I have a real physical system where I have a moving coil inducing in a conductive (non-laminated) metal core.
So I decided to model it, at each position, as being a system of 2 coils, one coil being the one where I can apply a voltage signal and the other one being shorted.
I wrote these differential equations to (try to) clarify my head:
(1) -M⋅dI2/dt-L1⋅dI1/dt + I1⋅R1 + V0⋅cos(wt)= 0
(2) -M⋅dI1/dt-L2⋅dI2/dt + I2⋅R2 = 0
They describe KVL in each coil, being M the mutual inductance, L1 and L2 their self inductances, I1 ad I2 the currents in each and V0⋅cos(wt) is the voltage signal I'm applying in coil 1.
The problem is that this alone gives me no idea of how would I adjust M, L1 and L2 to mimic the measures I have, which only inform me the apparent inductance of the coil 1 for different frequencies.
And I couldn't think of any way of solving this system of equations analytically, If I did that and got a closed-form expression I could make a fitting function and fit my measured data.
So, is it possible? Is there an analytical solution for such system?