Electronic – Output impedance of a series shunt feedback amplifier

There are two "tricks" to answering those questions with transistor amplifiers.

The first trick is to understand that the current-transfer ratio (h_FE) of a transistor effectivly multiplies the resistance at its emitter. So, to find the impedance looking into the base, you calculate the effective resistance at its emitter, multiply this by h_FE, and then add the internal base resistance.

The second trick is to realize that the result of the calculation above is typically orders of magnitude larger than the other resistances connected to the base of the transistor and can therefore be ignored. In other words, the input impedance of a transistor amplifier is usually very close to the impedance of its bias network alone. Indeed, bias networks are very often designed so that this is the case.

The output impedance is a question of how much the output voltage changes with output current: ΔV/ΔI. The transistor itself is essentially a current source, and whatever current it is passing is shared between the various resistances connected to the emitter. Therefore, the output impedance is equal to the net emitter resistance, not including the resistance of the external load.

The "gain" formula as given by you is the transconductance of a transistor in common-emitter configuration WITH feedback.

1.) Without feedback (RE=0) this transconductance is simply $$gm = \frac{hfe}{hie}$$

2.) With feedback we have $$A= \frac{gm}{1+gm*R_E}= \frac{hfe}{hie+R_E}$$

From this, we can derive the loop gain Aloop=-gm*Re. Note that gm=hfe/hie.

3.) Based on the loop gain expression (and knowing that the output resistance will be increased using feedback) we can expect (approximation):

$$Z_{out}= Z_o*(1+gm*R_E)=r_{ce}*(1+gm*R_E)$$

4.) Comment: The above expression for Zout is an approximation because it is derived under the assumption of IDEAL voltage feedback. However, the feedback voltage is provided across RE (not an ideal voltage source). More than that, also the influence of the input circuitry was neglected. However, we should not forget, that during practical operation of this circuit the large value of Zout is paralleled with the collector resitor Rc (which is much smaller that Zout). Hence, the approximation seems to be acceptable.