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## Homework Statement

Find ##a)\ H(s)=\frac{V_o}{V_s}## of the filter in the circuit below.

##b)\ ##The center frequency ##\omega_0##

##c)\ ##The band-width B

Correct answers:

##a)\ H(s)=\frac{6s}{12s^2+11s+1}##

##b)\ \omega_0=0.28\frac{rad}{s}##

##c)\ B=0.71\frac{rad}{s}##

## Homework Equations

Ohm's law: ##V=IR##

##s=j\omega##

For series resonance:

##\omega_0=\sqrt{\omega_1\omega_2}##

##B=\omega_1-\omega_2##

##\omega_1=-\frac{R}{2L}+\sqrt{(\frac{R}{2L})^2+\frac{1}{LC}}\ ##, Where R is in Ohm's, L in Henry's, C in Farads

##\omega_2=-\omega_1##

## The Attempt at a Solution

Part ##a)\ ##I used KVL to find currents ##I_1\ ##(Left loop) and ##I_2\ ##(Right loop)

##C_2=2F\ \Rightarrow\ \frac{1}{2s}##

##C_1=3F\ \Rightarrow\ \frac{1}{3s}##

Left loop: ##V_s=(1+\frac{1}{2s})I_1-\frac{1}{2s}I_2\ ## (Eqn. 1)

Right loop: ##\frac{-1}{3s}I_2-2I_2+\frac{1}{2s}I_1=0##

##\Rightarrow\ -(\frac{1}{3s}+2)I_2=\frac{-1}{2s}I_1##

##\Rightarrow\ I_1=2s(\frac{1}{3s}+2)I_2##

##\Rightarrow\ I_1=(\frac{2}{3}+4s)I_2\ ## (Eqn. 2)

Plugging (Eqn. 2) into (Eqn. 1): ##V_s=(1+\frac{1}{2s})(\frac{2}{3}+4s)I_2-\frac{1}{2s}I_2##

Distributing and solving for ##I_2##:

##I_2=\frac{V_s}{\frac{8}{3}+4s+\frac{1}{3s}-\frac{1}{2s}}##

Now, ##V_0## using Ohm's Law is ##V_0=2I_2##

Therefore:

##H(s)=\frac{V_o}{V_s}=\frac{2}{\frac{8}{3}+4s+\frac{1}{3s}-\frac{1}{2s}}##

Multiplying by ##\frac{s}{s}##:

##H(S)=\frac{2s}{\frac{8s}{3}+4s^2+\frac{1}{3}-\frac{1}{2}}##

Simplifying:

##H(s)=\frac{2s}{4s^2+\frac{8s}{3}-\frac{1}{6}}##

Parts ##b)\ ##& ##c)##

Two issues: 1) Circuit is not solely series or parallel 2) There is no inductor

1) I can handle series and parallel RLC circuits, but this circuit is mixed. Can I turn this into a series or parallel circuit?

2) My equations for center frequency and band-width involve an inductor value and I'm assuming I shouldn't just ignore it.

Any assistance is greatly appreciated.