How does mutual inductive coupling affect transformer operation?

[tim9000]

Hi,
I have a couple of questions regarding re-wiring a three phase transformer.
Say you have a three phase TX with a ratio of 415:22 and you want to use it as a single phase step-up TX (excite it with a variac or something), this is a thought-experiment.

If you excited only the middle leg of the many turn side (leaving the other two outer 415 coils open), and series-ed up the outer two legs of the 22 coils, as in the RHS of the picture, you would have Φ flux equally through each secondary, so it'd be a 415:44. (Because you left the middle 22 coil open). However, if you also series-ed up the middle leg secondary in reverse (compared to the outer two, as in the LHS of the picture), because it has 2xΦ flux as the outer two, I assume it would have double the voltage. So what would the resulting turns ratio be? 415:66? Or 415:88?

My second question is about the mutual coupling coefficient, between all the coils on a TX core, single or three phase, does this 'M' have to equal a total maximum of '1' between all the coils? That is to say, even if you have multiple identical coils on the same core, coupled well, between any two the coupling is only at most 0.5? For example on a three phase core, the middle coil will link to the other two 0.5 each.
And a side question about this, is that if you were actually looking at the coupling of one limb to another, given that the coils are not wound one on top of the other on the same limb. The flux has to travel through the core to it. How much less will the coupling coefficient be, compared to if the coils were both wound over each-other (say, bifilar) on the same limb? (How much increased leakage flux, very generally speaking.)

Thanks very much

 

[jim hardy]

If you excited only the middle leg of the many turn side (leaving the other two outer 415 coils open), and series-ed up the outer two legs of the 22 coils, as in the RHS of the picture, you would have Φ flux equally through each secondary, so it'd be a 415:44.

??
It'd be a 415::22 . Each outer 44 volt winding has half flux so makes only 11 volts.

However, if you also series-ed up the middle leg secondary in reverse (compared to the outer two, as in the LHS of the picture), because it has 2xΦ flux as the outer two, I assume it would have double the voltage. So what would the resulting turns ratio be? 415:66? Or 415:88?

As you drew it it'd be 415::0 . That's because, if you look at direction you connected them, center is in series opposing not aiding the outer two. Flux goes UP it and DOWN the others. Your dot is on wrong end.
If you swap ends so it's in series aiding you'll get 415::44.

 

[tim9000]

??
It'd be a 415::22 . Each outer 44 volt winding has half flux so makes only 11 volts. View attachment 206518

Hah! you're right, one 415 Φ through a 22 coil would only be 11V, so yeah, 22V secondary.

As you drew it it'd be 415::0 . That's because, if you look at direction you connected them, center is in series opposing not aiding the outer two. Flux goes UP it and DOWN the others. Your dot is on wrong end.
If you swap ends so it's in series aiding you'll get 415::44.

View attachment 206518

Yes, you're right that the middle limb flux is going up, while the outer limb fluxes are going down. I did know that and I meant to draw the coil winding itself reversed on that limb of the core diagram. That's why I drew the dot on the top for that one. But I see I made a mistake. Assuming that the middle limb coil is connected opposite to the outer coils, and given the above quote, wouldn't it be 415:44 as you say?

But how is the dot on the electrical diagram drawn the wrong way?Thanks Jim!

 

[jim hardy]

But how is the dot on the electrical diagram drawn the wrong way?

well, look how you drew the dots they're in series aiding
your schematic drawing shows most of them upside down from your physical drawing adding to the confusion

assume a current direction and do right hand rule, one aids flux while others oppose it, or vice versa

 

[tim9000]

well, look how you drew the dots they're in series aiding
your schematic drawing shows most of them upside down from your physical drawing adding to the confusion

assume a current direction and do right hand rule, one aids flux while others oppose it, or vice versa

Ah, I think I see what my issue was. I should have reversed the winding, which keeps the dot the same as the others, rather than reversing the dot.

So an analogy would be like this:
(*attached*)

 

[jim hardy]

I should have reversed the winding, which keeps the dot the same as the others, rather than reversing the dot.

Well , depends on whether you want consistency in your drawing between directions indicated for windings and dots and polarities.

Being consistent helps our thinking.

 

[tim9000]

Well , depends on whether you want consistency in your drawing between directions indicated for windings and dots and polarities.

Being consistent helps our thinking.

Indeed. So in conclusion, you can add the EMF from coils, just like you would sum the voltages from a 11V + 22V + 11V = 44V series of batteries?
Thanks

 

[jim hardy]

Indeed. So in conclusion, you can add the EMF from coils, just like you would sum the voltages from a 11V + 22V + 11V = 44V series of batteries?
Thanks

So long as you write the signs in accordance with the (properly drawn) dots. Right hand rule rules.

 

[tim9000]

So long as you write the signs in accordance with the (properly drawn) dots. Right hand rule rules.

Okay, great, so no issue with imbalance of counter-emf, etc. I didn't think so, but wanted to make sure.

How about if the coils are wound on the same core but different limbs? How much worse is the mutual coupling coefficient:

And a side question about this, is that if you were actually looking at the coupling of one limb to another, given that the coils are not wound one on top of the other on the same limb. The flux has to travel through the core to it. How much less will the coupling coefficient be, compared to if the coils were both wound over each-other (say, bifilar) on the same limb? (How much increased leakage flux, very generally speaking.)

Also, for any set of coils wound on a core, does the mutual coupling coefficient for the whole set have to add up to a (maximum) total of 1? This is to say, for instance if you had three coils on a 'E' core, between each coil M would be = approximately 1/3 ?

Thanks!

 

[jim hardy]

How about if the coils are wound on the same core but different limbs? How much worse is the mutual coupling coefficient:

With the flux being able to shift around among the three limbs i don't know how you'd calculate M.
Wouldn't it vary with load on the individual limbs?

 

[tim9000]

With the flux being able to shift around among the three limbs i don't know how you'd calculate M.
Wouldn't it vary with load on the individual limbs?

Say that you were considering it more from a geometric perspective, without load. ...well Okay, say at rated load, operating peak flux at knee point. I'm just wondering, roughly speaking, how much of a difference would bifilar winding make to the amount of leakage flux, and how much is going to couple through the core anyway. I'd think at most 5% difference, at a guess. But someone with more industry experience might say 'oh winding the coils together is a must, for performance'...

 

[tim9000]

With the flux being able to shift around among the three limbs i don't know how you'd calculate M.
Wouldn't it vary with load on the individual limbs?

I can't find my inductance calculations book (still renovating house), does the mutual coupling coefficient for a series of coils all on the same core add up to a total of 1? For example, if you were looking at at three phase Isolation transformer, ignoring self-coupling, what would the coupling coefficient between one primary coil to another be approximately? Because I tried simulating at three phase TX on LTSpice and I had issues making the coefficients near 1 (it tells me I've made an impossible relation). Intuitively, I thought it'd be about M = 1/2 between each adjacent primary pair of coils, and the same between adjacent primary-secondary pairs, with each of the primary-secondary pairs sitting on the same limb at about M = 1.Thanks