Bounds for the number of DC operating points of transistor circuits

A transistor circuit consisting of linear positive resistors, q exponential diodes, and p Ebers-Moll modeled bipolar transistors has at most (d+1)/sup d/2/sup d(d-1)/2/ isolated DC operating points where d=q+2p. If, instead of bipolar transistors, the circuit employs Shichman-Hodges modeled field-effect transistors, then it can have at most 2/sup p/ 3/sup 2p/(4p+q+1)/sup q/2/sup q(q-1)/2/ isolated DC operating points. Bounds are also obtained for the number of DC operating points in circuits using other transistor models.