Please help! Mutual inductance (M) and coupling coeffcient (k)

[gazrog]

Hi Guys,

I am really confused about how to caluclate the mutual inductance (M) and coupling coeffcient (k).

I am using two pairs of indentical square loops, so number of turns 1, each on their own piece of FR4 substrate so the structures are planar. The structures are resonant, through the use of a gap at the top of the loop giving a capacitance and forming a RLC circuit. As I understand two resonant structures will give better coupling.

Nearly everything I have read requires k to calculate M or M to calculate k and its driving me nuts. For example:

For mutual inductance, measure the inductance of the primary and secondary in series, and then interchange the connections of one winding for a second reading. Then apply the equation:

M = 1/4 (Lseries(+) - Lseries(-))

But I have read that for a coupling application you cannot just add up the inductances and use that value, you have to add (2M) and minus (2M) respectively when you caluclate Lseries(+) and Lseries(-) in the equation above.

Now I assume in the equation above that M is the mutual inductance of both loops. So I assume you need a formula for the individual loops M's, in order to use them in the series inductance calculation?

Im so confused, any help would be great. Also any good formulas for planar loop structures with respect to coupling would be good as its mostly for loops of wire and transformers.

Many Thanks!

 

[WimRFP]

Did you also find:

Ls/Lo = 1-k^2 ?

Ls is primary inductance when the secundary is short circuited, Lo is the primary inductance when the secundary is left open and k = flux linkage factor (inductor coupling in PSPICE). IF you resonate the primary, you can determine k based on the shift in resonance frequency.

also: k = sqrt( M12*M21/(Ls*Lp) ). s = secondary, p = primary.

 

[gazrog]

Hello,

So are M12 and M21 just the inductance's of the respective coils when the other coil is short circuited? I don't really understand your post.

Many Thanks

 

[WimRFP]

You know that in general: V = L*dI/dt, or in frequency domain, V = jw*L*I. w = omega = 2*pi*f.

Mutual inductance follows the same reasoning, except that the voltage is taken across another inductor (that does not carry the source current). So if you have two inductors (L1 and L2), one can say:

V2 = M21*dI1/dt, or V2 = jw*M21*I1, assuming that current through L2 is zero.

You have also an M12 and of course the self inductances (you may think of M11 and M22 similar to S11 and S22 in scattering parameters).

The complete voltage across L2: V2 = M21*dI1/dt + L2*dI2/dt.

Of course you may read L1 as Lprim and L2 as Lsec (or vica versa)

If I made some typo, I hope somebody will correct me.