Macromodeling of Electrical Interconnects and Packages
Macromodeling of Electrical Interconnects and Packages

Lumped 3D Interconnects
Modeling of 3D interconnects like connectors, junctions, packages, and vias is one of the most challenging tasks in systemlevel characterization of highspeed electronic systems. Such structures behave as short circuits at low frequencies, but have a significant influence on the signals with the high data rates that are currently used for digital transmission. A careful Signal Integrity analysis requires accurate macromodels for all these structures. However, due to the 3D nature with complex geometry/material configurations (see the outline of a connector metal in the picture), the modeling procedure must rely on fullwave characterization or on direct measurement. This research activity aims at translating the large amount of data (usually port responses in time or frequency domain) that is obtained by either process into macromodels characterized by low complexity and good accuracy. These macromodels should be available in a form which is directly usable within standard SPICElike circuit solvers. Several approaches have been followed to reach this goal, from Subspacebased StateSpace System Identification (4SID) projection techniques to new formulations of the wellknown Vector Fitting algorithm.

Transmission Lines
Transmission lines structures at chip, multichip, package, and board level constitute one of the most critical parts for the signal integrity of all electronic systems. Nonetheless, an accurate and efficient transient simulation of transmission lines is still a challenging task even in the most advanced circuit solvers. This is due to the intrinsic difficulties in the design of stable algorithms for the timedomain analysis of structures with frequencydependent parameters. Indeed, it is well known that accurate interconnect models must take into account metal and dielectric losses, which lead to possibly large attenuation at increasing frequency. The underlying physics is best captured using a frequencydomain approach, leading to constitutive parameters with a complex dependence on frequency. A robust approximation is therefore required for the conversion to time domain of the constitutive line equations and the subsequent generation of a line macromodel to be employed in a transient simulation. This research activity is focused on the transmissionline macromodeling based on the Generalized Method of Characteristics (MoC). The key ingredients are delay extraction and rational approximation. One of the main results of this activity is the highefficiency implementation named TOPLine for IBM internal circuit solver (PowerSPICE).

Passivity of Macromodels
Standard macromodeling procedures adopt some kind of rational approximation in order to synthesize lumped electrical equivalents for given interconnect structures. Depending on the particular approximation process being adopted, the macromodel may result passive or nonpassive. Passivity may be defined in a loose sense as the inability of a given structure to generate energy. Nonpassive models may lead to instabilities in transient simulations, depending on the termination networks being adopted (see figure). This situation can be avoided by checking the models for passivity and by trying to compensate the passivity violations. Our research in this field led to investigate the spectral structure of socalled Hamiltonian matrices associated to the macromodel. The algorithms and tools that we have developed allow to pinpoint all the passivity violations and to eliminate them by an iterative spectral perturbation algorithm.

Model Order Reduction and PEEC
As opposed to the macromodeling approaches from port responses, other important techniques are available. In particular, the wellknown Partial Element Equivalent Circuit (PEEC) method can be used to translate a threedimensional interconnect into an equivalent circuit. The latter is derived through a discretization of the Maxwell equations in integral form. Unfortunately, the resulting equivalent circuits are usually too large to be effectively used in systemlevel simulations. Therefore, Model Order Reduction techniques are usually employed to provide for smaller approximate circuits. One of the advantages of this approach is the guaranteed passivity of the final macromodels. We experimented this approach by applying Krylovbased order reduction to PEEC models to several test problems and to avionic equipment.