Question

For the given circuit, which one of the following is the correct state equation?

• A. $\frac{d}{dt}\left[\frac{v}{i}\right]=\left[\begin{array}{cc}-4& 4\\ -2& -4\end{array}\right]\left[\frac{v}{i}\right]+\left[\begin{array}{cc}0& 4\\ 4& 0\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\end{array}\right]$
• B. $\frac{d}{dt}\left[\frac{v}{i}\right]=\left[\begin{array}{cc}-4& -4\\ -2& -4\end{array}\right]\left[\frac{v}{i}\right]+\left[\begin{array}{cc}4& 0\\ 0& 4\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\end{array}\right]$
• C. $\frac{d}{dt}\left[\frac{v}{i}\right]=\left[\begin{array}{cc}-4& -4\\ -2& 4\end{array}\right]\left[\frac{v}{i}\right]+\left[\begin{array}{cc}4& 4\\ 4& 0\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\end{array}\right]$
• D. $\frac{d}{dt}\left[\frac{v}{i}\right]=\left[\begin{array}{cc}4& -4\\ -2& -4\end{array}\right]\left[\frac{v}{i}\right]+\left[\begin{array}{cc}0& 4\\ 4& 4\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\end{array}\right]$

Answer 1

The correct option is A $\frac{d}{dt}\left[\frac{v}{i}\right]=\left[\begin{array}{cc}-4& 4\\ -2& -4\end{array}\right]\left[\frac{v}{i}\right]+\left[\begin{array}{cc}0& 4\\ 4& 0\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\end{array}\right]$

From source transformation

KVL in loop 1,
$2i_1=2i+0.5\frac{di}{dt}+V$
$\frac{di}{dt}=-2V-4i+4i_1...\left(i\right)$
KCL at node (a),
$i=0.25\frac{dV}{dt}+\frac{V-i_2}{1}$
$\frac{dv}{dt}=-4V+4i+4i_2...\left(ii\right)$

$\frac{d}{dt}\left[\frac{v}{i}\right]=\left[\begin{array}{cc}-4& 4\\ -2& -4\end{array}\right]\left[\frac{v}{i}\right]+\left[\begin{array}{cc}0& 4\\ 4& 0\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\end{array}\right]$