The standard diode equation for current-voltage characteristics of pn junction is
(At room temp, e/kT ≈ 40 V-1.) npn and pnp transistors have two of these junctions. A model for an npn is
The collector current is sum of the negative of the BC ideal diode current and the BE transport current. Similar for emitter current.
Use relation between reverse saturation currents,
Can rewrite collector and emitter currents
Convert to SPICE transistor parameters by changing subscripts of each α, E to F(forward) and C to R(reverse), and using current gain β = α/(1 - α). For example, for 2N3904, Is=6.7e-15 A, βF=416, and βR=0.737
After a bit of algebra it is possible to find the base current
