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LCR Circuit analysis using complex impedances

Please refer to the attachment for questions complete with circuit diagrams, and provide full answers.

1a) Derive an expression for the transfer function Vout/Vin of the circuit below (assume that no current flows to the output terminals AB).

1b) State whether the circuit can be made to produce an output signal with zero phase shift. If a zero phase shift is possible, describe the conditions under which it occurs.

2a) Calculate the Thevenin's equivalent e.m.f. and impedance as seen between terminals AB (assume C2 is the load at AB).

2b) Using the Thevenin's equivalent circuit derived in Part a above, calculate the voltage VOUT.

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Solution Preview

1a) Lets work out the impedance of the || combination of the resistance (R) and the inductance (L) at an angular frequency w. Lets call this impedance Z(out) which is given by the || impedance rule as

1/Z(out) = 1/R + 1/jwL (1) {where w is angular frequency of operation}

We will substitute s = jw in (1) so

1/Z(out) = 1/R + 1/sL

Z(out) = sLR/{R + sL} (2)

Impedance of the series capacitor is given by

Z(cap) = 1/jwC = 1/sC (3)

Total impedance of the circuit is given by Z(tot) where

Z(tot) = Z(cap) + Z(out) (4)

Putting in expressions for Z(out) & Z(cap) as obtained in (2) & (3) into (4) we get

Z(tot) = 1/sC + sLR/{R + sL}

Z(tot) = ({R + sL} + (s^2)*LCR)/sC(R + sL) (5)

Now by the voltage divider rule we can say that the transfer function
Av = V(out)/V(in) = Z(out)/Z(tot) (6)

Av = Z(out)/{Z(tot)} (7)

Av = sLR/{R + sL}*sC(R + sL)/({R + sL} + s^2*LCR)

Av = sLR*sC*(R + sL)/(R +sL)*({R + sL} + s^2*LCR) (8)

Reducing to

Av = s^2*LCR/({R + sL} + s^2LCR) ...

Solution Summary

An LCR circuit is presented and the transfer function of Vout/Vin is determined based on known passive values. An expression for the phase is determined and questions about zero phase are asked and answered. The Thevenin equivalent circuit is deduced and the Thevenin emf and input impedance worked out