Necessary and sufficient condition for uniqueness of solutions of certain piecewise linear resistive networks containing transistors and diodes

Equations of the form F(x)=f(x)+Ax=y where f(x)=(f₁(x₁) f₂(x₂) . . . f_n(x_n))^T, A is a P₀ matrix and for all i=1,2, . . ., n, f_i(x_i), are monotonic and piecewise linear but not necessarily strictly monotonic, are studied. Such networks are shown to have a unique solution if the Jacobian determinant has the same sign in all the regions. This is much more general than several sufficient conditions available in the papers by Sandberg and Willson (1969) and by Chien (1977). Furthermore it is not possible to improve this any further as the condition is both necessary and sufficient. A new sufficient condition is proposed to quickly check the sign condition on the Jacobians.