# Second Order Laplace Trasform For RLC Circuit

• Engineering

## Homework Statement

The circuit below (please see attactment) shows an RLC filter circuit whereby RC = 1/2 and LC = 1/16. Determine the differential equation that describes the relationship beween the input voltage Vi(t) and output voltage Vo(t). If the initial conditions of the capacitor and inductor are zero, determine the laplace frequency response H(s) = Vo(s)/Vi(s) of the filter and calculate the magnitude of |H(s)| s=jWc at the frequency Wc = 1/sqrt(LC).

KVL

## The Attempt at a Solution

So Ive worked out the differential equation as:

Vi(t) = LC(Vo(t))'' + RC(Vo(t))' + Vo(t)

Here is my attempt at the laplace transform and subsequent frequency response:

Vi(s) = LC[s2Vo(s) - sVo(0-) - Vo'(0-)] + RC[sVo(s) - Vo(0-)] + Vo(s)

Vi(s) = LCs2Vo(s) + RCsVo(s) + Vo(s)

Vi(s) = (LCs2 + RCs + 1)Vo(s)

Vi(s)/Vo(s) = 1 + RCs + LCs2

Vo(s)/Vi(s) = 1/ (1 + RCs + LCs2) = H(s)

In which case given 1/sqrt(LC) = 4 and RC = 1/2 and LC = 1/16
at the frequency 1/sqrt(1/16) |H(s)| is:

H(s) = 1/ (1 + RCs + LCs2)

H(s) = 1/ (1 + 1/2(4) + 1/16(16)

H(s) = 1/ (1 + 2 + 1)

H(s) = 1/4 = 0.25

Does this solution look correct or have I gone wrong somewhere in the laplace transform. ie I have assumed Vo'(0) = 0 given that the question stipulates that the capacitor and inductor have 0 charge initially.

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