- #1
Captain1024
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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.