---
title: "CE small-signal model, gain and input impedance"
subject: "Electronics Devices and Circuit"
chapter: 2
key: "semester_3/electronics_devices_and_circuit:ce-small-signal-gain-input-impedance"
source: https://www.bctengineer.com/q/semester_3/electronics_devices_and_circuit/ce-small-signal-gain-input-impedance
site: BCT Engineering
---

# CE small-signal model, gain and input impedance

Electronics Devices and Circuit · Chapter 2: The Bipolar Junction Transistor

- Asked in: 2081 Baisakh, 2+3+3 marks
- Video: [Small Signal Analysis of BJT](https://www.youtube.com/watch?v=NESchIntkR8) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electronics_devices_and_circuit/ce-small-signal-gain-input-impedance
- Whole chapter: https://www.bctengineer.com/md/semester_3/electronics_devices_and_circuit/chapter-2

## Question

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.

## 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.

## Answer

#### Small-signal (hybrid-π) model
With the emitter bypass capacitor the emitter is at ac ground, and the coupling capacitors are short circuits.

```text
 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.

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

#### Input impedance
The input sees R1, R2 and rπ in parallel:
```text
Zin = R1 || R2 || rπ
```

#### Voltage gain
The output voltage is across the collector load:
```text
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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the emitter bypass capacitor do in the ac model?
   A. It reverses the gain sign
   B. It adds RE to the base resistance
   C. It puts the emitter at ac ground
   D. It opens the emitter circuit
   Answer: C. CE shorts RE at signal frequency, so the emitter is at ac ground and coupling capacitors are shorts. (Key point: Effect of bypass capacitor)

2. What is Zin of the bypassed CE amplifier?
   A. RC || RL
   B. R1 + R2 + rpi
   C. (beta+1)(re + RE)
   D. R1 || R2 || rpi
   Answer: D. The base sees the bias resistors and rpi in parallel. (Key point: Input impedance)

3. What is Av with ro neglected?
   A. Av = +gm RC
   B. Av = -gm (RC||RL)
   C. Av = rpi/RC
   D. Av = -RC/RE
   Answer: B. vo = -gm vpi (RC||RL) and vpi = vi. (Key point: Voltage gain)

4. How are rpi and gm related for IC = 1 mA, beta = 100?
   A. gm = 100 mA/V, rpi = 1 kΩ
   B. gm = 40 mA/V, rpi = beta/gm = 2.5 kΩ
   C. gm = 4 mA/V, rpi = 25 kΩ
   D. gm = 40 mA/V, rpi = 25 Ω
   Answer: B. gm = IC/VT = 40 mA/V and rpi = beta/gm = 2.5 kΩ. (Key point: rpi and gm)

## Related in Electronics Devices and Circuit

- [Bypass capacitor significance; CE small signal analysis](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Ace-bypass-capacitor-small-signal) — Electronics Devices and Circuit, chapter 2, asked 2×
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- [CE amplifier input resistance and overall gain](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Ace-input-resistance-gain-680k) — Electronics Devices and Circuit, chapter 2
- [CE amplifier input, output resistance and gain (RE1, RE2)](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Ace-partial-bypass-numeric) — Electronics Devices and Circuit, chapter 2
- [CE amplifier Rin, Rout, Av, Ai (470k, 2.2k, 0.56k)](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Ace-emitter-resistor-470k-numeric) — Electronics Devices and Circuit, chapter 2

## Related in other subjects

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