Two Problems on Minimality in RLC Circuit Synthesis

A (one-port) RLC circuit is an interconnection of resistors and reactive elements (inductors and capacitors) with a pair of driving-point terminals. The circuit may be characterised by its impedance, which is the transfer function from an applied current to the induced voltage across the drivingpoint terminals. A given circuit has a unique impedance which may be routinely calculated from information about the constituent elements and their interconnections. It is well known that this impedance is necessarily positive-real [2], and that the McMillan degree of the impedance is less than or equal to the number of reactive elements in the circuit [9]. RLC circuit synthesis is concerned with obtaining RLC circuit realisations for a given impedance. Several key contributions to the synthesis problem were made in the first half of the twentieth century [1, 2, 5], but work on the subject petered out in the early 1970s. There has been a recent resurgence of interest in the field due to the introduction of a new mechanical component—the inerter [21]—which permits an exact analogy between electrical and mechanical circuits. The application of the inerter to mechanical vibration problems (e.g. vehicle suspensions) motivates a fresh look at questions of minimality and efficiency of realisation that have remained unresolved in the field. In this article, we discuss two such open problems.