Analysis on the Centroid Offset of Coupling Beam in Array-composite Resonator due to Process Variations

Abstract

Ideally, a mechanically coupled array-composite resonator can improve linearity and motional resistance by a factor equal to the number of constituent resonators used in the array. Owing to fabrication nonidealities, however, random variations in the dimensions of resonators and coupling beams often compromise the actual increase in output current. As a result, previous array composites have fallen short of their expected impedances and linearity improvements, particularly when the number of constituent resonators used in the array is large. Therefore, in this paper, we present a detailed theoretical analysis on the centroid offset of beams caused by process variations and some related substantive issues. Specifically, an equivalent circuit model for a coupling trapezoid beam is first deduced in the case of an offset centroid. Then, a model for a coupled double-disk resonator including a beam with an offset centroid is established by combining the equivalent circuit model with an existing circuit model of a radial-contour mode disk resonator. Finally, numerical results are obtained by simulations using ANSYS and PSpice with the mechanical and electrical models, respectively. It is shown that the resonant frequency varies by 45.7, 110, and 250 ppm when the beam centroid shifts by 6, 12, and 18‰, respectively. When the centroid shifts towards one disk by 18‰, the output current amplitude of this disk decreases by 38.7‰ to 1.657 μA (compared with 1.724 μA at zero offset), while the output current amplitude of the other disk increases by 35.9‰ to reach 1.785 μA. In addition, when there is an offset, the Q of the resonator decreases. In particular, when the centroid has an 18‰ offset, Q decreases by nearly 19‰.

Numerical Study of the Influence of Horizontal Spatial Distribution of Macropores on Water Infiltration

“…The position of the sectional centroid [65,66] of the macropores at the soil surface represents the macropore position in this study. The concept of weighted mean distance is used to establish the spatial dispersion.…”