BJT Alpha–Beta Relationship

α = β/(β+1), β = α/(1−α)

Quick answer

This calculator solves α = β / (β + 1), β = α / (1 − α) for Common-Base Gain α. With Common-Emitter Gain β 1.1, the result is 0.5238. It solves in 2 directions, so you can enter the values you have and get the one you need.

Calculator

Common-Base Gain α
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Formula

α = β / (β + 1), β = α / (1 − α)

Description

A bipolar transistor has two current gains: alpha (common-base, collector/emitter current) is just under 1, and beta (common-emitter, collector/base current) is large. They describe the same device and convert directly into one another. A small change in alpha near unity produces a large change in beta, which is why beta is poorly controlled in manufacturing.

Variables

  • α — Common-base current gain (Ic/Ie, < 1)
  • β — Common-emitter current gain (Ic/Ib)

Practical Notes

α = 0.99 gives β = 99; α = 0.995 gives β = 199. This sensitivity is why circuits are designed to be insensitive to β using emitter degeneration and feedback.

Symbols and units

Symbol Meaning Default unit Accepted units
alpha Common-Base Gain α — —
beta Common-Emitter Gain β — —

Step by step

  1. 1. What you know: Common-Emitter Gain β 1.1
  2. 2. The relation: α = β / (β + 1), β = α / (1 − α)
  3. 3. Substitute: beta = 1.1
  4. 4. Result: Common-Base Gain α = 0.5238

Worked examples

Minimum range

Given: Common-Emitter Gain β 0.055

Result: Common-Base Gain α = 0.05213

Low range

Given: Common-Emitter Gain β 0.55

Result: Common-Base Gain α = 0.3548

Typical

Given: Common-Emitter Gain β 1.1

Result: Common-Base Gain α = 0.5238

High range

Given: Common-Emitter Gain β 1.76

Result: Common-Base Gain α = 0.6377

Maximum range

Given: Common-Emitter Gain β 17.6

Result: Common-Base Gain α = 0.9462

Solving for each variable

Solve for Common-Base Gain α

Common-Emitter Gain β 0.055 → 0.05213

Solve for Common-Emitter Gain β

Common-Base Gain α 0.025 → 0.02564

How Common-Base Gain α changes with Common-Emitter Gain β

Common-Emitter Gain β (input) Common-Base Gain α (result)
0.11 0.0991
0.275 0.2157
0.55 0.3548
0.825 0.4521
1.1 0.5238
1.65 0.6226
2.2 0.6875
4.4 0.8148
7.7 0.8851
11 0.9167

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Frequently asked questions

How do you calculate BJT Alpha–Beta Relationship?

Use α = β / (β + 1), β = α / (1 − α). With Common-Emitter Gain β 1.1, Common-Base Gain α works out to 0.5238.

What is the formula for BJT Alpha–Beta Relationship?

The formula is α = β / (β + 1), β = α / (1 − α). A bipolar transistor has two current gains: alpha (common-base, collector/emitter current) is just under 1, and beta (common-emitter, collector/base current) is large.

How do you find Common-Emitter Gain β when you know the other values?

Rearrange for Common-Emitter Gain β: with Common-Base Gain α 0.025, Common-Emitter Gain β is 0.02564.

Related Concepts

All Discrete Amplifiers formulas →
Thermal Voltage Small-Signal Emitter Resistance Transistor Transconductance Diode Dynamic Resistance Common-Emitter Voltage Gain BJT Alpha–Beta Relationship Darlington Pair Current Gain Emitter-Follower Input Impedance

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