Synthesis of Linear Time-Varying Passive Networks

A state-space synthesis procedure is given for linear timevarying passive impedance matrices. The synthesis uses only passive components. I N THIS PAPER, we consider linear lumped finite networks composed of interconnections of time-variable and passive resistances, capacitances, inductances, gyrators, and transformers. The main problem considered is to pass from an input-output, or port, description of the network (in terms of its impedance matrix) to an internal description (in terms of a set of element values, and a scheme for interconnection). \ Amidst prior work on this and related problems, we note especially the work of Spaulding, e.g., [I], [2], who obtained some necessary conditions for a prescribed impedance matrix to be passive, and necessary and sufficient conditions for a prescribed impedance matrix to be the impedance of a network containing all lossless elements. He also obtained a synthesis procedure for this latter class of impedances. Another impedance synthesis procedure for lossless element networks was derived by Saeks [3], paralleling the Cauer synthesid:, while [4] presents a lossless synthesis based on the scatiering matrix. Further necessary conditions on an impedance matrix f r it to be associated with a passive element network were 8 erived in [5], and more recently, some characterizations of passivity, using a state-spape description of a prescribed impedance, were obtained in [6] and [7]. The material closest to that presented in this paper is, however, [8]. In [8], sybthesis procedures were obtained given the validity of a certain conjecture; this conjecture was known to be true for a limited class of impedances, and the claim of the conjecture was its truth for all passive iinpedances. Much of this paper, in effect, amounts to an examination of this conjecture and a delineation of when it is true. The broad structure of the paper is as follows. We assume there is given the state-space equations of a time-varying impedance matrix Z(t,z), which is related to F(.), G(.), H(.), and J ( . ) by Manuscript received June 29, 1973; revised November 9, 1973. This work was supported by the Australian Research Grants Committee. The authors are with the Department of Electrical Engineering, University of Newcastle, N.S.W., Australia. In (I), v ( . ) and u(.) denote, respectively, the port voltage and current vectors of some network with impedance Z(.;). In (2), 6(.) and I ( . ) are, respectively, the unit impulse and the unit step function, and @(.,a) is the transition matrix of F(.). The synthesis problem is one of passing from Z ( . , . ) to a network with impedance Z(., a ) . (Slightly more complex Z(-, .) will be considered in the sequel; for this discussion, though, (2) will be adequate.) If Z(.,.) alone is known, rather than F(.), G(. ) , H ( . ) , and J( .) separately, it is a standard procedure of linear system theory to find F(.), G(.), H(-), and J(.) from Z(.,.); therefore, we shall assume such matrices are all known a priori. The impedance Z(.,.) corresponds to a passive network if one has for all times to and t,, to < t, and all u(.): (The double integral on the left is the energy supplied to the network over [to,tl] with the network initially in the zero state.) The first major step is to use the passivity property (3) to conclude the existence of at least one nonnegative definite symmetric matrix P(.) such that -PF F'P $' H PG [ (H PG)' J + J' ] r O. (4) A complete controllability condition on [F,G] must be assumed, and existence of P(.) is actually guaranteed in the first instance if J + J' is nonsingular. In case J + J' is singular, we show that P(.) satisfies a slightly weaker condition than (4), but may satisfy (4) also. In case [F,H] is completely observable, P(.) is nonsingular. Following proof of the existence of P(.), we show how to compute such a P(-). In so doing, we give conditions applying to the case of J + J' singular for P(.) to satisfy (4) rather than the slightly weaker condition. The matrix P(.) is now used to define a coordinate basis transformation; the new state-space equations then allow both reactance extraction and resistance extraction syntheses. The idea of these syntheses in the time-invariant case is discussed in [9]-[ll], and their use in time-varying problems appears in [8] and [12]. The computation of P(-) when J + J' is singular can, if desired, proceed via a series of' steps which, from the synANDERSON AND MOYLAN: PASSIVE NETWORKS 679 thesis point of view, correspond to extraction of series inductor and shunt capacitor elements, such as occurs in the "preambles" of many classical time-invariant synthesis procedures. This interpretation will be made clear subsequently. 11. NECESSARY AND SUFFICIENT CONDITIONS FOR THE