# Interdisciplinary Applied Mathematics

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SW

dW    dW c

~x—oqt + -—bxt dqt    dxt

vtdqt + Ftdxt,

(17.2)

where vt is the voltage between the plates and Ft is the mechanical force acting on the movable plate. Using equations (17.1), (17.2) and the expression for capacitance, we obtain

vt(qt ,xt)

dW

dqt

qt(d + xt) to A

and Ft(qt,xt)

dW

dxt

qt

2eoA

+k(xt-xr).

Since the equations are nonlinear, they are linearized using the Taylor’s series expansion around some bias point (x0, q0). The constitutive equation, describing the linear relations between the incremental or small signal effort variables    and    the    state    variables,    for    voltage    at    the    bias    point    (x0,q0)    is

given by

v(q, x)

Transformer

FIGURE 17.4. Decomposition of the transfer matrix into elemental matrices for circuit representation.

di’t

dqt

dvt 4+ 7—

0    dXt

x

0

(d + x0) CqA

«+7f’

q    vo

+ 75-X

Co    (d+xo)

(17.3)

Using the    constitutive    equation    for    force and    the    expression    for    Ft,    we

obtain the final expression for the force, i.e.,

F (q,x)

m

dqt

, dFt

q + 7—

o    dxt

x

o

-AFq + kx

eoA

i’o

(d + xo)

q + kx.

(17.4)

The constitutive equations and the final expressions for v and F as given by equations (17.3) and (17.4), respectively, can be used to construct the transfer matrix as shown in Figure 17.3. The transfer matrix can be decomposed into elemental matrices in several ways, giving rise to many feasible circuit representations of the device. Figure 17.4 shows one decomposition of the transfer matrix and the corresponding circuit representation. There are several other circuit representations possible for the same device (see (Tilmans, 1996), for details) including some with pure capacitive circuits. Typically, the designer chooses the most appropriate circuit representation based on the application.

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