33 State-Space Modeling of Nonlinear Electrostatic Transducers and Experimental Characterization Using LDV 261 Fig. 33.2 A cross-sectional schematic of the electrostatic MEMS transducer [15] Fig. 33.3 Diaphragm deflection profiles measured using the scanning LDV at zero volts, at an intermediate voltage less than the collapse voltage, and at a voltage greater than the pull-in voltage the diaphragm and the relatively high residual tensile stress, the diaphragm is stress-dominated rather than bending-stiffness dominated, such that the diaphragm behaves as a membrane rather than a plate. In an initial measurement, the diaphragm is actuated with a low-frequency triangle waveform that ramps from 0 V to 20 V and back down. The result of this experiment is presented in Fig. 33.3 which is a segmented scan of the transducer’s diaphragm displacement obtained at different voltages. The images are captured using a Polytec MSA-050 scanning laser doppler vibrometer (LDV) system. Image (a) shows the diaphragm status at 0 volts. Image (b) shows the diaphragm at an intermediate voltage, less than the pull-in voltage, and Image (c) presents the diaphragm profile after pull-in. In this pull-in state, approximately 70 percent of the diaphragm is in contact with the backplate and the diaphragm has a flat profile across the central region. Pull-in for this transducer occurs at 18 V and subsequent snapback at 13 V [15]. Next, the transducer is driven with a 4-cycle square wave burst with the signal value ranging from 0 V to +20 V and with frequency equal to 4 kHz, 36 kHz, and 96.8 kHz. Figure 33.4a, b, and c presents the drive signal and the measured and simulated center-point diaphragm displacement. The high-frequency oscillations in the square wave excitation signal are experimental artifacts and are not significant. Interesting and noteworthy features are as follows. In Fig. 33.4a, for each excitation cycle the diaphragm traverses the full 1.8 .μm diaphragm-backplate gap, pulls in, and subsequently rings down when released. In Fig. 33.4b the period of the excitation is shorter, and the diaphragm’s ring-down is interrupted by each subsequent pull-in cycle. In Fig. 33.4c the half-period of the excitation is shorter than the transient pull-in time of the diaphragm, and the diaphragm therefore never pulls into contact with the backplate. The model predicts this phenomenon accurately. Furthermore, it is interesting to note that the diaphragm displacement is slightly different in each excitation cycle, showing some degree of ramp-up. This subtlety is also accurately predicted by the model. Figures 33.4d, e, and f presents the diaphragm displacement in response to excitation signals that are the complement of the those in (a), (b), and (c). The actuation voltage begins and ends at 20 V and is held at 20 V in between burst cycles. Comparing image (a) against (d) and (b) against (e), the only significant difference is that the ring down of the diaphragm at the end of the burst is prevented in the complementary drive case, as the diaphragm is pulled-in against the backplate to prevent ring-down. Comparing (c) against (f), however, the results are quite different. The peak-to-peak displacement in response to the waveform in (c) is approximately 1.μm while that of complementary tone burst is approximately 2.μm. In the complementary case, the diaphragm begins the cycle in a pulled-in state with stored elastic potential energy, and this energy results in larger amplitude oscillations when the diaphragm is released from pull-in. Here again, the model does an exceptional job at predicting these important characteristics. A second commercial MEMS microphone structure was also used as an actuator. This microphone is a Knowles model SPM0687LR5H-1. Figure 33.5 presents a scanning electron micrograph (SEM) of the chip which comprises two diaphragms electrically connected in parallel. The transducer was driven as an ultrasound transmitter near 70 kHz using a unipolar
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