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CE small-signal model, gain and input impedance

Short answer

The hybrid-π model shows Zin = R1||R2||rπ and Av = -gm(RC||RL). With rπ = β/gm and gm = IC/VT, for IC=1mA, β=100, rπ=2.5kΩ, Av≈-120 with 3kΩ load.

Draw the small signal model of CE amplifier and derive the expression for voltage gain and input impedance of CE amplifier with emitter bypass capacitor.

Small-signal (hybrid-π) model

With the emitter bypass capacitor the emitter is at ac ground, and the coupling capacitors are short circuits.

 vs--Rs--+----B--rπ--E          C--+--+--
         |        |    \          |  |
       R1||R2     vπ    gm vπ ----+  RC||RL
         |                          |
        GND                        GND

Components: rπ between base and emitter, controlled source gm vπ (collector to emitter) in parallel with ro, load RC || RL; bias resistors R1 || R2 at the input.

gm = IC/VT ,    rπ = β/gm = β VT/IC ,    re = VT/IE

Input impedance

The input sees R1, R2 and rπ in parallel:

Zin = R1 || R2 || rπ

Voltage gain

The output voltage is across the collector load:

vo = -gm vπ (RC || RL || ro) ,    vπ = vi

Av = vo/vi = -gm (RC || RL)   (ro neglected)

The gain with source resistance Rs is Avs = Av · Zin/(Zin + Rs).

Example

IC = 1 mA, β = 100: gm = 40 mA/V, rπ = 2.5 kΩ. With RC || RL = 3 kΩ, Av = -120 and Zin ≈ 2.2 kΩ (if R1 || R2 = 20 kΩ).

✓ Answer checked by Claude Opus 5.5 (high effort)

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