On the resonance frequencies of microbridges

High-Q HF microelectromechanical filters

“…12 and 13). For the resonators of this work, is dominated by anchor step-up and finite elasticity effects [17], [18], which are predictable using finite element analysis (FEA).…”

“…Pursuant to this, the voltage-to-displacement transfer function at a given location (see Fig. 5) at resonance is first found using phasor forms of (4)- (6), (8), and (9) and integrating over the electrode width to yield (17) Using the phasor form of (1), the series motional resistance seen looking into the drive electrode is then found to be (18) Inserting (17), factoring out , and extracting yields (19) Note that the effective integrated stiffness defined in (6) can also be extracted from (17), yielding (20) The transformer turns ratio in Fig. 6 models the mechanical impedance transformation achieved by mechanically coupling to the resonator at a location displaced from its center.…”

VHF free-free beam high-Q micromechanical resonators

“…For these reasons, this paper uses the expression from [6], which for convenience and later use is repeated as (9) where the variable now represents the resonance frequency including electromechanical coupling, and and are the effective stiffness (including adjustments due to external coupling) and mass [5], [6], respectively, at any location on the resonator beam, indicated in Fig. 3, is a fitting parameter that accounts for beam topography and finite elasticity in the anchors [6], [10], [11], and is the mechanical stiffness of the resonator at location , similar to , but this time, for the special case when V (i.e., no electromechanical coupling) and given by (10) where is the resonance frequency of the free-free beam sans electromechanical coupling, obtained from (1) or (5). In (9), is a parameter representing the combined electrical-to-mechanical stiffness ratios integrated over the electrode width , and satisfying the relation [6] ( 11) where is the permittivity in vacuum, is the electrode-toresonator gap spacing, which varies as a function of location along the length of the beam due to -derived forces that statically deflect the simply supported (by dimples) beam (cf.…”

Frequency-selective MEMS for miniaturized low-power communication devices

“…24; 4) it has been assumed that ; 5) is the permittivity in vacuum; 6) the function models the effect of an electrical spring stiffness that arises when a bias voltage is applied across the electrode-to-resonator gap, and that subtracts from the mechanical stiffness ; 7) is the electrode-toresonator gap spacing as a function of location, which changes due to beam bending under a static load; and 8) is a scaling factor that models the effects of surface topography. For the of this work, is dominated by anchor step-up and finite elasticity effects [30], [31], which are predictable using finite-element analysis (FEA). In practice, assuming a set value for , designing for a specific frequency amounts to setting geometric dimensions , , and via computer-aided design (CAD) layout since all other variables are determined at the outset by fabrication technology.…”

“…Using (18) and (24), and recognizing that (30) for parallel-plate capacitively driven devices, for the HF micromechanical filter of Fig. 26 takes on the form (31) Using (31) with (19) and (29), the relevant dependencies under a given frequency scaling can be comparatively written out as (32) (33) (34) (35) Using these equations, if scales by , also scales by , , and scale by , and must scale by to maintain a constant .…”