Symbolic Simplification by means of Graph Transformations ZDENEK KOLKA 1) , MARTIN VLK 1) , DALIBOR BIOLEK 2) , VIERA BIOLKOVA 1) 1) Department of Radioelectronics, Brno University of Technology 2) Department of Electrical Engineering, University of Defence, Brno CZECH REPUBLIC kolka@feec.vutbr.cz, dalibor.biolek@unob.cz, biolkova@feec.vutbr.cz Abstract: The paper deals with an effective method for approximate symbolic analysis of linear circuits based on nontrivial transformations of voltage and current graphs. The method is based on eliminating the low- voltage branches from “high-voltage” loops and the low-current branches from “high-current” nodes, which simplifies circuit equations and the final symbolic formula. This allows reaching a higher degree of simplification in comparison with the simple edge deletion or contraction used in previous methods. The paper describes how the branches with low voltage or current can be identified and removed. Key-Words: Symbolic analysis, Topology reduction, Two-graph method, Linear circuits, Circuit theory, CAD tools 1 Introduction Recently, we have seen growing interest in the symbolic analysis of large multi-physics systems; see [1], [2], and references therein. The applicability of exact symbolic analysis is constrained to relatively small linear systems, as the size of the resulting expression grows exponentially with the number of components. If we appropriately restrict the range of frequency and network parameters, the majority of symbolic terms can be removed from large expressions without any significant numerical error [3]. Negligible symbolic terms are identified numerically, based on the known parameters of circuit components. The simplification methods can be divided into three classes according to the stage of analysis at which the simplification is performed: Simplification Before Generation (SBG), Simplification During Generation (SDG), and Simplification After Generation (SAG) [3]. The SAG methods are simple but very expensive in terms of computation and storage. Pure mathematical methods of the SDG type may have problems with the interpretability of resulting expressions [4]. The SBG methods simplifying the circuit equations or graphs are the most effective ones, as they work with a relatively small number of circuit equations [4]. A method, published as Sifting Approach [5], is based on a heuristic algorithm consisting in device parameter elimination from numerator and denominator submatrices separately. Another method [6], called Two-graph Simplification, modifies the voltage and current graphs constructed for the numerator and the denominator separately. The simplification consists in deleting or contracting edges representing a network parameter with sensitivity-based control strategy. The matrix-based method from [7] is based on removing individual matrix elements to obtain a simplified solution. However, in some cases the matrix simplification may surprisingly add symbolic terms that were not present in the original expression. We have proposed a different approach based on the two-graph method. Instead of simply deleting or contracting the graph edges it modifies the graph structure in order to decrease the number of common spanning trees. The respective graph-theoretical proofs were published in [8]. This paper deals with a practical method for identifying low-voltage or current branches that can be removed from high- voltage loops or high-current nodes. 2 Application of Two-Graph Method 2.1 Basic Principle The basic idea of the two-graph method consists in formulating separately Kirchhoff’s Voltage Law and Kirchhoff’s Current Law by means of two graphs – the voltage graph G V and the current graph G I [9]. Each circuit element of the admittance type (RLCg m ) is represented in each graph by just one edge. Other elements are modeled using equivalent models [9]. The determinant of the nodal admittance matrix is ∑ ∩ ∈ = ) ( ) ( ) ( I V ) ( det G T G T t t Y t ε Y (1) where Y (t) is the tree-admittance product of tree t, T(G V ) and T(G I ) are the sets of all spanning trees of voltage and current graphs. The intersection ) ( ) ( I V G T G T ∩ represents the common spanning Recent Researches in Circuits, Systems and Signal Processing ISBN: 978-1-61804-017-6 160