The Feasibility of Numerical Calculations of Vias Using the Matrix-Penciled Moment Method

Abstract

The feasibility of numerical calculations of a through hole via using the Matrix- Penciled Moment Method (MP-MM) proposed by Show-Gwo Hsu and Ruey-Beei Wu is investi- gated. It shows that the MP-MM can't be used to the via which is near the source port, and there is a lower bounded operating wavelength beyond which MP-MM is not applicable. The conclusions are proven by both theory and the simulation results gotten by HFSS. The miniaturization and large scale integration of electronic devices places increasing demand on the multilayer interconnection geometry. Design of high-speed electronic systems requires careful consideration of the high frequency problems due to parasitic efiects from these interconnects. In multi-layer structures, there are parallel lines for signal transmission as well as vias to connect the signal lines in difierent layers. Several studies involving the electrical characterization and design of the signal lines have been reported in previous literature. The equivalent circuits for some basic via structures have been presented by several investigators on the basis of quasi-static analysis (1). Using the flnite-difierence time-domain (FD-TD) method (2), presented a full wave analysis for the propagation characteristics of a through hole via. In (3), three-dimensional mode matching technique was used to model the via hole grounds in microstrip. In (4,5), flnite-elment time-domain method was used to analyze through hole. In the last decade, Show-Gwo Hsu and Ruey-Beei Wu proposed full wave characterization of a through hole via using the Matrix-Penciled Moment Method (6). Based on the Pocklinton Equation, the method calculates the current distributions on via discontinuities by MoM, extracts poles by using Matrix-penciled method, gets the complex coe-cient of every mode by solving Vandermonde equations, and flnally, obtains the scattering parameters of via from superposition principle. The analytical and computational process is more complex than any other computational electromag- netic methods, but it has predominance in computational accuracy and speed for using the right basis and testing function and transforming the three dimensional problem to one dimensional by equivalent thin wire approximation. The MoM is a computational electromagnetic method with high computational accuracy in most scattering and antenna radiation problem, but the method in Show-Gwo Hsu's literature contained two equivalences: 1) The shape of the microstrip/strip lines in horizontal segment (shown in Figure 1 is ignored and treated as circular cylinder. This equivalence is accurate when the length of transmission lines is inflnite, but in actual microwave circuits, due to the restriction of trace and the measure of microwave circuit boards, the microstrip/strip line can't extend to inflnity. The equivalence theory and its limitation will be presented in Sections 3 and 4, respectively. 2) MP-MM is based on Pocklinton Equation obtained by considering the antenna as thin wire. Section 5 will discuss the condition in which the transmission line can be seen as thin wire and propose a lower bounded operating wavelength of this method. There is no question about the predominance of this method in computing the electromagnetic characterizations of via, but it is not a general-purposed method for it is not based on the real structure and rigorous equation which has no approximation. At present, most literatures about this method introduce its principles (6) and/or some partic- ular calculational examples (7), on the contrary, neglecting the investigation about the application spectrum in which the method can be used, that is the emphasis of this paper. The structure investigated in this paper is a through via with one ground plane. The tool used for simulation is HFSS.