Inductance of Parallel Coils on an uncut toroid magnetic core

[dilectron]

Hi,
I have a toroid of a soft magnetic material (any). I wind the complete toroid with single turn coils and connect all their terminals in parallel. Then I wind a second layer of single turn coils on the previous layer and so on.

I want to calculate the equivalent inductance L. I calculate it for a vacuum core (Kappa vac or µ0) and can multiply my end result with the relative permeability (µr).

So I first calculated the self-inductances of the single coils, then I use these self-inductances to calculate the mutual inductance of coils that lay on top of each other with M=k SQRT(Lx*Ly) and used a value of 1 for k since normally coils on top of each other are considered to have perfect coupling. Then with Le=(L1*L2)-M² / L1+L2 - 2M i calculated the inductance of EACH SET of 2 coils in such pancake stack. This yields almost always zero (or numbers E-19 and smaller) because Lx and Ly (given moderated stack of 5 layers with thin wire) are almost identical. So, no matter how many layers I have on top of each other, the equivalent inductance is L, the inductance of one coil of one turn.

Now I consider such a parallel pancake stack as a single coil, and my toroid is filled completely with them so that I have the maximum possible pancakes on it. This keeps their mutual angles and inter-distance short and the k value for a toroid is always one (perfect coupling).

So we have again a situation of perfectly coupling, identical coils in an aiding configuration. So technically, here also the inductance L should be the L of one single pancake since Lx*Ly will again be equal to M². Until here my conclusion was that the inductance L for such toroidal core filled with parallel pancake coils is the L of one single turn of coil. As a result the inductive reactance (XL) will be very low and the ohmic resistance (R in parallel) will be very low too.

Let's call this a short-circuit coil configuration.

Since the above approach is relatively heavy I started to look for better and faster ways to calculate the total L of a toroid with only single turn parallel windings.

I found Grover's, "Tables for the Calculation of the inductance of circular coils of rectangular cross section." and I used formula 4 (for pancake coils: L=0.001 N² a P f) and used the tables in the book to make the corrections the way the book explains.

I also made the calculation with (µ0N²h/2π)*ln(b/a) (from MIT http://web.mit.edu/viz/EM/visualizations/notes/modules/guide11.pdf).

In both cases I get a result for L that is 3 to 4 orders of magnitude LOWER then the calculation for the pancakes above. However, if I use the total number of turns (as if the toroid with all pancakes was a single coil) then my end total is 2 orders of magnitude bigger then a single turn of coil. I did take into account that the formula from Grover uses centimeters and yield micro Henry and that the one for MIT is in SI units. So the only explanation I can see so far is that the N² makes the difference since in the approach via individual pancakes N=1, hence 1² = 1, and is neutral in multiplication.

So my question is, does the type of formulas as mentioned in the previous paragraph (Grover & MIT) apply if all the turns are actually in parallel.

Those books say nothing about that and the formula's don't give it away either. One could think that for inductance calculation it isn't important whether you use a single coil of X horizontal turns and y vertical layers high OR one of X times 1 horizontal turn and Y times 1 vertical layer high connected in parallel, because all dimensions remain the same and the total number of turns remains the same. At best the current distribution (if we see the rectangular winding as a current sheet) would be better with the parallel version.

Can someone help me to clarify what approach is the best and possible explain why.

Thanks.

 

[rude man]

S

I'm not sure what you're doing. But if you're winding one coil around the toroid core so you have 2 terminals, then you wind a second coil around the core, again 2 terminals, doesn't matter if the 2nd coil is on a second layer or on the same layer, then you connect the dotted coil ends together and also the undotted ends together, then the result is double the uncoupled inductance. (I hope you know what #dotted ends" means).

But if you connect each coil's dotted end with the other coil's undotted end then you'd get zero inductance, i.e. a short.

Let $L_1$ be the 1st coil's inductance with the other coil unconnected, and same for second coil's $L_2$. $L_T$ is the total inductance of the two coils connected to each other.

Generally, $L_T = L_1L_2(1+k)/(L_1+L_2).$. k is the coupling coefficient (% one coil's field seen by other coil). k is + for in-phase coils and - for out-of-phase coils. -1 < k < +1.

You now have a new inductance $L_T$ and you can add as many extra coils as you like, each time combining the new $L_T$ with the next L_i , i=3,4 etc .

Come to think of it, why would you want to do that in the first place? Your max inductance is with just 1 coil.

 

[dilectron]

rude man, thank you for you reaction.

The approach that you suggest is the one I already use and is in my OP with the conclusion that indeed inductance remains that of one coil. The coils are single turn, per layer identical, aiding (=dot aligned), perfect coupling (k=1, or -1 if you take the phase into account) . The toroid is completely filled with 100 horizontal single turn coils, and 5 layers of single turn coils exactly on top of it. All these 500 individual coils are connected in parallel to two rails (+/-).

1) the question was, would the alternative formula's that I mentioned work (because they are a faster approach and may capture the slight difference in circumference of the coil turns at different layers). Those formula would have the merit to make 1 calculation per toroid rather then daisy chaining multiple calculations. I tried them and they gave 3 to 4 order of magnitude different results compare to the daisy chain method. However, the mentioned book (Grover's) says to expect precisions that are 1/1000 to 1/10000 better then most formula's (Rosa, Lyle, Dwight, Butterworth, Spielrein, etc), so these magnitude differences would make sense. I am actually seeking for confirmation.

2) Usage: I try to compare this 100x5 layers parallel single turns to a single 100x5 layers coils on an identical toroid. Then I look at B in the center (which is different), the proximity effect between the turns (which is different), the skin effect @ Hz (which seems to be the same), the heat (which is different), etc.

Then step by step I want to develop an insight for myself to understand which small change is responsible for what and try to explain for each one why that is. And, I would love to ones have an inductance calculation of a non-simple coil that results in exactly what is measured, and not just approximations with sometimes quite large error margins (1-3%).

My intuition tells me that there must be a single general formula that can be developed that allows the precise calculation of any type of coil instead of choosing the formula that is best suited for a specific type of coil (as in the above mentioned names) such as coils, pancake, solenoid, toroid, spiral, banked, etc.Thank you for replying.

 

[berkeman]

then you connect the dotted coil ends together and also the undotted ends together, then the result is double the uncoupled inductance.

Double or 4x?

The inductance is proportional to the square of the number of turns in the coil, assuming full flux linkage.

https://en.wikipedia.org/wiki/Inductance

 

[dilectron]

berkeman, thank you.

I think it is not that simple in this case. All coils are single turn coils that share a same magnetic core are perfectly coupled (tiedly next one other and on top of each other).
I followed this page https://www.electronics-tutorials.ws/inductor/parallel-inductors.htmland since my coils are in aiding (dotted sides aligned) configuration and k=1 the inductance L is :

L12= (L1*L2) - M² / (L1+L2) - 2M where M is k * SQRT(L1*L2)
And so daisy-chaining with L12 & L3, then L123 & L4, then L1234 & L5, etc.

The end result is that L is the same as the self inductance of one single coil of one turn, no matter how many single turn coils are added, provided they are added by full layers to keep k=1 (short inter-distance, almost no angle)

I also made the inductance calculation with :

(µ0N²h/2π)*ln(b/a)

from the toroid specific formula in this MIT document http://web.mit.edu/viz/EM/visualizations/notes/modules/guide11.pdf.

It yields a result that is several orders of magnitude different, but not necessarily wrong possibly just more precise, although it is a formula for a COIL of a SINGLE WINDING of x horizontal * y vertical layers. So NO parallel connections since it is a single coil with only 2 terminals.

My question was (seeking confirmation) whether this formula would be a valid alternative for the daisy-chain calculation used above because it takes only one calculation rather then 'n' where n is the number of single turn coils.

Thank you.

 

[berkeman]

Sorry, I'm not tracking what you are saying. Why are you connecting all of those single-turn coils in parallel? What is the application?

 

[dilectron]

berkeman, I am comparing two coils in detail. Both toroidal magnetic cores of same dimensions.

One with a single coil of 100 horizontal turns and 5 vertical layers (N=1*500=500) and one with 100 horizontal single layer coils and 5 layers high (N=(1*100) * 5=500) where ALL 500 single turn coils are connected in parallel (aiding = dots aligned). Both fill the toroid completely (hence perfect coupling because in both cases the turns are next to each other and on top of each other as close as possible.

I then look at the difference as explained in my answer to rude man. There is not an application. In a first step I want to calculate things in the most precise and shortest way.

Hence my question of the MIT approach in my post would be a valid equivalent for the daisy-chain approach which is cumbersome. The MIT formula works for sure for the first coil (1*500) and I seek confirmation if it would work for the second coil ((1*100) * 5) with identical dimensions and turns BUT with all single turns couple in parallel.

Thank you.

 

[rude man]

Double or 4x? https://en.wikipedia.org/wiki/Inductance

?
You have 2 coils wound on the toroid, ergo 4 terminals. 2 dotted, 2 not.

 

[rude man]

I

My intuition tells me that there must be a single general formula that can be developed that allows the precise calculation of any type of coil instead of choosing the formula that is best suited for a specific type of coil (as in the above mentioned names) such as coils, pancake, solenoid, toroid, spiral, banked, etc.

Assuming known, stable and linear (B as function of H) core permeability $\mu$, calculations for a toroid can be pretty close to reality since there is little flux leakage; the B flux is quite well contained within the core.

This is far from the case with e.g. solenoids, especially short ones where flux leakage is appreciable. As an extreme example: to compute the inductance of a 1-turn coil is well-nigh impossible, involving advanced integrals ("elliptic") and including the diameter of the wire, and even then the computations are approximations.

Also, have you considered the effect of variable permeability of the core? Unless the core is gapped there will be appreciable errors due to $d\mu/dt$, $d\mu/dT$, saturation etc. A gapped core mitigates such errors greatly but at the cost of radically reduced inductance.

So abandon hope for a universal inductance formula. Gapped toroids can come fairly close though.

Thank you for replying.

 

[berkeman]

?
You have 2 coils wound on the toroid, ergo 4 terminals. 2 dotted, 2 not.

Sure. But connecting two coils in series on the same core gives 4x the magnetizing inductance, no? The inductance of a coil is proportional to the number of turns^2 like in the wikipedia link that I posted. So by connecting two of the coils in series, that doubles the number of turns...

 

[rude man]

Sure. But connecting two coils in series on the same core gives 4x the magnetizing inductance, no? The inductance of a coil is proportional to the number of turns^2 like in the wikipedia link that I posted. So by connecting two of the coils in series, that doubles the number of turns...

The OP stated from the outset that the coils are wired in parallel, not series.

 

[berkeman]

Okay, I guess that explains my misunderstanding. Still a strange problem statement IMO.

 

[dilectron]

The OP stated from the outset that the coils are wired in parallel, not series.

Yes, correct. All coils are connected in parallel and all are single turn whether they are in the 1st, 2nd,...5th layer.

 

[rude man]

Still a strange problem statement IMO.

Fer sure!