COMPUTATION OF SIMPLE SYMBOLIC POLE EXPRESSIONS F. Constantinescu, M. Nitescu, C.V. Marin, D. Marin Department of Electrical Engineering “Politehnica” University Bucharest, Romania Abstract Some approximate symbolic pole expressions of a fourth order circuit are computed with a relative error of 10 -3 in a design point. A very simple form (a ratio of second degree polynomials) of these expressions is obtained. It follows that these formulas are valid for a large domain in the design space and simple enough to be used by designers. I. INTRODUCTION Some symbolic analysis programs are able to find approximate symbolic expressions of the poles and zeros using the pole splitting technique, analytical formulas [1,2] for the roots of a polynomial up to the fourth degree, and symbolic polynomial deflation [2,3,6]. The sifting approach using device parameter elimination, determinant simplification and factorization, cancellation of factors between numerator and denominator is used in [4] for the computation of symbolic pole expressions. All these procedures are heuristic approaches i.e. they don’t necessarily lead to symbolic pole/zero expressions for an arbitrary circuit. In [5] a characteristic polynomial whose order is reduced to 1 or 2 is computed by approximating the matrix pencil (A,B) where the circuit equations are written as (A-sB)x=0. A detailed review of these techniques is given in [7]. In [13] the time constant matrix is used for finding approximate pole/zero expressions. Another approach allowing the determination of the approximate symbolic expressions for all poles/zeros of an arbitrary circuit is presented in [8]. This procedure uses the LR algorithm [9] converging to the eigenvalues of a circuit matrix (the state matrix in the case of pole computation). By this way are obtained rational expressions in terms of circuit parameters for real eigenvalues and expressions using at most one square root for complex eigenvalues. Even though the convergence is ensured, the computational complexity of the LR algorithm may limit the usefulness of this method [10]. To overcome this difficulty a procedure for computation of the symbolic expressions for a part of the eigenvalues (belonging to a “cluster” in the complex plane) only has been developed [11]. By this way the computational complexity is reduced, the LR algorithm working on a reduced matrix corresponding to the “non-separable” eigenvalues (belonging to the “cluster”) only. Formulae obtained with this method are valid in a vicinity of the design point. The usefulness of pole/zero expressions is given by their simplicity and their domain of validity in the design space. An algorithm for pole expression simplification after computation is presented in this paper. An evaluation of the symbolic formulae validity in the design space is done. In Section II the pole/zero computation algorithm is presented. Section III contains the algorithm for formula simplification. Section IV contains an example. A discussion and some conclusions and a future work plan are presented in Section V. II. POLE/ZERO COMPUTATION The main steps of our algorithm are: 1. Derivation of the state matrix in symbolic form 2.Computation of numerical eigenvalues corresponding to nominal values of circuit parameters. 3. Cluster identification 3.1. A maximal number of minimum module poles are eliminated; 3.2. A maximal number of maximum module poles are eliminated; 3.3. The remaining poles form the “cluster”. 4. Ordering of the state matrix so that |a 11 |≥|a 22 |≥...≥|a nn |. 5. LR iterations The reduced state matrix A=[a ij ] being given the LR iterations are: A=L 1 R 1 , R 1 L 1 =A 1 , A 1 =L 2 R 2, R 2 L 2 =A 2 , and so on (a sequence of decompositions and multiplications). In the LR decomposition L is a lower triangular matrix with unitary diagonal and R is an upper triangular matrix. The program goes out from the LR iteration loop after the step p if the smallest module eingenvalue satisfies an error condition (to be defined in the following). If λ n is real the LR iteration will be continued with an (n-1) • (n-1) matrix A* obtained from A p by removing the last row and the last column. In this case λ n = a nn . If λ n is complex A* is obtained by removing the last two rows and columns. In this case λ n and λ n-1 are the eigenvalues of the 2 • 2 lower right -hand block of A p . Considering the numerical eigenvalue e i * as “exact”, the relative error of the symbolic eigenvalue s i is given by: e e e i i i - * * where e i is a numerical value 35 ISBN: 973-652-674-7