passive networks - Finding state equations for an RL circuit with inductors in T configuration?

I'm working on an exercise which asks to "Identify state variables and write apropriate state equations" for the following circuit.

^{simulate this circuit – Schematic created using CircuitLab}

Trying to find the answer I get the following system

$$ \begin{cases} v = v_S-R_1i_1-(L_1-M){i_1}', \\v=(L_2-M){i_2}' + R_2i_2, \\v=M{i_3}', \\i_1=i_2+i_3. \end{cases} $$

from which I can't properly derive the requested state equations - I end up with equations like \$0=0\$

The difficulty seems to me to be purely mathematical but as the problem is related to circuit design I guess this forum is better place to ask this question than the mathematical one.

So, I need some help.

Thanks

Answer

In comments you said that what you're really after is to "Identify state variables and write appropriate state equations", rather than solve the circuit. I'll answer how to write the state equations.

In an analog circuit, the state variables are inductor currents and capacitor voltages. So here, the state variables are \$i_1\$, \$i_2\$, and \$i_3\$.

To find the state equations, first, substitute your 3rd equation into your first two:

$$Mi'_3 = v_s -R_1 i_1 -(L_1-M)i'_1$$ $$Mi'_3 = (L_2-M)i'_2 + R_2 i_2$$

and re-write your fourth equation in terms of derivatives:

$$i'_1 = i'_2 + i'_3$$

Now move all the time-derivative terms to the left and other terms to the right:

$$ \begin{align}-(L_1-M)i'_1 + Mi'_3 &= -R_1 i_1 + v_s\\ (L_2-M)i'_2 + Mi'_3 &= R_2 i_2\\ i'_1-i'_2-i'_3 &= 0\end{align}$$

Now you have a system with the form

$${\bf M}\left(\begin{matrix}{i'_1\\i'_2\\i'_3}\end{matrix}\right)={\bf A}\left(\begin{matrix}{i_1\\i_2\\i_3}\end{matrix}\right)+{\bf B}v_s$$

Now you just have to pre-multiply each side of the equation by \${\bf M}^{-1}\$, or, equivalently, do algebra to eliminate all but one of the three derivative terms in each line, and you'll have the expected state equations:

$$\left(\begin{matrix}{i'_1\\i'_2\\i'_3}\end{matrix}\right)={\bf M}^{-1}{\bf A}\left(\begin{matrix}{i_1\\i_2\\i_3}\end{matrix}\right)+{\bf M}^{-1}{\bf B}v_s$$

Finding \${\bf M}^{-1}\$ will be tedious, so I'll leave that to you since this was your homework.