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200                                                                  Inertial Sensors

                 electrostatic forces. An automatic gain control (AGC) control loop ensures that the
                 oscillation amplitude is constant. In the presence of a rotation about the axis normal
                 to the sensing element plane, energy is transferred to the inner gimbal structure,
                 which starts vibrating at the same frequency at an amplitude proportional to the
                 angular spin rate. Maximum sensitivity is achieved when the drive frequency of the
                 outer structure is equal to the resonant frequency of the inner gimbal. The sensing
                 element could be operated in a force-balance mode. Electrostatic forces generated by
                 voltages on the feedback electrodes counterbalance the movement of the inner gim-
                 bal. The fixed electrodes above the inner and outer gimbal structure were fabricated
                 by an EDP wet-etch that removes sacrificial silicon dioxide. The lower electrodes
                 underneath the structure were implemented as p-type buried electrodes and are elec-
                 trically isolated by a reverse biased p-n junction from the substrate. The gap between
                 the fixed electrodes and the movable on the resonators is between 8 and 10 µm. To
                 increase the mass of the inner resonator, an inertial mass made from gold, of 25-µm
                 height, was electroformed.
                    The first polysilicon surface-micromachined vibratory rate gyroscope was pre-
                 sented in 1996 by Clark and Howe [51]. It is a direct implementation of the lumped
                 model presented in Figure 8.22. Standard comb drive actuators were used to excite
                 the structure to oscillate along one in-plane axis (x-axis), which allows relatively
                 large drive amplitudes. Any angular rate signal about the out-of-plane axis (z-axis)
                 excites a secondary motion along the other in-plane axis (y-axis). The sensing ele-
                 ment is shown in Figure 8.24 and consists of a 2-µm-thick polysilicon structure. In
                 this reference quadrature error is discussed in detailed and it is shown that a mis-
                 alignment of the primary oscillation axis with the ideal x-axis of only one part in 3.6
                 million will result in a quadrature error equal to the signal of a 1°/sec rotation about
                 the z-axis. No fabrication process can be accurate to such a degree, and hence, elec-
                 trostatic tuning is used to alleviate this problem. The quadrature error is propor-
                 tional to the position of the primary oscillation, whereas the Coriolis acceleration is
                 proportional to the velocity of the primary oscillation; hence, the resulting forces are
                 90° out of phase (this explains the term quadrature error). The inner interdigitated




                                        Structural anchor
                                        to substrate



                                  Input
                                  Rotation                   Sense
                                                             Mode
                                                                         Comb drives to
                                                    Driven Mode          sustain oscillation







                                 Interdigitated comb finger
                                 deflection sense capacitors
                 Figure 8.24  Surface-micromachined gyroscope. (After: [51].)
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