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A general method based on the Laplace expansion for determining the transfer function of a wide variety of linear electronic circuits is discussed. The technique developed requires only the calculation of a number of driving-point resistances to specify the coefficients of the transfer function. Dominant-pole techniques are used and extended, making the procedure useful in both analysis and design. As computation only involves resistance networks, complex arithmetic is not required in determination of the response.

IN part I of this paper1 methods for the synthesis of the optimum nonlinear bang-bang control system with complex roots was presented. In this, the scope and utility of this synthesis are verified by analytical studies of the dynamic response capabilities of this systems.

A numerical procedure is described for processing the data of a microwave measurement in which the measured points are distributed in a form of a circle in a complex plane. Instead of plotting the measured data on a Smith chart and analyzing them by graphical methods, the data are analyzed by the method of least squares. The result of this analysis consists of three complex numbers K, L, and M, which define the bilinear transformation in question. The procedure is illustrated on the example of impedance versus bias measurements on a varactor diode which was recently described by E. W. Sard. The necessary formulas are derived for computation of elements of the equivalent circuit from the above constants K, L, and M. The procedure is well-suited for programming a digital computer.

From an abstract point of view, both electric circuits and switching circuits may be considered as weighted nonoriented graphs. There are two main differences between electric and switching circuits from this point of view. The first of these is the algebra to which the weights belong. In electrical network theory the weights belong to the complex number field or to the field of rational functions of a complex variable since they are impedances or admittances. In switching theory they are Boolean functions. The second difference is that one is interested, in electrical network theory, in the circuits or the "loops" in the system, whereas in switching theory one is interested in the paths. This paper seeks to relate the two theories by means of topological considerations. Formulas are derived relating the switching function and the driving point admittance function of a two-terminal network. Certain relations between dual networks are also established. The paper concludes with a synthesis procedure for a type of switching circuit of academic interest, the single contact switching circuit, and statements of some important unsolved problems.

This paper considers computational problems arising in the solution of the linear optimal regulator problem. The proposed solution for the constant feedback gain matrix is an adaption of the eigenvector solution proposed by many authors. Techniques are given which are numerically stable and do not require complex arithmetic. These techniques offer considerable savings in computation time and eliminate many of the problems that plague more conventional methods.

A transfer-function synthesis procedure using distributedactive RC circuits is presented. The distributed element is a rectangular distributed RC network with contoured electrodes, and the active element is an operational amplifier. These two elements comprise the entire circuit required for voltage transfer-function synthesis. Explicit realizability conditions are derived for bilinear and biquadratic transfer functions as well as for their degenerate forms. These conditions define two planes with regions of realizability within which the poles and zeros of the voltage transfer function must lie. For the case of complex roots the realizability region is defined in the complex frequency plane. For pairs of real roots the realizability region is depicted on a plane whose coordinates are both real. An example demonstrating the procedure is presented.

In automatic control and other branches of engineering it is frequently necessary to find the roots of an algebraic equation. One of the authors has presented iteration methods for solving for both real and complex roots by right-to-left synthetic division. These methods are convergent for many, but not all, classes of roots. This paper presents a study of the convergence of the methods and extends the methods to cover the regions of nonconvergence. In addition, an improved array is presented for checking the Routh-Hurwitz stability criterion. The use of the methods in analysis and design of automatic control systems is demonstrated by numerical examples.

A MATHEMATICAL theory, suitable for appraising and controlling directive properties of linear antenna arrays, can be based upon a simple modification of the usual expression for the radiation intensity of a system of radiating sources. The first step in this modification is closely analogous to the passage from the representation of instantaneous values of harmonically varying quantities by real numbers to a symbolic representation of these quantities by complex numbers. The second step consists in a substitution which identifies the radiation intensity with the norm1 of a polynomial in a complex variable. The complex variable itself represents a typical direction in space. This mathematical device permits tapping the resources of algebra and leads to a pictorial representation of the radiation intensity.

The general solution to the slightly damped network is expressed in terms of the undamped solution by means of series expansions. The first part of the paper gives a method for evaluating the complex roots of the determinantal equation, and the second part shows how the expansions of the first part may be correlated with the Heaviside formula to form the complete approximate solution. An example illustrates the application to a simple network.

The necessary procedure involved in the transition from the Laplace transformation to the solution of linear differential equations is summarized, and a particular form of partial-fraction expansion which may be advantageous in special cases is noted. The remainder theorem with regard to algebraic polynomials is restated, and it is shown how this theorem may be applied to the evaluation of high-degree polynomials for real and complex numbers. A numerical example is treated to illustrate application to a typical transfer function. Finally, the method is shown to be useful in the evaluation of the frequency response of such a transfer function.

By programming pairs of amplifiers to represent the real and imaginary parts of complex variables on an analog computer, techniques are developed which 1) permit the mechanization of the method of steepest descents, for finding roots of complex functions, in Cartesian coordinates; 2) allow the generation of complex functions utilizing the Cauchy integral theorem; 3) enable a programmer to scan the z plane efficiently when electronic mode control and track-and-hold circuitry are available. Examples are given which show that these techniques frequently provide more efficient programs, in terms of equipment, than previous methods.

A generalized Barker sequence is a finite sequence \{a_{r}\} of complex numbers having absolute value 1 , and possessing a correlation function C(\tau) satisfying the constraint |C(\tau)| \leq 1, \tau \neq 0 . Classes of transformations leaving |C(\tau)| invariant are exhibited. Constructions for generalized Barker sequences of various lengths and alphabet sizes are given. Sextic Barker sequences are investigated and examples are given for all lengths through thirteen. No theoretical limit to the length of sextic sequences has been found.

The vector method is just as useful in solving problems involving transient conditions in electric circuits as it has proved to be when the currents and potentials are steady sinusoids. As far as the writer is aware, the vector method for determining transients in rotating electric machines was first used by L. Dreyfus. Previously the method had been applied to fixed combinations of resistances, inductances and capacitances by Kennelly and others. By making certain assumptions that are, however, quite reasonable in many cases, the transient currents in nearly all of the common types of electric machinery are damped sinusoids. Fortunately the damping is exponential and is thus readily accounted for. It is interesting to trace the development of the method. In the solution of all problems in direct currents the potentials, currents, and circuit constants are real numbers. In the corresponding problem in which the applied potentials are steady sinusoids, these quantities are all represented by complex numbers. In all other respects the working out of the solution is identical with that followed in the direct-current case. When the currents are damped sinusoids, they and the potentials and the circuit constants can still be represented by complex numbers. There is this difference, however; the vectors which represent the currents and potentials shrink exponentially as they rotate and the values of the circuit constants depend not only upon the frequency of the current, but also upon its rate of shrinking. Again the solution of any problem follows the same procedure that would the corresponding one in which the currents are steady sinusoids. In both the steady and damped sinusoidal cases the circuit constants depend upon the angular velocity of the vectors which represent the currents. In the former, the angular velocity is purely imaginary while in the latter it is complex, the real part being the rate at which the current vector shrinks and the imaginary portion, its angular velocity. In electric machinery in which rotating magnetic fields are produced, these fields shrink exponontially as they rotate when the currents are damped sinusoids. If these rotating magnetic fields are represented by vectors, the vectors will have a complex angular velocity just as do the currents. The e. m. f. which is produced by a steady sinusoidal variation of flux lags the flux by 90 degrees, whereas if the flux variation is a damped sinusoid, the angle of lag is less than 90 degrees, depending upon the damping. The mathematical relation, however, is the same, vis., the e. m. f. is proportional to the negative of the product of the flux and its angular velocity. It is then readily appreciated that the form of the solution for the transient state is the same as that which is used for the steady state. Before the method can be expected to give as accurate results as are obtained when predicting the steady operation, considerable experimental data must be obtained in order to determine the best methods of measuring the necessary constants, for these may be somewhat different during the transient period than during steady operation.

A network containing switches is equivalent to a number of networks that differ in the values of their components, in the arrangement of the components, or in both respects. When analyzing or synthesizing such a network, one may treat each different network by itself, and then combine the results. This paper describes a method by which the different aspects of a switchable network may be treated simultaneously. The mathematics by which the network is treated is a combination of ordinary field algebra (complex numbers) and Boolean algebra. The mathematical foundation is first laid out, then interpreted in terms of switchable network elements. The paper is concluded with some examples of analysis and synthesis of switchable networks.

A method of implementing two single-bit adders is discussed. These adders can be used individually to realize the conventional functions of serial addition and serial multiplication on a pair of operands, or they can be cascaded to allow the serial addition of three operands for forming the product of complex numbers. In either case, the circuits will detect the occurrence of an overflow or the generation of the number minus one, and they will allow an addition to be rescaled by outputting the correct bits during the additional shifts, whether the addition overflowed or not.

ENGINEERING, and particularly electrical engineering, has become a science whose future progress depends largely on the extensive use of mathematics and physics. In the writer's early experience it was not unusual to hear prominent engineers state that the place for an integral sign is the violin. The use of complex numbers in circuit analysis, the introduction of operational and transform methods, and the application of tensor analysis in electrical engineering have brought to our profession the realization that not only the integral sign, but also other mathematical hieroglyphics, are indeed just as essential to the engineer as they are to the mathematician.

In the first part of this paper we consider an ensemble of complex numbers (a set of signals in time, the field scattered by different configurations of scatterers, etc.) and obtain relations among various average functions that may be defined for such an ensemble. In particular, we consider the average total intensity, the coherent intensity, and the incoherent intensity (absolute squared functions of the ensemble); the coherent phase, average phase and average-square phase; the variances of the real and imaginary components of the ensemble and their covariance (or equivalently, the second moments of any phase-quadrature components of the field), as well as the higher moments. The second moments are represented in terms of the incoherent intensity and the real and imaginary parts of an "asymmetry function" P ; if P=0 , then the variances are equal and the covariance is zero. In the second part of the paper, we briefly sketch the formalism that leads to scattering function representations for the intensities and coherent phase, and then develop the corresponding scattering representation of the new function P . We illustrate the development by explicit results for "gas-like" random distributions of large tenuous scatteres. Thus we obtain direct relations between the various statistical functions mentioned above and the fundamental parameters of the scattering problem. Since the parameters enter differently in the various averages, and since these functions can be measured simultaneously and independently, the results facilitate inverting measured data.

An iterative procedure is derived for computing time controls for regulator systems described by vector differential equations of the form \dot{x}=Ax+bu where A and b are constant |u| \leq 1 . The iterative computational procedure is based on the parameterization of the minimal time control in terms of the initial conditions of the system's adjoint differential equation. A function is derived relating the initial state of the system to the adjoint system's initial conditions and to the time required by the optimal control to drive that state to zero. Mapping properties of this function, on which the control computation depends, are discussed. To demonstrate the results of this computational procedure, an example is included in which a time-optimal control for a fourth order system with two pairs of complex roots is computed.

The following system is considered: \dot{x}= Ax + Bu y = Cx where x is an n vector describing the state of the system, u is an m vector of inputs to the system, and y is an l vector ( l \leq n ) of output variables. It is shown that if rank C = l , and if (A,B) are controllable, then a linear feedback of the output variables u = K*y, where K*is a constant matrix, can always be found, so that l eigenvalues of the closed-loop system matrix A + BK*C are arbitrarily close (but not necessarily equal) to l preassigned values. (The preassigned values must be chosen so that any complex numbers appearing do so in complex conjugate pairs.) This generalizes an earlier result of Wonham [1]. An algorithm is described which enables K*to be simply found, and examples of the algorithm applied to some simple systems are included.

An absolute bound on limit cycle oscillations in fixed-point digital filter implementations due to roundoff errors is presented. Periodicity of the limit cycles is assumed in the derivation. Useful design results are explicitly given for the case of second-order filter sections. In addition it is shown that this bound is equal to the rms bound of Sandberg and Kaiser for real roots, and is never more than a factor of two greater than the rms bound for complex roots for second-order filters.

The scattering matrix is normalized to complex n-port loads, and some of the elementary properties of this generalized scattering matrix are discussed. The normalized scattering matrix is obtained in a straightforward manner from both the current-basis and voltage-basis scattering matrices. These matrices are presented from the intuitively attractive viewpoint of measures of the deviation of actual circuit behavior from optimal circuit behavior. The optimal matching condition employed is the specialization to the timeinvariant case of that obtained for complex, time-varying n-port networks excited by arbitrary waveforms.

ALMOST ALL OF THE WORK published on the synthesis of optimum nonlinear bang-bang or relay control systems has dealt with the case where the controlled system is characterized by real roots. For such systems there have been numerous papers devoted to the derivation of the optimum controller. Outside of Bushaw's work1,2 there has not been much in the literature on the explicit definition of the optimum nonlinear controller in the phase plane, or phase space, in the case where the controlled system is characterized by complex roots. Even in the case of Bushaw's work, the solution for the controller is presented only for regulator-type systems and his solution is not applicable to servomechanism-type systems. These points are discussed in more detail in the first part of this paper. There have been other very fine contributions to the literature on this problem.3?5 However, in these contributions the question of the complete optimum nonlinear controller as defined in the phase plane or space is still left unsolved.

A new, simpler derivation is presented for Stansfield's formulas for the most likely position ion and the distribution of its error when a number of bearings are measured from known locations. The simplification results from the use of complex numbers. Simple results are obtained for the case in which bearings are measured at regular, short intervals along a straight reconnaissance flight path, providing asymptotes for Butterly's curves.

A computer program has been developed to find the partial fraction expansion of a ratio of two polynomials with real coefficients. The program is based on a method suggested by P. Henrici (Eidenossische Technische Hochschule, Zurich, Switzerland) and utilizes real numbers only, thus dispensing with complex arithmetic. The program assumes nonrepeated roots for the denominator polynomial. Henrici's technique may be extended to ether situations.

The zeros of f(z) = (az - b)^{n} \pm (cz - d)^{n} are found to lie on a circle of radius |(ad - cb)/(|a|^{2} - |c|^{2})| with its center at z = (a^{\ast}b - c^{\ast}d)/(|a|^{2} - |c|^{2}) , where a, b, c , and d are complex numbers and n is assumed real. When |a| = |c| the locus of the zeros is a straight line perpendicular to the line joining the points b/a and b/c and intersecting it at z = 0.5(b/a + d/c) . The zeros are found analytically and constructed geometrically.

The far scattered fields by a semi-infinite conductor are composed of the geometic-optics and the diffraction field. By applying the technique of complex number integral, the separation of these two fields in the neighborhood and other region of the geometrical shadow and reflection boundaries is clarified. This was ambiguous in the above paper.

Formulas are presented for the direct calculation of the scattering parameters of a linear two-port, when it is measured by an imperfect network analyzer. Depending on the hardware configuration of the test set, the measurement system is represented by one of two flowgraph models. In both models presented, leakage paths are included. The error parameters, i.e., the scattering parameters of the measuring system, are six respective ten complex numbers for each frequency of interest. A calibration procedure, where measurements are made on standards, will determine the error parameters. One of many possible calibration procedures is briefly described. By using explicit formulas instead of iterative methods, the computing time for the correction of the scattering parameters of the unknown two-port is significantly reduced. The addition of leakage paths will only have a minor effect on computational complexity while measurement accuracy will increase.

Three test values at most indicate whether a fourth-order equation with real coefficients has one or two pairs of conjugate complex roots or exclusive real roots.

It was recently shown that companion matrices possess a recursive property which allows their function to be computed efficiently and accurately. In this letter, an additional recursive property of companion matrices is derived. These two properties are then combined to suggest a substantially improved algorithm for the evaluation of functions of companion matrices. It is also indicated that by using this algorithm, the complex arithmetic operations stemming from the presence of complex eigenvalues in real matrices are reduced to a bare minimum.

Smith chart conversion of complex numbers from rectangular form to polar form, and vice versa, often involves the construction of one or more constant magnitude circular arcs. If the magnitude is being sought, it is frequently necessary to resort to graphical interpolation between the arcs. In any event, should the magnitude, given or desired, be between approximately one-fourth and four, the centers of the constant magnitude arcs inconveniently lie off the conventional Smith chart. This paper presents geometric methods with brief proofs for keeping all magnitude constructions on the commercial chart and for increasing the accuracy of results by referring interpolations to the accurately printed scalings on the horizontal axis.

FREQUENCY response from experimental transient-response data is a most sought-after transformation. Basically, the transient response of a linear system is governed by the roots of its characteristic equation. This suggests that the transient response may be used to find these roots when the equation for the system is not known. If the roots so found are real, simple graphical procedures yield the corresponding frequency response. Complex roots are not considered here.

The paper considers three separate special cases of functions for which the zeroes are determined graphically. First, the spring constant of a structural member is replaced by a complex number to allow for hysteresis losses during vibration. The corresponding characteristic function has non-conjugate zeroes which are useful in calculating frequency response, but have no meaning for transient response. Secondly, vibration acting along the input axis of a pendulous accelerometer causing displacement of the proof mass; an in-Phase cross vibration will cause a rectified torque. The real part of the compliance function is needed; knowing the zeroes is useful but not necessary in evaluating the integral over the frequency range by use of residues. Third, sampled-data systems have z transforms which often involve the sum of reciprocals of vectors. Finding real values of zeroes is simplified because only magnitudes need be considered; some graphical techniques can simplify the general vector problem. For each of the above problems alternate methods of calculation can be used; those presented are most apt to be of value to those engineers who are used to working with zeroes and poles. This paper will be published by the AIEE.