Analogue Electronics/BJTs/Active Mode/ß dimensional Analysis

This page will show that &beta;, the common-emitter current gain of a BJT has no units.

&beta; is given by:



\beta = 1/\left( {\frac + \frac{1} {2}\frac } \right)$$

where
 * Dp and Dn are the hole and electron diffusivity, in cm2 s−1
 * ND and NA are the donor and acceptor doping concentrations, in cm−3
 * W is the base width, in cm
 * Lp is the hole diffusion length in the emitter, in cm
 * &tau;b is the minority carrier lifetime in the base, in s

So we have:


 * $$\left[ \beta \right] = \left( {\frac

+ \frac } \right)^{ - 1}$$

Notice that the first term in the addition is a ratio of two quantities with identical dimensions. This leaves us with:


 * $$\left[ \beta \right] = \left( {\frac

} \right)^{ - 1} = \left( {\frac } \right)^{ - 1}$$

We now have the reciprocal of a ratio of identically dimensioned quantities. Therefore, &beta; is dimensionless.