Calculating the mutual coupling coefficient of E cores

[tim9000]

Hi,
I can't find my inductance calculations book. Say that you were considering magnetic flux linkage more from a geometric perspective:
-Does the mutual coupling coefficient for a series of coils all on the same steel core add up to a maximum total of ~1?
Because I tried simulating a three phase 'E' core TX on LTSpice, and I had issues making the coefficients between coils near 1 (it tells me I've made an impossible relation or something).

-For example, if you were looking at a three phase Isolation transformer, ignoring self-couplings, what would the coupling coefficient between one primary coil to another primary coil be, approximately?

Intuitively, I thought it'd be about M = 1/2 between each adjacent primary pair of coils, and the same mutual coupling coefficient between adjacent primary-secondary pairs, with each of the primary-secondary pair which are sitting on the same limb at about M: ~1.

However, given the LTSpice, maybe it's more like M = 1/3 between adjacent limb coils?

Similarly, I'm wondering how much of a difference (improvement) would a bifilar winding make to [reducing] the amount of leakage flux, and how much is going to couple through the core to another coil anyway (because the steel permeability is so high).

Thank you!

 

[berkeman]

What is your coil construction? Is it similar to this 3-phase transformer?

http://conceptsofelectricaltheory.blogspot.com/2016/03/transformers-2.html

 

[jim hardy]

-Does the mutual coupling coefficient for a series of coils all on the same steel core add up to a maximum total of ~1?

Why would it ?
What's definition of mutual coupling?

 

[tim9000]

With regards to the coupling:

What is your coil construction? Is it similar to this 3-phase transformer?

Yes, it is exactly a comparison of this core with the various winding typologies that I would like to discuss. How would the coil windings of the above picture compare to that of a bifilar wound "HV/LV" TX (in keeping with the labels of the picture). I presume there is the least leakage flux (self-inductance) with bifilar winding. Similarly, if you had (for instance) a single phase core-type core, where the primary and secondary are on opposite sides of the magnetic path, how much poorer is the coupling coefficient between the coils (very generally speaking)? I presume the leakage flux would be worst here, comparatively.

As I recall M = k√(L1*L2), but I'm not sure how to evaluate this for a three phase core...

Why would it ?
What's definition of mutual coupling?

I don't necessarily think it should, I'm my query was spurred on by LTSpice not permitting it.Thanks guys

 

[jim hardy]

I don't necessarily think it should, I'm my query was spurred on by LTSpice not permitting it.

Good, i don't either. Draw a picture. Any flux that only links one coil can't be mutual.

 

[berkeman]

How would the coil windings of the above picture compare to that of a bifilar wound "HV/LV" TX

I don't think I've ever seen a high voltage / low voltage bifilar wound transformer. The whole point of bifilar winding is to lower the leakage inductance (while sacrificing higher Cww). How could you achieve high voltage isolation in a bifilar wound transformer?

 

[tim9000]

Good, i don't either. Draw a picture. Any flux that only links one coil can't be mutual.

Not one coil, core.

Cheers

 

[tim9000]

I don't think I've ever seen a high voltage / low voltage bifilar wound transformer. The whole point of bifilar winding is to lower the leakage inductance (while sacrificing higher Cww). How could you achieve high voltage isolation in a bifilar wound transformer?

I'm sure there aren't HV/LV bifilar, that's why I put quotation marks around the label.

Cheers

 

[jim hardy]

Not one coil, core.

?

-Does the mutual coupling coefficient for a series of coils all on the same steel core add up to a maximum total of ~1?

if you were looking at a three phase Isolation transformer, ignoring self-couplings, what would the coupling coefficient between one primary coil to another primary coil be, approximately?

 

[tim9000]

?

You said any flux that only links one coil; I'm not talking about any flux only linking one coil.

I brought up Bifilar because I wanted to in a way use it as a comparison standard between coil magnetic couplings, as it is the best possible.

As I said, I wanted a bit of a general sense of how effective various winding arrangements are. Something I should have done in the first instance, here is an illustration for your comparison:

Keeping in mind that each coil is linked to each other coil because they're all on the same core.

A) -> Comparison of coils on adjacent limbs: Imagine that each colour has HV and LV windings. What would the mutual coupling between, say, red and green primaries, or red and blue primaries, be?
A) -> Comparison of coils on same core: say that there are HV and LV coils comprising each red, green and blue. What would the mutual coupling the HV and LV red coils be?

B) Comparison of coils on the same core: Say red is the primary, and orange is the secondary winding, green is the primary winding, grey is the secondary winding etc. What would the coupling of the Red to orange coils be? Similarly, what would the coupling of the Red to grey, red to green, be?

C) Red is wound bifilar with orange, etc: what is the coupling of red to orange, what is the coupling of red to green?I hope this clarifies my original line of inquiry.

Thanks

 

[jim hardy]

A) -> Comparison of coils on adjacent limbs: Imagine that each colour has HV and LV windings. What would the mutual coupling between, say, red and green primaries, or red and blue primaries, be?

You worked that years ago in your magamp thread.
The magnetic circuit apportions flux according to the mmf(amp-turns) in each leg. It's a circuit that you solve with mmf's and reluctances.

 

[tim9000]

You worked that years ago in your magamp thread.
The magnetic circuit apportions flux according to the mmf(amp-turns) in each leg. It's a circuit that you solve with mmf's and reluctances.

There must be differences in leakeage fluxes between the three winding types. Also:
If I were to treat this like a simple NI -- reluctance circuit, then would not: each of the A), B) and C) all have the same coil coupling relationships, comparatively? Because certainly bifilar winding, is has far superior coupling to that of B) where the primary and secondary are on the same limb, but occupying different parts of it.

Note: there are primary and secondary coils in A) but you can't see them because the outer RED coil obscures the ORANGE secondary coil underneath.

As per my last post, I have my own personal expectations of what the coupling order of best to worst would be. [For example, I expect bifilar coils wound on the same section of core like red to orange as in C), would have better coupling to that of the HV/LV (primary/secondary) coils in A), which itself would have better coupling of red to orange than in C), which would have the poorest coupling]. But there are a couple of things I don't understand, for instance:
When you have a better coupling of primary to secondary on the same limb (like in the case of bifilar) is there also an impact on the other coils on the other limbs, like coupling between adjacent limb primary-primary coils gets worse? This is to say, would the coupling of the red to green primaries, be the same for B) as they would for C)? After all, the magnetic path hasn't changed, just the winding geometry. Or has improving the coupling of red to orange between those coils also decreased the coupling of red to green? etc.

[That flux generated by red passes through orange, then it passes through grey/green and purple/blue, but if you've improved the way it couples through orange, does this also effect how well it passes through grey/green and purple/blue, or can you have your cake and eat it too. If so, as long as the insulation was high enough, bifilar would be win-win]

Thank you

 

[jim hardy]

[That flux generated by red passes through orange, then it passes through grey/green and purple/blue, but if you've improved the way it couples through orange, does this also effect how well it passes through grey/green and purple/blue, or can you have your cake and eat it too. If so, as long as the insulation was high enough, bifilar would be win-win]

You posited single phase.
In that case
Flux generated by red passes through orange then divides between the other two legs according to MMF's from current in their respective windings and of course reluctance.

 

[tim9000]

You posited single phase.

Regarding the A), B), C) illustration, single or three phase doesn't matter, at this point I'm just trying to step-wise discuss if there are any differences in leakage flux and mutual coupling between windings on the same limb or adjacent limbs.

Flux generated by red passes through orange then divides between the other two legs according to MMF's from current in their respective windings and of course reluctance.

If I were to treat this like a simple NI -- reluctance circuit, then would not: each of the A), B) and C) all have the same coil coupling relationships, comparatively?...This is to say, would the coupling of the red to green primaries, be the same for B) as they would for C)? After all, the magnetic path hasn't changed, just the winding geometry.

Thanks

 

[jim hardy]

if there are any differences in leakage flux and mutual coupling between windings on the same limb or adjacent limbs.

Why look for word crutch rules to memorize and mis-apply? Just go back to basics. It saves a lot of trouble.

What's definition of mutual coupling?

http://www.electronics-tutorials.ws/inductor/mutual-inductance.html

Calculate Φ₁₂ for all the coil pairs on your E-core.
Does it change with amp-turns in L₂?

 

[tim9000]

Why look for word crutch rules to memorize and mis-apply? Just go back to basics. It saves a lot of trouble.
http://www.electronics-tutorials.ws/inductor/mutual-inductance.html
View attachment 207427

Calculate Φ₁₂ for all the coil pairs on your E-core.
Does it change with amp-turns in L₂?

Sorry for the delayed reply.
I understand that Φ₁₂ will vary ever so slightly than Φ₁₃ due to a slight proximity difference b/w ends of the core (if 2 is the limb in the middle of the 'E') but will basically be the same. (I.e. the small extra reluctance will divide the flux not quite 50/50 between coils 2 and 3)
But my point at this very moment is that the flux (example from coil 1 primary) Φ₁₂ travels through the core on it's way to coil 3 primary, via coil 1's secondary in slightly different ways. The secondary could be tightly coupled as in C) or over the top of it as in A) or in series with it as in B). So does this make no difference to the coupling of coil 1 to coil 3?

Thanks

 

[jim hardy]

So does this make no difference to the coupling of coil 1 to coil 3?

I'd think not. What happens in leftmost leg stays in leftmost leg.

 

[tim9000]

I'd think not. What happens in leftmost leg stays in leftmost leg.

Can you elaborate on that? :p

 

[jim hardy]

What happens in the leftmost leg stays in the leftmost leg.

Can you elaborate on that? :p

Back to the basics ?

https://www.princeton.edu/ssp/josep...ide-to-maxwells-equations-D.-FleischLEISC.pdf

which my undergrad course instructor simplified to

Closed loop Integral around any closed line path of MMF dot d_length equals the current enclosed .
°∫ H ⋅ dl = I_enclosed

I_enclosed is of course amp-turns..

Applying that,

so MMF is not changed by rearranging the coils.

 

[tim9000]

Back to the basics ?

https://www.princeton.edu/ssp/josep...ide-to-maxwells-equations-D.-FleischLEISC.pdf


View attachment 207836



which my undergrad course instructor simplified toI_enclosed is of course amp-turns..

Applying that,

View attachment 207838so MMF is not changed by rearranging the coils.

Ah geez, I did know how to derive all of Maxwell's laws once. I don't have my formula exercise book on me at present.
It's the right hand side I'm hazy on: was it something like all the electric charge that moves due to the magnetic field on the surface cancels out due to symmetry or something, so it's zero?

As per your previous post, if I'm not mistaken l_enclosed is zero, because no H is enclosed, because H is parallel to the plane of the loop. However, if the l_enclosed was on the same plane as the coil turns I_enclosed would be the integral of the flux.

I think this is too crude a method of analysis to use Ampere's law precisely due to one of the factors I'm trying to examine, namely leakage flux.

A) -> Comparison of coils on adjacent limbs: Imagine that each colour has HV and LV windings. What would the mutual coupling between, say, red and green primaries, or red and blue primaries, be?
A) -> Comparison of coils on same core: say that there are HV and LV coils comprising each red, green and blue. What would the mutual coupling the HV and LV red coils be?

B) Comparison of coils on the same core: Say red is the primary, and orange is the secondary winding, green is the primary winding, grey is the secondary winding etc. What would the coupling of the Red to orange coils be? Similarly, what would the coupling of the Red to grey, red to green, be?

C) Red is wound bifilar with orange, etc: what is the coupling of red to orange, what is the coupling of red to green?

Based on your last reply I'll hazard some guesses. Now, the amount of flux that flows through the core might not change,(I'm leaning towards this being the case). However, for example I'm highly skeptical that the amount of flux that flow from red to orange will be the same in B) as it is in C). (Due to a poorer coupling than when bifilar).

This may have other consequences beyond these immediate primary-secondary coils as I previously queried, or it may not. I.e. conceivably the coupling to adjacent limb coils is the same throughout A) B) & C) are the same because the respective leakage flux doesn't change throughout the three arrangements(?) But that's what I'm seeking clarity for.

Thanks as always

 

[jim hardy]

Now you've changed the question.

However, for example I'm highly skeptical that the amount of flux that flow from red to orange will be the same in B) as it is in C). (Due to a poorer coupling than when bifilar).

You asked about coupling between windings on different limbs.
Now you're talking about windings on the same limb.

 

[tim9000]

Sorry, ten day late reply.

Now you've changed the question.

You asked about coupling between windings on different limbs.
Now you're talking about windings on the same limb.

Actually, I attempted to communicate that the question was always how the flux would vary between coils on the same limb and on adjacent limbs.

A) -> Comparison of coils on adjacent limbs...
B) Comparison of coils on the same core: Say red is the primary, and orange is the secondary winding, green is the primary winding, grey is the secondary winding etc. What would the coupling of the Red to orange coils be? Similarly, what would the coupling of the Red to grey, red to green, be?

So, I think thus far we clarified that the flux between adjacent limbs would be the same relative in either A), B) or C).
But that there would be differing leakage flux between coils on the same limb relative to the A), B) or C) winding arrangement. Thus, I would conclude that this amount of leakage flux doesn't impact the coupling between limbs, just merely coils on the same limb, which is where the advantage lies.

Did anyone know roughly how much of an impact these A, B and C variations would have to the mutual coupling coefficients in this regard?

Cheers

 

[jim hardy]

Actually, I attempted to communicate that the question was always how the flux would vary between coils on the same limb

I took this

if you were looking at a three phase Isolation transformer, ignoring self-couplings, what would the coupling coefficient between one primary coil to another primary coil be, approximately?

to mean primary coils on different limbs.
I didn't think about multiple voltage primary windings which might be tapped or might be separate coils..
Sorry about that..

 

[tim9000]

I took this

to mean primary coils on different limbs.
I didn't think about multiple voltage primary windings which might be tapped or might be separate coils..
Sorry about that..

No worries mate. I struggle to communicate at the best of times. I have near limitless patience with PF because I'm asking people to help me out with understanding these scientific things, and when they reply it's nice of them, because they have no obligation to me to read my long posts. So if I need to repeat the odd thing, or try explaining something another way, then it's a price I'm happy to pay for people taking time out of their lives to help me.

That's my view anyway.

Have a think about it and get back to me if you can.

Cheers

 

[jim hardy]

conceivably the coupling to adjacent limb coils is the same throughout A) B) & C) are the same because the respective leakage flux doesn't change throughout the three arrangements(?) But that's what I'm seeking clarity for.

I believe that would be so.

for example I'm highly skeptical that the amount of flux that flow from red to orange will be the same in B) as it is in C). (Due to a poorer coupling than when bifilar).

I too would expect some small difference . .

It is difficult to communicate with precision, isn't it ?

The art of reasoning is nothing more than a language well arranged

"But, after all, the sciences have made progress, because philosophers have applied themselves with more attention to observe, and have communicated to their language that precision and accuracy which they have employed in their observations: In correcting their language they reason better."

my old friend Lavoisier

 

[tim9000]

I believe that would be so.

View attachment 209027

I too would expect some small difference . .

It is difficult to communicate with precision, isn't it ?

my old friend Lavoisier

Hi Jim,
thanks for the reply, great, so I can put this to bed concluding: The amount of leakage flux on the same limb can vary depending on the winding arrangement, but the amount of flux coupling between adjacent limbs will be the same regardless of winding arrangement.

If I may 'pivot' this thread slightly; regarding the real transformer model:

https://upload.wikimedia.org/wikipedia/commons/3/39/TREQCCT.jpg

Let me see if I've got this correct; for a fixed frequency the EMF on the secondary is based on how much flux flows through the core, and consequently the size and composition of the core. So for larger E_s and E_p voltage you need a larger core, which unfortunately means that the R_c will be lower and core losses will be higher.

Some other aspects for verification:
The more turns on the primary coil means the larger X_m is.
The voltage on the magnetising branch is, in part, due to the secondary coil resistance and leakage reactance.

With all this in mind, if you make a core bigger than it needs to be, so it's will under saturation, it would seem to me that R_C is smaller than it needs to be and that the losses are larger than they need to be. Are there any benefits though?

Thanks!

 

[jim hardy]

So for larger Es and Ep voltage you need a larger core,

for larger Es and Ep you need a larger core either to carry more flux or to encircle more turns or both.

The more turns on the primary coil means the larger Xm is.

Sure. L is proportional to N²

The voltage on the magnetising branch is, in part, due to the secondary coil resistance and leakage reactance.

When secondary is carrying current, sure.

With all this in mind, if you make a core bigger than it needs to be, so it's will under saturation, it would seem to me that RC is smaller than it needs to be and that the losses are larger than they need to be. Are there any benefits though?

It runs cooler and there's less third harmonic current in the primary. That's because it never gets so close to the knee. Recall Steinmetz - iron loss is proportional to B^1.4 or B^1.6 depending what author you read.
So as you increase volume your increasing pounds is linear but watts lost per pound goes down as roughly pounds^1.5
If you consider lifetime energy cost instead of initial price you might actually be ahead to oversize the core.
I just helped my friend Harry fix up a pre-WW2 (or maybe pre WW1 !) GE repulsion motor. It's 1hp but big as a modern 7 hp. It runs just barely warm to the touch.

 

[tim9000]

for larger Es and Ep you need a larger core either to carry more flux or to encircle more turns or both.Sure. L is proportional to N²When secondary is carrying current, sure.It runs cooler and there's less third harmonic current in the primary. That's because it never gets so close to the knee. Recall Steinmetz - iron loss is proportional to B^1.4 or B^1.6 depending what author you read.
So as you increase volume your increasing pounds is linear but watts lost per pound goes down as roughly pounds^1.5
If you consider lifetime energy cost instead of initial price you might actually be ahead to oversize the core.
I just helped my friend Harry fix up a pre-WW2 (or maybe pre WW1 !) GE repulsion motor. It's 1hp but big as a modern 7 hp. It runs just barely warm to the touch.

I wish I understood Steinmetz's equation better. So from what I gather, it's more about observation rather than most base level scientific explanation.

I didn't consider the core X_m would have one impedance to the fundamental and another to the 3rd harmonic, so there you go, I learned something. It would be better in that regard.

So is that Steinmetz power equation specifically related to heat and magneto-striction?

But I'm still confused, isn't core loss based on R_c which will be lower the larger the core is?

I have another side question which I wanted to wait to ask you, but I can't; that being: If you were using a supply to a transformer or motor, which wasn't a pure sineusoid, but was instead a half parabola above and below the axis, so it resembled a sineusoidal wave. What would the impact of this be? Would it be any better or any worse than a regular sine wave? (I'm deliberately leaving 'better' and 'worse' vague because I don't even know how to define them, they're unknown-unknowns to me)

Thanks!

 

[jim hardy]

But I'm still confused, isn't core loss based on Rc which will be lower the larger the core is?

of course.
P = E²/Rc
If you operate a larger core at same flux density that's same loss per pound in more pounds so loss will go up. But larger core at same flux density is more flux hence more volts per turn.

If you operate that same larger core at same total flux to give same volts per turn, that's less flux density so loss per pound goes down.

So might Rc be related to B ?

 

[tim9000]

of course.
P = E²/Rc
If you operate a larger core at same flux density that's same loss per pound in more pounds so loss will go up. But larger core at same flux density is more flux hence more volts per turn.

If you operate that same larger core at same total flux to give same volts per turn, that's less flux density so loss per pound goes down.

So might Rc be related to B ?

Hi Jim,
Great reply, as usual. Thanks.
Assuming the total flux does remain constant while the core is increased in mass, how could R_c go up?
I can't quite see how B relates to Rc, I thought Rc would be geometrical and based on the conductivity (e.g. steel additives etc.) and lamina thickness of the core.
Could you please enlighten me?

Thanks

 

[tim9000]

You've left me on a cliff-hanger :p

 

[tim9000]

I was having another think today about R_c and B, and how increasing the core cross sectional area and volume, could reduce losses when it would seem that R_c should go up.

I still don't quite understand it but I'm thinking (without evidence) that maybe (I say cautiously because I know you can't speculate on PF) that the losses are based from more than just R_c?

See, I was thinking core losses would be simply I_RC²*R_c
However, maybe it's a combination of this AND the B^1.6 ??
Because that way losses could be based off R_c which would go down (so I_RC would go up, and consequently it's contribution to power) but B might decrease by a factor of more than 1.6, so the total loss actually goes down?

Please let me know if there is some formula or explanation I'm missing.

Thanks

 

[jim hardy]

Steinmetz's original paper is out here someplace on internet, i found it once and linked to it.
...................................
Think about it.
The transformer equivalent circuit is a math model describing the physics not the actiual physics itself. We can refine it ad infinitum.
Iron loss depends on how hard you push the core and it's not linear . It's B^^{1.4 or 1.6 or something.}
You know if you double volts per turn you double flux. If you halve one you halve the other.

(1/2)^^1.6 = 0.32

In the transformer model
P = E²/R_core
Halving volts quarters the right side but only lowers P by 0.32 not 0.25 .
R_core = E² / P

So R has to change.

I'm sure you can work out a R vs E relation using logarithms but that makes equations not intuitive-at-a-glance for me. I have to work first in simple arithmetic then build toward elegance. I expect a roughly square root relation because in dividing you subtract exponents and 2 - 1.6 = 0.4 .

Appreciate any corrections - did i make a thought blooper ?

 

[tim9000]

Steinmetz's original paper is out here someplace on internet, i found it once and linked to it.
...................................
Think about it.
The transformer equivalent circuit is a math model describing the physics not the actiual physics itself. We can refine it ad infinitum.
Iron loss depends on how hard you push the core and it's not linear . It's B^^{1.4 or 1.6 or something.}
You know if you double volts per turn you double flux. If you halve one you halve the other.

(1/2)^^1.6 = 0.32

In the transformer model
P = E²/R_core
Halving volts quarters the right side but only lowers P by 0.32 not 0.25 .
R_core = E² / P

So R has to change.

I'm sure you can work out a R vs E relation using logarithms but that makes equations not intuitive-at-a-glance for me. I have to work first in simple arithmetic then build toward elegance. I expect a roughly square root relation because in dividing you subtract exponents and 2 - 1.6 = 0.4 .

Appreciate any corrections - did i make a thought blooper ?

Thanks Jim,
Yes, if you either halve the volts/turn or double the area of the core, B is halved.
I found this:

Which gives a better explanation than what I saw earlier (more easy for me to digest), what do you think of this? It doesn't have the ^{1.4 or 1.6}, do you suppose they just rounded it up to 2?
I assume the P_core = P_eddy + P_hysteresis ?

So I was thinking that a manufacturer would give you a 'watts / pound' for your steel sheet, (that this would be related to R_C and that the more steel you used, the more watts / pound the core would be consuming. Would the eddy current constant K_e basically just be the conductivity of the steel and the cross-sectional area of the core?

(Sorry, I'm writing this reply very rushed)

Cheers

 

[jim hardy]

It doesn't have the 1.4 or 1.6, do you suppose they just rounded it up to 2?

They place it between 1.5 and 2.5 for hysteresis looks like 2 for eddy current heating

So I was thinking that a manufacturer would give you a 'watts / pound' for your steel sheet,

https://www.mag-inc.com/Products/Ta.../Square-Permalloy-80-Material-Property-Curves