Evaluation of microstrip Green function

Numerical Aspects in the Calculation of the Transient Lightning Electromagnetic Radiation Over Lossy Ground

“…It is found that there is a peak at a particular u value in the integrand in Eqs. (10), (12) and (14). The appearance of such a peak in the integrands is apparent due to the following factor, k ｜, so that the reason for having a peak is entirely due to u 1 .…”

Green's functions in lossy layered media: integration along the imaginary axis and asymptotic behavior

“…In addition, other authors have deformed the real axis into a path going through the upper half complex plane to avoid the poles of the spectral domain Green's functions, using rectangular contours [8], or alternatively elliptic integration paths [9]. While all these techniques are indeed efficient for relatively small source-observer distances, they usually require the integration of functions exhibiting abrupt variations and fast oscillating behaviors when large source-observer distances are involved [2], [10]. However, many current practical problems involve distances of several tens or even hundreds of wavelengths.…”

“…However, the imaginary axis algorithm developed in [2] is only valid for lossless layers. Related to this technique, it is interesting to mention the work derived in [10], which utilized the imaginary axis algorithm to develop a series representation of the Green's functions, by using the Bessel function argument-multiplication theorem. An additional advantage of the technique in [10] is that the dependence with the source-observer distance ( ) is extracted from the basic Sommerfeld integrals, so that numerical integration need not to be repeated for all 's.…”

“…Related to this technique, it is interesting to mention the work derived in [10], which utilized the imaginary axis algorithm to develop a series representation of the Green's functions, by using the Bessel function argument-multiplication theorem. An additional advantage of the technique in [10] is that the dependence with the source-observer distance ( ) is extracted from the basic Sommerfeld integrals, so that numerical integration need not to be repeated for all 's. The technique derived in [10], although efficient, remains valid only for lossless layers, since it is based on the original technique described in [2].…”

“…An additional advantage of the technique in [10] is that the dependence with the source-observer distance ( ) is extracted from the basic Sommerfeld integrals, so that numerical integration need not to be repeated for all 's. The technique derived in [10], although efficient, remains valid only for lossless layers, since it is based on the original technique described in [2].…”