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:


\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.