Please show all work for this problem thanks Find the s-type transfer function o

Question

Please show all work for this problem thanks

Find the s-type transfer function of the circuit shown Consider Im as the input and Vc as the "output". You MUST show work to get credit Simplify your expression for H(s) so that the denominator is a simple polynomial with no coefficient before the highest-order s term in the denominator. How many zeroes does this transfer function have? If it has 1 or more, express them (probably in terms of R1, R2, R3, L and C). How many poles does this transfer function have?

Explanation / Answer

APPLY LAPLACE TRANSFORM TO THE CIRCUIT AND REPLACING

Inductance with SL, Capcitance with 1/CS

SL and R2 are in series with R3 in parallel the resultance impedence is

Z=(R2+SL)//(R3)= (S*L*R3+R2*R3)/(S*L+(R2+R3))

Apply current division rule for finding current through capcitance

Ic=Iin(s)*Z(s)/(Z(s)+1/c*s)=Iin(s)*(S*L*R2+R2*R3)*C*S/(S^2*C*L*R3+C*S*R2*R3+1)

Voltage across the capcaitor =Ic(s)*(1/C*S)=Iin(s)*(S*L*R2+R2*R3)/(S^2*C*L*R3+C*S*R2*R3+1)

Transfer function = Vc(s)/Iin(s)=(S*L*R2+R2*R3)/(S^2*C*L*R3+C*S*R2*R3+1)

Vc(s)/Iin(s)={(S/C)+(1/C*L)}/{(S^2+S*(R2/L)+(1/L*C*R3))}

Zeros of the transfer function are the numerator poles = s=-1/L

No of Poles of the transfer function are thedenominator poles =2

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