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

• D_p and D_n are the hole and electron diffusivity, in cm² s⁻¹
• N_D and N_A are the donor and acceptor doping concentrations, in cm⁻³
• W is the base width, in cm
• L_p is the hole diffusion length in the emitter, in cm
• τ_b 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.