Common collector amplifier Ri, Ro and gain

Circuit and ac equivalent

Common collector (emitter follower) with voltage divider bias: input at the base through C1, output from the emitter through C2 to RL, collector at ac ground (VCC).

 Vs--Rs--||--+--------B--\ 
             R1||R2       |> (ib)  re model
                          E--+--||--RL
                             RE
                            ---

At ac, the supply is ground: R1, R2 go from the base to ground; RE and RL (from the emitter) are in parallel. Let R'E = RE || RL and RB = R1 || R2. Use the T-model: emitter current ie = (β+1) ib, emitter resistance re = VT/IE.

Input resistance

Looking into the base: vb = ie (re + R'E) = (β+1) ib (re + R'E), so

Rib = vb/ib = (β+1)(re + R'E)  ≈ β (re + R'E)
Rin = RB || Rib = R1 || R2 || (β+1)(re + RE||RL)

This is large because the small emitter resistance is multiplied by (β+1).

Voltage gain

vo = ie R'E ,     vi = ie (re + R'E)

Av = vo/vi = R'E / (re + R'E)  ≈ 1     (since R'E ≫ re)

With a source of resistance Rs: Avs = Av · Rin/(Rin + Rs). The gain is positive (no phase shift), slightly less than 1.

Output resistance

Set vs = 0 and look back from the emitter (RL removed). The base sees Rs || R1 || R2 = R's, so

Rout = RE || [ re + R's/(β+1) ]   ≈ re + R's/(β+1)

It is small: tens of ohms for typical values.

Current gain

Ai = io/ii = Av · Rin/RL  ≈ (β+1) · RB/(RB + Rib) · RE/(RE + RL)

Summary

Hence the stage is a buffer: high input and low output resistance with unity voltage gain.