---
title: "Hybrid pi and T model, rπ and re"
subject: "Electronics Devices and Circuit"
chapter: 2
key: "semester_3/electronics_devices_and_circuit:hybrid-pi-t-model"
source: https://www.bctengineer.com/q/semester_3/electronics_devices_and_circuit/hybrid-pi-t-model
site: BCT Engineering
---

# Hybrid pi and T model, rπ and re

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

- Asked in: 2080 Baisakh, 3+3 marks
- Video: [Transistor Models | Hybrid π Model | T Model or Re Model | Transistor  Equivalent AC circuit](https://www.youtube.com/watch?v=Vo-EK4JKRnk) — Gautam Varde
- 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/hybrid-pi-t-model
- Whole chapter: https://www.bctengineer.com/md/semester_3/electronics_devices_and_circuit/chapter-2

## Question

Draw Hybrid π and T model. Derive the relationship between rπ and re.

## Short answer

rπ is base-emitter resistance in hybrid-π; re is emitter-base resistance in T-model. Since ie = (β+1)ib and vbe same, rπ = vbe/ib = (β+1)(vbe/ie) = (β+1)re. Also rπ = β/gm, re = VT/IE.

## Answer

#### Hybrid-π model
The hybrid-π model is used for small signals in the active region. It has rπ between the base and the emitter, a voltage-controlled current source gm vπ between the collector and the emitter, and (optionally) ro in parallel with the source.
```text
    B o-----+                +---------+-----o C
            |  +             |         |
           rπ  vπ         gm vπ (↓)    ro
            |  -             |         |
    E o-----+----------------+---------+-----o E
```
gm = IC/VT ,  rπ = vπ/ib = β/gm ,  ro = VA/IC.

#### T-model
The T-model is used when the base is the common terminal or when there is a resistor in the emitter. It has re between the emitter and the base, and a current source α ie between the collector and the base.
```text
    C o-------( )  α ie
              |
    B o-------+
              |
             re
              |
    E o-------+
```
re = VT/IE = α/gm.

#### Relation between rπ and re
rπ is the resistance seen at the base: rπ = vbe/ib. re is the resistance seen at the emitter: re = vbe/ie. The same vbe appears in both, and ie = (β+1) ib. Hence
```text
rπ = vbe/ib = vbe/[ie/(β+1)] = (β+1)(vbe/ie)

        rπ = (β+1) re
```
Also, rπ = β/gm and re = VT/IE = α/gm, so rπ/re = β/α = β + 1. For IC = 1 mA, β = 100: gm = 40 mA/V, rπ = 2.5 kΩ, re ≈ 24.8 Ω (rπ = 101 × 24.8 Ω).

## Check yourself

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

1. Which elements form the hybrid-pi model?
   A. re between E and B, alpha ie source
   B. Only a resistor RE
   C. Two diodes and two sources
   D. rpi between B and E, gm vpi source between C and E, optional ro
   Answer: D. The hybrid-pi model has rpi at the input, a voltage-controlled current source gm vpi and optional ro. (Key point: Hybrid-pi elements)

2. Which elements form the T-model?
   A. rpi and gm vpi
   B. ro only
   C. Two diodes
   D. re between emitter and base and a source alpha ie between collector and base
   Answer: D. The T-model uses re in the emitter path and the alpha ie source. (Key point: T-model elements)

3. What is the relation between rpi and re?
   A. rpi = (beta+1) re
   B. rpi = beta/re
   C. rpi = re
   D. rpi = re/(beta+1)
   Answer: A. rpi = vbe/ib and ie = (beta+1) ib with the same vbe, so rpi = (beta+1) re. (Key point: Relation rpi and re)

4. Why does the factor (beta+1) arise?
   A. Because gm = beta VT
   B. Because ie = (beta+1) ib while the same vbe appears across both
   C. Because re = beta/gm
   D. Because ic = ib/beta
   Answer: B. Resistance at the base is vbe/ib, and ib is (beta+1) times smaller than ie. (Key point: Why it follows)

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