**1. Introduction**

154 Microelectromechanical Systems and Devices

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of (100) Si surface etched in KOH and TMAH solutions. *Sensors and Actuators A*,

Micro-electro-mechanical Systems (MEMS) are components with micron-scale moving parts based on materials and processes of microelectronics fabrication. This is a good example of on-chip integration of electronics, microstructures, microsensors and microactuators. Accurate simulation of MEMS requires precise modeling of all effects of mechanical and damping forces, electrostatic forces and inner stresses, heat transfer, thermal expansion, piezoelectric stresses etc.

Modern methodology of MEMS design implies that the entire MEMS can be investigated only at higher abstraction levels such as **schematic and system** ones, where accurate macromodels can be used [1]. On the other hand, at component or device levels the physical behavior of three-dimensional continuums is described by **partial differential equations** (PDE) easily solvable by Finite Element or Finite Difference Element Methods (FEM or FDM) [2,3], available in ANSYS –like software. Component level simulations are classified in single - domain and coupled - domain simulations, both being very computer time- consuming.

The goal of this chapter is to consider methods of automatically obtaining macromodels of MEMS and their mechanical or non-electric components from ANSYS models as equivalent electric circuits or low order differential ordinary equations for further use in circuit design software. This can be done by using different model order reduction techniques developed in recent years.

When dealing with the modern MEMS, the possibility for using a single environment to simulate objects, where different physical processes such as electrical, mechanical, optical, thermal etc. take place, plays an important role. Here we have to represent different subsystems of the initial MEMS as equivalent models of the same physical nature permitting to combine them for solution in a single computational process. After that, the complete behavioral model of the entire MEMS and its subsystems can be compiled either in VHDL-AMS language (as sets of ODE) or in SPICE-like language (as equivalent electric circuits).

The Microsystems design exploits various analytical and numerical methods for virtual prototyping of MEMS. It also demands for libraries of electromechanical, optical models and microfluid components, including springs, bulks, buffers, capacitors, inductances, operational amplifiers, transistors etc. Three basic possible approaches of MEMS design procedure are illustrated below: FEM/FDM Model, Reduced Order Model (ROM), Coupled system-level model.

Macromodels of Micro-Electro-Mechanical Systems (MEMS) 157

The FULL file contains all the information about the system: the system element matrices, Dirichlet boundary conditions, equation constrain and the load vector. It is generated using ANSYS partial solver, which enables to assemble system element matrices for the desired analysis without solving them and it therefore computationally fast. The speed of the reading operation has been optimized taking into account that the element matrices are sparse. The load vector directly gives the matrix-vector product *Bf* and thus describes the distribution of all loads being applied. In order to obtain the *B* matrix, and thus being able to modify the inputs singularly, it is necessary to repeat the partial solution for each input of

It would be easier and more intuitive for the designer to explore the design space if the MEMS model had only a few variables with a clear relationship between them and the overall device performance. *Reduced-order models (ROM)*, also called *macromodels,* lend themselves very well to these purposes. The main idea behind the reduced order model is that the number of ordinary differential equations (ODE) needed to simulate the system has been reduced from perhaps many thousands in the case of the full FEM simulation, to just a

Such the macromodel simulation can be very efficient computationally compared to the FEM model. A designer can use the FEM model for different component geometry and materials trying and the ROM model for investigation of different input forces effect (fig.2).

interest.

**3. ROM (Reduced Order Model)** 

few basis function coordinates (fig.1).

Fig. 1. Reduced order model illustration

Fig. 2. Compact reduced order model in MEMS design [33]
