Extension of the normal tree method

The results of this paper are applicable to linear electrical networks that may contain ideal transformers, nullors, independent and controlled sources, resistors, inductors, and capacitors, and, under a topological restriction, gyrators. A relation between summands of some expansion of the network determinant and pairs of conjugate trees is proved, which uncovers the equivalence of known criteria on generic solvability based on matroids and those based on pairs of conjugate trees. New criteria on the solvability of active networks are given. A method to obtain complete sets of generic state co-ordinates is established, which includes the following extension of the wellknown normal tree method: The generic order of complexity equals the sum of the number of forest capacitors and the number of co–forest inductors in any normal pair of conjugate trees, the latter term being introduced in this paper. The voltages across the forest capacitors together with the currents through the co-forest inductors may be given initial values independently from each other. Further, a systematic method of augmentation that yields networks of generic index 1 is proposed. All results are expressed in terms of network determinants as well as in terms of network graphs, and all given criteria may be checked by efficient algorithms. Copyright © 1999 John Wiley & Sons, Ltd.

[1]  E. J. Purslow,et al.  Solvability and analysis of linear active networks by use of the state equations , 1970 .

[2]  P. Bryant,et al.  The order of complexity of electrical networks , 1959 .

[3]  Leon O. Chua,et al.  On the Dynamic Equations of a Class of Nonlinear RLC Networks , 1965 .

[4]  András Recski.  Unique solvability and order of complexity of linear networks containing memoryless n‐ports , 1979 .

[5]  E. Kuh,et al.  The state-variable approach to network analysis , 1965 .

[6]  Leon O. Chua,et al.  Computer-Aided Analysis Of Electronic Circuits , 1975 .

[7]  U. Feldmann,et al.  Computing the generic index of the circuit equations of linear active networks , 1996, 1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World. ISCAS 96.

[8]  David P. Brown.  Derivative-explicit differential equations for RLC graphs☆ , 1963 .

[9]  William H Cunningham,et al.  Improved Bounds for Matroid Partition and Intersection Algorithms , 1986, SIAM J. Comput..

[10]  L. Chua,et al.  Uniqueness of solution for nonlinear resistive circuits containing CCCS's or VCVS's whose controlling coefficients are finite , 1986 .

[11]  M. M. Milic.  Explicit formulation of the state equations for a class of degenerate linear networks , 1971 .

[12]  J. Tow.  The explicit form of Bashkow's A matrix for a class of linear passive networks , 1970, IEEE Transactions on Circuit Theory.

[13]  Mirko M. Milić.  Some topologico‐dynamical properties of linear passive reciprocal networks , 1977 .

[14]  L. Chua,et al.  Topological criteria for nonlinear resistive circuits containing controlled sources to have a unique solution , 1984 .

[15]  M. Chandrashekar,et al.  On the existence of solutions to linear active networks: A state‐space approach , 1974 .

[16]  E. J. Purslow,et al.  Order of complexity of active networks , 1967 .

[17]  Marc Fosseprez,et al.  Non-linear Circuits: Qualitative Analysis of Non-linear, Non-reciprocal Circuits , 1992 .

[18]  András Recski,et al.  Sufficient conditions for the unique solvability of linear networks containing memoryless 2-ports , 1980 .

[19]  Martin Hasler,et al.  Non-linear non-reciprocal resistive circuits with a structurally unique solution , 1986 .

[20]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[21]  P. Bryant,et al.  The Explicit Form of Bashkow's A Matrix , 1962 .

[22]  Takao Ozawa,et al.  Topological conditions for the solvability of linear active networks , 1976 .

[23]  M. Milic,et al.  General passive networks-Solvability, degeneracies, and order of complexity , 1974 .

[24]  Dominique de Werra,et al.  A discrete model for studying existence and uniqueness of solutions in nonlinear resistive circuits , 1994, Discret. Appl. Math..

[25]  Herman E Koenig,et al.  Analysis of Discrete Physical Systems , 1967 .

[26]  Brent R. Petersen,et al.  Investigating solvability and complexity of linear active networks by means of matroids , 1979 .

[27]  J. Tow,et al.  Order of complexity of linear active networks , 1968 .

[28]  Takao Ozawa,et al.  Order of complexity of linear active networks and a common tree in the 2-graph method , 1972 .

[29]  K. Murota.  Systems Analysis by Graphs and Matroids: Structural Solvability and Controllability , 1987 .

[30]  Kazuo Murota,et al.  Systems Analysis by Graphs and Matroids , 1987 .

[31]  J. Tow,et al.  The A-matrix of linear passive reciprocal networks☆ , 1972 .

[32]  A. Willson,et al.  A fundamental result concerning the topology of transistor circuits with multiple equilibria , 1980 .