# 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.