## Mutual inductance problem

05.04.2008 17:02

A few days ago I was browsing through my notes from the first year of study and I stumbled upon this interesting problem I've never managed to solve. I've discussed this a couple of times with the late professor Valenčič and we couldn't find a flaw in my line of reasoning. So, if you know what's wrong, please drop me a mail.

In professor's book, there is an introduction to the principles of mutual inductance that goes like this:

Imagine two coils (designated 1 and 2) in some arbitrary relative position to each other. Current i1 that flows through coil 1 will cause a magnetic flux through coil 1 Φ11 = i1 ⋅ L1 (according to the definition of inductance). However some of the flux will also flow through coil 2, designated Φ21.

Mutual inductance between coils 1 and 2 is then by definition:

M21 = Φ21 / i1

Obviously, the magnetic flux Φ21 is less or equal to Φ11, so we define a coupling coefficient k ≤ 1 so that:

Φ21 = k ⋅ Φ11

Now due to principle of reciprocity, the same holds true if the current flows through coil 2 and we calculate the flux through coil 1. Coupling coefficient stays the same:

Φ12 = k ⋅ Φ22

Now multiply both mutual inductances:

M21 M12 = k2 Φ11 / i1 Φ22 / i2

Use definition of inductance:

M2 = k2 L1 L2
M = k sqrt(L1 L2)

Now this final formula is present in a lot of literature and it's certainly correct. It's also certainly true that from the principle of reciprocity M21 = M12. However this way of deriving the formula seems dubious - from steps above you can also see that:

M21 = k Φ11 / i1
M12 = k Φ22 / i2

And since M12 = M21:

M = k L1 = k L2

L1 = L2

Which would mean all pairs of inductances are equal, which is certainly false since we didn't impose any restriction on the geometry of the two coils.

If I would have to guess, I would say something is wrong with the first application of the reciprocity to the fluxes through the coils, however from what I know this should be correct.

So yeah, if you are fluent in electromagnetic theory, I would love to hear your opinion.

Update: see this follow-up post for a solution to this problem

Posted by | Categories: Ideas

I don't know if someone answered this already, since it was from a long time ago, but taking a look at it, I would say that the assumption that the coupling coefficient is the same is a little dubious.

Take for instance a configuration in which a smaller loop in sitting on the same plane inside a larger loop. If current is put through the larger loop, only a portion of the magnetic flux goes through the smaller loop, making k less than one. However if current is put through the smaller loop, all of this magnetic flux would be within the larger loop, making k = 1.

Posted by Laura

Yes, two coils on the same plane inside of each other is a good case to test the assumptions above. But not for the reason you gave.

If current goes through the smaller loop, indeed all of the magnetic flux inside the smaller coil would go through the larger loop. But a part of the flux in the opposite direction would also go through the gap between the coils, again making k < 1.

Posted by Tomaž

I think the problem is here:
You suppose that ( F is the symbol for flux )
F21 = k F11
F12 = k F22

but I think this two "k" should not be identical, and should be different from the coupling coefficient k. Let's rewrite them as
F21 = k21 F11 ,or M21 = k21 L1
F12 = k12 F22 ,or M12 = k12 L2

then from M12 M21 = k12 k21 L1 L2
compare with the definition of coupling coefficient equation M^2 = k^2 L1 L2 ,
it clearly shows that
k12 k21 = k^2
so k12 and k21 are reciprocal but not equal.

Posted by kaworuweb

Yes, that assumption was indeed the problem here. See my follow-up post (there's a link above)

Posted by Tomaž

Φ12=Φ21 ( are reciprocal)
but
Φ11, Φ11 not identical
so k12 and k21 not equal if L1,L2 are different

Posted by Alex Luhovsky