An upper bound for the number of solutions of a piecewise-linear equation related to two-transistor circuits

This paper discusses the number of solutions for a class of piecewise-linear equations related to two-transistor circuits composed of transistors, linear passive resistors, and DC sources. By using Ebers-Moll model, a transistor is replaced by two nonlinear resistors and two linear current-controlled current sources. We assume in this paper that the nonlinear resistors are ideal diodes. Then a circuit equation for the above circuit is a piecewise-linear equation Ti + Gv = b where each component i_k and v_k of the vectors i and v respectively are subject to v_ki_k = 0, v_k ≤ 0, and i_k ≥ 0. We show that the number of solutions for the equation is at most 5.