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

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

*β* is given by:

where

*D*and_{p}*D*are the hole and electron diffusivity, in cm_{n}^{2}s^{−1}*N*and_{D}*N*are the donor and acceptor doping concentrations, in cm_{A}^{−3}*W*is the base width, in cm*L*is the hole diffusion length in the emitter, in cm_{p}*τ*is the minority carrier lifetime in the base, in s_{b}

So we have:

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

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