What is the transfer function for the network shown below $\frac{V_{c_{2} (s)}}{V_{i (s)}}$

\frac{V_{c 2 (s)}}{V_{i (s)}} = \frac{1}{(s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}) (1 + s C_{1} R_{1} + s^{2} C_{1} L_{1}) + s C_{2}}

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\frac{V_{c 2 (s)}}{V_{i (s)}} = \frac{s C_{1}}{(s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}) (1 + s C_{1} R_{1} + s^{2} C_{1} L_{1}) - s C_{2}}

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\frac{V_{c 2 (s)}}{V_{i (s)}} = \frac{s C_{1}}{(s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}) (1 + s C_{1} R_{1} + s^{2} C_{1} L_{1})}

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\frac{V_{c 2 (s)}}{V_{i (s)}} = \frac{1}{(s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}) (1 + s C_{1} R_{1} + s^{2} C_{1} L_{1})}

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Solution

The correct option is B $\frac{V_{c 2 (s)}}{V_{i (s)}} = \frac{s C_{1}}{(s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}) (1 + s C_{1} R_{1} + s^{2} C_{1} L_{1}) - s C_{2}}$

Applying KVL in loop-1,
$V_{i} (s) = R_{1} I_{1} (s) + \frac{1}{s C_{1}} (I_{1} (s) - I_{2} (s)) + s L_{1} I_{1} (s)$ ...(i)
$V_{i} (s) = I_{1} (s) [R_{1} + \frac{1}{s C_{1}} + s L_{1}] - \frac{I_{2} (s)}{s C_{1}}$ ...(ii)

Applying KVL in Loop-2

$\frac{1}{s C_{1}} (I_{2} (s) - I_{1} (s)) + V_{c_{2}} (s) + R_{2} I_{2} (s) = 0$ ....(iii)

$V_{c_{2}} (s) - \frac{I_{1} (s)}{s C_{1}} + I_{2} (s) (\frac{1}{s C_{1}} + R_{2})$ =0 ...(iv)

$V_{c_{2}} (s) = \frac{I_{2} (s)}{s C_{2}}$ ...(v)

$∴ I_{2} (s) = s C_{2} V_{c_{2}} (s)$ ...(vi)

From equation (vi) and (iv), we get

$V_{c_{2}} (s) - \frac{I_{1} (s)}{s c_{1}} + s C_{2} V_{c_{2}} (\frac{1}{s C_{1}} + R_{2}) = 0$ ...(vii)

$I_{1} (s) = s C_{1} V_{c_{2}} (s) (1 + s C_{2} (\frac{1}{s C_{1}} + R_{2}))$ ...(viii)

$I_{1} (s) = s C_{1} V_{c_{2}} (s) (\frac{s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}}{s C_{1}})$ ...(ix)

$I_{1} (s) = V_{c_{2}} (s) (s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2})$ ....(x)

Substitue equation (vi) and equation (x) in equation (ii), we get

$V_{i} (s) = V_{c_{2}} (s) (s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}) (R_{1} + \frac{1}{s C_{1}} + s L_{1}) - \frac{s C_{2} V_{c_{2}} (s)}{s C_{1}}$ ....(xi)

$V_{i} (s) = V_{c_{2}} (s) \frac{(s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}) (1 + s C_{1} R_{1} + s^{2} L_{1} C_{1})}{s C_{1}} - \frac{s C_{2} V_{c_{2}} (s)}{s C_{1}}$ ...(xiii)

The transfer function is

$\frac{V_{c_{2} (s)}}{V_{i (s)}} = \frac{s C_{1}}{(s C_{1} + s C_{2} + s^{2} C_{1} C_{2} R_{2}) (1 + s C_{1} R_{1} + s^{2} C_{1} L_{1}) - s C_{2}}$

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