The transient excitation of a straight thin-wire segment is analyzed with the aid of a one-dimensional integral equation for the current along the wire. An almost exact derivation of that equation, in which only the radial current on the end faces is approximated, is given. The integral equation obtained turns out to be identical to the reduced version of Pocklington's equation. On the basis of this derivation, existing and new numerical solution techniques are critically reviewed. Pocklington's equation and Hallen's equivalent form are solved directly by marching on in time as well as indirectly via a transformation to the frequency domain. For Pocklington's equation, a conventional moment-method discretization leads to a Toeplitz matrix that is inverted with Levinson's algorithm. For Hallen's equation, the Toeplitz structure is disturbed, and the frequency-domain constituents are determined with the aid of the conjugate-gradient-FFT method. Illustrative numerical results are presented and discussed. >

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Transient excitation of a straight thin-wire segment: a new look at an old problem

We consider the transient scattering of an electrically polarized, two-dimensional, pulsed wave by a two-dimensionally inhomogeneous, lossy dielectric cylinder embedded in free space. The problem is formulated in terms of a contrast-source domain integral equation over the interior of the cylinder. This integral equation is solved numerically, with the aid of the "marching-on-in-frequency" method recently proposed in [1]. To this end, a new discretization procedure is introduced, which is more accurate than the usual piecewise-constant moment-method approach. This discretization preserves the convolution-type structure of the continuous equation, which enables the repeated application of the "conjugate-gradient FFT" method. The initial estimates required in this method are generated by marching on in frequency, using a special extrapolation scheme. Numerical results are presented and compared with the results obtained by applying alternative techniques.

Transient scattering by a lossy dielectric cylinder: marching-on-in-frequency approach

A new approach was recently proposed for the numerical solution of integral equations describing transient scattering problems The idea is to first discretise in space, and then solve the resulting system of linear time-domain equations by carrying out a temporal Laplace transformation This approach combines the efficiency of the marching-on-in-time method with the stability and the accuracy of frequency-domain techniques The repeated solution of the Laplace-transformed system of linear equations thus obtained with the aid of iterative technique is considered Initial estimates are generated by marching-on-in-frequency and using a new extrapolation scheme The feasibility of this idea is demonstrated by solving electric-field integral equations for the total current along a straight thin-wire segment and the current density along a perfectly conducting cylinder

Marching-on-in-frequency method for solving integral equations in transient electromagnetic scattering

The transient excitation of a straight thin wire segment parallel to a plane interface between two homogeneous dielectric half spaces is analyzed by the continuous-time, discretized-space approach. The analysis is carried out in three steps. First, the subproblem of a single wire embedded in a homogeneous dielectric medium, excited by a voltage source and an incident electric field, is studied with the aid of a time domain integral equation. This equation is discretized in space and transformed to the frequency domain. This procedure results in a system of linear equations of a fixed dimension which is solved by marching on in frequency. Second, the subproblem of a horizontal, pulsed dipole over the interface between two homogeneous dielectric half spaces is considered. The reflected field in the upper medium is computed by spectral techniques. With the aid of a time domain Weyl representation, a fixed, composite Gaussian quadrature rule is derived for the semi-infinite integral involved. The angular integral is evaluated in closed form. Third, for the complete problem, the reflected field in the upper medium is expressed as a superposition of contributions from dipole sources on the wire axis, and subsequently treated as a secondary incident field in the integral equation for the current on the wire. This integral equation is then solved by combining the techniques developed for the two subproblems. Representative numerical results are presented and discussed.

Transient excitation of a straight thin wire segment over an interface between two dielectric half spaces

The transient excitation of two identical, straight, thin wire antennas above a plane interface between two homogeneous dielectric half spaces is analyzed. The two wires are located parallel to each other and to the interface, and one of them is excited by a voltage source. By applying symmetry considerations, the problem is decomposed into two single-wire problems, for which a method of solution is available from previous work by the authors [Rubio Bretones and Tijhuis, 1995]. The problem is solved in two steps. First, the configuration of two wires in a homogeneous medium is studied. The electric field integral equation for the total current on the wires is derived directly in the time domain and subsequently solved by using the continuous-time discretized-space approach. This results in a linear system of equations of a fixed dimension which is solved by marching on in frequency. Subsequently, we consider the complete configuration. As in our previous work, the field reflected by the interface is treated as a secondary incident field in the integral equation for the currents on the two wires. This leads to an integral equation of a form similar to the one describing the currents on the two wires in free space. In this equation the response of a pulsed dipole source in the two-media configuration occurs as a Green's function. The spatial Fourier inversion involved is carried out with the aid of a fixed composite Gaussian quadature rule. This again leads to a system of equations of a fixed dimension, which can be solved by marching on in frequency. Finally, some representative numerical results are presented and discussed.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/96RS02502

Transient excitation of two coupled wires over an interface between two dielectric half spaces

The authors proposed a hybrid method that combines the advantages of the frequency-domain techniques and the marching-on-in-time method while avoiding their disadvantages. As in the marching-on-in-time method, the time-domain integral equations are discretized in space. This results in a system of linear time-domain equations of a fixed dimension. This system, in turn, is solved by the frequency-domain procedure of subjecting it to a temporal Laplace or Fourier transformation. Using this approach, the authors solved three scalar scattering problems for which benchmark results obtained by alternative methods were available.<<ETX>>

R. Wiemans

Papers

A hybrid method for solving time-domain integral equations in transient scattering