View source for Analogue Electronics/BJTs/Active Mode/ß dimensional Analysis

{{Analogue Electronics Page}}
This page will show that ''&beta;'', the common-emitter current gain of a BJT has no units.

''&beta;'' is given by:

::<math>
\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)</math>

where
*''D<sub>p</sub>'' and ''D<sub>n</sub>'' are the hole and electron diffusivity, in cm<sup>2</sup> s<sup>−1</sup>
*''N<sub>D</sub>'' and ''N<sub>A</sub>'' are the donor and acceptor doping concentrations, in cm<sup>−3</sup>
*''W'' is the base width, in cm
*''L<sub>p</sub>'' is the hole diffusion length in the emitter, in cm
*''&tau;<sub>b</sub>'' is the minority carrier lifetime in the base, in s

So we have:

::<math>\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}</math>

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

::<math>\left[ \beta  \right] = \left( {\frac{{L^2 }}
{{L^2 T^{ - 1} T}}} \right)^{ - 1}  = \left( {\frac{{L^2 }}
{{L^2 }}} \right)^{ - 1}</math>

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