Realization of n-port resistance matrices

The problem of realization of a given real symmetric matrix R of order n as an open-circuit resistance matrix of some resistive n-port network having nullity n is considered in this paper. The solution presented is in the form of a realization technique which implies the necessary and sufficient conditions for this type of realization. Derived from fundamental topological considerations in the analysis of resistive n-port networks, the realization technique presented here is found to be superior to the existing ones.     

The existence and the determination of terminal equations for hybrid connected N-ports

A new procedure for the existence and the determination of terminal equations for the hybrid interconnection of two n-port networks is given. It is shown that this procedure generalizes the previous works of Duffin, Trapp and Mitra for the hybrid sum of terminal impedance matrices which are either Hermitian positive semi-definite (HSD) or arbitrary matrices with complex entries. For the application of the new procedure it is shown that each n-port need not be terminal solvable or well defined. In fact, even in the case of active RLC n-ports considered in sinusoidal steady-state, the previous results of Duffin, Mitra and Trapp established for HSD matrices or arbitrary complex matrices can still be utilized with the use of modified terminal equations, to obtain the terminal equations of the resulting n-port networks. It is also noted that the method introduced in this paper is independent of the choice of generalized inverses.     

Short-circuit admittance matrix synthesis with rc common-ground networks and a minimum number of grounded finite-gain phase-inverting voltage-controlled voltage sources

A new procedure for the synthesis of active RC networks when grounded finite-gain phase-inverting voltage-controlled voltage sources serve as active elements is developed. It is proved that an arbitrary n × n matrix of real rational functions of the complex frequency variable can be realized as the short-circuit admittance matrix of a grounded transformerless active RC n-port network containing (n+1) grounded finite-gain phase-inverting voltage-controlled voltage sources (VCVSs). In general all the (n+1) grounded VCVSs are necessary. The structure proposed to prove a general theorem is later simplified for the realization of a restricted but important class of real rational matrices to obtain considerable savings in the computation volume and in the number of passive components used for the realization of the network. Examples are given to illustrate presented synthesis procedures.     

Resistance and capacitance of 4 × n cobweb network and two conjectures

A classic problem in electric circuit theory studied by numerous authors over 160 years is the computation of the resistance between two nodes in a resistor network, yet some basic problem in m × n cobweb network is still not solved ideally. The equivalent resistance and capacitance of 4 × n cobweb network are investigated in this paper. We built a quaternion matrix equation and proposed the method of matrix transformations in terms of the network analysis. We proposed a brief equivalent resistance formula and find that the equivalent resistance is expressed by cos(kπ/9) in a series of strict calculation. Meanwhile, an equivalent resistance of infinite networks is gained. Using the inverse mapping relation between capacitance parameters and resistance parameters, the equivalent capacitance formula is also given for the 4 × n capacitance cobweb network. By analyzing and comparing the equivalent resistances of the 1 × n, 2 × n, 3 × n and 4 × n cobweb networks, two conjectures on the equivalent resistance and capacitance of the m × n cobweb network are proposed. Copyright © 2013 John Wiley & Sons, Ltd.     

A class of nonrealizable paramount matrices

Although paramountcy¹ is a necessary and sufficient condition for the realizability of a 3 by 3 matrix as either an open-circuit impedance matrix or a short-circuit admittance matrix of a resistive 3-port,² Weinberg's 4 by 4 example^3,5 shows that paramountcy, although necessary for open-circuit impedance realization, is not sufficient. Here we exhibit a class of 4 by 4 completely irreducible paramount matrices, including Weinberg's example, no member of which is realizable as an open-circuit impedance matrix. This is accomplished by examining those topological constraints inherited by a realizing network when algebraic conditions are imposed on the paramount matrix which the network realizes. Our development deals with an n-port network, for arbitrary n, and also provides new criteria guaranteeing nonrealizability of an irreducible¹ paramount matrix.     

Energy-related concepts for non-linear time-varying n-ports: Passivity and losslessness

Definitions of passivity and losslessness are presented that apply to n-port networks which are not necessarily linear, time-invariant, or lumped; in fact, these definitions apply to any n-port that has an abstract dynamical system representation. For lumped, non-linear n-port networks that can be mathematically represented by a finite-order dynamical system, conditions for passivity and losslessness are formulated in terms of properties of the state equation function, the output function, etc. These conditions can be verified without solving the state equation, and can be viewed as non-linear generalizations of the well-known time-domain and frequency-domain passivity and losslessness conditions for linear time-invariant lumped n-port networks.     

On the number of conductances required for realizing Y and K matrices

Upper bounds are established on the number of conductances required for realizing a real symmetric matrix Y as the short-circuit conductance matrix of a resistive n-port network containing no negative conductances, and for the realization of a real matrix K as the potential factor matrix of a similar network without negative conductances. These results are the consequence of the properties of the modified cut-set matrix of an n-port and a theorem in the theory of linear programming.     

Lossless 2-ports between non-reactive terminations

Lossless reciprocal 2-ports between terminations with even impedance functions are considered. They are the unifying concept for the synthesis of FDNR and leapfrog RC-active filters in their most general form. The scattering matrix of lossless reciprocal 2-ports with respect to such terminations is defined, its canonic form deduced and the transfer of active power discussed. This scattering matrix is related to the usual scattering matrix of the same 2-port but with respect to unit resistance terminations. In such a way the precise structure of the 2-port as an LC ladder can be deduced from properties of the original transfer function and the termination impedances and the LC synthesis can be performed by conventional methods and programs. An exhaustive enumeration of the different cases that may arise is given. It can give a general outlook on the different RC-active filters that may be obtained.     

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.     

On the realization of gyrators by nullors and resistors

It is proved that for the realization of a gyrator by nullors and resistors, at least two nullors and at least two resistors are needed. A generalization is the result that any n-port network possessing both an impedance and an admittance matrix and consisting of nullors and resistors must contain at least n resistors. It is conjectured that a realization of a gyrator with only two nullors requires four resistors and a realization of a gyrator with only two resistors requires three nullors.     

(n +p)-node realizability of Y-matrices of (n + 1)-node resistance n-port networks

Sufficient conditions are established for the (n +p)-node, p>2, realizability of Y-matrices of (n + 1)-node resistance n-port networks. It is shown that these conditions are a generalization of the previous known results for p = 2 and p =n.     

Representation of noise in a particular type of linear n-port network

A noisy linear N-port network that can be represented by a noise-free linear N-port network with voltage and current noise sources at the ports is considered. If the linear N-port network has a particular type of admittance matrix, it is possible to show that as many as N/2 when N is even and (N – 1)/2 when N is odd of the ports may be cleared of noise sources. The total number of unique representations of such a network is given for both the general case and for the case when a particular port or ports must be free of noise sources.     

Synthesis of n-port active rc networks with a minimum number of capacitors

A new and practical synthesis procedure is presented for the realization of an arbitrary n × n matrix of real rational functions of the complex frequency variable as the short-circuit admittance matrix of a transformerless active RC n-port network. The realization requires a theoretically minimum number of capacitors p, where p is the degree of the matrix, and no more than (n + p + 1) grounded finite-gain phase-inverting voltage-controlled voltage sources. All the capacitors and ports are grounded. The freedom implicit in the synthesis procedure allows the inclusion of constraints on the passive element values. Furthermore, in special cases the realization is achieved with a reduced number of conductances and voltage-controlled voltage sources. The synthesis procedure is simple to apply and can readily be implemented on a digital computer. Several examples are given.     

Topologie des kopplungsfreien Reaktanz-2n-Poles mit geringstem Aufwand an Speichern

Übersicht In dieser Arbeit werden Kriterien abgeleitet, die cs erlauben, an einem vorgelegten kopplungsfreien Reaktanz-2n Pol die Zahl der überflüssigen Spulen und Kondensatoren zu ermitteln. Ausgangspunkt der Untersuchungen ist die im 2. Kapitel dargestellte Maschenimpedanzmatrix eines Reaktanz-2n-Poles und deren lineare Transformation in die Matrix einer äquivalenten Schaltung. Im 3. Kapitel wird ein Zusammenhang zwischen der geringsten Zahl von Spulen und Kondensatoren und topologischen Größen aufgezeigt. Im letzten Kapitel werden Erweiterungen der gewonnenen Sätze dargestellt.

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Summary Criteria will be deduced, which permit to find out the number of superfluous coils and capacitors in a linear and losslessn-port without magnetic couplings. The investigation starts with the loop impedance matrix of a losslessn-port and its linear transformation in the matrix of an equivalent circuit. This is described in the second chapter. In the third chapter relations are to be presented between the minimum number of coils and capacitors and topological structures. Some extensions of the won theorems, especially on the invariance of the minimum number of coils and the minimum number of capacitors in equivalentn-ports, are given in the last chapter.

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Mit 2 Textabbildungen     

Formulae of resistance between two corner nodes on a common edge of the m × n rectangular network

This paper deals with the equivalent resistance for the m × n resistor network in both finite and infinite cases. Firstly, we build a difference equation driven by a tridiagonal matrix to model the network; then by performing the diagonalizing transformation on the driving matrix, and using the auxiliary function tz(x,n), we derive two formulae of the equivalent resistance between two corner nodes on a common edge of the network. By comparing two different formulae, we also obtain a new trigonometric identity here. Our framework can be effectively applied in complex impedance networks. As in applications in the LC network, we find that our formulation leads to the occurrence of resonances at frequencies associated with (n + 1)?_t = kπ. This somewhat curious result suggests the possibility of practical applications of our formulae to resonant circuits. At the end of the paper, two other formulae of an m × n resistor network are proposed. Copyright © 2014 John Wiley & Sons, Ltd.     

A state space approach to the tangential interpolation problem with bounded real matrices

This paper presents a global solution of the tangential interpolation problem via state space concepts. the solution is obtained in two steps. the first step consists of factorization of the Nevanlinna-Pick matrix and calculating the state space vectors of a passive n-port network N. In the second step, using the state space vectors and the interpolation data, a passive minimal state space realization of N is derived from which the desired interpolating bounded real matrix is obtained. the presented approach involves only standard algebraic matrix procedures and provides the constructive algorithm for the interpolating matrix.     

An approach to modelling DC-DC converter circuits using graph theoretic concepts

An approach to modelling DC-DC converter circuits based on graph theoretic concepts is discussed in this paper. the DC-DC converter circuits are treated as networks containing switches, with the magnitudes of their associated eigenvalues much smaller than the switching frequency. the procedure for modelling this class of networks essentially involves separating the original network into two N-port networks, one containing those branches responsible for all phenomena peculiar to the switching action and the other containing the remaining branches of the network. the two N-port networks thus formed lead to a systematic and convenient way of developing low-frequency circuit models for DC-DC converter circuits.     

Cascade decomposition of a passive 2n-port network

This paper deals with a cascade decomposition of a passive 2n-port network. after a basic discussion relative to complex- and unit-normalized scattering matrices, necessary and sufficient conditions are presented under which given lossless 2n-ports D₁ and D₂ are simultaneously extractable from the front- and back-ends of the passive 2n-port N, respectively, to leave a passive 2n-port remainder Ne. Based on these conditions, some relations between the individual and simultaneous extractability of D₁ and D₂ from N for both degenerate and non-degenerate remainders N_e are discussed. the results are presented in terms of scattering as well as impedance matrices and can be considered as an extension of those given in Reference 1.     

Synthesis of n‐port resistive networks containing 2n terminals

This paper is concerned with the realizability problem of n‐port resistive networks that contain 2n terminals. A necessary and sufficient condition for any real symmetric matrix to be realizable as the admittance of an n‐port resistive network containing 2n terminals is obtained. This condition is based on the existence of a parameter matrix. Furthermore, the values of the elements are expressed in terms of the entries of the admittance matrix and the parameter matrix. Finally, a numerical example is used to illustrate the results. Copyright © 2013 John Wiley & Sons, Ltd.     

General interpretation of sensitivity functions

A general interpretation of sensitivity functions of linear networks is given in order to determine any nth order sensitivity function with respect to any parameter. The calculation uses an indirect method: ‘transfer function products’ give the desired sensitivities instead of derivation. The interpretation makes it possible to construct a computer program, too, which needs only one matrix inversion for a complete sensitivity analysis (including first- and higher-order sensitivities). It is shown that the adjoint network approach is superfluous and more complicated than the given method.