Subgradient Techniques for Passivity Enforcement of Linear Device and Interconnect Macromodels

This paper presents a class of nonsmooth convex optimization methods for the passivity enforcement of reduced-order macromodels of electrical interconnects, packages, and linear passive devices. Model passivity can be lost during model extraction or identification from numerical field solutions or direct measurements. Nonpassive models may cause instabilities in transient system-level simulation, therefore a suitable postprocessing is necessary in order to eliminate any passivity violations. Different from leading numerical schemes on the subject, passivity enforcement is formulated here as a direct frequency-domain ${{cal H}_infty}$ norm minimization through perturbation of the model state-space parameters. Since the dependence of this norm on the parameters is nonsmooth, but continuous and convex, we resort to the use of subdifferentials and subgradients, which are used to devise two different algorithms. We provide a theoretical proof of the global optimality for the solution computed via both schemes. Numerical results confirm that these algorithms achieve the global optimum in a finite number of iterations within a prescribed accuracy level.