Necessary and Sufficient Conditions for the Existence of ±RC Networks

Necessary and sufficient conditions for the existence of networks containing positive resistors (+R), negative resistors (-R), and positive capacitors (C) are presented. It is shown that the elements of the open-circuit impedance matrices of R, C -ports can have poles only on the real axis of the complex frequency plane and these poles must be simple. The matrices of residues in all the finite poles must be positive semidefinite, and if there are any poles at infinity the matrices of residues in them must be negative semidefinite. These conditions are sufficient as well as necessary. Further, any matrix satisfying the above conditions can be characterized by a passive RC -port, in a modified frequency variable, with a negative resistor added in series at each of the ports. As a consequence of these conditions, driving-point impedance functions of R,C networks can have only simple zeros and poles alternating on the real axis and the residues in all the finite poles must be positive.