**5. Conclusion**

186 Microelectromechanical Systems and Devices

But if the insulation layer is rather thick its restrictive effect should be taken into account. If the maximal center displacement is equal to initial thickness air gap the moving plate touches the insulator top surface of the electrode when the input *Vn* voltage reaches the value of (*V*max may be calculated from simulation). But as soon as the input voltage will be decreased under this value the plate will leave the hard stop. So, its capacitance does

*w*max

( ,) 2 *L*

Parameter *τ* in (50) defines an ultrasonic transducer frequency band and can be calculated

through the plate displacement and its velocity , which are defied

*w t* ( ,) ( ,)

The coefficient 3 appears in (51) due to the fact that a capacitance recharge to 98% for a time

*w tv t*

<sup>1</sup> ( , )/ ( , ) 32 2 *L L*

It is possible to see two included procedures in according to fig.18: one for development of ROM for an ultrasonic transducer plate, where a deflection and speed of central point`s deflection of transducer plate is calculated for the value *Vn*, and second - for determine of MEMS system- equivalent capacity value (SLM), using values of plate central point co-

Instead of using two sequence procedures mentioned above it is possible using functional possibilities of the circuit simulator NetALLTED to built a single system-level equivalent circuit model for an ultrasonic transducer by introducing directly into the equivalent- circuit ROM of mechanical MEMS part the additional arbitrarily connected element (a Depended Source) with an informative function which is determined by equation (50) [42]. Optimization procedures of NetALLTED allow getting the desirable values of this transducer capacity and through it to get a desirable value of output signal of an ultrasonic transducer system-level model by the changing ROM parameters, which, in turn, are depended upon an ultrasonic transducer construction sizes and used material

(51)

2 2 *L L v tw t*

not demonstrate the hysteretic behavior (fig.17).

*V*max

from ROM equations in the following way:

value which is equal approximately 3τ.

ordinates .Then the cycle of calculations recurs whereupon.

Fig. 18. ROM- system-level model coupled simulations

properties.

In this chapter, the methods and issues encountered in the development of MEMS macromodels at the system level have been presented. System level modeling is the highest and most abstract level of modeling. This level requires various devices` linking of MEMS *component level models* – both electronic and micromechanical – into a micro-electromechanical system. *System-level models* of MEMS components are needed to allow a fast and sufficiently exact investigation of their behavior to simulate entire MEMS.

 Starting point for the extraction of a reduced order model (ROM or a macromodel) is already its description with a large ODE system, which is typically derived using physical modeling techniques based on Finite Element Method (FEM) which is rather time consuming. Macromodels application allows the extraction of lower order ODE system that reproduces the input/output behavior with good accuracy. Particular attention has been posed in the chapter on the possibility to get a macromodel circuit presentation.

There are special methods for generating ROM for MEMS components and entire MEMS based on FEM descriptions. To derive macromodels of smaller sizes different approaches (*Modal decomposition, Moment matching, Equivalent circuit presentation)* were developed. Usage of the reduced MEMS components models allows applying successfully modern circuit simulators in workflow for MEMS design on system level.

Three automatic procedures to generate device reduced order macromodels, being based on full FEM/FDM models, were demonstrated in this chapter. Two of them are suitable for simulators with possibilities to get input information in the equation forms (ODE or OAE). The third one in opposite produces macromodels in circuit presentation and so it is more suitable for circuit simulators. The Modal ROM approach is based on using natural (modal or resonant) frequencies of MEMS structure and it is spread mostly in the USA and Asia. The Moment matching ROM approach is based on using the Krylov subspace for transfer functions and it is popular in Western Europe and Asia. The Equivalent circuit ROM approach is based on using a capacitive-inductive-resistive circuit model for mass, damping and stiffness matrices and it is used mostly in Eastern Europe. It is worth to notice that the Modal ROM approach requires some full ANSYS runs to perform a proper orthogonal decomposition during basic functions determination in opposite to the Moment matching and the Equivalent circuit ROM approaches for which it is enough to use ANSYS only for FEM model matrices building.

It seems to be interesting and perspective trying to combine mentioned approaches, for example, to start with Krylov/Arnoldi reduction of ODE dimension, then to build the proper equivalent circuit for obtained ODE systems and finally to apply Y/Δ transformation or n-port transformation for further reducing macromodel order.
