Guitar Pickup Theory #6a: Coil shape Part 1 – DCR and inductance

This entry will examine the effect of the shape of a pickup coil on the tone of the pickup, as well as the underlying causes, and provide empirical evidence as support. More specifically, I want to address tall+narrow coils vs short+wide coils. As with many aspects of audio-related electronics design, general consensus tends toward being ‘in the right ballpark’ in terms of JUST the bottom line, but wanders into hearsay when it comes to specifics, and is fairly lacking in terms os publicly-accessible empirical test data, apart from a few brave souls.

A FEW THINGS BEFORE I BEGIN:
1) This went a lot longer than I expected because of all the rabbit holes I ended up having to follow, so I may have to break this post up into several entries. I also double and triple-checked some of the results because they seemed unintuitive in terms of commonly propagated (and some of my personal) beliefs. Some of the math also ended up being a MASSIVE undertaking, and I ended up with dozens of graphs and sound clips, so for the most part, I will cut out all those but the most helpful.  Consider this an overview, and I will try to hit on the specifics of each parameter in future specialized posts.  I also took so long with this post that I lost some of the graphs in the depths of my overstuffed hard drive, so I will come back and post those when I come across them.

2) It became obvious, soon after I started testing, that the conventional thinking was either wrong or lacking, so I might inadvertently be wandering into a contentious area. I hardly ever have a desire to ‘weigh-in’ on arguments online since I rarely see them as constructive, so I hope that I don’t step on any toes with these posts… they are merely a public log of my attempts to analyze and explain as thoroughly as I can, for the benefit of anyone else that may be barking up the same tree. This is also intended to be useful for hobbyists that don’t have the cash/knowledge to buy/build test gear… hopefully I can save them a bit of time and frustration.  And as always, I will try to detail my methods so that the tests can be repeated by anyone who wants to ‘check my math’ so to speak.

3) This post deals with trying to understand how the shape of the coil effects the FREQUENCY RESPONSE… not the OUTPUT EFFICIENCY. The two seem to be mathematically at odds with each other at times, so I wanted to clear that up before continuing, just to avoid starting any bad information through misunderstanding. I will briefly touch on some of the effects of shape on output before I begin, but I will cover it more thoroughly later in “Guitar Pickup theory #4a: Optimizing pickup output (part 2)” or whatever I end up calling that post:

4) There are many many other factors that effect the sound of a pickup (some of those have been touched on here in this blog, others will be dealt with in the future), so remember that we are only dealing with factors directly relevant to coil shape in this installment. There are also areas that I did not touch on here because either the effect was extremely minor, or the effect was difficult to translate into a blog format, or I do not have the gear to test accurately, etc.  An example is skin effect/ac resistance, wire layer bulge due to rectangular coil shape, etc.  So this is not absolutely comprehensive, but hopefully it hits the biggest influencing factors.

INTRO:
We know already that coil can be broken down electrically into three basic major, components: inductance, resistance, and capacitance. These three components are inter-related in a coil to the point that physical dimensions are closely tied to them in a way that we don’t see in many other places in electronics. In a guitar, we also have to consider the tone/volume/switching electronics, as well as cable capacitance, input impedance, and input shunt+Miller capacitance, etc. We can use a lump Thevenin equivalent (which is not entirely accurate since these properties are distributed, but we’ll get to that later) for much of that to determine the input response/tone, but for the sake of this discussion we are going to isolate the pickup, and only refer to the electronics as a point of reference. The internal resistance, capacitance and inductance form an RLC tank, which is fed to the electronics to determine the resonant frequency, cutoff frequency, and the Q of the peak, which are all related.

The general consensus seems to be that given the same number of turns of the same wire, taller+narrower coils are brighter than shorter+wider coils (all other materials, electronics, and construction being equal). This is not necessarily in question here. What IS in question how and why geometry effects tone (although I MUST state that I specifically used the qualifiers for a reason… outside those parameters, there is NO reason that a short+wide coil cannot be designed to be bright, and no reason that a tall+narrow coil cannot be designed to be dark — both are quite possible).

I have seen 3 explanations reiterated time and time again on the inter webs: one strikes me as vague (or confused), one strikes me as partially-true (i.e. there is an effect, but it is nearly inconsequential or it is applied improperly to the issue), and one strikes me as mostly true (i.e. it explains the majority of the effect, but not all). Those popular explanations are that a wider coil is darker than a narrower coil with the same number of turns because:
theory 1) (mostly true) – the shape gives more inductance and DCR per turn, which makes for a lower resonant frequency with a lower Q value
theory 2) (partially true) – the additional length of wire used creates a greater capacitance, which shifts the resonant frequency
theory 3) (conditionally true or false) – a wider coil senses a wider string area, which sounds warmer/fatter

Let me ALSO mention that other builders have discovered that some of those explanations may be less than true (or at least insufficiently fleshed-out), and they have said as much publicly, so I am not really breaking new ground here or making any new claims — i am just providing my own test data. I will be testing all 3 of those theories. So without further ado… onto the testing.
TEST #1 – INDICTANCE AND DCR:
PREFACE:
I ran these tests with 4 different setups: Inductance meter, breadboard op-amp jig against a known capacitor and crossover inductor, DIY ’dead bug’ RF jig against known capacitor and inductor, and DIY ’dead bug’ RF jig using the ‘dual resonance’ method. The coil I calibrated the jig with is quoted as being accurate to 3%, and the capacitors that I used to test against were polypropylene capacitors quoted as 2.5% accurate. I also tested the capacitor against groups of their own type, and picked the ones with values right in the middle of the distribution to use in the tests. The most accurate method seemed to be the dual resonance method with the RF jig, so I will be using that in the future, but even that method was a few percent off, and the higher-eddy-loss cores with a low Q value made life difficult, so I had to fudge some of those numbers and make some guess-timations, but the air coil numbers should be pretty close.

jig

Again, we are looking for trends more than exact math. Also, it should also be stated that (as many builders have discovered) measurements are quite effected by temperature, so I tried to do these tests within a close period of time to minimize that. ALSO, with a high-permeability cores, the inductance of a coil becomes wildly frequency-dependent, which further complicated measurement, so I try to both measure at a few frequencies, and use the same frequencies from one test to another. Higher frequencies are more stable, so I use 5kHz and 10kHz when possible, and then use 1kHz as a common measurement. In my experience, anything below that gets sketchy.

The point of the first test was to empirically examine the difference in DCR and inductance between two shapes of coil with an equal number of turns. Determining DCR and inductance is simple for a round-core spiral coil, and could have been calculated rather than built… but what fun is that? I wound 2 coils for the initial tests, each with 1000 turns of #38 wire with a low-capacitance (but controlled and even) winding pattern with medium tension (relatively speaking). The target was for one coil to be half the height of the other. I let the other parameters fall where they may.

THE COILS:
Here are their physical parameters (height x diameter):
-Coil1(T/N) = tall+narrow: 5/8”H x 3/8”D
-Coil2(S/W) = short+wide: 5/16”H x 1/2”D
(both cores are 1/4” diameter and made of silicon steel laminations, with the magnetomotive force source being formed by a neodymium magnet under the pole piece).

(coil picture)

Here are their electrical parameters:
Coil1(T/N)
-DCR 50.6 ohms
-air inductance = 2.55mH
-alnico inductance = 4.286mH
-steel inductance = 31.74mH

Coil2(S/W)
-DCR 72.3 ohms
-air inductance = 7.62mH
-alnico inductance = 8.644mH
-steel inductance = 38.58mH

After measuring, I checked these measured values against the hand-math, and they are within reasonable range for a looser, low-capacitance wind.

I followed this up with another test with larger coils, tighter winding pattern, more turns, and rectangular cores. Again, I used #38 wire, but this time with 2500 turns, and again, the height of the tall+narrow coil is 2x the shorter+wider coil. This was to see if the general trends scale to larger coils and oblong shapes. I also added silicon steel laminations in the form of 3 small laminations and 5 long laminations.

(coil picture)

Here are their physical parameters (height x width1 x width2):
-Coil3(T/N) = tall+narrow: 1/2”H x 3/8”W1 x 5/8”W2
-Coil4(S/W) = short+wide: 1/4”H x 3/4”W1 x 1 ”W2

Here are their electrical parameters:
Coil3(T/N)
-DCR 198 ohms
-air inductance = 35mH
-alnico inductance = 47mH
-steel inductance = 85mH
-3 short lams = 95mH
-5 long lams = 192mH

Coil4(S/W)
-DCR 290 ohms
-air inductance = 81mH
-alnico inductance = 87mH
-steel inductance = 116mH
-3 short lams = 127mH
-5 long lams = 196mH

And as a real-world example, I used some pickups that exemplify the same principle: a P-90 and a strat pickup. Just the coils of each were used. That way the effects of the core material, mounting hardware, covers and metalwork do not effect the results. P-90s seemed to traditionally be wound by turns count by hand (about 10,000 for bridge) which puts the other specifications in a rather wide range depending on the other variables, so there is quite a bit of variation from one to another, but here are some measured values of what is on hand.

(p90 and strat pic)

Coil5(strat bridge)
-DCR 6k
-air inductance = 2.2H
-alnico inductance = 2.6H
-steel inductance = 4H

Coil6(P-90 neck)
-DCR 8k
-air inductance = 4.6H
-alnico inductance = 5.2H
-steel inductance = 7.4H

The general trend of the results are pretty similar to the trends of the smaller coil test, but the effect was not as great as the permeability of the core increased with the various materials. Again, I checked these measured numbers against a coil inductance calculator, and the measured results are within 17% of the calculated values (which is nice to know since It validates my measurement jig) BUT the calculators tend to be overly-optimistic about winding pattern, so they usually return unrealistically-good values anyway.

Just for kicks and giggles, here are some computer-simulated coils (reminder: these are values for an air-core coil):

Simulated strat:
-Inductance = 1.675H
-DCR = 5.88k ohms
-Turns = 7993
-Winding thickness = .141”
-Winding height = .46”
-wire length = 3437’

Simulated P-90:
-Inductance = 4.75H
-DCR = 8.19k ohms
-Turns = 10418
-Winding thickness = .32”
-Winding height = .256”
-wire length = 5080’

The dimensions of the simulated coils’ thickness and wire length are a bit unrealistic since the program simulates perfect machine winding layup (and perfect wire diameter and zero stretch, etc), but apart from that, the specs are not too far from reality, and since both coils are idealized, it is a nice addition to the test data to help us see the trends. SO here are some more simulations (again, these are values for an air-core coil):

Strat (i.e. taller+narrower) wound to P-90 DCR:
-DCR = 8.15k
-Inductance = 3.33H
-Turns = 10866
-Winding thickness = .2”
-winding # of layers = 68+
-Wire length = 4761’

Strat (i.e. taller+narrower) wound to P-90 inductance:
-DCR = 9.66k
-Inductance = 4.75H
-Turns = 13050
-Winding thickness = .226”
-winding # of layers = 80+
-Wire length = 5702’

These numbers should illustrate further the difference between what is possible from the two coil shapes. If the simulation is close, winding a strat-shaped coil to P-90 inductance numbers would require over 13,000 turns and might not fit on a strat bobbin (maybe… but it would be close with the #42 wire I simulated unless you machine-wound it, which would up the self-capacitance greatly… more on that in the next set of tests).

DCR:
I won’t dwell on DCR (DC resistance) much because it is obvious that since the diameter of each layer increases more for the shorter coil, the DCR will increase at a greater rate for the same number of turns as compared to the thinner coil. For the coils that I wound, the DCR of the shorter coil was nearly 50% higher. That will obviously increase as you add winds and increase the outer turn diameter more. This won’t effect our resonant frequency, but it WILL effect the Q of the peak since it is damped by our coil’s resistance, which is one reason that hot/high-turn-count pickups have much lower resonant peak Q values. BUT this is HIGHLY influenced by the load impedance i.e. the value of the volume pot, tone control, and amp/pedal’s value. We already know that using lower-ohm-value controls (i.e. 250k volume pots) will flatten your resonant peak more than higher-ohm-value controls (i.e. 1Meg pots). So, for example, depending on your winding specs, doubling your DCR with 250k controls may cause the Q value to decrease by 3dB or so, whereas doubling your DCR with 1Meg pots may only decrease the Q peak by 1dB or so, etc, since your peak is less damped by the 1Meg control load.

Higher DCR will also (to a much lesser extent) decrease the output voltage with regard to the load impedance as it forms the top leg of a voltage divider, but since DCR usually means that you used more windings, in practical terms, that means that each successive turn will increase the output less than the previous turn… which again varies according to your load impedance.

Here is an example of A 5H/100pF coil simulation with a 500k||1M load, where the coil DCR is increased by 2x, from 8k to 16k.  You can see that the Q of the resonant peak has been damped by about 1.75dB or so, and the overall output reduced by about 0.4dB.  The -3dB cutoff is changed from about 3.45kHz to about 3.34kHz, so the overall change is slight, but likely noticeable:

DCR Q

INDUCTANCE:
We can see a few trends in terms of inductance in terms of these tests:
-the air-core inductance of the shorter coil is much higher than the taller coil (almost 3x in this case)

-this difference between the two coils decreases somewhat as the permeability of the core increases (2x greater for alnico and 22% greater for laminated steel)

-regardless of the core, the inductance of the shorter+wider coil is higher by some degree for the range of core permeability values we are likely to see in a guitar pickup (which is likely a range of ~50% higher to ~200+% higher for the range of alnico poles, steel slugs/screws/blades, and laminated poles/blades, etc).

-another important note is that a longer core allows the coil to take better advantage of the core material’s permeability, as does how centered the coil is on the core, so for the same MATERIAL the taller coil will see a larger effective permeability from the core, offsetting the effect of the short+wide coil’s additional per-turn-inductance somewhat

Conclusion of TEST #1:
a shorter+wider coil will have a higher mean DCR-per-turn and higher mean inductance-per-turn as compared to a taller+narrower coil, with the difference in inductance between the two being progressively greater with progressively lower core permeability values (in other words… the difference in inductance will be greater with alnico cores than steel cores, i.e. the difference will be greater from say, a strat coil to an equivalent Jazzmaster coil, than from a single-coil with steel poles (like an SDS-1 and a P-90).

So in terms of the first of our three theories (that a short+wide gives more inductance and DCR per turn, which makes for a lower resonant frequency with a lower Q value), that tests out as true. It has been shown that the short+flat coil is more efficient in terms of inductance-per-turn (about 20-50% for an air core, according to the various tests here). It has also been shown that the shorter+wider coil has a higher DCR-per-turn as well, so you will see a lower/wider resonant peak. The lower peak, combined with the higher inductance will serve to shift the peak further toward the midrange-frequencies and away from the pick-attack frequencies, dampen (and by extension widen) the Q of the peak, and lower the high frequency bandwidth cutoff.

The degree of effectiveness of these effects can be seen here:

So all else being equal (which it is not… but stick with me here) doubling our inductance as we did above, we shifted the peak frequency about half an octave, which is significant.

strat vs p90 sweep cropped

If you remember, waaaay back in the intro, that I said that this theory is MOSTLY true… that is because there are some other issues which I will touch on in the next round of testing in terms of other parameters. See you there!

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About alexkenis

Guitarist, philosopher, tinkerer

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