A linear circuit theory conservation theorem using the numerical range
April 25, 2022
by Lance

The theoretical work of which I am most proud was on a linear circuit theory conservation theorem that used the numerical range and produced tight bounds on the performance of circuits composed of linear elements. The basic paper was titled “Frequency limitations in circuits composed of linear devices.” I followed it by some work that built on that foundation, including “The stability and passivity of MOSFET device models that use nonreciprocal capacitive elements.” Prof. John Wyatt of M.I.T. was my coach and mentor for this whole set of activities.

This paper investigates limitation on the frequency response of networks constructed out of components specified by their small signal models. Tellegen’s theorem is used to find tight bounds on the maximum frequency of oscillation. The problem is reduced to deciding whether zero is in the numerical range of a complex non-Hermitian matrix. A decision method is presented, and transistor and negative resistance amplifier examples are developed.


In this paper we examine the activity and stability of circuits built from device models formed by a linear active or passive multiport resistor in parallel with a positive definite, but nonreciprocal, multiport capacitor. Numerical range methods are used to determine the maximum frequency of oscillation and maximum exponential growth rate of the solutions for both conservation of real power and conservation of complex power. We also examine the stability and activity of these devices when a positive-definite multiport resistor is added in series. It is shown that with the inclusion of the resistor, the circuit becomes passive at high frequencies even though the capacitor is nonreciprocal. The implications of these results for MOSFET device modeling are discussed.