Showing papers in "IEEE Transactions on Circuit Theory in 1970"

A second-generation current conveyor and its applications

Abstract: A recent publication [l] introduced the concept of current conveying and an implementation in the form of a circuit building block termed the current conveyor (CC). This block has proven to be useful in many inst.rumentation applications, some of which have already been test)ed and reported [2], [3], while others are still under investigation. This correspondence introduces another new building block embodying the current conveying concept, but with different and more versatile terminal characteristics. This new block is considered to be a secondgeneration current conveyor,1 and h.ence is termed CC II.2 Application of CC II to the areas of active network synthesis and analog computation will be considered..

Lumped-circuit models for nonlinear inductors exhibiting hysteresis loops

Abstract: A new mathematical model of dynamic hysteresis loops is presented. The model is completely specified by two strictly monotonically increasing functions: a restoring function f(.) and a dissipation function g(.). Simple procedures are given for constructing these two functions so that the resulting model will simulate a given hysteresis loop exactly. The model is shown to exhibit many important hysteretic properties commonly observed in practice such as the presence of minor loops and an increase in area of the loop with frequency. In the case of an iron-core inductor, the mathematical model is shown to be equivalent to a lumped-circuit model, consisting of a nonlinear inductor in parallel with a nonlinear resistor. Extensive experimental investigations using different types of cores show remarkable agreement between results predicted by the model with those actually measured. The most serious limitation of this dynamic model is its inability to predict dc behaviors. For the class of switching circuits where dc solutions are important, a special dc lumped-circuit model is also presented.

Application of Volterra series to intermodulation distortion analysis of transistor feedback amplifiers

Abstract: This paper analyzes the intermodulation distortion of a solid-state feedback amplifier. Each transistor of the amplifier has been represented by a frequency-dependent model incorporating four nonlinearities. The Volterra series is used as the analysis tool. The primary motivation for this study is that intermodulation distortion is a critical problem in high-frequency long-haul solidstate systems. A computer program that calculates the amplifier intermodulation distortion has been developed. This program can be used to select the optimum bias point of each stage, optimum interstage and feedback networks, and optimum load and source impedances. Good correlation between calculated and measured results was obtained. The closed-form expressions derived show that the feedback is fully effective in reducing the open-loop second-order distortion and that the feedback may not significantly reduce the open-loop third-order distortion, if the "second-order interaction" phenomenon is dominant. The second-order interaction is explained in detail in this paper.

Very high-Q insensitive active RC networks

Abstract: A new circuit realizing high stable Q 's is presented. The frequency dependence of the operational amplifiers, which has been the principal limiting factor for high Q at higher frequencies for networks presented in the literature, has been taken into account. The performance of the new network is shown to be superior to the realizations thus far presented in the literature for the following reasons. The Q realized by the new circuit is independent of the operational-amplifier bandwidth, which results in a stable response. The realization is also insensitive to manufacturing tolerances and, therefore, it can be readily tuned for the desired Q and center frequency without encountering any stability problems.

A generalized form of Tellegen's theorem

Abstract: Among the theorems of circuit theory, Tellegen's theorem is unusual in that it depends solely upon Kirchhoff's laws and the topology of the network. The theorem therefore applies to all electrical networks that obey Kirchhoff's laws, whether they be linear or nonlinear, time-invariant or time-variant, reciprocal or nonreciprocal, hysteretic or nonhysteretic; the excitation is arbitrary, and the initial conditions are also immaterial. When specific assumptions are made concerning the network elements, the excitation, and the initial conditions, Tellegen's theorem reduces to many useful network theorems. In this paper a generalized form of Tellegen's theorem that allows the efficient derivation of new results is presented. A special formthe "difference form"-of this theorem is shown to be of particular value, and also capable of simple expression in terms of wave variables. The application of the generalized form of Tellegen's theorem is illustrated by an example.

On digital single-sideband modulators

Abstract: This paper concerns digital single-sideband modulators with analog inputs and outputs. A group of 12 modulators is treated as a single system, with analog-to-digital converters at 12 input ports and a digital-to-analog converter at a single output port. Tradeoffs are obtained between computational parameters. Substitution of digital counterparts for the phase shifters and product modulators in 12 analog Hartley modulators gives a simple program. However, operation in real time requires quite a large number of multiplications per second. A similar digitalization of 12 Weaver modulators requires even more multiplications. However, the 12 Weaver modulators can be transformed as a system, to reduce the multiplication rate for the Hartley system by a factor 4 . The cost is an increase in scratch pad storage and a more complex program.

Connectivity considerations in the design of survivable networks

Abstract: The problem of constructing networks that are "survivable" with respect to branch damage is considered. The networks are modeled by linear graphs and a square symmetric "redundancy" matrix R'=[r'_ij] is specified. Algorithms are given to construct an undirected graph G with a minimum number of branches such that 1) G contains no parallel branches, and 2) for all i, j there are at least r'_ij branch disjoint paths between the ith and jth vertices. These algorithms are complicated but may easily be applied to construct graphs with several hundred vertices.

Quantitative analysis of simple and interconnected systems: Stability, boundedness, and trajectory behavior

Abstract: In practice one is not only interested in the qualitative type of information obtainable from the stability (in the Lyapunov sense) of a dynamic system, but also in quantitative data, such as specific trajectory bounds and specific transient behavior. A system could, for example, be stable and still be completely useless because it may exhibit undesirable transient characteristics (e.g., it may exceed certain limits imposed on the trajectory bounds). In order to develop a meaningful quantitative theory for the analysis of dynamic systems, stability is defined here in terms of subsets of the state space that are prespecified in a given problem and, in general, may be time varying. The properties of these subsets yield not only information about the stability of a system, but they also yield estimates of trajectory bounds and of trajectory behavior. The theory developed here is general enough to include autonomous and nonautonomous systems, linear and nonlinear systems, simple systems and interconnected systems. The composite, or interconnected systems considered are analyzed and treated in terms of their subsystems. After stating various definitions of stability and instability, theorems that yield sufficient conditions for stability and instability are stated and proved. These theorems involve the existence of Lyapunov-like functions which in general do not possess the usual definiteness requirements on V and V. In order to demonstrate the developed theory, several examples are considered.

On graphs of invulnerable communication nets

Abstract: Graph theoretic techniques provide a convenient tool for the investigation of the vulnerability of a communication network to either natural or enemy-induced damage. In this paper, the communication network is represented by a nonoriented linear graph, and the problem of synthesizing networks to provide optimal. protection against damage is investigated. Damage is associated with the removal of a set of nodes that disconnects the graph or by the removal of a set of edges that disconnects the graph. It is assumed that fixed cost for edges may be envisioned as prescribing the total number of edges allowed to connect a given number of nodes. A class of graphs that provides optimum damage protection for a fixed cost is then derived. Bipartite graphs are investigated in detail. It is shown that the complete bipartite graph is also an optimal damage-resistant net and that it can be decomposed into Hamiltonian cycles to yield optimal graphs of lower order. Another possible criterion of optimality is introduced and studied. This new measure is the minimum number of edges that must be deleted in order to isolate a given number of nodes. It is then shown that the complete bipartite graph is also optimal with respect to this alternative measure of vulnerability. This paper also contains proofs of several new and interesting graph theoretic results.

Realization of the Chua family of new nonlinear network elements using the current conveyor

Abstract: In a recent paper, Chua [l] introduced three new network elements, namely, the mutator, the scalar, and the reflector. These, together with the rotator that he introduced earlier [2], form a family of four basic elements that prove to be useful in the area of nonlinear network synthesis as well as suggesting many unusual circuit applications. This correspondence extends the work of Chua by providing a more unified treatment of the circuit realization of the four elements. The current conveyor [3] will be used to obtain implementations that are in closer match to the ideal equations defining the elements.

Nonideal performance of two-amplifier positive-impedance converters

Abstract: Positive-impedance converting networks may be used to realize driving-point inductance and frequency-dependent negative resistance. The Q factors of these immittances are evaluated over a wide frequency range in terms of the relevant parameters. Three distinct types of Q- factor behavior are derived and it is shown that extended wide-band operation may be achieved such that Q factor is virtually independent of amplifier phase shift. The use of design-value asymptotes is explained as a means of determining the behavior of practical circuits.

Novel RC-active-network synthesis using generalized- immittance converters

Abstract: A general synthesis procedure capable of realizing any stable transfer function is described; generalized-immittance converters are used in which the input immittance is proportional to s^{2} \times load immittance. The resulting realizations are shown to have low sensitivity to element variations. The sensitivity in realizations of second-order transfer functions is independent of the selectivity and it is shown to be much lower than the sensitivity in realizations using positive-gain amplifiers or negative-immittance converters. A unique feature is that in some cases the same circuit can be used as a low-pass as well as a high-pass filter. The synthesis procedure is illustrated by examples and experimental results.

A constructive graph-theoretic solution of the Shannon switching game

Abstract: A simple graph-theoretic solution to the Shannon two-person switching game is given. The solution is constructive in that algorithms have been formulated to determine if a game played on any given graph is a short, cut, or a neutral game. The proof makes use of a result due to Kishi and Kajitani, who showed that the edges of any linear graph G can be decomposed into a partition containing three blocks. From this partition one constructs three graphs that form a principal partition denoted by the ordered triple (D2, G2, H2). It is proved that the game is a short (cut) [neutral] game if and only if a distinguished edge e belongs to G_{2}(H_{2}) [D_{2}] . Strategies for playing each game are given. Finally, duality theory is used to prove that a global strategy exists for a cut game as well as for a short game, where by a global strategy with respect to a short game is meant that the short player can win with respect to any edge spanned by the pair of cospanning trees without knowing which of these edges is the distinguished edge with respect to the game being played.

Maximum response with input energy constraints and the matched filter principle

Abstract: A method is developed for determining the maximum response of a system under various constraints of the input. The solution is based on appropriate extensions of the reasoning leading to the matched filter principle. Special cases include a number of problems treated elsewhere. However, the available solutions involve integral equations and the calculus of variations. In the following approach the results are obtained in simple algebraic terms.

Integration system of a nonlinear transient network-analysis program

Abstract: Users of nonlinear transient network-analysis programs, which use explicit integration procedures, have reported that excessive computer time is required to compute the transient response of networks containing widely separated time constants. Several recent papers have shown that for these problems significant computation-time reductions can be obtained by using implicit integration formulas. This paper is concerned with the selection and implementation of an implicit integration formula used in the CIRPAC nonlinear transient network-analysis program. Time-comparison data presented here indicate that the integration system that is described in this paper is significantly faster than the integration system of a network-analysis program that uses an explicit integration algorithm.

Distributed multimode oscillators of one and two spatial dimensions

Abstract: Large 1-dimensional (ID) and 2-dimensional (2D) arrays of almost-linear and almost-lossless networks are investigated using the equivalent linearization technique of Kryloff and Bogoliuboff. The active element is assumed to be a distributed cubic nonlinear shunt conductance. It is demonstrated that an arbitrary number of nonresonant (or asynchronous) modes can be stably excited on a 2D oscillator, but only a single mode on a ID oscillator. Experiments confirm that a variety of multimode oscillations, resonant as well as nonresonant, are observed on 2D oscillators but not on ID oscillators.

On the factorization of transfer matrices of lossless two-ports

Abstract: The problem of factorizing the transfer matrix is treated in a rigorous way for all lumped finite lossless two-ports, including those containing gyrators and imaginary resistances. It is shown how, for a given order in which the factorization should be realized, all solutions can be obtained in a simple way once an acceptable solution of a certain set of fundamental equations, consisting of a certain number n of linear homogeneous equations in n + 2 unknowns, has been found. A simple procedure for establishing these fundamental equations is given.

Survivable communication networks and the terminal capacity matrix

Abstract: The problem of constructing networks that are "survivable" with respect to branch damage is considered. A square redundancy matrix R = [r_{i,j}] is specified. Algorithms are given to construct a graph with minimum number of branches so that for all i, j there are 1) at least r_{i,j} undirected branch disjoint paths between the ith and the j th vertices, or 2) there are exactly r_{i,j} undirected branch disjoint paths between the ith and the jth vertices. These algorithms are closely related to the optimal realization of terminal capacity matrices. Two of the algorithms are extended to the optimal realization of terminal capacity matrices for symmetric or pseudosymmetric graphs.

Topological degrees of freedom and mixed analysis of electrical networks

Abstract: A new concept of topological degrees of freedom is introduced as a generalization of the concepts of rank and nullity. This number.corresponds to the minimum number of independent variables taken in mixed analysis of electrical networks, which is a generalization of well-known node-pair and loop analysis. An explicit formula giving topological degrees of freedom and a graph-theoretical algorithm for finding all the minimum sets of independent variables are presented. Another new concept of extremal tree is introduced and shown to play a central role in obtaining the results.

Solvability and analysis of linear active networks by use of the state equations

Abstract: A study of the reduction process used to obtain the state equations for a network containing capacitors, resistors, inductors, and controlled and incontrolled sources of all types, gives the necessary and sufficient conditions for such a network to possess a solution. These conditions involve both the topology of the network and its element values. The formation of the state equations also depends both on the network's topology and its element values. It is shown that the state variables can be chosen from the variables associated with the controlled sources, and any set of state variables chosen for the passive network, formed by removing all the sources. A topological restriction on the active network, which ensures that the maximum order of complexity of the active network is the same as that of this passive network, is given.

Linear transformation converter and its application to the synthesis of nonlinear networks

Abstract: A generalized approach for synthesizing a prescribed resistive nonlinear n-port is presented with the help of a linear 2nport called the linear transformation converter (LTC). It is proved that a large class of voltage-controlled (current-controlled) resistive nports can be realized by using only one LTC and an appropriate homeomorphic nonlinear n-port. The LTC is shown to be decomposable into the cascade connection of a"rotator," "reflector," and "scalor." This factorization provides not only a general realization technique, but also a geometrical interpretation of the capabilities of the 2-port LTC to transform a monotonic characteristic curve into a prescribed multivalued curve. In particular, necessary and sufficient conditions are given for realizing a multivalued two-terminal resistor by connecting an appropriate monotonic resistor across port 2 of a 2-port LTC.

Impedance and admittance matrices of distributed three-layer N-ports

Abstract: The port impedance and admittance matrices of threelayer structures of linear, passive, isotropic materials are defined and their properties briefly investigated. Discrete equivalent circuit representations are given, together with two examples.

Power-series equivalence of some functional series with applications

Abstract: In this paper, we show that the Laplace transform of the expansion h(t) = \sigma_{n = 0}^{\infty} c_{n} g_{n} (t) for some important sets g_{n} (t) is equivalent to a power-series expansion. Techniques based on this result are presented for obtaining the coefficients c. as those of a power series; also, methods are presented for obtaining the functional series inverse. The set of Laguerre functions is discussed in detail and, using the power-series equivalence, the truncation error is obtained. The application of the power-series equivalence to the summing of series is shown and illustrated with the Neumann series. Finally, the extension of the power-series equivalence to the expansion of functions of several variables is given. The areas for which the techniques developed are relevant include the analysis and design of signals and the identification and synthesis of processes and systems.

Average energy storage by a one-port and minimum energy synthesis

Abstract: The average energy stored by the elements of a linear passive time-invariant one-port is examined. Except in special cases, the energy storage is not determined by the impedance function Zo(p) alone. However, it is shown that for a given Zo(p) there is a minimum energy storage for sinusoidal excitation. Further, this minimum energy can be achieved at all frequencies as a minimum energy synthesis.

Generalized interdigital linear phase networks with optimum maximally flat amplitude characteristics

Abstract: The closed-form solution is presented for microwave bandpass nonminimum phase transfer functions that exhibit an optimum passband maximally flat amplitude characteristic while maintaining a maximally flat group-delay response about band center. There are two basic transfer functions obtained depending upon whether the degree of the resulting network is even or odd. The transfer characteristics are shown to be capable of being realized by resistively terminated symmetrical generalized interdigital networks. Synthesis procedures are presented for the case of n odd corresponding to the input and output ports being situated at the same physical end of the generalized interdigital line and for the case n even where the ports are at opposite ends. These procedures are based upon the one-port synthesis of the even- and odd-mode forms. Finally, the synthesis procedure is illustrated for the case of n = 6 , and typical transfer characteristics are shown for this example.

Equivalent networks with nullors for positive immittance inverters

Abstract: Some equivalent networks are presented for positive immittance inverters and converters. The simplest configurations are described with two and three nullors that correspond to sixty-two different inverter circuits and sixty-six different converter circuits. The theorem of "twin" transformation of networks with nullors, which was used in the development of some equivalent networks, is outlined.

Capacitance and resistance minimization in one-port RC networks

Abstract: A complete topological characterization is given for RC networks that realize an RC driving-point impedance with minimum total capacitance or resistance. In particular, it is shown that for a constant input voltage, maximizing the steady-state capacitor voltages minimizes the total capacitance, and for a constant input current, maximizing the steady-state resistor currents minimizes the total resistance. It then follows that one of the Foster canonical forms is a minimum total capacitance structure and the other is a minimum total resistance structure. Explicit formulas for the minimum total capacitance and resistance are given.