Targeting inductance, wind count formula for bobbins?

[nuke]

TL;DR#

Is there a formula for wind count to hit a certain inductance in a humbucker coil set? 


In reference to my opening thread here about the MIJ 1984 Fender Esprit pickups (aka, "those weird Schaller")  I've pulled them from the guitar and measured and plotted them more carefully.  The neck is wound to 5.84H, the bridge to 9.1H, a lot of inductance. I don't know the reasons, perhaps Fender wanted a strong single-coil sound in coil cut mode, or since D'Aquisto was the product manager for the Fender Master series, which included the D'Aquisto jazz-boxes with the same pickups, and he wanted something mellow and jazzy. 

The Bode plot of the bridge shows a 0.8dB peak @ 1.55khz, basically flat as a ruler.

For my Esprit, they're pretty dark and I'm contemplating re-winding them. The pickups are mechanically unique, to say the least. Even the bobbins, fillister screws and slugs are not the same as other common Schaller humbuckers. The top of the plastic pickup cover is radiused with about a 25" arc. The Slugs are in 3 different lengths to match the curve of the top so they sit flush, and the fillister-head screws have a uniquely tall head, and the ridges molded in the holes of the screw bobbin are in graduated heights, again, to make the screw heads flush with the covers. 

I'm going to try a magnet swap, if I can get some machined down to metric 59mm x 12.03mm x 2.97mm  (yeah, I measured a bunch of Schaller magnets). I might try various A5 and A8 to see how they do with the existing coils. 

I think I'm going to wind up rewinding these bobbin sets to hit close to what I'm looking for. 

My inclination is to wind them down to about the same as the Duncan SH-2 Jazz set, or a SH-2/SH-4 Jazz/HB combo. 

Fender eventually went to SD 59 neck and JB bridge in the Robben Ford model (re-launch of the Esprit), also a classic combo. 

[stratotarts]

There could be such an engineering formula, but it would have to account for so many factors, as to require a lot of specific data, obtained by standardized methods, also the complete disclosure of manufacturing data such as turns count which is generally held proprietary if it is known at all.

What is known, is that there is a very tight relationship between turns count and inductance, all other things being equal. Your problem is, you don't know what that turns count is. If you make a guess and wind it yourself, you will at least know that, and then you can rewind it based on that pretty reliable relationship. Unless, you can somehow count turns when you take the wire off.

Or, you might get lucky the first time.

[antigua]

Based on my observation of factory made pickups versus individually made pickups, I think you could only map out turns to inductance for a given form factor and metal composition, with a given winding machine, like the Tanac AX3 or something that makes perfect coils each time. For hand guided coils, the end results are always all over the place.

I think you might be discovering a new problem that hasn't been much encountered with combining pickup making with pickup analysis, and that's difficulty with reproducibility. You can analyze what you made and see what you ended up with, but it's still difficult to make something to a target spec. 

Could you possible wind the coil a ways, then stop, sand away some insulation from the wire and measure the inductance of the coil with an LCR meter while it's still on the mandrel? With a Fender single coil I think you'd get a good reading, but with PAF bobbins, the inductance will increase by a good amount once they screws or slugs are introduced to the bobbin core. 

[JohnH]

There are formulae and online calculators for coils of various geometry and cores, but they are for regular inductors. It would take a lot of study and testing to adapt them reliably to apply them to elongated bobbins with particular core elements.

[nuke]

Yeah, I'm sure they've have their own internal data, which I don't have of course. 

In looking around, the winding formula is relatively simple for solenoid coils, 

Inductance = (turns^2 * area * mu) / length

I presume that "area" is also true of the elongated circle form of a pickup bobbin. 

The particular bobbins I'm looking at are 68mm long, by 17mm wide, area of (51 * 17) + (Pi 8.5^2) = 1094mm^2 I'd estimate there's about 5mm between the rim of the bobbin and the passages for screws/slugs, which gives a center core 58mm long by 7mm for 395mm^2, or call it 400mm^2. The actual area of the core will vary with the number of layers of wire wrapped around and the actual size of the center of the bobbin, that I don't know without cutting the wire out.  So just wild-arsed guess on my part.

The space is about 6.2mm tall between the inside of the rims.  

With no screws/slugs, air/plastic relative permeability approx = 1. 

turns^2 = (Inductance * length) / (4Pi * 10 E-07 * area )

Let's say, we want 1.1H on a single bobbin with no slugs/screws:

turns = sqrt(  (1.1H * 6.2 E-03 ) / ( 4Pi * 10 E-07 * 400 E-06 ) )

=3623 turns

I happen to have a similar disassembled humbucker, the screw side bobbin without screws, measures 1.095H and 4000 ohms DCR. 

If they used #44 AWG, yes, that fits turns for a tight machine wind and resistance checks out according to some pickup winding estimators. So, kinda in the ballpark, for guessing anyway.  

The one with slugs still in it from the same pickup, indicates 1.9H. So the permeability is some kind of conglomerate of air and iron slugs.  with an estimated relative permeability of 1.73. 

Back plugging that in, if the desired per coil inductance is 2.5H and the assumed relative mu is 1.73:

turns = sqrt( (2.5H * 6.2 E-03) / (1.75* 4Pi *10 E-07 * 400 E-06)) 

= 4197 turns per bobbin, or about 500 more turns per bobbin. 

Before I took it apart, the fully assembled pickup measured 4.45H and 8.1k DCR. 

Seems in realm of possibilities anyway. Not having actual measurements of the bobbin core and the likelihood I screwed up the math somewhere not withstanding.