Efficient numerical method for computing per-unit-length impedance of transmission lines with lossy substrate

For recent VLSI circuits, the clock frequency and integration density have continuously increased. Thus, accurate estimate of the per-unit-length resistance and inductance of interconnect has become important for maintaining the signal integrity [1]. In order to design for the transmission lines successfully, it is necessary that correctness of the selected formulas or numerical methods can take into account AC loss due to the skin and proximity effects [2]. The integral equation method is an effective tool for calculating the per-unit-length impedance or current density distributions in the rectangular cross section of metallic lines [3]. Considering that the increase in resistance due to the eddy current flowing in lossy semiconductor substrate has attracted attention nowadays [4], it is advantageous if we can extend the powerful integral equation solutions to this case. In view of the above, we propose an efficient numerical method for computing per-unit-length impedance of multiconductor transmission lines having rectangular cross section backed by lossy substrate. This is a straightforward extension of the present authors' recent work [5]. In the process of the moment method, discretization of the cross section was performed according the skin effect by the use of nonuniform grid, which remarkably saves the computational cost. With keeping the advantage of this solution, we generalize the geometry of the line with regard to the number of conductors and the existence of lossy semi-infinite substrate. The effect of the substrate is taken into account in terms of a correction to the kernel function of the logarithmic type. The appeared Sommerfeld integral in the kernel can be analytically evaluated using the Struve and Neumann functions in the quasi-static regime.