Interdisciplinary Applied Mathematics

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quantities across subsystem boundaries. The components can be decomposed further by either (a) interconnection of basic components (structural modeling) or (b) description by a set of algebraic equations (behavioral modeling). The models of reusable components are stored in a block library.

Behavioral description of devices and subsystems when supported by hardware description languages (HDL) leads to the approach of macromodeling based on HDLs. Some examples of HDLs are HDL-A25 (Mentor Graphics, 1995), MAST26 (Mantooth and Vlach, 1992), and SpectreHDL27 (Cadence, Ltd., 1997). The features of these HDLs are (i) Multidomain description, (ii) clear distinction between interface and algorithmic kernel of the model, (iii) interface to embedded C programs, and (iv) mechanisms for handling nonlinear ODEs to be solved by the internal algorithm.

17.2.1 Equivalent Circuit Representation

In the equivalent circuits approach, the microsystem comprising the mechanical, electrical, and fluidic components is represented by their electrical equivalents (Tilmans, 1996). The approach is based on the mathematical analogy between electrical and mechanical systems; specifically, the formal similarities between the integrodifferential equations governing the behavior of electrical and mechanical systems is the basis of the analogy. Newton’s second law of motion relates the force F acting on a body of mass m by the relation


where u and x are the velocity and displacement of the mass. This is analogous to the constitutive equation of an electrical inductor

where V is voltage, i is current, q is charge, and L is the inductance in the electrical circuit. Here F plays the same role as the voltage V, the velocity u as the current i, and the displacement u as the charge q. The mass m in mechanical systems represents the inductance L in an electrical circuit. However, while force is a “through variable” in mechanical systems, voltage (its representative in the electrical system) is an “across variable” in electrical systems. In general, it can be seen that a through variable in the mechanical system is represented as an across variable in an electrical system and vice versa. Hence, in the equivalent circuit representation, series connections in mechanical systems are represented by parallel connections in electrical systems and vice versa.

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