where PC cancels in the difference. The elimination of this arbitrary parasitic capacitance PC allows measurements to be repeatable across different facilities, using different capacitance meters. 4.3 CALIBRATION OF DISPLACEMENT The measured changes in capacitances in closing the two gaps can be written as 1 1 1, 2 2 layout h h C N gap N gap gap g g and (6) 2 2 1, 2 2 layout h h C N gap N ngap gap g g (7) where the unknown parameters , h, g, N, and are the permittivity of the medium, layer thickness, gap between comb fingers, number of comb fingers, and fringing field factor. The unknown quantity gap is the difference in gap-stop size between layout and fabrication, which must be found to calibrate displacement. This difference in geometry between layout and fabrication was exemplified in Figure 1. The quantities 1,layout gap and 2, 1, layout layout gap ngap in (6) and (7) are exactly known from layout, where 1 n is a layout parameter chosen such that 1 2 gap gap . In Equations (6) and (7) we have assumed that the movement of the comb drive shown in Figure 3 is in plane and in the direction of x. By Equation (6) by (7), we cancel the all unknown quantities except gap . We have 1, 1 2 1, layout layout gap gap C C ngap gap . (8) Solving Equation (8) for gap yields 1 2 1, 1 2 1 1 layout n C C gap gap C C . (9) Equation (9) is the geometrical difference between layout and fabrication in terms of measured changes in capacitance and exactly-known layout parameters. Therefore, the EMM-measured gap-stops are 1 1,layout gap gap gap and 2 1,layout gap ngap gap , where we have assumed that both gap-stops experience the same change in geometry in going from layout to fabrication because the gaps are within close proximity to each other. That is, we have assumed that during fabrication, the two gaps experience the same lithography, deposition, and etching reaction processes. Now the ratio of the displacement 1 gap to its corresponding change in capacitance 1C yields a property that we define as the comb drive constant , 1 1 C gap . (10) 71
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