For the system shown in below figure- what is the transfer function VosVis-Given R1-R2-C1-C2-1 unit

After taking Laplace transform, the series combination of R_1 and nbsp;C_1 gives: R_1+1sC_1 Similarly, the parallel combination of nbsp;R_2 and nbsp;C_2 ...

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Control Systems

Definition of Transfer Function

For the syste...

Question

For the system shown in below figure, what is the transfer function $\frac{V_{o} (s)}{V_{i} (s)}$ ?

(Given $R_{1} = R_{2} = C_{1} = C_{2} = 1 u n i t$ )

\frac{V_{o} (s)}{V_{i} (s)} = \frac{s}{s^{2} + 3 s + 1}

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\frac{V_{o} (s)}{V_{i} (s)} = \frac{1}{s^{2} + 3 s + 1}

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\frac{V_{o} (s)}{V_{i} (s)} = \frac{s^{2}}{s^{2} + 2 s + 1}

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\frac{V_{o} (s)}{V_{i} (s)} = \frac{s + 1}{s^{2} + 2 s + 1}

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Solution

The correct option is A $\frac{V_{o} (s)}{V_{i} (s)} = \frac{s}{s^{2} + 3 s + 1}$
After taking Laplace transform, the series combination of $R_{1}$ and $C_{1}$ gives:
$R_{1} + \frac{1}{s C_{1}}$

Similarly, the parallel combination of $R_{2}$ and $C_{2}$ gives:
$\frac{\frac{R_{2}}{s C_{2}}}{R_{2} + \frac{1}{s C_{2}}}$

Substituting, $R_{1} = R_{2} = C_{1} = C_{2} = 1$ makes the above combinations $1 + 1 / s and \frac{1 / s}{1 + 1 / s}$

On applying the voltage divider rule, we get

$V_{o} (s) = \frac{\frac{1 / s}{1 + 1 / s}}{1 + \frac{1}{s} + \frac{1 / s}{1 + 1 / s}} V_{i} (s)$

$\frac{V_{o} (s)}{V_{i} (s)} = \frac{\frac{1}{s + 1}}{\frac{s (s + 1) + (s + 1) + s}{s (s + 1)}}$

$\frac{V_{o} (s)}{V_{i} (s)} = \frac{s}{s^{2} + 3 s + 1}$

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For the system shown in below figure, what is the transfer function Vo(s)Vi(s)? (Given R1=R2=C1=C2=1 unit)

For the system shown in below figure, what is the transfer function $\frac{V_{o} (s)}{V_{i} (s)}$ ?

(Given $R_{1} = R_{2} = C_{1} = C_{2} = 1 u n i t$ )