# Common Emitter Transistor Gain Equations & Theory

## Transistor gain theory with derivations for transistor gain starting with Kirchoff’s principles applied to the common emitter transistor configuration.

It is possible to gain some more insight into the operation of the transistor by looking a little further into the mathematics behind the operation of the device.

First the theory and gain will be examined for the common emitter circuit, where the emitter is common to both input and output circuits.

## Transistor gain derivation

It is sometimes useful to use some simple transistor theory to derive the gain calculations and formulas.

The most widely used circuit is the common emitter where the emitter is common to both input and output circuits.

Also figures for transistor gain including Β and H Hfe, hfe refer to the common emitter circuit.

For any transistor circuit we find that the following simple formula expresses the currents flowing into the transistor:

$-\mathrm{Ic}=\alpha \cdot \mathrm{Ie}+\mathrm{Ico}$

Where
Ic = collector current
α = Fraction of emitter carriers reaching collector, typically in the region 0.95 to 0.99
Ie = emitter current
Ico = Reverse current from base to collector

Eliminating Ie we are able to develop the following formula:

$\mathrm{Ic}\left(1-\alpha \right)-\alpha \cdot \mathrm{Ib}+\mathrm{Ico}=0$

$\mathrm{Ic}=\frac{\alpha }{1-\alpha }\mathrm{Ib}-\frac{\mathrm{Ico}}{1-\alpha }$

The factor α / (1 - α) is called Β Beta which is the common emitter current gain. For example if α = 0.98, then the transistor Beta Β can be calculated to be 49.

Replacing it is possible to determine the collector current, Ic in the equation below:

$\mathrm{Ic}=Β\cdot \mathrm{Ib}-\left(Β+1\right)\mathrm{Ico}$

Assuming the reverse current is low and can be ignored we see:

$\mathrm{Ic}\approx Β\cdot \mathrm{Ib}$

## Large & small signal gain theory

The gain of a transistor varies slightly according to the signal applied and where it is on the operating curve.

As a result of this, the calculations and theory can be adapted to accommodate this. In fact often two figures are quoted. Both Hfe, hfe are seen in data sheets for transistors, Hfe being the DC gain, and hfe being the small signal gain.

The datasheets normally define the test conditions under which the tests were made, and in this way it is possible to interpret the performance and compare different devices.

The equations for the different conditions can be given as:

$\mathrm{hfe}=\frac{\mathrm{\delta Ic}}{\mathrm{\delta Ib}}$

$\mathrm{Hfe}=\frac{\mathrm{Ic}}{\mathrm{Ib}}$

It can be seen that hfe describes the small signal changes whereas Hfe uses the total current levels.

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