Sufficient conditions for the unique solvability of linear networks containing memoryless 2-ports

Combinatorial necessary and sufficient conditions for the unique solvability of linear networks containing n-ports are well known for the ‘general’ case. They are only necessary if relations among n-port parameters are also taken into consideration. In the present paper combinatorial sufficient conditions are presented for linear networks containing RLC elements and memoryless 2-ports. The somewhat surprising result is proved that whether a 2-port can cause certain types of singularities can be predicted before the interconnection. A concept, similar to the normal tree (which intersects ideal transformers by one, gyrators by two or no edges) is introduced for arbitrary 2-ports. Its existence implies unique solvability. Relations to previous results and algorithmical aspects are also discussed.     

Unique solvability of RLC nullor networks

Sufficient conditions for the unique solvability of linear active networks modelled as RLC nullor networks are given. an interpretation of these sufficient conditions in terms of subnetworks of the network being studied is also given. A simple procedure for calculating the matrices involved in these sufficient conditions is described and illustrated by example.     

Topological conditions for the solvability of linear active networks

The solvability problem of a linear active network is approached from a purely topological point of view using the two-graph method. It can be said that a topological condition for the solvability is the existence of a common tree of the voltage and current graphs. A few conditions for the existence of a common tree are derived. If there exists no common tree, subgraphs which cause the nonexistence can be distinguished, and a partition of two-graphs can be introduced. The partition has similar properties to the principal partition of a graph or the canonical form of a bipartite graph, and a structure of two-graphs represented by a partial ordering of sets of edges can be defined. An algorithm to find the partition and a common tree, if one exists, or if no common tree exists, a tree of one of the graphs which has as many common edges as possible with a tree of the other graph, is given. The decomposition of the coefficient matrix accompanying the structure is discussed, and algorithms to determine the decomposition is given.     

Topological conditions for the unique solvability of linear time-invariant and time-varying networks

The problem of knowing whether the non-unique solvability depends on the particular values of the components or on their topological interconnections is studied for linear networks with arbitrary, time-invariant as well as time-varying n-ports. Within every network, the topological notions of its sockets and of their independence are introduced. Networks with independent sockets are shown-at least when there are no relations among the non-zero coefficients, nor repetitions of the same coefficient are allowed, i.e. under suitable generality assumptions-to be uniquely solvable. Networks with dependent sockets are shown to be never uniquely solvable. Polynomially bounded algorithms, requiring only integer arithmetic, to test independence are available. When independence fails, a topological configuration of components which shows fewer topologically independent variables than equations, is proved to exist.     

Necessary and sufficient conditions for absolute stability of linear n-ports

A set of necessary and sufficient conditions is stated and proved for the absolute stability (under any passive terminations), in the ‘bibo’ sense, of a linear n-port characterized by its open-circuit impedance matrix. A more explicit set of such conditions is derived for the special case n = 3. In the process of deriving this set of conditions, some results and theorems concerning finding the zero sets of two and three parameter functions, are stated and proved. Also derived is a set of necessary and sufficient conditions for the absolute stability of two-ports characterized by the scattering parameters. Such sets have already been derived previously, but the present set is considerably simpler.     

Order of complexity of rlc networks containing a reactive gyrator

Using a particular expansion of the network determinant, a simple formula is derived giving the total number of natural frequencies of a passive RLC network containing a reactive gyrator. The order of complexity is expressed in terms of the degrees of the polynomials in the gyration impedance and the alteration in the network topology due to gyrator embedding. Quantitative conditions for the order of complexity of the active network exceeding that of the network without the gyrator are obtained. Formulas are also derived for the number of zero and non-zero natural frequencies.     

Existence of hybrid matrices for active n-ports encountered in the analysis of transistor networks

It is shown that hybrid matrices exist for certain types of active n-ports constructed using resistors and controlled sources. Furthermore, it is shown that the existence of these hybrid matrices implies that a certain well-known set of necessary and sufficient conditions, which guarantees the uniqueness of solutions to the nonlinear DC equations of transistor networks, can be simplified. Additional results are presented which show how the amount of computation required to verify these conditions can be reduced.     

Canonic forms of Brune and Darlington 2n-ports

The Brune and Darlington syntheses for one-ports have been extended to n-ports by several authors. the synthesis algorithms are well established and are not discussed in this paper, which is essentially concerned with the minimal realizations (in the total number of elements, including ideal transformers) of the 2n-ports occurring as building-blocks in the synthesis. A unified view of that subject is presented, including many new results and a critical survey of the literature. an appendix dealing with the contragradient reduction of a pair of rectangular matrices is of independent algebraic interest.     

Pre‐arrangement of solvability,complexity, stability and quality of GPC systems

A design problem of generalized predictive control (GPC) of plants characterized by discrete‐time models with a non‐zero transportation‐delay is considered. Conditions for the existence of the solution of the problem in terms of relevant design parameters (tuning knobs) are proposed and the system characteristics are discussed. Several design hints suitable for such plant models are proposed. Consequently, it is shown that by a suitable choice of the GPC tuning knobs a definite pre‐arrangement of the solvability, complexity, stability and quality (including robustness) of the resultant GPC control systems can be achieved. Some numerical examples are also given to illustrate the design problem and proposed methodology. Copyright © 2002 John Wiley & Sons, Ltd.     

On the solvability and computation of D.C. transistor networks

A study of non-linear d.c. networks containing transistors, diodes, linear resistors, independent voltage and current sources and linear controlled voltage and current sources, described by Tf(x) + Gx = b is presented. A few theorems concerning both the solvability and numerical computation of d.c. transistor networks are proved. The conditions sufficient for the existence of at least one solution and of a unique solution are defined. A method for numerical computation of the networks without the need to determine inverse matrices is presented and the convergence of the iterative technique is analysed. Suggestions are formulated regarding applicability of the method for computation of an approximate solution (close to the exact solution) which may subsequently be quickly corrected using the Newton-Raphson method.     

Existence and uniqueness of solutions for DC networks containing non‐linear ideal op amps

In this paper we are concerned with networks obtained by connecting independent sources, linear resistors and non‐linear ideal op amps. A necessary and sufficient condition for the existence and uniqueness of solutions for every positive output saturation voltage of the op amps and every value of the independent sources is found. Copyright © 2001 John Wiley & Sons, Ltd.     

Electrical characterization of linear and non‐linear random networks and mixtures

A material composed of a mixture of distinct homogeneous media can be considered as a homogeneous one at a sufficiently large observation scale. The problem of mixture characterization has been exactly solved in case of linear random mixtures, that is, materials for which the various components are isotropic, linear and mixed together as an ensemble of particles having random shapes and positions. In the present work the authors briefly review the linear theory and then consider mixtures of non‐linear media. In particular they give formulas for obtaining their constitutive equations for current density, electrical displacement, and magnetic induction. These relations have been derived by means of heuristic considerations on random networks and they have been verified with simulations obtaining a high degree of accuracy. Copyright © 2003 John Wiley & Sons, Ltd.     

The solvability and monotonical properties of a certain class of D.C. non-linear networks

The problem of the existence and uniqueness of solutions for a certain class of non-linear networks, including locally active elements and controlled sources, is considered in the paper. the conditions under which the input characteristics of the network are monotonically increasing are also analysed. Two theorems relating to the problem in question, containing some results in terms of the network topology, are formulated.     

A recursive power flow method for radial distribution networks: Analysis,solvability and convergence

A new Recursive Power Flow (RPF) algorithm for balanced/unbalanced radial distribution systems is proposed in this paper. Four generalized constant matrixes, that are calculated based on the graph theory, are used to obtain the power flow solution. These matrixes completely describe topology of a distribution system and models of its elements. In this method, each branch of a distribution system is considered as a bipolar. The method is able to consider detailed model of the elements. Solvability and convergence properties of the method have been mathematically presented. The method has been studied on the standard 16, 33 and 69-bus balanced distribution systems. In addition, this method is tested on 4 and 13-bus unbalanced standard distribution systems. Also, the method is tested on a 1337-bus real distribution system in Lorestan Electrical Power Distribution Company, Iran. The simulation results demonstrate the accuracy, computational efficiency and robustness of the proposed method.     

Identification of linear systems with time delay and unknown order

This paper proposes a system identification method for linear systems with time delay and unknown order. We hypothesize a number of candidate models, which have different order, because system order is unknown. In each model, we estimate the model parameters based on the maximum likelihood method by using nonlinear optimization technique. Then both local optimization technique and global search method are used because the estimated parameters may fall into a local minimum. After all candidate model's parameters are estimated, one model is selected among these models to estimate the system order by using a posteriori probability based on Bayes's theorem. Finally, the validity of the proposed method is demonstrated by numerical examples. © 2003 Wiley Periodicals, Inc. Electr Eng Jpn, 145(3): 61–68, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.10166     

Some topologico-dynamical properties of linear passive reciprocal networks

The paper considers the complete solvability and the order of complexity of passive RLCT (T = multiwinding ideal transformer) networks. A topological approach based on the determinant polynomial of the matrix of hybrid equations, formed as a set of 1st-order differential and algebraic equations, reveals the structure of the formulation tree and the subnetworks accountable for degeneracies. Topological and algebraic degeneracies are defined. Necessary and sufficient conditions for complete solvability are derived, and two algorithms are given to determine the order of complexity topologically, i.e. without having an explicit state-space representation.     

On complete stability of linear and quadratic programming neural networks

The paper addresses complete stability (CS) of the important class of neural networks to solve linear and quadratic programming problems introduced by Kennedy and Chua (IEEE Trans. Circuits Syst., 1988; 35 : 554). By CS it is meant that each trajectory converges to a stationary state, i.e. an equilibrium point of the neural network. It is shown that the neural networks in (IEEE Trans. Circuits Syst., 1988; 35 : 554) enjoy the property of CS even in the most general case where there are infinite non‐isolated equilibrium points. This result, which is proved by exploiting a new method to analyse CS (Int. J. Bifurcation Chaos 2001; 11 : 655), extends the stability analysis by Kennedy and Chua (IEEE Trans. Circuits Syst., 1988; 35 : 554) to situations of interest where the optimization problems have infinite solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Synthesis of passive networks containing periodically operated thyristors

This paper gives a new approach to the designing of power electronic networks. A method for the determination of passive time-varying networks is given when time functions of the excitation and response are given. These time functions are periodic. the period is divided into intervals. the network is linear and time-invariant in each interval. A sequence of the linear time-invariant networks is realized using switches. Switched windings of a multiport transformer are defined. Such elements enable one to realize the sequence of the linear time-invariant networks. Conditions for the substitution switches for thyristors are formulated. A structure of a thyristor converter is presented as an example.