Interdisciplinary Applied Mathematics

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3. The characteristic equations, which relate the effort variables as a function of state variables, and the transfer matrix, which relates the effort-flow variables at the electrical port directly to those at the mechanical port, are derived.

4. The transfer matrix is used to obtain the equivalent circuit representation. Typically, the equivalent circuit representation may not be unique. So a practical situation is chosen.

We now illustrate the equivalent circuit representation of a parallel plate electrostatic actuator (in the absence of air-damping) using the above procedure. Figure 17.2(a) shows the parallel plate electrostatic actuator consisting of a rigid mass suspended by two flexible beams. A potential difference is applied between the ground plane and the mass, giving rise to attractive electrostatic forces. The mass moves down, and the beams bend due to these forces. The schematic representation of the device is given in Figure 17.2(b). Since the mass is rigid, the suspension structure can be modeled as a mass-spring system. The two identical beams represent the two springs.    The    spring constant    (k    =    2ki)    can be derived    from    the    beam

flexure formula. We use the energy method as described earlier to generate

FIGURE 17.3. Transfer matrix computation for the parallel plate actuator using lumped parameters.

the lumped parameters. The total energy W of the system, consisting of electrical (We) and mechanical (Wm) energies, is given by

W = We(qt,xt) + Wm (qt,xt)

It , 1,,

2C(xt) + 2k{Xt Xr)


where qt and xt are the charge and displacement at the time instant t, and xr is the equilibrium position of the mass; C is the capacitance of the system at the time instan, which is a function of xt and is given as C(xt) = eoA/(d+xt); e0 is the permittivity of vacuum and A denotes area. Taking the total differential of the energy represented by equation (17.1), we obtain

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