Scattering Matrix Synthesis Via Reactance Extraction

This paper considers the problem of rnultiport passive rietwark synthesis for a rational hounded real scatter in^ matrix S ( p ) using state-space ideas. The technique is to extract reactances from a network synthesizing the prescribed S ( p ) to yield a resistive coupling network that may contain transformers, resistors, and if there is no reciprocity constraint, gyrators. Here a minimum number of reactive elements are always sn5cient to give a synthesis, reciprocal and nonreciprocal. The synthesis method relies on an algebraic criterion for a class of rational matrices termed "discrete bounded real:' that solves the passive synthesis problem. Starting with an arbitrary coordinate basis, a state-space description is found for a discrete bounded real matrix derived from the given hounded real S(p). A coordinate transformation of the state-space is found from the solution of an Manusoript received June 25, 1969; revised February 9, 1970. This work was supported by the Australian Research Grants Committee and the Colombo Plan. Part of the work of B.D.O. Anderson was carried out at the Center for Control and Information Sciences, Institute of Technology, Southern Methodist University, Dallas, Tex. The authors are with Department of Electricel Engineering, University of Newcastle, New South Wales, 2308, Australia. akebraic equation, which vields almost immediately a oassive synthesis for ~ ( p ) . When i synthesis for a symmeiiic biunded real S ( p ) is required to use no gyrators, a further state-space basis .. . -. transformation is found that preserves passivity and simultaneously inserts reciprocity. The contributions of this paper, to scattering matrix synthesis, are twofold: first, to sohe the nonreciprocal synthesis problem including the mimimal reactive element equivalence problem via calculations that need not rely on application of any classical ideas, and second, to present for the 6rst time a reciprocal synthesis of a passive symmetric scattering matrix with a minimum number of reactances using state-space ideas. N MANY recent papers [I]-171, [19], [ZO], the problem of multiport passive network synthesis using stateA space ideas has been oonsidered. The synthks of a rational bounded real scattering function or matrix is discussed in [I], while [2]-[7] discuss the synthesis of a rational positive real impedance function or matrix, and [I91 and [20] consider syntheses of networks with two-variable and several-variable matrices, respectively. In Youla and Tissi's paper [I], a new approach is considered for the synthesis o f a passive n-port network %ith a rationa1 bounded realscattering matrix. Here a network N synthesizing a rational bounded real S(p) is regarded as a cascade connection of two networks N , @i d M,, where NAv, is memoryless and may contain transfo&iw, resietom,'&d if ttiere is no reciprocity constraint, gymtom; M, is simply induotors and capacitors uncoupled fmm each other (see Fig. 1). Hence, we call the synthesis rneth4 a . reactance ..&xtraction synthesis. The number 5f ieaotive elements used is always a minimum, being the 8amc as "oe degree [8] of S(p). The synthesis procedure af [I] is not a t all simple; the process of bordering a certain matrix h t o a parauplitary matrix using the theory of Oono and Yasqswa 191 is necessary. $n idis paper, the same technique of reactance extraction ipthesis is adopted. The method involves a more direct approach to soivkg the problem, .becaye here an algebrho criteriosfor discrete bounded real rational matrices (to be defined later) is developed, which finds immediate ipplication to the passive synthesis problem. Both the aonreeiprocal i n d t h e reci.procal synthesis use a minimal n u b e r o f reactive elements. The state-space approach to network synthesis seem to have originated with [I]. As remarked above, the bchirique of [I] for synthesizing a passive scattering matrix relied on application of some classical ideas, going beyond even the notion of a matrix spectral factorization. No proce6ure @odd be given for reciprocal syuthesis. >in I2J and 131, the impedance synthesis problem was considered, and the ex&& to which non-state-space ideas was wed was small. A certain matrix required for a coordinate basis change needed to be calculated, and one procedure suggested for this was spectral factorization. Since that time though, other ~alculation procedures have been 8tudied, relying on the solution of a quadratic matrix ineqmlity [[10]. 'Reference [I.@] coupled with [2] and [3] $hen constitutes a complete purely state-space solution (including computational procedures) of the synthesis problem for passive impedances, and even solves the equivalence problem. Neither [2] nor [3] solves the reciprocal synthesis problem. Reference [4] provided a very good statement of various ways of how the reciprocal synthesis problem may have been attacked, but the first nonclassical solutions of the reciprocal synthesis problem are given in 151-[7] all for impedances. Reference [5] descrikes a synthesis using a minimal number of reactances, and some of the ideas of this reference proved of great help in dev12loping the reciprocal synthesis section of this paper. Reference [6] outlines two techniques, both nonminimal in the number of reactive elements, while one' of these is described .in detail in 171. The synthesis of [7] is a state-space parallel of the classical Bayard synthesis, and uses a minimal number of resistors. The contributions of this paper, to scattering matrix synthesis, are really twofold: first, to reduce the nonmns TE~WSACTIONB ON CIRCUIT TBEORY, NOYEMBER 1910 reciprocal problem to calculations independent of any classical notions while using minimal nuniber of reactances, and second, to present for the first time a reciprocal synthesis of a passive symmetric scattering matrix using a minimum number of reactances via state-space techniques. 11. REVIEW OF QTATE-SPACE D SCRIPTION FOR RATIONAL MATRICES In [Ill, it is shown that any rational n X n matrix W(p) with W(m) finite possesses a decomposition of the form W(p) = J + H1(pI F)-'G (1) where F, G, H, and J are real constant matrices (the prime denotes transpose). Any quadruple IF, G, H, J ) for which this formula holds is termed a realization of W(p), and a minimal realiiation if F has minimal dimension. Several important properties of minimal realizations are 1) if IF, C , N, J ) is one minimal realization of W(p), others are given by j TFT-I, TG, (T-')'H, J ) for arbitrary nonsingular T; 2) if IF, G, H , J J is minimal, the realization is completely controllable and observable, and rank [G, FG, . . . , Fk-'GI = rnnk [N, FIE, . . . , (PI)'-'HI = k where F is k X k; 3) the minimal dimension k of F is the McMillan degree of the matrix W(p), see 1121, which is also the minimal number of reactive elements used in any passive synthesis of a bounded real (or positive real)