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                                                 Resonant Micromechanical Systems

                                                             Resonant Micromechanical Systems  265

                                              2
                                             Ȧ z         F 0
                                              r 0
                                                     2
                                       x ˙˙ = í   íȦ x +    sin(Ȧt )
                                        i    ʌȝ x ˙  r i  m      i
                                               i i
                                                                                          (5.95)
                                                    2
                                                  Ȧ z               F
                                                            2
                                       x ˙˙  = í    r 0   íȦ x    +  0  sin(Ȧt  )
                                        i +1    ʌȝ   x ˙    r i +1  m      i +1
                                                 i +1 i +1
                                Because the sensing is performed electrostatically, the microdevice displace-
                                ment can be assessed at any time by Eq. (5.90) in terms of the measured
                                capacitance, namely,
                                                         gC i
                                                     x =     í l                          (5.96)
                                                      i   İl   0x
                                                           z
                                By combining Eqs. (5.94) and (5.95), the following two equations are pro-
                                duced:
                                                        2
                                                       Ȧ z         F
                                                              2
                                  x ˙  = x ˙ + ǻt (1 íȕ ) í  r 0  íȦ x +  0  sin(Ȧt )
                                   i +1  i        1    ʌȝ x ˙  r i  m     i
                                                        i i
                                                   2
                                                  Ȧ z              F 0
                                                   r 0
                                                           2
                                       + ǻt ȕ  í         íȦ x    +   sin(Ȧt  )
                                           1   ʌȝ   x ˙    r i +1  m      i +1
                                                 i +1 i +1
                                                                                          (5.97)
                                                                2
                                                               Ȧ z
                                                  (ǻt) 2        r 0   2    F 0
                                  x   = x + ǻtx ˙ +   (1 íȕ ) í    íȦ x +    sin(Ȧt )
                                   i +1  i     i   2      2   ʌȝ x ˙  r i  m      i
                                                                i i
                                                     2
                                        (ǻt) 2      Ȧ z       2      F 0
                                                     r 0
                                       +    ȕ  í           íȦ x    +    sin(Ȧt  )
                                          2  2   ʌȝ   x ˙     r i +1  m     i +1
                                                   i +1 i +1
                                By starting with zero-displacement and zero-velocity initial conditions, it is
                                possible to determine the series Ș i  by working with Eqs. (5.97) in this se-
                                quence:
                                1. Determine the coefficient of dynamic viscosity Ș i+1  as a function of known
                                  displacements x i  and x i+1  , the velocity dx/dt at moment i, and the dynamic
                                  viscosity coefficient at the previous time moment Ș i  from the second of
                                  Eqs. (5.97).
                                2. Determine the velocity dx/dt at moment i + 1 from the first of Eqs. (5.97).
                                  The average coefficient of dynamic viscosity can eventually be determined
                                as the arithmetic average of the n values of Ș i .

                                The motion about the x direction can be enabled, as illustrated in
                              Fig. 5.40b, in “parallel-plate” transduction where the mobile plate moves





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