On relative error minimization in passivity enforcement schemes

This paper presents a new technique for the elimination of passivity violations in linear lumped macromodels. The main algorithm is based on the perturbation of imaginary eigenvalues of suitably-defined Hamiltonian matrices, as documented in the existing literature. We introduce a modification aimed at the minimization of the relative error in the model responses during the passivity enforcement. This strategy allows the accurate modeling of structures characterized by a large dynamic range, as typically found in microwave filters or advanced packaging applications. Introduction and motivations Linear macromodeling provides a flexible and effective solution for fast and accurate simulation of complex interconnects. However, macromodel identification from time-domain or frequency-domain tabulated data [1]-[8] often leads to nonpassive results. Though possibly accurate, non-passive models may lead to unstable behavior when used in a CAD tool for system design and verification. Therefore, passivity should be enforced in some way during the model identification process. Despite the intense research efforts that have been devoted to the subject, the passivity enforcement of linear macromodels still poses several challenges. Currently available solutions can be grouped into tree main classes. On one hand, methods based on convex optimization [9]-[12] are guaranteed to find the optimal solution. Unfortunately, these techniques are limited to small-scale models due to their large computational complexity. A second class is provided by a posteriori passivity correction techniques based on linear or quadratic programming [13, 14]. These methods are based on inequality constraints imposed at discrete frequency samples. They are applicable to larger-size models but are not guaranteed to fully enforce passivity. Finally, other methods exploit the theory of Hamiltonian matrices [15]-[18]. They are applicable to larger-size models and do provide a global passivity characterization and enforcement. However, the convergence is not always guaranteed and the solution they offer is only sub-optimal. All passivity enforcement techniques apply some perturbation to the model until its passivity is achieved. This perturbation is performed using special constraints insuring that the model accuracy is preserved. These constraints have always been formulated so that the absolute error in the responses is minimized, except for the very recent results in [19]. In this work, we present a method allowing for the systematic preservation of the relative error during the passivity enforcement. We show that the proposed technique leads to superior performance with respect to standard schemes in all cases characterized by responses with large dynamic range. This scenario is typical, e.g., in packaging applications and RF component modeling. Formulation The main formulation is here presented for a scalar (oneport) macromodel. The extension to multi-port structures will be fully documented in a forthcoming report, although some hints are provided in the next sections. We consider a one-port linear macromodel described by one of the following equivalent representations