This document provides the solution to finding the input impedance, output impedance, and voltage gain of a circuit. It first combines two resistors in parallel. It then uses Kirchhoff's current law and voltage divisions to derive an expression for the voltage gain as (r + R1 + gm r R1)/r. The input impedance is found to be r || Rf by shorting the output. The output impedance is found to be R1 || Rf by shorting the input.

1. For the following circuit Find the input impedance. Find the output impedance. Find the
voltage gain.
Solution
RD and ro are between the same two points - vout and ground. Therefore, they can be combined
in parallel.
R1 = RD || ro (parallel combination)
Let ir be the current through resistor r, irf be the current through resistor Rf and ir1 be the current
through resistor R1 (parallel combination of RD and ro)
By Ohm's law, we have:
ir = vGS / r = vin / r
irf = ( vin - vout ) / Rf
ir1 = vout / R1
By KCL, we have:
ir = ir1 + gm vGS = ir1 + gm vin
ir x r x R1 = ir1 x r x R1 + gm vin x r x R1
vin R1 = vin r - vout r + gm vin r R1
Av = vout / vin = ( r + R1 + gm r R1 ) / r
This is the voltage gain.
To find the input impedance, make vout = 0 and look from the input end. Shorting the output
would short R1. The only resistors which remain are r and Rf and both of them are in parallel to
each other.
Thus,
Rin = r || Rf = (r x Rf) / (r + Rf)
To find the output impedance, make vin = 0 and look from the output end. Shorting the input
would short r. The only resistors which remain are R1 and Rf , which are parallel to each other.
Thus,
Rout = R1 || Rf = ro || RD || Rf