Perturbation schemes for passivity enforcement of delay-based transmission line macromodels

This paper introduces two different algorithms for passivity enforcement of transmission-line macromodels based on the generalized method of characteristics (MoC). The first scheme corrects passivity violations via perturbation of purely imaginary solutions of a nonlinear Hamiltonian eigenvalue problem. The second scheme is based on a linearly constrained quadratic optimization formulated at carefully selected frequency samples. Both schemes can be viewed as extensions to the case of MoC-based macromodels of existing techniques that were only available for lumped macromodels. The resulting passive delay-based macromodels can be safely employed in system-level signal integrity simulations, due to their inability to produce any spurious energy gain. Several numerical examples illustrate the behavior of both schemes on transmission line structures of practical interest, highlighting their main differences and the open problems deserving further research work.