can someone explain mutual inductance?

I have a pile of textbooks here but I'm still having trouble wrapping my brain around this.

(here A is magnetic vector potential, a is area, J is current density, i is current, M is mutual inductance)

I'm modeling loops of wire in FEMM and trying to calculate self- and mutual inductance (because I can look these up.. I have a more complex model to work on next). I have it pretty much down but I am confused by the instructions for calculating mutual inductance, and I am also unsure of how it all comes into play.

The instructions for FEMM say that you can calculate the mutual inductance in a structure by integrating A over the cross-section of a conductor "with only one coil turned on." However, the authors make no effort to explain what "turned on" means. I can leave the secondary out of circuit, which produces a completely unreasonable value for M. I can also make the secondary out of air, or put it in its own circuit with i=0. These approaches come much closer to theoretical values, but I don't know which is correct, and I am sure there is a big difference at some point, because the field lines differ in these cases.

I've googled around for a better explanation but most forum posts seem to be pretty specific to one problem or another. Can someone else offer a source, or pen an explanation, of calculating/modeling self- and mutual inductance in a more generic sense? Even some hints would help.

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Self-inductance is defined as L=V/(dI/dt). Similarly, mutual inductance M between coils 1 and 2 is defined as M_21 = M_12 = V_1/(dI_2/dt) = V_2/(dI_1/dt). This definition requires that the induced voltage be measured in an open circuit, and that the current ramp is slow enough that capacitive effects have a chance to die away.

Say you have two coils with inductance L_1 and L_2 and mutual inductance M. If you wire them in series aiding, the inductance becomes L_1 + L_2

  • 2M, because dI/dt is the same in both coils, so besides the self-inductance contribution, you via mutual inductance, you get the current in L_1 contributing to the field in L_2 and vice versa, which is where the 2M comes from.

If you short out L2, then instead of M producing an open-circuit voltage, it produces the current that cancels out the net magnetic field in L2, which is I_1*(M/L_2) That current, which changes linearly just like I_1, in turn produces an open circuit voltage in L_1 of I_2*M**2/L_2. So the net inductance of the first coil becomes

L_1' = L_1 - M**2/L_2.

These things are easier to express in terms of the coupling coefficient k, which is defined as

k = M/sqrt(L_1*L_2)

and is always less than 1.

Mutual inductance is perfectly well defined in these situations, but it's much easier to calculate when L_2 doesn't perturb the fields due to L_1, which requires that L_2 be open-circuited, just as your manual suggests.


Phil Hobbs

Dr Philip C D Hobbs
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Phil Hobbs

Mutual inductance is essentially the voltage you see in one loop of wire when you change the current through another loop of wire.

I had trouble with it for years before I ran into the transformer equation

V1=3D L1.dI1/dt +M.dI2/dt

V2=3D M. dI1/dt + L2.dI2/dt

M is less than or equal to root L1.L2. . Transformers usually give you an mutual inductance that is more then 99% of the geometric mean of the primary and secondary inductance.

-- Bill Sloman, Nijmegen

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Bill Sloman

I was doing something basic not to long ago and some of this is currently at the top of my head.

M = Mutual Inductance.

Is basically the amount of inductance that is shared between multiple windings and works with the coefficient number of K which can range from lowww (.xxx) up to 1.0. Because there is virtually no such thing as perfect coupling between 2 or more windings, the K is used in the calculations and averages from .7 to .99

so M = K*sqrt(Lpri*LSec) which is the amount of L (inductance) that is shared between the windings.

If you would like a text on it, I think I have something kicking around on the PC..

Hoped that did something for you.

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Okay, this helps me understand where I'm falling short.

I'm not very experienced with FEMM, so I don't know for sure, but it appears to me that I can't really do this in FEMM. Because it models a coil as an "axisymmetric problem," I'm only looking at a cross-section of the wire and it looks like this problem can't be solved this way. If I have an open coil, then current is going to zero at the ends and have a maximum somewhere in the middle. Unfortunately this means I can't do the problem I'm trying to do as FEMM only works in two dimensions.

I have some more thoughts on this but I need to try some calculations first and confirm my suspicions. Thanks for the nudge.

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