Lossy Transmission Line Modeling and Simulation Using Special Functions

Abstract

A new algorithm for modeling and simulation of lossy interconnect structures modeled by transmission lines with Frequency Independent Line Parameters (FILP) or Frequency Dependent Line Parameters (FDLP) is developed in this research. Since frequency-dependent RLGC parameters must be employed to correctly model skin effects and dielectric losses for high-performance interconnects, we first study the behaviors of various lossy interconnects that are characterized by FILP and FDLP. Current general macromodeling methods and Model Order Reduction (MOR) algorithms are discussed. Next, some canonical integrals that are associated with transient responses of lossy transmission lines with FILP are presented. By using contour integration techniques, these integrals can be represented as closed-form expressions involving special functions, i.e., Incomplete Lipshitz-Hankel Integrals (ILHIs) and Complementary Incomplete Lipshitz-Hankel Integrals (CILHIs). Various input signals, such as ramp signals and the exponentially decaying sine signals, are used to test the expressions involving ILHIs and CILHIs. Excellent agreements are observed between the closed-form expressions involving ILHIs and CILHIs and simulation results from commercial simulation tools. We then developed a frequency-domain Dispersive Hybrid Phase-Pole Macromodel (DHPPM) for lossy transmission lines with FDLP, which consists of a constant RLGC propagation function multiplied by a residue series. The basic idea is to first extract the dominant physical phenomenology by using a propagation function in the frequency domain that is modeled by FILP. A rational function approximation is then used to account for the remaining effects of FDLP lines. By using a partial fraction expansion and analytically evaluating the required inverse Fourier transform integrals, the time-domain DHPPM can be decomposed as a sum of canonical transient responses for lines with FILP for various excitations (e.g., trapezoidal and unit-step). These canonical transient responses are then expressed analytically as closed-form expressions involving ILHIs, CILHIs, and Bessel functions. The DHPPM simulator can simulate transient results for various input waveforms on both single and coupled interconnect structures. Comparisons between the DHPPM results and the results produced by commercial simulation tools like HSPICE and a numerical Inverse Fast Fourier Transform (IFFT) show that the DHPPM results are very accurate.