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Circuit Theory, IRE Transactions on

Issue 4 • Date December 1960

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Displaying Results 1 - 25 of 26
  • Prof. Dr. Balthasar Van der Pol - In Memoriam

    Publication Year: 1960 , Page(s): 360 - 361
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  • A tribute [Into. to special issue deicated to Balthasar van der Pol]

    Publication Year: 1960 , Page(s): 362
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  • A Method of Analysis in the Theory of Sinusoidal Self-Oscillations

    Publication Year: 1960 , Page(s): 398 - 413
    Cited by:  Papers (11)
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    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. View full abstract»

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  • Reviews of Current Literature

    Publication Year: 1960 , Page(s): 554 - 555
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  • Differential Equations with a Small Parameter Attached to the Highest Derivatives and Some Problems in the Theory of Oscillations

    Publication Year: 1960 , Page(s): 527 - 535
    Cited by:  Papers (3)
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    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. View full abstract»

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

    Publication Year: 1960 , Page(s): 382 - 386
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    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. View full abstract»

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  • On Coding Theorems for General Simultaneous Channels

    Publication Year: 1960 , Page(s): 513 - 516
    Cited by:  Papers (1)
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    This paper deals with the lengths of codes for several simultaneous channels in a general formulation. The paper is almost self-contained, and requires for its reading no prior knowledge of information theory. View full abstract»

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  • On the Method of Averaging

    Publication Year: 1960 , Page(s): 517 - 519
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    The method of averaging of van der Pol was devised to obtain periodic and almost periodic solutions of quasi-linear systems of differential equations. A theorem is stated for a particular case where this method has been justified mathematically and an example is given to illustrate the results. View full abstract»

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  • Balth. van der Pol's Work on Nonlinear Circuits

    Publication Year: 1960 , Page(s): 366 - 367
    Cited by:  Papers (2)
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    First Page of the Article
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  • Analysis and Synthesis of Nonlinear Systems

    Publication Year: 1960 , Page(s): 459 - 469
    Cited by:  Papers (4)
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    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 Fréchet. 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. View full abstract»

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  • Multistable Circuits Using Nonlinear Reactances

    Publication Year: 1960 , Page(s): 432 - 440
    Cited by:  Papers (2)
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    This paper presents some results in the asymmetric resonant circuit, which is constructed by applying a dc voltage E_0 and a sinusoidal voltage E_1 \sin {\omega }t to a series resonant circuit containing a nonlinear capacitance. The circuit is analyzed using the Ritz-Galerkin method. Assuming that the first approximation for the charge q takes on the form Q_0 + Q_1 \sin({\omega }t + \theta) , the curves \omega - Q_1 , {E_1} - {Q_1} , and {E_0} - {Q_1} , are obtained. It is shown that the asymmetric circuit is a tristable circuit with respect to these parameters, while the usual ferroresonant circuit is a bistable circuit. The influence of the dc voltage E_0 upon the circuit is studied in particular, and an interesting phenomenon, called "residual jump phenomenon," may occur in the circuit only when E_0 is changed and others fixed. Furthermore, the study shows how to construct a multistable circuit using nonlinear reactances. Combining some nonlinear reactances with suitable bias voltages, circuits having two, three, four or five stable solutions are obtained. View full abstract»

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  • Theoretical Aspects of Nonlinear Oscillations

    Publication Year: 1960 , Page(s): 368 - 381
    Cited by:  Papers (2)
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    This review gives an outline of the modern theory of nonlinear oscillations. This field, originated about four decades ago by the late Prof. B. van der Pol, acquired a considerable momentum in recent years with a gradually increasing number of ramifications, some of which are still in a state of development. For this reason the selection of topics for this review is narrowed down to matters which have reached a state of a reasonable codification. It has been assumed that a certain amount of familiarity with these topics is available on the part of the reader so that only essential points are emphasized. Topological methods and analytical methods are written in a condensed manner, while special attention is given to the stroboscopic method, which is probably less known, and which is used extensively later. A survey of the principal nonlinear phenomena investigated under the assumption of near linearity is given. Phenomena of parametric excitation, subharmonic resonance and synchronization are outlined with some detail, but those of the so called asynchronous actions are treated less completely as their detailed presentation would lengthen the paper unduly. The final section of this review deals with the so called "piece-wise linear phenomena;" it transcends already purely analytic methods and belongs to recent developments concerning nonlinear and nonanalytic oscillations often encountered in the theory of automatic regulation and control. View full abstract»

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  • On Nonlinear Networks with Random Inputs

    Publication Year: 1960 , Page(s): 479 - 490
    Cited by:  Papers (2)
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    The paper attempts to combine the Wiener-Bose method for characterizing and synthesizing nonlinear systems with the Ku-Wolf method for analyzing nonlinear systems with random inputs. A simple partition theory is first presented. It is shown that a general nonlinear system can be partitioned into two portions: one linear portion with memory or storage, and one nonlinear portion which may also include linear elements. The partition method, the Taylor-Cauchy transform method, and the transform-ensemble method are developed, and illustrated by an example. It is shown that the output of a nonlinear system to a random input can be expressed as the summation of a_{n}q_{n}(t) , for n = 0, 1, 2, and so on, where q_{n}(t) depends upon the form of the functional representation of the modified forcing function or the actuating signal, and a. denotes a set of random variables which are related to the statistics of the random input. Wiener's theory of nonlinear systems is then reviewed. The Wiener-Bose method is outlined as follows. Let the output of a shot-noise generator be the standard probe for the study of non-linear systems. The standard random input is fed to a Laguerre network giving Laguerre coefficients u_{1},u_{2},\cdots . The output of the over-all system is then expressed as Hermite function expansions of the Laguerre coefficients. By the ergodic hypothesis it is then possible to express the output as the summation of A_{\alpha } V(\alpha ) e^{-u^{2}/2} By taking the time average of c(t) V(\alpha ) , where c(t) represents either the actual output or the desired output, we get the coefficients A_{\alpha } , which characterize the actual system or the system to be designed. Knowing A_{\alpha } , the synthesis procedure is obtained from the summation of A_{\alpha } V(\alpha ) . By combining the output c(t) , obtained from the Ku-Wolf analysis, with the output V(\alpha ) from the Laguerre network and Hermite function generator, we can get the-characterizing coefficients A_{\alpha } . It is suggested that the correlation of a_n , a set of random variables related to the random input, and A_{\alpha } , the characterizing coefficients, may sh- ed light on a unified approach for the analysis and synthesis of nonlinear systems with random inputs. View full abstract»

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  • Directions of Mathematical Research in Nonlinear Circuit Theory

    Publication Year: 1960 , Page(s): 542 - 553
    Cited by:  Papers (2)
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    In this paper we wish to present a potpourri of problems of some interest and difficulty arising in the field of nonlinear circuit theory. Perhaps the only sensible way to catalogue scientific problems is in terms of "solved or unsolved." Yet this classification is itself a very subjective one, dependent upon the times and the fashions. Recall the dictum of Poincaré that the solutions of one generation are the problems of the next. In this paper, we have attempted, for the sake of convenience, to group categories of problems under the headings of "descriptive," "control," "stochastic," and so forth. Convenient as some of this nomenclature is, it should be regarded with a certain amount of suspicion. Most significant problems blithely cut across these artificial boundaries within fields of specialization, and within science itself. In these days of rapidly and dramatically changing technology it would be rather brash to attempt to predict the type of mathematics that will be most urgently required even ten years from now. It is, however, fairly safe to look about and note the requirements of the present and of five years back. The difficulties that abound render a certain time lag inevitable, and it may well be that new scientific developments may render fields obsolete and mathematical solutions for problems within those fields unnecessary before they are even obtained. View full abstract»

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  • A Method for Determining Optimum Systems Using General Bayes Criterion

    Publication Year: 1960 , Page(s): 491 - 505
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    A method for obtaining an optimum system using general Bayes criterion is developed. This method is applicable to the cases in which a signal to be estimated depends on certain finite-dimensional vector U , and where the input is the sum of a function depending on the vector U and a normally distributed noise which is independent of the vector, or may be reduced to this form by some nonlinear transform. The method is also applicable to some problems in which U is a vector with countably infinite number of components. As a special case the method yields the solution previously given in which the input is a linear function of U . The method affords effective determination of optimum systems designed for the detection and estimation of signals in the presence of noise using various practically adequate criteria under rather general conditions. The optimum system given by the method is in general nonlinear, but in some special cases it may be linear. In particular, the optimum system is linear if the signal and the input are linear functions of components of the vector U , which is also normally distributed or is an unknown nonrandom vector which may assume any value, and the loss function is any function or functional of the difference between the signal and its estimate (i.e., of the system error). The application of the method to the problem of signal detection and to certain problems of signal estimation are given. View full abstract»

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  • Periodic Solutions of Forced Systems Having Hysteresis

    Publication Year: 1960 , Page(s): 423 - 431
    Cited by:  Papers (6)
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    In Section I, the motion of a mass on a material rod is discussed. The stress-strain law exhibits a hysteresis effect, so that the stress at any time is a function of the whole strain history. A "rupture restriction" which limits the magnitudes of the admissable displacements, velocities, and accelerations is also discussed. It is shown that the appropriate formulation of the phenomenon is in terms of function spaces and mappings of function spaces. The system is forced with a periodic forcing function of period \Omega and the problem we set ourselves is to find solutions of period \Omega . In Section II, there is presented first a brief elementary introduction to "Banach Space for Engineers." Then, a purely mathematical theorem is proved concerning the convergence and the speed of convergence of an iterative method for solving a problem in mappings of a Banach Space, and the existence and uniqueness of solutions to that problem. In Section III, the problem posed in Section I is identified with the various concepts introduced in Section II, thus furnishing rigorous proof of the existence and uniqueness of the periodic solution sought in Section I, a computational method for numerically evaluating the solution, and bounds on the errors of approximation. The problem dealt with is quite general, including, in addition to the hysteresis effects, as special cases, the equations of Hill, Mathieu, Duffing, van der Pol (with a forcing term), and a quite general class of nonlinear ordinary differential equations. The method is also applicable to partial differential equations. View full abstract»

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  • The Problem of Quality for Nonlinear Self-Regulating Systems with Quadratic Metric

    Publication Year: 1960 , Page(s): 469 - 473
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    The systems considered in this paper are characterized by differential equations of the form dot{x}_{k} = \sum _{\alpha } b_{k\alpha } x_{\alpha } + f_{k}(\alpha _1,\cdots ,x_n, t) (k= 1, \cdots , n) which are defined over a region N of Euclidean space E_n with metric R^2 = \sum _{i=1} x_i^2 , and with t ranging over some interval T . The f_k (k+1,\cdots , n) are assumed to be such that 1) f(0,\cdots ,0,t) \equiv 0 and 2) there exist positive constants L_k such that |f_k(x_1,\cdots ,x_n,t) | \leq L_k R for all points in N and all t in T . The problem of quality means the problem of determining the values of m adjustable parameters p_i,\cdots ,p_m in the b_{k\alpha } , and f_k in such a way as to result in a rapid return to equilibrium of the representative point in phase-space subject to a limitation on the amount of overshoot. This problem is formulated in precise terms, and a method of solution for it is indicated. As an illustration, the method is applied to a problem in regulation which was formulated by Bulgakov. View full abstract»

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  • Some Extensions of Liapunov's Second Method

    Publication Year: 1960 , Page(s): 520 - 527
    Cited by:  Papers (131)
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    >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. View full abstract»

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  • Statistical Theory of Systems Reducible to Linear

    Publication Year: 1960 , Page(s): 506 - 513
    Cited by:  Papers (1)
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    A statistical theory for nonlinear systems, whose operators are reducible to linear, is given. The analysis as well as the synthesis problems of such systems are considered. View full abstract»

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  • On the Number of Stable Periods of a Differential Equation of the van der Pol Type

    Publication Year: 1960 , Page(s): 535 - 542
    Cited by:  Papers (3)
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    Van der Pol's differential equation with forcing term can, for some values of the parameter b , have stable periodic solutions of two distinct periods. The paper examines whether, with a generalization of the equation not too unlike the original, there can be more than two distinct periods. The answer is affirmative. The example (with three periods-in principle, there can be many), while within the permissible limits, is very sophisticated for its kind. But if the phenomenon can happen at all, it may be presumed to happen in much simpler cases. View full abstract»

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  • Some Remarks on Oscillators Driven by a Random Force

    Publication Year: 1960 , Page(s): 476 - 479
    Cited by:  Papers (2)
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    A brief and partly heuristic discussion of the problem of spacing of zeros of the displacement of an oscillator driven by a random force is given. Although no explicit results have been obtained, the problem is reduced to a study of a certain integral equation and equivalently, of a partial differential equation with a peculiar boundary condition. The boundary condition has previously been conjectured by Uhlenbeck and Wang. View full abstract»

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  • On Automatic Controls

    Publication Year: 1960 , Page(s): 474 - 475
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    The problem of automatic controls has been treated more or less linearly by a number of Soviet authors (Lurye, Letov, Yacubovich). Here a treatment is presented taking into account the nonlinearity of the regulated system. Certain consequences of this nonlinearity are also made clear. View full abstract»

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  • The Transient Behavior of Nonlinear Systems

    Publication Year: 1960 , Page(s): 446 - 458
    Cited by:  Papers (1)
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    It is shown that the classical perturbation procedure for treating nonlinear systems leads to solutions expressed as Fourier-like series with slowly varying coefficients. These slowly varying coefficients contain the information about the long term behavior of the system. Inconsistently, the classical perturbation procedure expresses these coefficients as power series, a mode of expression which has notoriously poor long term validity. An operational procedure is presented for treating oscillations having slowly variable amplitudes and frequencies. An extension of the usual impedance concepts is presented for expressing the frequency characteristics of both linear and nonlinear elements when oscillations with many frequencies are present simultaneously and when these oscillations vary in both frequency and amplitude. From these methods, a perturbation procedure is devised which permits the behavior of systems to be computed with any order of accuracy, using only the algebraic processes which are characteristic of operational procedures. This procedure avoids expressing its results in terms of the local time. Instead, it expresses them in terms of the fundamental characteristics of the oscillations which axe present. As a consequence, the final solutions have the much desired long term validity and they may be used to obtain asymptotic estimates of the behavior of the system. The method is able to treat systems containing nonlinear perturbing elements and elements which we have described as moderately nonlinear. By means of examples it is shown that it is a straightforward process to treat systems to second order accuracy. This level of accuracy covers a large number of the intercoupling effects that characterize the more sophisticated nonlinear phenomena. View full abstract»

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

    Publication Year: 1960 , Page(s): 413 - 422
    Cited by:  Papers (4)
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    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. View full abstract»

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  • Amplification in Nonlinear Reactive Networks

    Publication Year: 1960 , Page(s): 440 - 446
    Cited by:  Papers (1)  |  Patents (4)
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    Sufficient conditions for validity of the Manley-Rowe equations in a classical nonlinear reactive network are shown to be conservation of energy and proportionality of average power with frequency. It is shown by counter example that conservation of energy alone is insufficient. A sufficient condition for a quantum mechanical maser model is found to be equilibrium of state populations. In this case conservation of energy follows as a consequence from the equilibrium condition, but proportionality of power with frequency is not required. View full abstract»

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