Application of the complex image method to multilevel, multiconductor microstrip lines

Application of Total Least Squares to the Derivation of Closed-Form Green's Functions for Planar Layered Media

“…Unfortunately, the introduction of this quasi-static term does not always suffice to solve the near-field convergence problems reported in [32]. The origin of these numerical problems is that the Hankel functions used in [32] and [34] introduce nonphysical singularities as , which is a well-known phenomenon [20], [23], [24], [29], [34]. If a quasi-static term is added to the Hankel functions, as in [34], the near-field problems are only eliminated in the particular cases where the Green's functions are singular as (e.g., shows this behavior when the source and field points are in the same horizontal plane, i.e., when ) since, in those cases, the quasi-static term is also singular, and this singularity dominates over the Hankel functions singularities [29].…”

“…In fact, and may have singularities as , and these singularities coincide with those of and (as stated above, the singularities may be present when the source and field points are in the same horizontal plane , but they never appear when the source and field points are in different horizontal planes ). The Hankel functions and of (13) and (14) also have singularities as , but these singularities are not shared by and , and this may have a detrimental effect on the accuracy of (13) and (14) as [20], [23], [24]. In the remainder of this section, it will be shown that the coefficients can be chosen in such a way that the sums of Hankel functions of (13) and (14) have a nonsingular smooth behavior in the vicinity of .…”

“…Thus, it is convenient to approximate as (18) where (19) The procedure designed for avoiding the effect of Hankel functions singularities in (13) can also be applied to (14). In this latter case, the analysis of the singularities suggests to use the following expansion of the functions as [43]: (20) Equation (20) must then be introduced in (14) in order to obtain the expansion of as (21) Equation (21) shows that the Hankel functions of (14) introduce a singularity of the type as . This singularity can be eliminated provided that is enforced.…”

“…In accordance with the results shown in [27] and [29], the discrete complex image method improves its stability when the complex exponentials used in the approximations of the spectral Green's functions are accompanied by both a quasi-static term contributing the near field and a surface-waves term accounting for the far field. This is the approach originally proposed in [16] and followed in [17], [20]- [24], [27], and [29]. The drawback of this approach is that the surface-waves term contains poles of the spectral-domain Green's functions, as well as residues of the Green's functions at these poles, and the accurate determination of both the poles and the residues requires time-consuming algorithms [24], [29].…”

“…The drawback of this approach is that the surface-waves term contains poles of the spectral-domain Green's functions, as well as residues of the Green's functions at these poles, and the accurate determination of both the poles and the residues requires time-consuming algorithms [24], [29]. It has also been stated that the Hankel functions of the surface-waves term introduce near-field singularities in the spatial-domain Green's functions that do not exist when source and field points are placed in different horizontal planes of a multilayered media (i.e., when ) [20], [23], [24]. Fortunately, this latter problem can be solved by using the error analysis technique proposed in [29].…”