# Stray inductance of mutual inductors in SPICE

Discussion in 'Electronic Design' started by Robert Latest, Dec 13, 2006.

1. ### Robert LatestGuest

I know: K goes up, Lstray goes down. But what is the precise
mathematical relationship between the two and the inductors?

Thanks: robert

2. ### Tom BruhnsGuest

Try the Wikipedia explanation at
http://en.wikipedia.org/wiki/Inductance, starting about 2/3 of the way
down the page at the part titled "mutual inductance." In short, two
inductors which share some magnetic field have mutual inductance, M.
Remember that in an isolated ideal inductor,

V = L * di/dt

(It may also be useful to recall that Faraday's Law says that there is
an electric potential around any closed loop which is proportional to
the rate of change of magnetic field enclosed by that loop...)

But when you add mutual inductance M between L1 and L2, where i1 is the
current in L1 and i2 the current in L2,

V(L1) = L1 * di1/dt + M * di2/dt.

Then the coupling coefficient k is given by

k = M/sqrt(L1*L2).

Clearly, if k=1, 100% of the magnetic field of L1 is shared by the
turns of L2, and vice-versa; and if k=0, none of the magnetic field of
either inductor is shared by the turns of the other. This gives some
clues about how to maximize k. Can you deduce that k is the fraction
of the magnetic field which is shared?

Cheers,
Tom