Avian’s Blog

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