NEW RESULTS IN NETWORK SIMULATION, SENSITIVITY AND TOLERANCE ANALYSIS FOR CASCADED STRUCTURES An useful design J.W. Bandler, M.R.M. Rizk and H.L. Abdel-Malek Group on Simulation, Optimization and Control Faculty of Engineering McMaster University, Hamilton, Canada L8S 4L7 ABSTRACT attractive, exact and efficient approach to network analysis for cascaded structures is for sensitivity and tolerance analyses, in particular, for a multiple of simultaneous parameter values. Introduction This paper presents a new and comprehensive treatment of computer-oriented cascaded network analysis. The approach permits efficient exact analysis, exact evaluation of first-order response sensitivities, exact evaluation of the effects of simultaneous large changes in any elements (as well as growing elements), exploitation of network structure: branches, symmetry, reciprocity, etc. All calculations are applied directly to the given network: no adjoint network is defined. All calculations involve at most the premultiplication of two by two matrices by row vectors or postmultiplications by column vectors. Response functions, sensitivities or large-change effects are represented analytically in terms of the parameters to be investigated. Theoretical Foundation Consider the two-port element depicted in Fig. l(a). The basic iteration, also summarized by Table 1, is ~ = A y, where A is the transmission or chain matrix,- y c6nt-sins the ‘output voltage and current and ~ the Corresponding input quantities. Forward ~.a#is (Fig. l(b), Table 1) consists of initializing row vector as either [1 01, [0 1] or a suitable lifiear combination and successively premultiplying each constant chain matrix by the resulting row vector until an element of interest, a reference plane or a termination is reached. Reverse analysis, similar to conventional analysis of cascaded networks, proceeds by initializing a ~ column vector as u above but uses postmultiplication. In summary, assuming a cascade of n two-ports ? = YO ❑ 4’!2... 4i ... !ni’ (1) and, applying forward and reverse analysis up to A=, this reduces to an expression of the form d=;l ‘–1 –“T “ “ y.c~~~lyl, (2) where Yn n =Cy (3) and c and d relate selected output and input variables of interest explicitly with ~ . The typical formula will, therefore, contain factors of the forms shown in Table 2. The (*) denotes either & ‘~, a~/a$, (where $ is a variable parameter contained in ~) or A! and the (t) denotes Q, 6Q, Q’ or AQ for function This work was supported by the National Research Council of Canada under Grant A7239 and by a Post- doctorate Fellowship to H.L. Abdel-Malek. presented. It is large changes in (a) Reference directions \forword+CT v+reverse I t-J-J.~...~ 1 (b) Forward and reverse analyses initiated from arbitrary reference planes to an element 1 I ., ‘sq3E1’@lplL (c) Typical analysis of a cascaded network with source and load impedances assumed constant I IFnlIl””II””nl#w (d) Subnetwork i cascaded with subnetworks k (at source end) and j (at load end) (e) Forward iteration for (d), transferring an equivalent source accounting for design variables from subnetwork k from one reference plane to the other funstion of ... . @ Y~ / It p reveres ifsralierr .. Y/ subnefwork I t II (f) Reverse iteration for (d) transferring an equivalent source accounting for design variables from subnetwork j from one reference plane to the other Figure 1 Notation and illustration of problems evaluation, first-order sensitivity, partial derivative or large-change sensitivity, respectively. A full reverse analysis taking [~~ v;] . [~1 e2] yields CH1355–7/78/0000-0079$00. 75 ~ 1978 IEEE 79