Output Impedance of a Transistor for the Emitter Follower (Common Collector)

The output impedance seen by the load (R_E in this example) is defined as:

Z_out = - dV_out / dI_out

The minus sign in the derivative comes from the fact the output impedance has the effect of decreasing V_out. The output current I_E, which is related to the base current.

V_in = I_BR_S + V_BE +V_E [V_out = V_E]

V_out = V_E = V_in – I_BR_S – V_BE [I_out = I_E and I_B = I_E/(β + 1)]

V_out = V_in – {I_E/(β + 1)}R_S – V_BE = V_in – {I_out/( β + 1)}R_S – V_BE

= - I_out{R_S/(β + 1)} + (V_in – V_BE)

So, Z_out = - dV_out/dI_out = -d/dI_out [ - I_out{R_S/(β + 1)} + (V_in – V_BE) ]

Z_out = R_S/(β + 1)

Thus we obtain the result that the impedance of the source, as viewed by the load, is reduced by the factor ~ 1/ β.

Zout = {RS/( β + 1) ≈ RS/β