# Mutual Inductance

Discussion in 'Electronic Design' started by Tim Williams, Mar 6, 2007.

1. ### Tim WilliamsGuest

So, I was wondering, what is the electrical model of mutual inductance?

It's explained in my physics textbook, with no electrical significance as
far as I can tell.

Leakage inductance is far simpler to deal with, but I noticed that my
textbook doesn't mention it and internet sites deal with cored examples.
Air core transformers have leakage inductance too!!

Purpose: wondering what the LL of coupled, concentric solenoids with
different diameters is.

Presumably, k (coupling konstant) is roughly the ratio of areas, but I can't
find where to get LL from that.

Tim

2. ### Rich GriseGuest

OK, don't quote me on this, as "officially" I'm "only" a Tech, but to
me it just seems intuitive that the mutual inductance is how much coupling
you have left after you subtract out the leakage inductance.

What that means in numbers, I have no idea. I have heard that they're
all relatively simple to measure - it has something to do with shorted
and open windings and schtuff like that.

Maybe google for "mutual inductance" or "coupling coefficient" or so.

Good Luck!
Rich

3. ### Guest

The transformer equations are

V1 = L1. d I1/ dt + M . d I2/dt

V2 = M. d I1/dt + L2. d I2/dt

where M is less than the geometric mean of L1 and L2, in proportion to
the extent that the flux from the two coils is imperfectly coupled.
For non-cored solenoids it would have to be close to the area ratio,
but then again, the aspect ratio of the solenoids is going to have to
come into it.

4. ### Helmut SennewaldGuest

Hello Tim,

We had a discussion in the LTspice group about coupling factors sometimes
ago.

Maybe in appendix-A of this paper is the answer.

http://focus.ti.com/lit/ml/slup171/slup171.pdf

Mr. Lloyd H. Dixon gives an example about the difference of
K12, K21 and "K". See Appendix A in this paper.

Best regards,
Helmut

5. ### Robert BaerGuest

I did not even get to appendix A; note on page 4, second column
"Thus. the "leakage" flux links the outer winding but not the inner
winding."
*Very* loud smack in the face that leakage inductance one way is
different than the other way.
So those "nice" symmetrical looking equations are all lies!

6. ### The PhantomGuest

You understand self-inductance, yes? Ignoring winding resistances, given
a two winding transformer, if you apply a current to one winding (or to a
single winding inductor), the voltage across the inductor is L di/dt. In
the case of a sinusoidal AC current i of frequency f in the inductor, the
voltage e across the inductor is
e = L*2*pi*f*i, or L = e/(2*pi*f*i). L is the ratio of the voltage across
the winding to the rate of change of the current in the winding.

Roughly, it's a relationship between the voltage across a winding and the
current producing it.

In the case of mutual inductance, it's the ratio of the voltage across a
winding to the rate of change of current in *another* winding.
Frederick Grover's book on inductance calculations
http://www.amazon.com/Inductance-Ca...5449707?ie=UTF8&s=books&qid=1173230870&sr=1-1
will tell you how to calculate the self-inductance of the solenoids and
also their mutual inductance. From this you can calculate the leakage
inductance. However, I will tell you that it will probably be easier to
build them and measure.

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