---
title: "Short note: π-models of BJT and MOSFET"
subject: "Electronics Devices and Circuit"
chapter: 2
key: "semester_3/electronics_devices_and_circuit:pi-model-bjt-mosfet-note"
source: https://www.bctengineer.com/q/semester_3/electronics_devices_and_circuit/pi-model-bjt-mosfet-note
site: BCT Engineering
---

# Short note: π-models of BJT and MOSFET

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

- Asked in: 2072 Kartik, 4 marks
- Video: [Hybrid-π Model](https://www.youtube.com/watch?v=yUEss0Dl6ww) — 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/pi-model-bjt-mosfet-note
- Whole chapter: https://www.bctengineer.com/md/semester_3/electronics_devices_and_circuit/chapter-2

## Question

Write a short note on π-models of BJT and MOSFET.

## Short answer

The hybrid-π model replaces the transistor with linear elements. For BJT: input resistance rπ = β/gm between base and emitter, a current source gm vπ, and output resistance ro. For MOSFET: infinite input resistance (open circuit), a current source gm vgs, and ro. gm = IC/VT (BJT) or 2 ID/(VGS–Vt) (MOSFET).

## Answer

The hybrid-π model is the standard small-signal equivalent circuit for transistors in their active (BJT) or saturation (MOSFET) region. It replaces the device with linear elements and a voltage-controlled current source.

#### π-model of the BJT
```text
  B o------+                +---------+------o C
           |  +             |         |
          rπ  vπ      gm vπ (↓)      ro
           |  -             |         |
  E o------+----------------+---------+------o E
```
- rπ = β/gm is connected between base and emitter (input resistance); vπ is the voltage across it.
- gm vπ is the collector current source, with gm = IC/VT.
- ro = VA/IC is the output resistance (Early effect).
- Parameters: gm = IC/VT; rπ = β VT/IC; re = α/gm ≈ 1/gm.

#### π-model of the MOSFET
```text
  G o---                    +---------+------o D
         +                  |         |
        vgs  (open)   gm vgs (↓)     ro
         -                  |         |
  S o-----------------------+---------+------o S
```
- The gate is insulated, so the input resistance is infinite: the gate-source port is an open circuit (rπ = ∞) at low frequency.
- The drain current source is gm vgs, with gm = 2K (VGS - Vt) = 2 ID/(VGS - Vt) = 2√(K ID) (K in A/V², for the convention ID = K (VGS - Vt)²; with ID = ½ kn'(W/L)(VGS - Vt)², the same result is written gm = √(2 kn'(W/L) ID)). The output resistance is ro = VA/ID = 1/(λ ID).
- The body effect adds a second source, gmb vbs.

#### Comparison
| Feature | BJT | MOSFET |
|---|---|---|
| Input resistance | rπ (finite) | ∞ (gate open) |
| gm | IC/VT | 2 ID/(VGS - Vt) |
| Current source | gm vπ (between C and E) | gm vgs (between D and S) |
| Output resistance | ro = VA/IC | ro = VA/ID |

At high frequency, the capacitances Cπ, Cμ (BJT) and Cgs, Cgd (MOSFET) are added.

## Check yourself

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

1. Which elements make up the BJT hybrid-π model?
   A. rπ between B and E, gm vπ source from C to E, and ro
   B. Only a resistor re
   C. A capacitor between B and E only
   D. rπ between C and E and a voltage source
   Answer: A. The model has rπ at the input, a current source gm vπ, and ro for the Early effect. (Key point: BJT π-model elements)

2. What is the input resistance of the MOSFET π-model at low frequency?
   A. rπ = β/gm
   B. Infinite, since the gate is insulated
   C. Equal to ro
   D. 1/gm
   Answer: B. The insulated gate draws no current, so the gate is an open circuit. (Key point: MOSFET input)

3. What gives the output resistance ro in the BJT model?
   A. ro = VT/IC
   B. ro = β/gm
   C. ro = VA/IC from the Early effect
   D. ro = RC
   Answer: C. Base-width modulation (Early effect) gives ro = VA/IC. (Key point: ro)

4. Which expression is gm for the MOSFET?
   A. IC/VT
   B. 2 ID/(VGS - Vt)
   C. VA/ID
   D. VT/ID
   Answer: B. For the square law device gm = 2 ID/(VGS - Vt), compared with IC/VT for a BJT. (Key point: MOSFET gm)

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