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

schematic

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

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    \$\begingroup\$ Are you looking for a DC steady state solution, AC steady state solution, or transient solution? \$\endgroup\$ – The Photon Jul 12 '17 at 2:54
  • \$\begingroup\$ @ThePhoton The form of the input voltage is not specified in the exercise. It says literally: "Identify state variables and write appropriate state equations". \$\endgroup\$ – Ruslan Jul 12 '17 at 3:35
  • \$\begingroup\$ In your exercise, does it say anything about jw or s? Because the first thing that comes to my mind is that I want to solve it by applying a Laplace transform. \$\endgroup\$ – Harry Svensson Jul 12 '17 at 4:06
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    \$\begingroup\$ Please edit the question to clarify whether your goal is to solve the system or to write the state equations. \$\endgroup\$ – The Photon Jul 12 '17 at 4:17
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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.

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  • \$\begingroup\$ Accomplishing the final steps in the solution you've proposed convinced me of my worst fears - that right hand side of each of the state equations includes multiple state variables... Fortunately, the course I'm following doesn't ask to solve further such equations (not in this chapter at least). Thank you very much for the help! \$\endgroup\$ – Ruslan Jul 13 '17 at 2:07

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