Foster's Reactance Theorem for a Multiport, and its Application to Q Factor Measurement

The classical Foster's reactance theorem relates the input impedance or admittance to the stored energy inside a lossless one-port network. Such relation has been successfully applied to the measurement of Q factor for a resonator given its one-port response. Sometimes, however, a resonator is naturally monitored at more-than-one ports. Although there are various methods for Q measurement given two-port network parameters, it remains open as for whether the Foster-based method can be extended to such scenario. In this paper, we first derive the Foster's theorem for the multiport case, providing the relation between the stored energy and the N-port Z or Y matrix. Based on the result, we then propose a new method for Q-factor estimation. A feature of this method is that we directly use the whole Z or Y matrices to estimate Q, without reducing it to a one- or two-port network. Two simulation examples are provided to demonstrate the feasibility as well as the detailed operation of this method.