Analogue Electronics/BJTs/Active Mode/ß dimensional Analysis
< Analogue Electronics < BJTs < Active ModeThis page will show that β, the common-emitter current gain of a BJT has no units.
β is given by:
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:
![{\displaystyle \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}}](../I/m/59acd80ad1ff360611eb8e676e4228a222e12532.svg)
Notice that the first term in the addition is a ratio of two quantities with identical dimensions. This leaves us with:
![{\displaystyle \left[\beta \right]=\left({\frac {L^{2}}{L^{2}T^{-1}T}}\right)^{-1}=\left({\frac {L^{2}}{L^{2}}}\right)^{-1}}](../I/m/a8e7ebb490ae3b5587ecc3f9cc85287c9688184b.svg)
We now have the reciprocal of a ratio of identically dimensioned quantities. Therefore, β is dimensionless.
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