# Misconceiving Mutual Inductance Coefficients

A commonly used formula for mutual inductance M between two nearby coils L1 and L2 is M = k√(L_{1}*L_{2}). This formula however assumes equal percentage flux linkages between the two coils. This requirement is often omitted in several of the references I have looked at.

That this requirement is not necessarily met can be seen by looking at fig. 1, which depicts a single-turn coil L2 imbedded within a long solenoid L1 with N_{1} turns. It is obvious that flux from L1 essentially totally links L2, whereas flux from L2 equally obviously is coupled only partially into L1.

As is well known, M_{21} = M_{12} = M. (The proof is not trivial; Feynman uses the magnetic vector potential while Skilling invokes an energy argument.)

The following fallacious reasoning can ensue, and I speak from experience! So M can be computed either way per the above. Obviously, it’s much easier to go with M_{21} since φ_{21} = φ_{11} . I won’t derive the result, which is easily done and is M = μ_{0}A/l in vacuum

where A = cross-sectional area of both coils and l = length of L1.

So now we say OK, there is 100% flux coupling between L1 and L2 so k = 1 and M = the (too!) familiar k√(L_{1}L_{2}). WRONG!

Were it right we could do an amazing thing: we could easily compute L_{2} which otherwise happens to be an exceedingly difficult task; L_{2} depends on A *and the wire radius R *and can only be derived from a special integral to some arbitrary order. But – we know L_{1}, we know M and we *think* we know k so L_{2} can easily be computed!

Not so! The result of this fallacious computation would imply that L_{2} is a function of L_{1}, even in the total absence of solenoid current i_{1}! And you could also ignore little inconvenient realities like L_{2} → ∞ as R → 0!

The thing is, there are really *two* coupling coefficients. Define

k_{1} = φ_{21}/φ_{11} and

k_{2} = φ_{12}/φ_{22}. Then we can perform a few substitutions based on the above to find that

M = √(k_{1}k_{2}L_{1}L_{2}).

Bottom line here is that we know k_{1} but not k_{2}, and L_{1} but not L_{2}. *There is one coupling coefficient k = k _{1} = k_{2} if and only if the fraction of flux from one coil coupling into the other is the same for both coils*.

Even if the single-turn coil were replaced by another long solenoid _{L2}, but of different length, wrapped on top of L1 so that A is the same for both, and L_{2} were as easily computed as L_{1}, still there are two different coupling coefficients involved, and the familiar formula L_{2}/L_{1} = (N_{2}/N_{1})^{2} is incorrect.

AB Engineering and Applied Physics

MSEE

Aerospace electronics career

Used to hike; classical music, esp. contemporary; Agatha Christie mysteries.

How does a coil work with matching box in an ion source chamber when producing a beam or making plasma,i am all confused as i am a student?

Nice work @rude man

Thank you!

I am not sure what you mean by a “matching box” and “making plasma”; could you give me a link that describes this setup?