An overview of standard macromodeling techniques (i.e., employ poles but no phase shifts) for transient simulation of high-speed interconnects is first presented. Then, the limitations of these standard macromodeling techniques (e.g., high model order and slow convergence) are discussed. In order to overcome these limitations, generalized method of characteristics (MoC) techniques include the physical phenomenology of phase shift (time delay) in addition to the system poles, thereby making it possible to model single and coupled transmission lines using far fewer terms than when standard macromodeling techniques are employed. Since MoC techniques incorporate the time delay into the model, causality is also guaranteed. In this paper, the MoC idea is extended by developing a hybrid phase-pole macromodel (HPPM) for the modeling of more complex interconnects with embedded discontinuities. Unlike other generalized MoC techniques that have only been applied to single and coupled transmission lines, the HPPM macromodel can be applied to larger portions of the system that contain multiple cascaded transmission lines and discontinuities. The HPPM parameters can be extracted from either measured or simulated transient data. Comparisons between a standard macromodel and the HPPM show that the HPPM has significant advantages in terms of reduced macromodel orders.

Analytical frequency-domain expressions for single and coupled transmission lines with triangular input waveforms are first developed. The inverse Fourier transform is then used to obtain an expression for the time-domain triangle impulse responses for frequency-independent transmission line parameters. The integral associated with the inverse Fourier transform is solved analytically using a differential-equation-based technique. Closed-form expressions for the triangle impulse responses are given in the form of incomplete Lipschitz-Hankel integrals (ILHI) of the first kind. The ILHI can be efficiently calculated using existing algorithms. Combining these closed-form expressions for the triangle impulse responses with a time-domain convolution method using a triangle impulse as a basis function, provides an accurate and efficient simulation method for very lossy transmission lines embedded within linear and nonlinear circuits.