Measuring Mutual Inductance between inductors

Hi everybody,

I'm trying to measure the mutual inductance between two inductors using a Network Analyzer. To do that, I follow the two-ports calibration procedure and than I measure the S and Z parameters. Considering the generic circuit of two coupled coils, I would expect that the mutual inductance value is "hidden" in the Z12 / Z21 parameter and that it can be found by using the formula:

M = Z12 / ( 2 pi freq)

, since in theory Z12 is purely imaginary. What I found is a real part which is almost constant in a wide range of frequency (≈500 Ω) and an imaginat part which increases as expected.

Can I neglect the Z12(Real) and use the formula to get the M parameter?

Thak you very much!

Matte

essentially M=imag(Z12)/angular freq . So the answer is yes.

Hello, could you please explain why M = Z12 / ( 2 pi freq) ? I am a novice in EMC. Please explain it detaily. Thank you very much.

laren said:

Hello, could you please explain why M = Z12 / ( 2 pi freq) ? I am a novice in EMC. Please explain it detaily. Thank you very much.

Click to expand...

Hi Laren,

the relationship comes from the definition of the impedance matrix Z of a circuit consisting in two coupled inductors. Since this is a 2-port system, you will have a 2x2 matrix describing the system: (V_1 V_2) = [Z] * (I_1 I_2) . If you write the relation between voltages and currents of the two port system you will find something like (neglecting parasitic capacitance and considering resistance):

V_1 = (R1 + jwL1)I_1 + jwM* I_2
V_2 = (R2 + jwL2)I_2 + jwM* I_1

Where R1 and R2 are the parasitic resistances of the two inductors, L* the auto inductance and w=2pif. Re-write this relation in "Matrix-mode" and you discover that Z12 = jwM and obviously M = Z12/2pifreq.

Note that Z12 = Z21 since mutual inductance between the inductors is the same.

Bests

Thank you so much. matte
I have a further question for you.
does Z11 Z12 Z21 Z22 refer to the S11 S12 S21 S22 respectively?
I am confused what S parameter exactly means. I hava a network analyzer at hand. But i don't know who to use it to extract mutual inductance. Could you please explain it for me? Thank you so much.

NO, definition of S and Z parameters is completely different. S-parameters are widely used in RF and they defined on the transmission line theory (reflection coefficient etc...). When the frequency goes up and the wavelength decreases, a network should be described by distributed parameters..

I suggest you to study, this is not the best place where learn this stuff.


Anyway, probably you network analyzer can give you also Z parameters (it was my case..try looking in options). If you have only S-parameters you can use conversion tables to calculate Z-parameters:

https://en.wikipedia.org/wiki/Impedance_parameters#Relation_to_S-parameters

have a nice day

I really appreciate what you have explained to me. Thank you. Have a nice day too