## Mutual inductance revisited

28.06.2008 18:54

Back in April I posted a question about the definition of the coupling coefficient. Well, I got the answer almost immediately via email from A. B., who confirmed my suspicions of the derivation given in the book and explained to me the correct way of defining k. Now I finally got around to writing a follow up post about this (and I can also at last clear all those books from my desk).

So, the correct way of arriving at the coupling coefficient and the formula for mutual inductance would go like this (using previous flux and current designations):

Mutual inductance between coils 1 and 2 is by definition:

M_{21} = \frac{\Phi_{21}}{i_1}

Due to principle of reciprocity, we can reverse the roles of coils:

M_{21} = M_{12} = \frac{\Phi_{12}}{i_2}

Now you can define the coupling coefficient k as:

k^2 = \frac{M_{21} M_{12}}{L_1 L_2} = \frac{\Phi_{21} \Phi_{12} }{\Phi_{11} \Phi_{22}}

From this definition you can see that the value of k only depends on the values of mutual inductance and self inductances of both coils, which themselves only depend on the geometry (i. e. shapes and positions of coils).

Finally, from this equation it also follows that:

M_{21} = M_{12} = M = k \sqrt{L_1 L_2}

So what is wrong with the derivation in the book? Basically it's this single assumption, that:

\Phi_{21} = k \Phi_{11}

and

\Phi_{12} = k \Phi_{22}

One clear case where this equation doesn't hold is when coils 1 and 2 have the same cross-sections and are placed one on top of another (so that both magnetic fluxes through their cross-sections are equal). By varying the number of turns of both coils, you can set an arbitrary ratio between Φ21 and Φ11.

This example shows that the ratio between both fluxes depends on self inductance of both coils. And indeed the correct equation that follows from the definition of k is:

\Phi_{21} = k \sqrt{ \frac{L_2}{L_1} } \Phi_{11}

So, thanks again to the anonymous reader for clearing this up.

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