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RLC transients -2

I1s and I2s a...

Question

$I_{1} (s)$ and $I_{2} (s)$ are the Laplace transforms of $i_{1} (t)$ and $i_{2} (t)$ respectively. The equations for the loop currents $I_{1} (s)$ and $I_{2} (s)$ for the circuit shown in the figure, after the switch is brought from position 1 to position 2 at t = 0, are

$⎛ ⎝ \begin{matrix} R + L s + \frac{1}{C s} & - L s - L s & R + \frac{1}{Cs} \end{matrix} ⎤ ⎦ (\begin{matrix} I_{1} (s) I_{2} (s) \end{matrix}] = (\begin{matrix} \frac{V}{s} 0 \end{matrix}]$

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⎛ ⎝ \begin{matrix} R + L s + \frac{1}{C s} & - L s - L s & R + \frac{1}{Cs} \end{matrix} ⎤ ⎦ (\begin{matrix} I_{1} (s) I_{2} (s) \end{matrix}] = (\begin{matrix} - \frac{V}{s} 0 \end{matrix}]

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⎛ ⎝ \begin{matrix} R + L s + \frac{1}{C s} & - L s - L s & R + L s + \frac{1}{C s} \end{matrix} ⎤ ⎦ (\begin{matrix} I_{1} (s) I_{2} (s) \end{matrix}] = (\begin{matrix} - \frac{V}{s} 0 \end{matrix}]

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⎛ ⎝ \begin{matrix} R + L s + \frac{1}{C s} & - L s - L s & R + L s + \frac{1}{C s} \end{matrix} ⎤ ⎦ (\begin{matrix} I_{1} (s) I_{2} (s) \end{matrix}] = (\begin{matrix} \frac{V}{s} 0 \end{matrix}]

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Solution

The correct option is A
$⎛ ⎝ \begin{matrix} R + L s + \frac{1}{C s} & - L s - L s & R + \frac{1}{Cs} \end{matrix} ⎤ ⎦ (\begin{matrix} I_{1} (s) I_{2} (s) \end{matrix}] = (\begin{matrix} \frac{V}{s} 0 \end{matrix}]$

When switch is in position 2,

KVL in loop (1),
$I_{1} (s) \cdot R + \frac{V}{s} + I_{1} (s) \cdot \frac{1}{s C} + [I_{1} (s) - I_{2} (s)] s L = 0$

$\Rightarrow I_{1} (s) [R + \frac{1}{s C} + s L] - I_{2} (s) \cdot s L = \frac{- V}{s}$
KVL in loop (2),

$[I_{2} (s) - I_{1} (s)] s L + I_{2} (s) R + I_{2} (s) \cdot \frac{1}{s C} = 0$

$\Rightarrow - I_{1} (s) \cdot s L + I_{2} (s) [R + s L + \frac{1}{s C}] = 0$
$⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} R + s L + \frac{1}{s C} - s L - s L R + s L + \frac{1}{s C} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ [\begin{matrix} I_{1} (s) I_{2} (s) \end{matrix}] = ⎡ ⎣ \begin{matrix} - \frac{V}{s} 0 \end{matrix} ⎤ ⎦$

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Similar questions

Q. In the given figure assume that the switch was in position 1 for a long time and thrown to position 2 at

t = 0

I_{1} (s)

and

I_{2} (s)

are the Laplace transform of

i_{1} (t)

and

i_{2} (t)

respectively. The equation for the loop currents

I_{1} (s)

and

I_{2} (s)

for the circuit after switch is brought form position 1 to position 2 at

t = 0

Q. In the circuit shown in Figure, switch

S

is closed at time

t = 0

. Let

I_{1}

and

I_{2}

be the currents at any finite time

t

, then the ratio

I_{1} / I_{2}

In the circuit shown figure switch S can be shifted to position A or Position B. Current through resistance is say i₁ when switch is in position A and i₂ when switch is in position B, then

Q. In the circuit shown, switch S is closed at

t = 0

. Let

i_{1}

and

i_{2}

be the current at any finite time t, then the ratio

i_{1} / i_{2}

In the circuit shown if switch S is closed at $t = 0$ , then $i_{1}$ and $i_{2}$ be the current at any finite time $t$ . The ratio $\frac{(i_{1})}{(i_{2})}$ is

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I1(s) and I2(s) are the Laplace transforms of i1(t) and i2(t) respectively. The equations for the loop currents I1(s) and I2(s) for the circuit shown in the figure, after the switch is brought from position 1 to position 2 at t = 0, are

$I_{1} (s)$ and $I_{2} (s)$ are the Laplace transforms of $i_{1} (t)$ and $i_{2} (t)$ respectively. The equations for the loop currents $I_{1} (s)$ and $I_{2} (s)$ for the circuit shown in the figure, after the switch is brought from position 1 to position 2 at t = 0, are