Consider the Topology I exposed in chapter 4 because of the advantages it provides: as explained in section 4.2, this topology reduces the parasitic feedthrough current and uses independent AC and DC voltage sources (more suitable for operating as a self-sustained oscillator [45]).
The beam parallel plane deformation is found to be not realistic enough, especially for large displacements of the middle point. Thus, the deflection profile of the electrostatically actuated cc-beam is considered in order to obtain an improved model. The electrostatic actuation can understood as a distributed phenomenon along the span that causes the beam deformation and at the same time is affected by it. Despite the fact that the distributed load caused by the electrostatic actuation is not uniform, especially for large deformations, the cc-beam is considered to be under a uniform load (q) accordingly to [78] (the deflection profile in this case is obtained as equation (3.12)). If the electrostatic coupling is not applied over the whole beam length, the elastic equation of the deflection profile is given by [78]:
ωq(x, y) =
where MA,B andRA,B are respectively the reaction forces and momentums in the clamped ends of the beam given by equations (5.2). The effective coupling length islcconsidering a symmetrical disposition of the electrode with respect to the beam.
MA=MB=−qlc
24l −l2c+ 3l2
; RA=RB= qlc
2 (5.2)
The maximum deflection occurs at the beam center (ωq(l/2)), whose displacement cor-responds to the x parameter (x = ωq(l/2)), and is used to find the normalized deflection profile for the cc-beam (5.3) (for the case of a uniform load applied to the whole beam span), that is, the deflection at each position in terms of such maximum. The expression obtained is independent of the load value.
ωq(x, y) = 384x Moreover, following what is exposed in section 3.1.1, the equation of the first mode shape of a cc-beam is [58]:
u1(x, y) =xC1[sinh (β1·y/l)−sin (β1·y/l) +α1(cosh (β1·y/l)−cos (β1·y/l))] (5.4) where α1 = (sinhβ1−sinβ1)/(cosβ1−coshβ1), β1 is the eigenvalue for the fundamental vibration mode (β1 =kn) , and C1 is a constant obtained by imposing that u1(l/2)
x = 1 (C1 = 1/[sinh (β1/2)−sin (β1/2) +α1(cosh (β1/2)−cos (β1/2))]), andx is the maximum deflection (at y=l/2). The beam deflection under a punctual force in the x-direction (Fx) applied at the center is found [78] to follow the polynomial equation (3.8). Following the same procedure as the one for equation (5.3), equation (3.8) can be expressed in terms of the maximum deflection point (and independent of the force value) as
ωF (x, y) =−48xy2 In figure 5-1a) the deflection profile obtained by FEM simulations of a cc-beam under electrostatic force is compared respectively with the shape of the deflection profile caused by either an uniformly distributed load (equation (5.3)) or lumped load applied to the center of the beam (equation (5.5)), and with the first mode shape of a cc-beam (equation (5.4)).
Equations (5.3), (5.4) and (5.5) are plotted in figure 5-1a) for the x value corresponding to the maximum deflection point given by FEM simulations. It is interesting to notice the good agreement between all these curves, and between these curves and the FEM obtained points. Given the accuracy of the deflection profile given by the curve (5.3), it is used as an
approximation of the real deflection profile. Moreover, it presents practical advantages: it is defined along the whole beam span (in contrast with equation (5.5) which is only defined for y ∈[0, l/2]), and it is a polynomial function (so it is simpler to operate with than the mode shape equation (5.4)).
Figure 5-1: a) Elatic deflection profiles comparison between the mode shape equation (5.4), the deflection under uniform load (polynomial equation (5.3)), the deflection under lumped and centered load (polynomial equation (5.5)) and the deflection profile provided by FEM simulation. b)Analytical (equation (5.3)) and FEM-obtained deflection profile, and the corresponding finite difference method (FDM) profile for N = 6 slices and N = 20 slices.
Both figures refer to an AMS 035 polysilicon cc-beam resonator whose dimensions are given in the first row of table 4.2 under a electrostatic load provided by a bias voltage of 34V.
As exposed in section 3.1.5, the electrostatic force acting on the beam is found as the negative gradient of the energy stored between the beam and each electrode, thus in the defined 1 DOF it is proportional to the x-derivative of the capacitance. The differential beam-electrode capacitance can be accurately approximated for the equation (5.6) as a function of the beam deflection (x) at the middle point and the position (y) on the beam.
dC(x, y) = 0thdy
s−ωq(x, y) (5.6)
The whole dynamic equation of the electromechanical system, posed as the dynamic equilibrium over the pointy=l/2, can be expressed in the terms previously introduced, as
an improvement of equation (4.2) :
where k1σ,k3σ are the linear and nonlinear stiffness coefficients that depend, a priori, on the residual fabrication stress, and kFF is the fringing field term, which provides a contri-bution to the coupling capacitance between the beam and the electrodes. These effects, the residual fabrication stress and the fringing field contribution to the coupling capacitance, and their influence on the parameters of the dynamic equation will be analyzed in following sections. The electrostatic force can be numerically approximated by integrating the differ-ential capacitance along the beam length, and subsequently by deriving it with respect to thexvariable. However, this numerical procedure results to be highly time-consuming and unfeasible in an electrical simulator. To conduct the described electromechanical system within IC design tools, only time derivatives can be contained in the model.
From the assimilation of the dynamic equation to the classic Duffing equation with para-metric excitation, and given the solution of the undamped and unforced Duffing equation based on Jacobi elliptic function explained in section 2.2.2, the resonance frequency of the system is found to present amplitude dependence on the nonlinear stiffness, following a function that can be assimilated to equation (2.40). The effect of the nonlinear stiffness on the resonance frequency and its dependence on the oscillation amplitude is illustrated by numerical simulations. In these simulations a control parameter ais introduced to modify the nonlinear stiffness when numerically solving the undamped and unforced Duffing system (meffx¨+k1x+ak3x3= 0) using dimensions and parameters corresponding to a 1MHz AMS 035 polysilicon cc-beam resonator (dimensions given in the first row of table 4.2). The max-imum power frequencies (namely the resonance frequencies) for each oscillation amplitude are represented in figure 5-2a). Those frequencies are compared and mainly agree with the analytically obtained ones from equation (2.40). Numerical simulations reported in figure 5-2a) show that the influence of the nonlinear stiffness on the resonance frequency is only important for oscillations whose values are significant fractions of the gap parameter. How-ever, in practice, the oscillation amplitude is limited by more critical effects like the pull-in effect. With this results, we can forecast that for the usual small oscillation amplitudes the
nonlinear stiffness will have a negligible effect on the resonance frequency.
Figure 5-2: a) Numerically and analytically obtained resonance frequency for each oscillation amplitude value, and for different values of a constant (a) which modifies the nonlinear stiffness. b) Average error between FEM simulations and the proposed model (based on FMD) for the values of capacitance (left) and its x-derivative (right) as a function of the number (N) of slices. In both figures the cc-beam resonator dimensions are the same and given in the first row of table 4.2.