noise, but have the signal from the motion of the strings still in phase. Now that's really good! This is how a Jazz Bass works or the in-between sounds on a Strat are quiet. A Gibson-style humbucker puts two of these coils right next to each other for the same effect.
But things are not perfect. The reversed polarity /reverse wind coil combo kills noise, but since each of the coils hears the string in a slightly different position, there's some cancellation of sound as well. This cancellation is why humbuckers don't sound as bright as single coil pickups.
Well, people are pretty darn clever, and somebody else figured out that if you wired two coils out of phase and didn't put any magnet in one of the coils, it wouldn't hear the string vibration but would still get rid of the noise - a quiet single coil with traditional single coil sound. This is what a Series bass humcancelling dummy pickup is - it's the coil with no magnet and is performing the noise reduction for both the pickups.
Theoretically, are almost perfect now, but one problem you now have is that you have to wind a coil that's precisely matched to the working pickup, but it can't pick up the instrument's sound! This is expensive in the case of the Series bass because there are active electronics. So you're seeing a great example of cost is no object perfectionism in the Series basses, which have an extra route for a dedicated dummy coil which even includes it's own dedicated preamp - what other bass has an active circuit which is there only to remove noise! In the case of a Series bass, the two pickups and canceller are carefully wound as a matched set to maximize noise reduction, and the electronics are tweaked to minimize noise without impacting sound (this is the low-noise mod that Ron does personally on the Series basses).
As anybody with a Strat or Tele will tell you, EM noise is often directional - you turn your body and the amount of buzz will change. So, you really want the humcanceller to be in the same orientation as the pickups, hence the Series humcanceller position between the pickups. But because the canceller doesn't need to be near the strings, you can put it else where in the body and it should work. The old PRS basses had three regular pickups plus a dummy coil that was exposed on the back of the bass; the Fender Elite had a fourth coil under the pickguard.
The Series design is non-compromised and expensive. One other extremely clever variation is an old one - Leo Fender's split coil Precision pickup from the late 50's. He was really clever on this on - there are two pickup sections that each sense two strings. They are matched and have reverse polarity and winding relative to each other. So, the coil under the E and A strings does double duty - it hears those two strings, but it also acts as a hum-canceller for the D and G strings - it can't hear the string motion, but does cancel the noise! Very, very clever! This kind of design works pretty well on a bass, but having a split coil pickup on guitar doesn't quite sound the same (some of Leo's G&L guitars had this kind of pickup).
David Fung