Network Synthesis with Negative Resistors

The development of new solid-state active elements, such as variable-capacitor diodes and tunnel diodes, has stimulated the network theorist to consider the negative resistor as an additional basic circuit element to be included in problems of linear network analysis and synthesis. In this paper it is first shown that if the negative resistor is added to the usual set of lumped passive building blocks, then it is possible to represent as a network any linear relation between n-port voltages and currents prescribed in terms of real, rational functions of a complex-frequency variable. This leads to the synthesis of some novel pathologic circuits which have neither immittance nor scattering representations, such as a one-port, which is simultaneously an open circuit and a short circuit (v=i=0, the "nullator"), and the linear network in which voltages and currents at the ports are completely arbitrary (the "norator," the unique, linear nonreciprocal, one-port). These elements are shown to be basic linear circuit building blocks. The second part of the paper considers the synthesis in the frequency domain of a real, rational n×n immitance matrix in which pole locations and pole multiplicities are completely arbitrary. It is shown that such a matrix can always be realized with lossless elements and at most n positive and n negative resistors.