# Analogue Electronics/BJTs/Active Mode/ß dimensional Analysis

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

β is given by:

$\beta = 1/\left( {\frac{{D_p N_A W}} {{D_n N_D L_p }} + \frac{1} {2}\frac{{W^2 }} {{D_n \tau _b }}} \right)$

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
• τb is the minority carrier lifetime in the base, in s

So we have:

$\left[ \beta \right] = \left( {\frac{{L^2 T^{ - 1} L^{ - 3} L}} {{L^2 T^{ - 1} L^{ - 3} L}} + \frac{{L^2 W^2 }} {{L^2 T^{ - 1} T}}} \right)^{ - 1}$

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

$\left[ \beta \right] = \left( {\frac{{L^2 }} {{L^2 T^{ - 1} T}}} \right)^{ - 1} = \left( {\frac{{L^2 }} {{L^2 }}} \right)^{ - 1}$

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