This Chapter contains three advanced topics in model order reduction (MOR): nonlinear MOR, MOR for multi-terminals (or multi-ports) and finally an application in deriving a nonlinear macromodel covering phase shift when coupling oscillators. The Sections are offered in a prefered order for reading, but can be read independentlty.
Section 5.1, written by Michael Striebel and E. Jan W. ter Maten, deals with MOR for nonlinear problems. Well-known methods like TPWL (Trajectory PieceWise Linear) and POD (Proper Orthogonal Decomposition) are presented. Development for POD led to some extentions: Missing Point Estimation, Adapted POD, DEIM (Discrete Empirical Interpolation Method).
Section 5.2, written by Roxana Ionutiu and Joost Rommes, deals with the multi-terminal (or multi-port) problem. A crucial outcome of the research is that one should detect "important" internal unknowns, which one should not eliminate in order to keep a sparse reduced model. Such circuits come from verification problems, in which lots of parasitic elements are added to the original design. Analysis of effects due to parasitics is of vital importance during the design of large-scale integrated circuits, since it gives insight into how circuit performance is affected by undesired parasitic effects. Due to the increasing amount of interconnect and metal layers, parasitic extraction and simulation may become very time consuming or even unfeasible. Developments are presented, for reducing systems describing R and RC netlists resulting from parasitic extraction. The methods exploit tools from graph theory to improve sparsity preservation especially for circuits with multi-terminals. Circuit synthesis is applied after model reduction, and the resulting reduced netlists are tested with industrial circuit simulators. With the novel RC reduction method SparseMA, experiments show reduction of 95% in the number of elements and 46x speed-up in simulation time.
Section 5.3, written by Davit Harutyunyan, Joost Rommes, E. Jan W. ter Maten and Wil H.A. Schilders, addresses the determination of phase shift when perturbing or coupling oscillators. It appears that for each oscillator the phase shift can be approximated by solving an additional scalar ordinary differential equation coupled to the main system of equations. This introduces a nonlinear coupling effect to the phase shift. That just one scalar evolution equation can describe this is a great outcome of Model Order Reduction. The motivation behind this example is described as follows. Design of integrated RF circuits requires detailed insight in the behavior of the used components. Unintended coupling and perturbation effects need to be accounted for before production, but full simulation of these effects can be expensive or infeasible. In this Section we present a method to build nonlinear phase macro-models of voltage controlled oscillators. These models can be used to accurately predict the behavior of individual and mutually coupled oscillators under perturbation at a lower cost than full circuit simulations. The approach is illustrated by numerical experiments with realistic designs.