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This paper presents the results of a theoretical study of the electromagnetic field in the vicinity of a rod antenna transmitting over a conductive ground plane. In the course of studying compatibility problems one occasionally finds it necessary to predict accurately the field strength within a few wavelengths of a radiating element. This problem i s more difficult than one might expect, especially if the length of the radiating element is greater than one tenth of a wavelength. In fact, the equations involved are sufficiently complex to effectively prohibit their use as a practical engineering tool. What is needed is a simple and rapid method for applying these equations. This paper presents such a method. The three general equations which describe the three components of the field are presented and discussed. A digital computer is used to obtain solutions for a specific antenna height and several specific frequencies. The graphical computer results are presented and discussed. A method for normalizing the three general equations is developed. It i s shown that the data obtained from the computer (for specific values of the variables) can be transformed into a normalized presentation in which all distances are measured in terms of wavelength. It therefore becomes possible to construct normalized field strength graphs. These graphs can easily be applied to obtain theoretical field strength values for the three components of the field over a wide range of parameters. Examples are presented, illustrating the method of application. Figure 1 is a typical normalized field strength graph. It is constructed for the case in which antenna height (h ) is equal to 0.685 wavelength. The horizontal axis represents horizontal distance from the base of the antenna, measured in wavelengths. Each of the curves on the graph corresponds to certain height above the ground plane, again measured in wavelengths. The vertical axis is field strength expressed in decibels above one microvolt per wavelength per ampere of antenna current. This graph i s for the vertical component of electric field. Similar graphs are constructed for the horizontal component of electric field, and for the magnetic field. The complete family of graphs is made up of a set of three graphs for each ratio of antenna height-to-wavelength to be evaluated.

Waveguides containing dielectric inserts which do not extend across either of the waveguide transverse dimensions find applications in many waveguide components. In the general case, there is no analytical solution available and a numerical approach has to be used. This paper presents the solution for lossy cases. It requires the solution of a partial derivative, eigenvalue, vector, complex wave equation.

You will find in the theoretical part (& II) of this paper, the formulas obtained by the geometrical theory of diffraction (G.T.D.) which allow to calculate the electric field in the near zone of a circular aperture. The numerical calculus (& III) made from these formulas, in the case of D/? = 27 allowed us to establish transversal sections and radiation patterns to 4 Rr, Rr and Rr/2. Their comparison with reference results established by the electromagnetic theory (E.M.T.) (& IV) authorizes us to draw a conclusion (& V) on the interest of the G.T.D. in the near zone of a circular aperture.

A class of singular quadratic minimization problems is examined by reducing the performance index to a quadratic form in the control variable. A necessary and sufficient condition for minimality is that the kernel of this quadratic form be non-negative definite, or equivalently, be a covariance kernel (function). Various properties of such functions are then used to obtain different sets of necessary and sufficient conditions. The present paper deals only with those properties of covariances that do not involve differentiability or integrability conditions. The properties include non-negative definiteness, continuity, symmetry, and factorizability. A study of these properties yields natural interpretations of a number of quantities and conditions which arise in singular quadratic minimization theory.

In this paper, a canonical form for state-space representation of time-delay systems will be derived. The approach taken here is to convert the state-space representation into a set of ordinary differential-difference equations of particular form and then synthesize such equations back to the canonical structure. Emphasis will be given to the simple cases of second and third order scalar time-delay systems.

In order to assess the viability of wedge guides, it is necessary to have quantitative data about the dispersion introduced by the truncation of the top. In this paper a boundary perturbation method for calculating the dispersion due to truncation or rounding of the wedge guide is presented. The results obtained with this perturbation method for quartz wedges, indicate that low dispersion can be obtained assuming reasonable tolerances for truncation or rounding of the top.

This paper presents a general engineering technique by which the shielding effectiveness of solid single-layer shields may be analytically evaluated. Shielding equations applicable to any homogeneous isotropic material are developed using the transmission-line approach of Schelkunoff. Corresponding limitations are discussed. Necessary information regarding near-field wave impedance is presented in normalized graphs. A useful computer program (FORTRAN IV) is given, together with application information. Calculated results are compared with corresponding test data.

With the growing popularity of integrated circuits the microstrip transmission line has received wide attention from the microwave community. A common feature that most known analytic solutions share is an assumption that the fundamental mode of propagation resembles a TEM wave closely enough to permit various electrostatic approximations such as conformal mapping, relaxation method, and variational principle. Such solutions although quite useful at low frequencies neglect the fact that the actual propagating modes cannot be TEM. The dispersion of non-TEM waves causes, at microwave frequencies, considerable deviations in effective dielectric constant and velocity of propagation from those values obtained by the electrostatic approximation. In this paper it is shown that the modes that exist on a microstrip transmission line must be hybrid in order that all boundary and continuity conditions be satisfied. In this rigorous analysis the hybrid modes are decomposed into sums of TE and TM (or LSE and LSM) space harmonics, each satisfying the wave equation and the external boundary conditions and their total satisfying the continuity conditions and boundary conditions on the strip. The final outcome of the analysis is a pair of coupled integral equations that are solved numerically. The results indicate that the fundamental hybrid mode is propagating at all frequencies and approaches the known static solutions at low frequencies. However, the correct solution deviates from the static approximation at microwave frequencies. Of particular interest is the fact that as the dielectric constant of the substrate increases the frequency range at which the TEM approximation is valid decreases. The results obtained show close agreement with available experimental data.

This paper is concerned with the realization of linearly separable switching functions where the realization has maximum stability to coefficient and logical signal fluctuations. A realization algorithm is presented for sequentially selecting the coefficients where each coefficient is as small as possible given the values of the previously selected coefficients and the requirement that the realization have unit gap length. As yet it is undetermined whether the algorithm always yields a realization for which the ratio of gap length to sum of coefficients is a maximum.

The analysis of many microwave devices, for example, an electromagnetic cavity containing inhomogeneous anisotropic medium, involves the solution of a vector field, boundary value problem in three dimensions. The method of moments (in particular, that of Galerkin or Rayleigh-Ritz) is a well known general technique of field computation. For a region of arbitrary shape, a basic difficulty of the method of moments is the generation of a set of expansion functions that are complete over the region and which converge not only inside the region but also on the boundary. This paper outlines a systematic technique for generating such a set with the aid of a computer.

The usual power loss method of evaluating the damping constant of cavities and the attenuation constant of waveguides, as caused by finite wall conductivity, breaks down in the case of degenerate modes and fails to predict the coupling between degenerate modes. Papadopoulos has treated the problem by means of a perturbation method involving an expansion of the fields in terms of the modes for the ideal cavity or waveguide. In this paper a variational formulation is presented that permits the eigenvalues for the lossy case to be readily computed. This formulation turns out to be a simple extension of the usual power loss method and in addition to giving the damping constant it also shows that there is an equal shift, in the resonant frequency in the case of a cavity, and in the phase constant for a waveguide. In addition the coupling between the degenerate modes is obtained. Furthermore it is shown that the new non-degenerate sets of coupled modes form an orthogonal set. The above properties all arise from the characteristics of the matrix eigenvalue problem which arises when the Rayleigh-Ritz technique is used in conjunction with the variational formulation of the boundary value problem.

The paper presents a graphical-numerical method for determining the transient stability limits of a two-machine system under the usual assumptions of constant input, no damping and constant voltage behind transient reactance. The method presented is based on the phase-plane criterion,1, 2 in contrast to the usual step-by-step and equal-area methods. For the transient stability limit of a two-machine system, under the assumptions stated, the sum of the kinetic energy and the potential energy, at the instant of fault clearing, should just be equal to the maximum value of the potential energy which the machines can accommodate with the fault cleared. The assumption of constant voltage behind transient reactance is then discarded in favour of the more accurate assumption of constant field flux linkages. Finally, the method is extended to include the effect of field decrement and damping. A number of examples corresponding to each case are worked out, and the results obtained by the proposed method are compared with those obtained by the usual methods.

The complementary variational principle developed in a Hilbert space setting provides a duality principle in the calculus of variations with dynamic constraints. This concept is adopted in this paper to investigate dual characterizations of optimal control systems. Systems under consideration include those with dynamics governed by linear ordinary differential equations, linear partial differential equations and non-linear ordinary differential equations.

The identification of parameters in the beam equation is investigated. The equations are assumed known to within a set of unknown constant parameters. The boundary and initial conditions associated with the Partial Differential Equation (PDE) are assumed known explicitly. This paper utilizes Galerkin's Criterion of the Method of Weighted Residuals (MWR) to transform the PDE problem into an ordinary differential equation problem. The step by step identification procedure is illustrated by the identification of parameters in the beam equation. A test apparatus was constructed and the parameters of the test beam were identified using the actual measurements as input data for the identification procedure.

This paper describes the formulation and testing of an analog computer model to predict the variation with time and depth of the Ekman drift velocity. The mathematical model used in the simulation is the one developed by Ekman. The model is tested on an analog computer to determine the response to a step-function wind assuming a uniform vertical eddy viscosity coefficient. This particular case treated provides a test for the analog model, as an analytical solution for this case already exists.

In this paper, Cauchy's [l] two-dimensional equations of coupled flexural and extensional vibrations of thin crystal plates are extended to the next hi&er order of approximation so as to accommodate the frequencies of the two lowest thickness-shear modes. The equations obtained are also extensions of previous equations [2] in which flexure and the same thickness-shear defomations were taken into account but coupling with contour modes was omitted. The coupled equations of higher order are dedwed fmm the three-dimensional equations of linear elasticity by a procedure based on Cauchy's series expansion and IUrchhoff's [3] variational methods. A system of five, secondorder, displacement-equations of motion is obtained, along with theorems of orthogoaality and uniqueness of solution. Comparison of the appropriate solution of the equations with Ekstein's [4] solution of the three-dimensional equations, for an infinite plate, reveals close agreement between the two frequency spectra, over the extended range of frequencies, for all five branches of the approximate equations. Solutions of the five equations are exhibited for rotated Y-cut rectangular quartz plates under various edge-conditions. One of these is a solution, found previously by another method [5], for contour modes, of rectangular plates with all edges free, not coupled with flexure.

In this paper the coupled mode equations governing the propagation of TE/spl deg//sub 0m/ waves in a tapered circular waveguide are derived. The derivation is based on the assumption that an elementary section of the taper can be thought of as an elementary truncated cone. This approach is superior to the cylindrical mode representation in that it eliminates the singularity in the coupled mode equations for the modal amplitudes at "cut-off " and also leads to a faster converging expansion. An iterative method of solving these equations is also outlined.