Steven L. Dvorak

Numerical computation of the incomplete Lipschitz-Hankel integral Je o ( a, z )

Abstract Two factorial-Neumann series expansions are derived for the incomplete Lipschitz-Hankel integral Je 0 ( a , z ). These expansions are used together with the Neumann series expansion, given by Agrest, in an algorithm which efficiently computes Je 0 ( a , z ) to a user defined number of significant digits. Other expansions for Je 0 ( a , z ), which are found in the literature, are also discussed, but these expansions are found to offer no significant computational advantages when compared with the expansions used in the algorithm.

Propagation of ultra-wide-band electromagnetic pulses through dispersive media

We develop an efficient method for the analysis of ultra-wide-band (UWB) electromagnetic pulses (e.g., double-exponential pulse) propagating through a waveguide or cold plasma (i.e., the ionosphere). First we show that the inverse Fourier-transform representations for the electric and magnetic fields satisfy second order, nonhomogeneous, ordinary, differential equations. These differential equations are solved analytically, thereby yielding closed-form expressions involving incomplete Lipschitz-Hankel integrals (ILHIs). The ILHIs are computed using efficient convergent and asymptotic series expansions. We demonstrate the usefulness of the ILHI expressions by comparing them with the fast Fourier-transform technique (FFT). Because of the long tails associated with UWB pulses, a large number of sample points are required in the FFT, to avoid aliasing errors. In contrast, the ILHI expressions provide accurate and efficient numerical results, regardless of the number of points computed. An asymptotic series representation for the ILHIs is also employed, to obtain a relatively simple, late-time approximation for the transient fields. This approximate late-time expression is shown to accurately model the waveform over a large portion of its time history. >

Series expansions for the incomplete Lipschitz‐Hankel integral Je0(a, z)

Bessel series expansions are derived for the incomplete Lipschitz-Hankel integralJe0(a, z). These expansions are obtained by using contour integration techniques to evaluate the inverse Laplace transform representation for Je0(a, z). It is shown that one of the expansions can be used as a convergent series expansion for one definition of the branch cut and as an asymptotic expansion if the branch cut is chosen differently. The effects of the branch cuts are demonstrated by plotting the terms in the series for interesting special cases. The Laplace transform technique used in this paper simplifies the derivation of the series expansions, provides information about the resulting branch cuts, yields integral representations for Je0(a, z), and allows the series expansions to be extended to complex values of z. These series expansions can be used together with the expansions for Ye0(a, z), which are obtained in a separate paper, to compute numerous other special functions, encountered in electromagnetic applications. These include: incomplete Lipschitz-Hankel integrals of the Hankel and modified Bessel form, incomplete cylindrical functions of Poisson form (incomplete Bessel, Struve, Hankel, and Macdonald functions), and incomplete Weber integrals (Lommel functions of two variables).

Exact, closed-form expressions for transient fields in homogeneously filled waveguides

It is well known that transient electromagnetic waves in waveguides exhibit dispersion. Exact, closed-form expressions, which involve Bessel functions of the first kind, have been derived for the impulse response of a waveguide, but exact, closed-form expressions for more complex pulses are absent from the literature. In this paper, it is demonstrated that incomplete Lipschitz-Hankel integrals can be used to represent transient pulses in homogeneously filled waveguides. A continuous wave pulse is investigated in this paper, however, this technique can also be applied to a number of other transient waveforms. The resulting expressions are verified by numerically integrating the pulse distribution multiplied by the known impulse response. >

Application of the fast Fourier transform to the computation of the Sommerfeld integral for a vertical electric dipole above a half-space

The fast Fourier transform (FFT) is used in conjunction with analytical techniques to obtain an approximate expression for the Sommerfeld integral associated with a vertical electric dipole above a half-space. The resulting expression, which is an analytic function of the source and observation locations, can be used in lieu of a numerical integration of the Sommerfeld integral. It can be used in either the near zone or the far zone, and it can even be applied for observation angles close to grazing. The accuracy of the expansion directly depends on the number of terms used in the FFT. Results obtained using the expansion are compared with those obtained both from a direct numerical integration of the Sommerfeld integral and the reflection coefficient method. >

Applications for incomplete Lipschitz-Hankel integrals in electromagnetics

Incomplete Lipschitz-Hankel integrals (ILHIs) are a class of special functions that appear in the analytical solutions for numerous canonical problems in electromagnetics. The reason that they appear so often in electromagnetics is because they provide solutions to the wave equation. The intent of the article is to provide an overview of the research that has been carried out in this area. After providing a brief introduction to the theory of ILHIs, applications for these special functions in problems involving diffraction, dispersion, sources radiating in layered media, and traveling-wave sources radiating in a homogeneous space, are discussed. >

Reaction analysis in stripline circuits

In this paper, a full-wave layered-interconnect simulator (UA-FWLIS), which is capable of simulating EM effects in packaging-interconnect problems, is introduced. Standard integral-equation-based method of moment (MoM) techniques are employed in UA-FWLIS. However, instead of using standard time-consuming numerical integration techniques, we have analytically evaluated the MoM reaction elements, thereby greatly improving the computational efficiency of the simulator. This paper illustrates the application of the simulator by employing it in the studies of coupling in a stripline structure and S-parameters for an interconnect.

Novel closed-form expressions for MoM impedance matrix elements for numerical modeling of shielded passive components

Recent advances in multi-layer insulating substrates have necessitated the development of efficient numerical modeling tools capable of handling complex three dimensional structures. In this paper, a closed-form solution is presented for the method of moments (MoM) impedance matrix elements obtained in the integral equation modeling of shielded passive circuits. This closed-form solution is written in terms of rapidly-computable special functions and has proven to be two orders of magnitude faster than direct numerical integration.

A study of the fields associated with horizontal dipole sources in stripline circuits

In this paper, we formulate closed-form expressions for the fields excited by an x-directed dipole source in both homogeneously-filled and inhomogeneously-filled stripline structures. These expressions are obtained by first employing spectral domain techniques, which yields a spectral-domain Greens function that involves simple algebraic and trigonometric functions. Then we take the inverse two-dimensional Fourier transform of that expression and represent it as a Sommerfield-type integral in the space domain. This Sommerfield integral has a highly oscillatory and slowly convergent nature. Therefore, we analytically evaluate this Sommerfield integral by applying residue theory. Summing over the residues evaluated at the pole locations in the complex plane yields a rapidly computable modal series expansion that is free from any numerical integration. The special functions that occur in the modal series expansions are rapidly computable Bessel functions. This method greatly improves the computational efficiency as compared with numerical integration. The field study carried out in this paper will help to extend the capabilities of a recently developed full-wave layered interconnect simulator (UA-FWLIS).

Lossy transmission line simulation based on closed-form triangle impulse responses

Analytical frequency-domain expressions for single and coupled transmission lines with triangular input waveforms are first developed. The inverse Fourier transform is then used to obtain an expression for the time-domain triangle impulse responses for frequency-independent transmission line parameters. The integral associated with the inverse Fourier transform is solved analytically using a differential-equation-based technique. Closed-form expressions for the triangle impulse responses are given in the form of incomplete Lipschitz-Hankel integrals (ILHI) of the first kind. The ILHI can be efficiently calculated using existing algorithms. Combining these closed-form expressions for the triangle impulse responses with a time-domain convolution method using a triangle impulse as a basis function, provides an accurate and efficient simulation method for very lossy transmission lines embedded within linear and nonlinear circuits.