Bias Circuit

C.2.1 Determination of the Linear Model Parameters
We can relate the values of the two parameters in (C.10) to the SPICE model parameters using
(B.7) with the substitution vBC = –(vCE – vBE). This is
Equation C.11
Differentiating (C.11) with respect to vBE (with vCE = VCE, Vce = 0), gm is found to be
Equation C.12
The approximate form only ignores a term on the order of IC/VAF, where VAF >> VT.
The output resistance is obtained with vBE = VBE (bias value) or Vbe = 0. This is
Equation C.13
where, from (C.11), IC(vCE = VBE) = ISexp(VBE/VT). For simplicity, the bias collector current,
from (C.11), IC = iC(VCE), is usually used for the calculation for ro. Finally, a relation for rπ
comes directly from (C.7), Ic = gmVbe, and (B.41), Ic = βacIb. Equating the two gives
Equation C.14
Hence, from the definition rπ = Vbe/Ib (with )
Equation C.15
Note that the right-hand side of (C.14) is the alternative current-dependent current source of
the linear model of Fig. C.2.
As discussed in Unit B.9, βac can be slightly different from βDC, but the distinction usually need
not be made in analysis or design. This is due to the fact that βDC tends to be quite variable
among devices and that most analog designs are based on making the results as independent