2.1 Transistor equations

There are several "figures of merit" for the operation of the transistor. The first of these is called the emitter injection efficiency , $\gamma$ . The emitter injection efficiency is just the ratio of the electron current flowing in the emitter to the totalcurrent across the emitter base junction:

If you go back and look at the diode equation you will note that the electron forward current across a junction is proportional to $N_d$ the doping on the n-side of the junction. Clearly the hole current will be proportional to $N_a$ , the acceptor doping on the p-side of the junction. Thus, atleast to first order

(There are some other considerations which we are ignoring in obtaining this expression, but to first order, and for most"real" transistors, is a very good approximation.)

The second "figure of merit" is the base transport factor, ${\alpha }_T$ . The base transport factor tells us what fraction of the electroncurrent which is injected into the base actually makes it to collector junction. This turns out to be given, to a very goodapproximation, by the expression

Where $W_B$ is the physical width of the base region, and $L_e$ is the electron diffusion length, defined in the electron diffusion length equation .

Clearly, if the base is very narrow compared to the diffusion length, and since the electron concentration is falling off like $e^{\frac{-x}{L_e}}$ the shorter the base is compared to $L_e$ the greater the fraction of electrons who will actually make it across. We saw before that a typical value for $L_e$ might be on the order of 0.005 cm or 50 μm. In a typical bipolar transistor, the base width, $W_B$ is usually only a few μm and so $\alpha$ can be quite close to unity as well.

Looking back at this figure , it should be clear that, so long as the collector-base junction remains reverse-biased, the collectorcurrent $I_{\mathrm{Ce}}$ , will only depend on how much of the total emitter current actually gets collected by thereverse-biased base-collector junction. That is, the collector current IC is just some fraction of the total emitter current $I_E$ . We introduce yet one more constant which reflects the ratio between these two currents, and call it simply" $\alpha$ ." Thus we say

Since the electron current into the base is just $\gamma I_E$ and ${\alpha }_T$ of that current reaches the collector, we can write:

Looking back at the structure of an npn bipolar transistor , we can use Kirchoff's current law for the transistor and say:

This can be re-written to express $I_C$ in terms of $I_B$ as:

This is the fundamental operational equation for the bipolar equation. It says that the collector current is dependent onlyon the base current. Note that if $\alpha$ is a number close to (but still slightly less than) unity, then $\beta$ which is just given by