Coupling factors between coils

[straylight]

I'm noticing a fairly consistent behaviour where I measure the inductances of each coil of a humbucker and togeter in series.

If each coil indicates an inductance of x, and they are independent, the combined inductance should be 2x, but I'm getting a result that's more like 2.2x.

Am I seeing the coupling factor in action here? Is there a way I can calculate this? I don't think it's an artefact of the test method as my cheap LCR meter is doing similar.

[antigua]

It is mutual inductgance. The section here "Mutually Connected Inductors in Series" describes it www.electronics-tutorials.ws/inductor/series-inductors.html , and in this given formula, "Ltotal = L 1 + L 2 + 2M"  2M is the extra 0.2 you're seeing. 

As for calculating it, I suppose if you see a 0.2 increase in inductance, that 0.2 can be worked into some algebra in order to yield the coupling coefficient. In fact, it would be nice to say "for a typical PAF, the mutual inductance between the two coils is X", so if anyone wants to solve a rainy day math problem...

[straylight]

Thanks!

What i really wanted, particularly to tie this back into SPICE models, is the coupling factor or coefficient of coupling as given by

k = M/SQRT(L₁L₂)

This was sneakily hiden on the previous page.

I suspect for a given construction of pickups, k will be thereabouts constant. I'm not sure i have enough raw data to support this, but it's time to automate the collection and calculation of key measurements across a collection of pickup and coil frequency response data...

[JohnH]

I think this matches results from Antiguas tests, that I used for making models in GuitarFreak (2017)

I took three of the humbucker tests, where series, single and parallel coil windings were measured. Once Id processed them to get the best (6 part) models that I could achieve to capture the responses, the resulting derived value of the main inductor was around 40 to 45% of L, for single coil and around 20% for parallel. L being the inductance for normal series wiring.

The parallel results are interesting. Is it magnetically feasible that if mutual inductance adds to series wirings, it does less or even reduces inductance in parallel wiring?

[antigua]

Aug 1, 2019 16:36:17 GMT -5 JohnH said:

I think this matches results from Antiguas tests, that I used for making models in GuitarFreak (2017)

I took three of the humbucker tests, where series, single and parallel coil windings were measured. Once Id processed them to get the best (6 part) models that I could achieve to capture the responses, the resulting derived value of the main inductor was around 40 to 45% of L, for single coil and around 20% for parallel. L being the inductance for normal series wiring.

The parallel results are interesting. Is it magnetically feasible that if mutual inductance adds to series wirings, it does less or even reduces inductance in parallel wiring?

Another page on that site describes what happens in parallel  www.electronics-tutorials.ws/inductor/parallel-inductors.html  :

Parallel Aiding Inductors




Parallel Opposing Inductors





"Where: 2M represents the influence of coil L 1 on L 2 and likewise coil L 2 on L 1." 

"Mutually connected inductors in parallel can be classed as either “aiding” or “opposing” the total inductance with parallel aiding connected coils increasing the total equivalent inductance and parallel opposing coils decreasing the total equivalent inductance compared to coils that have zero mutual inductance."

[straylight]

This is really helpful, thanks

[aquin43]

You can reduce the two parallel formulae to a single one:

L = (L1*L2 - M^2)/(L1 + L2 - 2*M)

and M can be either positive or negative.

M is a measure of the voltage induced in one coil by current flowing in the other as  a result of the coils sharing a magnetic flux.

If you consider the layout of a humbucker with the poles passing through the coils and a magnet joining the poles at the bottom, you have a U shaped magnetic circuit. If you imagine driving a current through one coil, which produces a flux going Down that coil, part of that flux will come Up the other coil, i.e. a reversal of the flux direction. But, the coils are wound in opposite directions so this cancels the inversion and makes the voltages induced in both coils have the same sign so M positive.

So, for two similar coils in parallel in a humbucker the net inductance is

(L^2 - M^2) / (2 * L - 2 * M))

which reduces to

(L+M)/2

Arthur

[antigua]

Aug 3, 2019 6:38:41 GMT -5 aquin43 said:

M is a measure of the voltage induced in one coil by current flowing in the other as  a result of the coils sharing a magnetic flux.

So if I apply one volt to the primary coil with a function generator, all I have to do is measure the voltage across the secondary to get M? Can the coupling coefficient be derived from that?

[stratotarts]

Aug 3, 2019 9:35:24 GMT -5 antigua said:

Aug 3, 2019 6:38:41 GMT -5 aquin43 said:

M is a measure of the voltage induced in one coil by current flowing in the other as  a result of the coils sharing a magnetic flux.

So if I apply one volt to the primary coil with a function generator, all I have to do is measure the voltage across the secondary to get M? Can the coupling coefficient be derived from that?

Not directly, because the output is load dependent. With a fixed resistive load, there might be a formula for it. I don't know what it would be.

[aquin43]

Aug 3, 2019 9:35:24 GMT -5 antigua said:

Aug 3, 2019 6:38:41 GMT -5 aquin43 said:

M is a measure of the voltage induced in one coil by current flowing in the other as  a result of the coils sharing a magnetic flux.

So if I apply one volt to the primary coil with a function generator, all I have to do is measure the voltage across the secondary to get M? Can the coupling coefficient be derived from that?

If the coil ends are accessible, it is possible to measure the mutual coupling by driving a small current through one coil and measuring the voltage induced in the other. The current should be small so as not to interfere with the magnets.

For example with a PAF style pickup, I placed a 1k resistor in series with one coil and applied 100 mV at 110 Hz. The voltage across the resistor measured 19mV, so the current was 19uA. The voltage across the other coil was 4.1mV.

The mutually coupled voltage is V = M * i * 2 * pi * f so M = V / ( i * 2*pi * f )

M = 4m1 / (19u * 2pi * 110) = 312mH

Knowing M and the coil inductances, K can be calculated

Arthur

[antigua]

According to the formula at the bottom of this PDF exality.com/files/Measuring%20Transformer%20Coupling%20Factor%20k.pdf , finding the coupling of a transformer is as simple as:

k = sqrt( 1 - ( L_shorted / L_primary ) )

In other words, you keep the primary connected to the meter, and observe how the inductance changes when the secondary is open or shorted. 

So I have a Seymour Duncan '59 neck, and using a DE-5000, the screw coil measures 1753.3H at 100Hz, and the value doesn't really change at all with the slug coil shorted. However, testing at 1kHz, I get 1635.2H from the screw coil, and 1596.0H with the slug coil shorted. In that case, k = 0.154831. Usually measuring the inductance at 1kHz doesn't provide a very accurate value, due to eddy current losses, but maybe in this application it's close enough, I don't know.


The same PDF above says that k can also be determined with   

k = M / sqrt( L1 * L2 ) 

So if I suppose that L1 and L2 for this pickup are both 2 henries, and supply aquin43's figure of 0.312H for M, the value comes out to... k = 0.156, two very similar outcomes despite using different data sources.

[aquin43]

Shorting one of the coils is exactly equivalent to the coupled coil eddy current loss model. The measured inductance goes from L at low frequencies to L - M^2 / L at high frequencies, the frequency dependence being determined by R/L of the shorted coil. Half of the inductance change is observed at ω=R/L. The method works well for transformers with low winding resistances but not so well for pickup coils where R/L of the coils represents too high a frequency for other losses to be ignored.

Arthur