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)(d)2(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(p) 3(2p)(4p + q + 1)(q)2(q(q- 1)/2) isolated de operating points. Bounds are also obtained for the number of de operating points in circuits using other transistor models.