1
\$\begingroup\$

I have no problem finding the transfer function with systems of 1 input and 1 output and then I can easily obtain the space state representation, but this exercise requires me to "Find the space state representation of the circuit, assuming the outputs are the currents in R1 and R2". How do I start thinking about this problem?

enter image description here

\$\endgroup\$
  • \$\begingroup\$ Write two equations, one each for i1 and i2 that depend on v1 and v2? \$\endgroup\$ – Daniel Feb 11 '17 at 17:31
2
\$\begingroup\$

A possible solution:

Dirceu Rodrigues Jr

Choosing state variables as current in the inductor and the voltage in the capacitor:

$$ x_1 = i_L = i_1 $$ $$x_2 = v_C $$

Applying KVL and KCL to the left mesh and top node, respectively:

$$ v_1 - R_1x_1 -L\dot{x_1}-x_2=0 $$ $$ -x_1 + C\dot{x_2} + \frac{x_2-v_2}{R_2} = 0 $$

Rearranging:

$$ \dot{x_1}= -\frac{R_1}{L}x_1 -\frac{1}{L}x_2 + \frac{1}{L}v_1$$ $$ \dot{x_2}=\frac{1}{C}x_1-\frac{1}{CR_2}x_2 + \frac{1}{CR_2}v_2 $$

In standard state space representation: $$ \mathbf{\dot{x}=Ax+Bu} $$ $$ \mathbf{y = Cx+Du}$$

And, since that \$i_2=\frac{x_2-v_2}{R_2}\$:

$$\begin{bmatrix}\dot{i_1}\\\dot{v_c}\end{bmatrix}=\begin{bmatrix}-R_1/L&-1/L\\1/C&-1/CR_2\end{bmatrix}\begin{bmatrix}i_1\\v_c\end{bmatrix}+\begin{bmatrix}1/L&0\\0&1/CR_2\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}$$

$$\begin{bmatrix}i_1\\i_2\end{bmatrix}=\begin{bmatrix}1&0\\0&1/R_2\end{bmatrix}\begin{bmatrix}i_1\\v_c\end{bmatrix}+\begin{bmatrix}0&0\\0&-1/R_2\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}$$

\$\endgroup\$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.