# Interdisciplinary Applied Mathematics

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Circuit simulation is today a mature subject, and various software packages exist for design purposes. The program SPICE, which is an acronym for Simulation Program with Integrated Circuit Emphasis, was developed in the 1970s at UC Berkeley (Nagel and Pederson, 1973), and since then it has become the unofficial industry standard among integrated circuit (IC) designers. SPICE is a general-purpose simulation program for circuits that may contain resistors, capacitors, inductors, switches, transmission lines, etc., as well as the five most common semiconductor devices: Diodes, BJTs, JFETs, MESFETs, and MOSFETs. SPICE has built-in models for the semiconductor devices, and the user specifies only the pertinent model parameter values. However, these devices are typically simple and can be described by lumped models, i.e., combinations of ordinary differential equations and algebraic equations. In some cases, such as in submicron devices, even usual semiconductor devices, e.g., MOSFET, simple modeling is not straightforward, and it is rather art than science to transfer from basic PDEs to approximated ODEs and algebraic equations.

###### 1.7.1 Reduced-Order Modeling

Reduced-order models or macromodels are often considered as a link between physical and systems level-modeling, as shown in Figure 1.32. In this level of modeling, the prime focus is model-order reduction, i.e., to reduce the number of degrees of freedom present in the physical level simulation to many fewer degrees of freedom. Reduced-order modeling essentially acts as a link between physical and system levels by projecting the results of detailed numerical simulations (physical level) onto spaces spanned by a very small number of appropriately chosen dynamical variables used in the system-level simulation. Reduced-order models are generally characterized by the following attributes (Senturia et al., 1997):

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