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GATE 2020 EC – Question 40

Control Systems · State Variable Model and Solution of State Equation · 2 marks · Multiple choice

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

Diagram for GATE 2020 EC question 40
  1. $\dfrac{d}{dt}\begin{bmatrix}v\\i\end{bmatrix}=\begin{bmatrix}-4&4\\-2&-4\end{bmatrix}\begin{bmatrix}v\\i\end{bmatrix}+\begin{bmatrix}0&4\\4&0\end{bmatrix}\begin{bmatrix}i_1\\i_2\end{bmatrix}$
  2. $\dfrac{d}{dt}\begin{bmatrix}v\\i\end{bmatrix}=\begin{bmatrix}-4&-4\\-2&4\end{bmatrix}\begin{bmatrix}v\\i\end{bmatrix}+\begin{bmatrix}4&4\\4&0\end{bmatrix}\begin{bmatrix}i_1\\i_2\end{bmatrix}$
  3. $\dfrac{d}{dt}\begin{bmatrix}v\\i\end{bmatrix}=\begin{bmatrix}4&-4\\-2&-4\end{bmatrix}\begin{bmatrix}v\\i\end{bmatrix}+\begin{bmatrix}0&4\\4&0\end{bmatrix}\begin{bmatrix}i_1\\i_2\end{bmatrix}$
  4. $\dfrac{d}{dt}\begin{bmatrix}v\\i\end{bmatrix}=\begin{bmatrix}-4&-4\\-2&-4\end{bmatrix}\begin{bmatrix}v\\i\end{bmatrix}+\begin{bmatrix}4&0\\0&4\end{bmatrix}\begin{bmatrix}i_1\\i_2\end{bmatrix}$

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Correct answer: (A) $\dfrac{d}{dt}\begin{bmatrix}v\\i\end{bmatrix}=\begin{bmatrix}-4&4\\-2&-4\end{bmatrix}\begin{bmatrix}v\\i\end{bmatrix}+\begin{bmatrix}0&4\\4&0\end{bmatrix}\begin{bmatrix}i_1\\i_2\end{bmatrix}$

Explanation

At the left node, $i_1=\frac{V_1}{2}+i$, so $V_1=2(i_1-i)$. The inductor gives $0.5\frac{di}{dt}=V_1-v=2i_1-2i-v$, i.e. $\frac{di}{dt}=-2v-4i+4i_1$. At the capacitor node, $i+i_2=0.25\frac{dv}{dt}+v$, i.e. $\frac{dv}{dt}=-4v+4i+4i_2$. These match option A.