Showing papers in "IEEE Transactions on Circuits and Systems I-regular Papers in 1960"

Some Extensions of Liapunov's Second Method

Abstract: >In the study of the stability of a system, it is never completely satisfactory to know only that an equilibrium state is asymptotically stable. As a practical matter, it is necessary to have some idea of the size of the perturbations the system can undergo and still return to the equilibrium state. It is never possible to do this by examining only the linear approximation. The effect of the nonlinearities must be taken into account. Liapunov's second method provides a means of doing this. Mathematical theorems underlying methods for determining the region of asymptotic stability are given, and the methods are illustrated by a number of examples.

Optimum Estimation of Impulse Response in the Presence of Noise

Abstract: The problem considered is that of estimating the impulse response of a linear system from records of its input and output during a limited interval of time when the system output is obscured by additive random noise. Standard results from statistical estimation theory are applied to derive least squares and Markov estimates which are optimum in the sense of having minimum variance among all linear unbiased estimates. No special assumptions are required concerning the form of the input. Expressions for the variances of the sampling errors are given. The relationships of these estimates to other methods of estimation which have been suggested are discussed.

Positive Real Functions of Several Variables and Their Applications to Variable Networks

Abstract: This paper is concerned with networks containing variable parameters. The concepts of multivariable positive real function, multivariable reactance function, and multivariable positive real matrix are introduced, and it is shown that these concepts are useful for dealing with a wide class of variable parameter networks. Several theorems on these functions are established. Extensions of several conventional procedures of circuit theory are presented for the synthesis of such networks and their limitations are discussed. The last section concludes with a discussion of the approximation problem of networks with one variable parameter.

Normalized Design of 90° Phase-Difference Networks

Abstract: The theoretical background exists for the design of 90° phase-difference networks. Continued interest in the design of such wide-band networks for single-sideband and other applications indicates the need for methods to ease the computational difficulties inherent in network design involving elliptic functions. Concise design data are given in the form of equations, curves and tables which simplify synthesis of LC or RC all-pass networks by eliminating the above computational difficulties. The normalized design curves presented cover a range of bandwidths of 2000 to 1 and permit the designer to write the response function of all-pass phase-difference networks with over-all complexity of up to 12 real pole-zero pairs.

On the Solution of Networks by Means of the Equicofactor Matrix

Abstract: In the solution of electrical networks, there arise matrices with the property that the sum of the elements of every row and of every column equals zero. On the node basis this is a direct consequence of Kirchhoff's current law coupled with the fact that the currents are invariant to a change of all node potentials by the same amount. As a consequence, all the first cofactors associated with determinants of such matrices are equal. The authors have named all such matrices equicofactor matrices and have based a general discussion of the solution of networks on these matrices. A new sign notation is introduced and problems of admittance-impedance conversion are treated. A proof is given of a theorem-called by the authors Jeans' theorem-which relates to the second cofactors associated with the determinant of the equicofactor matrix. This theorem is a consequence of the fact that, in the solution of networks, it is immaterial which node (mesh) is taken as reference and which equation is considered superfluous and suppressed from the given set, since the final answer must be the same. The theorem also shows that only (n-1)^2 coefficients associated with an n -node ( n -mesh) network are independent, regardless whether the network be described on an admittance or impedance basis. It is therefore concluded that there is perfect duality between the admittance and impedance description of networks, whatever their complexity.

Terminal and Branch Capacity Matrices of a Communication Net

Abstract: A communication net consists of branches representing communication channels with the weight of each branch being a positive real number which represents the capacity of transferring information through the branch (called a "branch capacity"). The terminal capacity between the vertices i and j of a communication net is the capacity of transferring information between the vertices i and j by considering the net as a whole. To indicate the terminal capacities between all possible pairs of vertices in a net, a terminal capacity matrix is defined. Then the necessary and sufficient conditions for a terminal capacity matrix are given. To represent the structure of a communication net, a branch capacity matrix is defined. Then the synthesis of a communication net from a given terminal capacity matrix is to obtain a branch capacity matrix from the given terminal capacity matrix.

Noise in Negative-Resistance Amplifiers

Abstract: Negative-resistance amplifiers that consist of a noisy negative resistance imbedded in a lossless three-terminal-pair linear network to make a two-port amplifier are analyzed. It is shown that Haus and Adler's noise measure M_{e} , is not constrained to lie between two eigenvalues of a noise matrix, as is the case for bona fide two-port amplifiers. Instead, it is always equal to its optimum value, and is independent of the (lossless) imbedding network used. As a corollary of this, the noise figure of such an amplifier fails to equal its optimum value only insofar as the exchangeable gain is not high. The single value of noise measure may be computed from any simple lossless circuit at hand, or else from the exchangeable noise power of the noisy negative resistance.

On the Analysis and Synthesis of Single-Element-Kind Net-works

Abstract: This paper presents a method for the realization of a given nth order node conductance (or capacitance or reciprocal inductance) matrix as an (n + 1) -node network with all positive elements and no ideal transformers. The first part establishes simple relations between branch conductances and elements in the given matrix which are assumed to be based upon a linear tree. These relations are analogous to the well-known ones pertinent to a so-called "dominant" matrix based upon the starlike tree. In an analysis problem they enable one to compute a single driving-point or transfer impedance with a minimum of computations. The second part of the paper develops a method whereby one can readily determine whether a given G matrix is based upon a tree (the necessary condition for its realization) and find the pertinent geometrical tree configuration when one exists. Once the latter is established, realization is simple and straightforward. The entire process requires no repeated trials and proceeds toward the desired goal with a minimum of effort.

Takahasi's Results on Tchebycheff and Butterworth Ladder Networks

Abstract: A Japanese paper published by H. Takahasi in 1951 gives formulas for element values of ladder networks with Tchebycheff characteristics. For a resistance-terminated low-pass ladder with a series inductance as the first reactance, these formulas are given by L_1= \frac{R_{1} s_{1}}{(k-k^{-1}) - (h - h^{-1})} K_{r,r+1}=\frac{s_{2r-1}s_{2r+1}}{b_r} (r = 1, 2, \cdots , n-1 ) where R_1 , is the input resistance and K_{r,r+1}= L_{r} C_{r+1} if r is odd; C_{r}L_{r+1} if r is even. b_r = \xi^{2} - c{2r}\xi \eta + \eta^{2} + s_{2r}^2 s_r = 2 \sin \frac{\pi r}{2n} c_r = 2 \cos \frac{\pi r}{2n} \xi = k - k^{-1} \eta = h - h^{-1} The positive constants k and h are related to the zeros and poles of the squared magnitude of the reflection coefficient |\rho (j \omega )|^{2} ; more specifically, the poles are \alpha_{2m+1}= k \epsilon^{2m+1} + k^{-1} \epsilon^{-(2m+1)} for m = 0, 1, 2, \cdots , 2n - 1 and \epsilon = e^{\frac{j \pi}{2n}} , and the zeros are \beta_{2m+1}= h \epsilon^{2m+1} + h^{-1} \epsilon^{-(2m+1)} for m = 0, 1,2, \cdots , 2n -1 . The final reactance can also be related to the output resistance RP so that the elements can be determined by starting from either the first or last element: L_n = \frac{R_{2} s_{1}}{(k-k^{-1})+(h-h^{-1})} if n is odd C_n = \frac{ s_{1}}{R_{2}[(k-k^{-1})+(h-h^{-1})]} if n is even The formulas for the Butterworth characteristic are derived from those for the Tehebycheff characteristic by a limiting process. A proof is also furnished by Takahasi. These results, which have been previously unknown to most network theorists, anticipate the work of many authors in the field. In the present paper Takahasi's results are given and his concise proof is expanded so that its potential application to presently unsolved problems may be more easily investigated.

Periodic Solutions of van der Pol's Equations with Large Damping Coefficient lambda = 0 sim 10

Abstract: A method of computing a periodic solution of van der Pol's equation is devised reducing the problem to the solution of a certain equation by means of Newton's method. For computing the value of the derivative necessary to apply Newton's method, the properties of variation of the orbit in the phase plane are used and, for step-by-step numerical integration of differential equations, a somewhat new method based on Stirling's interpolation formula combined with an ordinary Adams' extrapolating integration formula is used. The periodic solutions are actually computed for \lambda = 0 \sim 10 and the minute but important change of the amplitude described by van der Pol's equation is found.

Two-Stroke Oscillators

Abstract: A four-stroke oscillator, whether symmetrical or moderately dissymmetrical, is one in which the linear, conservative elements receive energy from the source two times per period; such are typically a Class A vacuum tube oscillator, a Class C push-pull oscillator, or a clock with anchor escapement. This paper presents a number of nonlinear, second-order differential equations which correspond to two-stroke oscillators in which the source is active or strongly active only once per period; such as a Class C oscillator. It is hoped that this second type will lead to a better understanding of a number of oscillations which it has not been possible to interpret so far by means of the four-stroke model.

On the Brune Process for n-Ports

Abstract: The Brune process for n -ports, established by Oono, McMillan, and Tellegen, is derived by an alternative method based on successive series and shunt extractions of passive n -ports containing fictitious complex resistances, which are finally eliminated by a simple equivalence transformation. The method is also applied to nonreciprocal n -ports, thus establishing that such an n -port of degree m is realizable with not more than m reactive elements. For reciprocal n -ports, the new derivation of Brune's process clarifies the physical structure of the reactive 2n -port generalizing Brune's section and yields explicit formulas for its element values.

A Method of Analysis in the Theory of Sinusoidal Self-Oscillations

Abstract: The article is devoted to the theory of weak resonance action on self-oscillating systems. The theory is based on the method of secondary-simplification of "shortened" equations for amplitudes and phases of the self-oscillating process. A series of new results is obtained. These results are taken as a basis for the creation of new radiotechnical devices.

Analysis and Synthesis of Nonlinear Systems

Abstract: The input-output relation of linear systems (convolution integral) is generalized to a class of nonlinear systems. This class is represented by analytic functionals as studied by Volterra and Frechet. The analysis can be performed by measuring the response of nonlinear systems to series of impulse functions. The synthesis involves linear systems, zero-memory nonlinear systems and multiple multipliers in the general case, noninteracting linear and zero-memory nonlinear systems in many practical cases. Physically, the class of analytical functionals describes systems obtained by cascading noninteracting linear and zero-memory non-linear systems in open or closed loop configuration. Orthogonal representations of nonlinear systems are considered; for bounded signals and in particular for sinusoidal signals, the Tchebycheff polynomials representation is shown to be especially convenient.

A Computational Method for Network Topology

Abstract: This paper deals with methods for listing all possible trees and multitrees, and for determining the relative signs of tree determinants and multitree determinants, which are necessary for the analysis of networks. A connected linear graph having n elements and all of its possible subgraphs has been studied by mapping on an n -dimensional vector space having n base vectors, and a method for finding out a number of new trees from one tree, or a set of trees, has been derived. Also, a method of computation for determining the relative signs of tree determinants has been developed. These computations can be performed easily and effectively by a digital computer provided with bitwise logical instructions.

Theoretical Bounds on the Performance of Sampled Data Communications Systems

Abstract: A general derivation is given for the minimum mean squared error in the smoothed output of a sampled data communication system. The message which is to be sampled and transmitted is assumed to be a stationary random function of time and is not strictly bandlimited. The results presented here are more general than previous calculations because the message spectrum has an arbitrary cut-off rate, the sampling duration can be finite, and "white" noise is assumed introduced into the system by transmission. Filtering of the message before sampling is shown to allow a smaller minimum error than without such filtering, even though the error is determined with respect to the original unfiltered message. The transfer function of the optimum presampling filter is derived as well as the corresponding optimum smoothing filter.

Modes in Linear Circuits

Abstract: The purpose of this review paper is to show that the concept of modes does apply to any linear time invariant circuit. This is accomplished by reducing the network equations to the standard vector form \dot{y} = Ay + f . In particular, it is shown that 1) any free oscillation of a linear circuit can be thought of as a superposition of non-interacting modes, 2) in the case of free oscillations, the amount of excitation of each mode can easily be expressed in terms of the initial conditions, 3) any forcing function excites each mode independently, and finally, 4) the resonance phenomenon is easily interpreted and the importance of the proper type of excitation is made obvious. Vector notation is used throughout. Examples of RC, RLC and active circuits are included.

Frequency Entrainment in a Self-Oscillatory System with Ex-ternal Force

Abstract: This paper deals with forced oscillations in a self-oscillatory system of the negative-resistance type. When no external force is applied, the system produces a self-excited oscillation. Under the impression of a periodic force, the frequency of the self-excited oscillation falls in synchronism with the driving frequency within a certain band of frequencies. This phenomenon of frequency entrainment also occurs when the ratio of the two frequencies is in the neighborhood of an integer (different from unity) or a fraction. Under this condition, the natural frequency of the system is entrained by a frequency which is an integral multiple or submultiple of the driving frequency. In this paper, special attention is directed toward the study of periodic oscillations as caused by frequency entrainment. The amplitude characteristics of the entrained oscillations are obtained by the method of harmonic balance, and the stability of these oscillations is investigated by making use of Hill's equation as a stability criterion. The regions in which different types of entrained oscillation, as well as beat oscillation, occur are sought by varying the amplitude and frequency of the external force. The theoretical results are compared with the solutions obtained by analog-computer analysis and found to be in satisfactory agreement with them.

Transfer Function Synthesis of Active RC Networks

Abstract: A general method is found to synthesize voltage transfer functions with complex poles and zeros using RC elements and a practical transistor amplifier. The method allows specified source and load resistances. The over-all network has a common ground and is economical in terms of the number of elements.

Physical Realizability Criteria

Abstract: The main objective of this paper is to present a brief survey of the theory of linear, passive, time-invariant n-ports both from an axiomatic standpoint and a synthesis point of view hthe first approach the n-port is defined in terms of its behaviour in real-time and its steady-state behaviour is deduced as a logical consequence In the second approach the network is restricted, apriori, to be of a special type, e g , lumped etc and methods of realizing it in terms of a finite number of known building blocks are investigated, To the author's best knowledge no definitive study of the active n-port on either basis is available However, under certain assumptions, an algebraic treatment appears to yield some non-trivial results This method -will be applied to the synthesis of n-ports containing both positive and negative resistors Several theorems concerning n-port stability are also given together with a discussion of nonreciprocity and its relation to gyrotropic media

Differential Equations with a Small Parameter Attached to the Highest Derivatives and Some Problems in the Theory of Oscillations

Abstract: This paper presents a brief review, for the most part, of the authors' results concerning systems of differential equations of the form \epsilon \dot{x}^1 = f^i(x^1,\cdots,x^k,y^1,\cdots,y^l) i =1, 2,\cdots,k \dot{y}^i = g^i(x^1,\cdots,x^k,y^1,\cdots,y^l) j =1,2,\cdots, where \epsilon is a small positive parameter. The emphasis is on periodic solutions of such systems which are close to discontinuous solutions. Such periodic solutions are mathematical representations of relaxation oscillations which are encountered in various mechanical, electrical and radio systems.

Unit Real Functions in Transmission Line Circuit Theory

Abstract: A new class of functions is introduced which has a direct physical significance in transmission line theory. These are called "unit real" (u. r.) and are derivable by bilinear transformations from positive real (p. r.) functions. The complex reflection coefficient is a unit real function of the "line vector" exp(-2j\theta) , where \theta is the electrical length of a section of line in a resistor-transmission line circuit. Just as in lumped constant circuit theory the impedance is a p.r. function of the complex frequency. U.r. and p.r. functions are compared. A new proof and a discussion of Richards' theorem are also presented.