# State Space Model for Variable Mutual Inductance

This is a hell of a request, but today I have recently introduced myself to the concept of the 'State Space' model. I've understood, to a basic level, how state space models work and I am convinced that state space is the correct method to progress with my current problem (maybe I'm wrong?).

The practicality of the situation is essentially me trying to recreate/model an age old Physics experiment known as the Thomson Coil Induction Ring Experiment and I wish to simulate the experiment via the coupled electrical equations that the ring and coil possess. The experiment was traditionally carried out with a capacitor bank discharging into the coil which in turn produces a changing current which in turn generates a changing magnetic flux density that produces current on the surface of the conductive ring.

Here is the question. I have the following model of two differential equations that I believe successfully describe the relationship between both LRC circuit and the ring. Differential Equation 1 is an LRC discharge circuit that possesses mutual inductance with another RL circuit (represented by Differential Equation 2) that moves away due to the magnetic flux density produced in the LRC circuit from Differential Equation 1, hence there is variable mutual inductance. The equations are as follows:

$$Vc = L_1 \frac{di_1}{dt}+i_1 R_1 - \frac{d}{dt}\left[M i_2\right]$$ and $$0 = L_2 \frac{di_2}{dt}+i_2 R_2 - \frac{d}{dt}\left[M i_1\right]$$

These both expand to the following: $$Vc = L_1 \frac{di_1}{dt}+i_1 R_1 - i_2\frac{dM}{dt}-M\frac{di_2}{dt}$$ and $$0 = L_2 \frac{di_2}{dt}+i_2 R_2 - i_1\frac{dM}{dt}-M\frac{di_1} {dt}$$

Since I am new to the concept of state space, a situation like this is most challenging for someone who has only just got to used to the concepts behind state space. To complicate matters further, I am interested in the evolution of the Mutual inductance, not with time, but rather with position, e.g.:

$$\frac{dM}{dt}\cdot\frac{dz}{dz} = \frac{dz}{dt}\cdot\frac{dM}{dz}= v\frac{dM}{dz}$$

As you can see this is a very very large request for assistance with deriving a state space model. I am also aware that this is non-linear and hence, takes more effort to produce results for due to the complexity.

Kind regards, Vhaanzeit