# Are there analytical solutions for systems involving self-inductance and mutual inductance between two coils?

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?

• substitute $I_1=e^{-\gamma t} (A_1 cos(\omega t) + B_1 sin(\omega t))$ and $I_2=e^{-\gamma t}(A_2 cos(\omega t) + B_2 sin(\omega t))$ and solve for the unknowns $A_1,A_2, B_1, B_2, \gamma$ given $V_o$ and $\omega$ – hyportnex May 11 at 11:24