Device description:
The resonator beam device used for this analysis is described in detail the following classic paper by Clark T. C. Nguyen:
VHF Free-Free Beam High-Q Micromechanical Resonators, Journal of Microelectromechanical Systems, Vol. 9, No. 3, September 2000, by K. Wang, A. C. Wong, and Clark T.C. Nguyen.
The objective of the paper was to demonstrate that it is possible to fabricate vibrating beam micromechanical resonators with high Q factors by placing support beams at the node points for the fundamental flexural mode. The support beams are subjected to torsion when the resonator beam flexes. This arrangement creates a beam that is essentially unrestrained (free-free) in the flexure mode. Beams with fundamental flexural mode frequencies of 30 MHz, 50 MHz, 70 MHz, and 90 MHz were fabricated and tested. The results are reported in the paper. This ANSYS Multiphysics evaluation focused on that geometry of the 50 MHz beam.
Analysis Capabilities:
This work demonstrates ANSYS Multiphysics’s capabilities in the following areas:
Analysis Methods:
A parametric solid model and FEA mesh were created using ANSYS Multiphysics Release 6.1. This parametric model can be used for the 30 MHz, 50 MHz, 70 MHz, and 90 MHz geometries. An FEA mesh was constructed for each geometry, and a modal analysis performed. Additionally, for the 50 MHz geometry: prestressed modal, electrostatic pull-in voltage calculation, and a prestressed harmonic evaluation was performed.
Design dimensions were used in all cases, except for the beam thickness and the initial gap. The measured beam thickness was used, and the extrapolated initial gap was used. The material density was not provided in the paper, so 2.333 e-15 kg/um3 was assumed.
For the prestressed harmonic analysis, a constant damping ratio (DMPRAT) was assumed. The assumed damping ratio is the reciprocal of twice the measured Q factor (DMPRAT = 1/(2*Q)).
The transmission gain calculated in the paper is based on the transresistance amplifier gain (Ramp) and the impedance (Zx). However, Zx is not provided. Therefore, instead of the transmission gain, the following transfer function was calculated :
Transfer Function = 20*log[0.10*SQRT((ir^2+ii^2)/V)] in dB
Where:
ir = real component of current (pA)
ii = imaginary component of current (pA)
V = amplitude of harmonically applied voltage (V)
MODAL ANALYSIS:
The full model was used for the modal analysis. The resonator beam, supporting beams, dimples, and anchors were modeled using the 8-noded SOLID45 element. The mesh contained 15,252 elements and 20,229 nodes. The bottom of the anchors were restrained in all directions (UX, UY, and UZ). The Block Lanczos solver was used to find the natural frequencies.
ELECTROSTATIC-STRUCTURAL PULL-IN:
A voltage difference applied between the resonator beam and drive electrode creates electrostatic forces in the resonator beam. A distributed array TRANS126 elements (see red arrow above) were used to model the air between the resonator beam and drive electrode.
A ¼ symmetry model was used for the electrostatic pull-in calculation. The resonator beam, supporting beams, dimples, and anchors were modeled using SOLID45 elements. Surface-to-surface contact elements were used to recognize contact between the dimple and the ground plane. The bottom of the anchors were restrained in all directions (UX, UY, and UZ).
A voltage difference was placed across the TRANS126 elements. The top nodes of the TRANS126 elements were set to zero voltage (ground), and the input voltage was applied to the bottom nodes.
TRANS126 is a fully coupled two-node, one-dimensional, electromechanical element that relates the electrostatic and structural responses. It has UX (or UY or UZ) and Volt DOFs. TRANS126 elements can be used in the “lumped” sense to represent the overall behavior of a device or in a “distributed” sense to represent an electrode.
The input for TRANS126 is a capacitance versus stroke curve. Where the stroke is the relative displacement of the ends of the transducer element. The capacitance versus stroke curve can be specified using a series of discrete data points or a polynomial expression. A simple 1/r expression was used, which is valid for this application. The EMTGEN macro was used to automatically generate the distributed array of TRANS126 elements. The TRANS126 elements share their top node with the resonator beam. The resonator beam is connected to the ground plane at the anchors. Thus, the top of the TRANS126 elements and the resonator beam are electrical ground. The bottom nodes of the TRANS126 elements represent the drive electrode. The drive electrode was not explicitly included in the model.
PRESTRESSED MODAL ANALYSIS:
A DC bias voltage between the resonator beam and the drive electrode produces electrostatic forces which create stress (prestress) in the resonator beam. The prestressed modal analysis was performed in two steps. The first step was a static analysis with the DC bias voltage applied and prestressing active (PSTRES,ON). The second step was a modal analysis with the prestressing still active. The DC bias voltage was applied using the distributed array of TRANS126 elements.
The full model was used for the prestressed modal analysis. The resonator beam, supporting beams, dimples, and anchors were modeled using SOLID45 elements. The air between the resonator beam and drive electrode was modeled using a distributed array of TRANS126 elements. The bottom of the anchors were restrained in all directions (UX, UY, and UZ) for both the static prestress analysis and the subsequent modal analysis. The bottom of the dimples were displaced to contact the ground plane for the static prestress analysis and the subsequent modal analysis. This applied displacement represents bonded contact between the dimples and the ground plane. That is, the dimples cannot separate from the ground plane.
For the static prestress analysis, a DC bias voltage of 86 Volts was applied across the TRANS126 elements.
PRESTRESSED HARMONIC ANALYSIS:
DC bias voltage between the resonator beam and the drive electrode produces electrostatic forces which creates prestress in the resonator beam. The prestressed harmonic analysis was performed in two steps. The first step was a static analysis with the DC bias voltage applied and prestressing active (PSTRES,ON). The second step was a harmonic analysis with prestressing still active.
The harmonic analysis was performed with an harmonically varying voltage from 0 to 1 Volts. The magnitude of the applied voltage is not important, because the calculated transfer function is a ratio of the output current to the input voltage. Therefore, a unit voltage is used. A constant damping ratio equal to 1/(2*Q) was assumed.
The ¼ symmetry model was used for the prestressed harmonic calculation. The resonator beam, support beam, dimple, and anchor were modeled using SOLID45 element. The air between the resonator beam and drive electrode was modeled using a distributed array of TRANS126 elements.
The bottom of the anchors were restrained in all directions (UX, UY, and UZ) for both the static prestress calculation and the harmonic analysis. The bottom of the dimples were displaced to contact the ground plane for both the static prestress calculation and the harmonic analysis. As with the prestressed modal analysis, this boundary condition prevents the dimples from lifting off the ground plane, and thus slightly ‘stiffens” the beam.
A DC bias voltage of 86 Volts was placed across the TRANS126 elements for the static prestress calculation. A harmonically varying voltage from 0 to 1 volt was applied across the TRANS126 elements for the harmonic analysis.
The lowest natural frequency dominated by flexure of the resonator beam is mode number 15. Its calculated value is 51.60 MHz. Modes 1 through 14 are dominated by deformation of the supporting beams. The following image shows the deformed shape of the flexural mode:
According to the paper, the measured flexural frequency was 50.35 MHz with the DC bias voltage applied. The calculated frequency without the DC bias applied was 51.15 MHz using the Timoshenko method . The ANSYS Multiphysics calculated frequency (51.60 MHz) without the DC bias voltage is close to the Timoshenko frequency. Modal evaluations for the 30 MHz, 70 MHz, and 90 MHz geometries were also performed. All calculated flexural frequencies were within 2% of the measured flexural frequencies
The calculated dimple down voltage was 37 Volts.
The calculated catastrophic pull-in voltage was 271 Volts.
A contour of plot of the vertical displacement under the dimple down voltage is shown above. The dimple down displacement is 0.028 um. The contour plot demonstrates that the entire resonator beam is displaced downward approximately 0.028 um. That result demonstrates that the resonator beam displaces without experiencing significant bending, which is consistent with the expected behavior of a free-free beam.
According to the paper, the dimple down voltage is 25.3 Volts and the catastrophic pull-in voltage is 521 Volts. However, the dimple down and catastrophic pull-in voltages were not measured. They were calculated using closed-form analytical methods. The accuracy of these methods is unclear. Also, the paper did not attempt to precisely calculate the dimple down or catastrophic pull-in voltages. These values were only calculated to ensure that the DC bias voltage was sufficient to seat the dimple against the ground plane, but not large enough to cause catastrophic pull-in. The applied DC bias voltage (86 Volts) is within the acceptable range using either the estimates in the paper or the values calculated by ANSYS Multiphysics.
The lowest natural frequency dominated by flexure was mode number 15. Its calculated value is 54.46 MHz. Modes 1 through 14 are dominated by deformation of the supporting beams. The DC bias voltage increases the flexural mode frequency by approximately 5.5% .
According to the paper, the measured flexural frequency was 50.35 MHz with the DC bias voltage applied. According to the estimated results in the paper, the DC bias voltage (prestress) shifts the frequency downward. That is, the fundamental flexural mode frequency decreases when the DC bias voltage is applied (51.16 MHz versus 50.83 MHz). However, according to ANSYS, the DC bias voltage shifts the frequency upwards. That is, the fundamental flexural mode frequency increases when the DC bias voltage is applied (51.60 MHz versus 54.46 MHz). For either case, the frequency shift is not significant and the ANSYS Multiphysics results are similar to the paper’s results.
ANSYS Multiphysics predicts an upward shift in the flexural frequency, because of the assumed boundary conditions. By restraining the dimples at the ground plane, ANSYS Multiphysics creates a stiffer beam, which therefore has a higher flexural frequency. If the dimples are not restrained, the electrostatic load “softens” the beam which causes a decrease in the flexural frequency.
From the measured results presented in the paper, it appears that the electrostatic load “softens” the beam slightly. It is not completely clear why the actual beam does not experience the slight “stiffening effect” predicted by ANSYS. It is possible that the dimples were not fully seated on the ground beam. To duplicate the ANSYS results, the dimples must be completely seated on the ground plane with sufficient electrostatic force to prevent lift-off. Note that slight differences between the paper’s results and ANSYS Multiphysics results are expected due to differences in the geometry (design dimensions versus actual dimensions) and/or material properties (assumed density).
The results are summarized in the next set of images. All results appear to be consistent with the expected behavior at resonance.
The displacement of the center of the resonator beam increases dramatically at the flexural frequency. The phase angle shifts from 180 degrees to zero degrees at the flexural frequency. The real and imaginary components of the output current both increase dramatically at the flexural frequency. The transfer function decreases at the flexural frequency.
The paper does not provide a plot of the displacement, phase angle, or current versus frequency. Therefore, the calculated ANSYS results cannot be compared to the measured results. The only result provided in the paper is a plot of the measured transmission gain versus frequency. A comparison of the transmission gain versus frequency plot from the paper and the calculated transfer function versus frequency plot is shown on the next slide. The shape of the plots and the absolute values of the transmission gain and transfer function are similar. To achieve a more precise correlation, more information about the transmission gain is needed. The shape and the plots and the magnitude of the transfer function are a function of the assumed damping. However, the results are not extremely sensitive to small changes in damping.
The ANSYS Multiphysics results correlate well with expected results. The modal results correlate well with the measured flexural frequency of the resonator beam. The prestressed modal results are slightly different from the measured results. The ANSYS Multiphysics results indicate that prestressing slightly increases the fundamental flexural mode frequency. This slight difference is probably caused by the assumed boundary conditions.
The ANSYS Multiphysics calculated dimple down voltage is greater than the predicted dimple down voltage and the calculated catastrophic pull-in voltage is less than predicted catastrophic voltage. However, the dimple down voltage and the catastrophic pull-in voltage were not physically measured. They were calculated using a closed-form analytical methods. The accuracy of these methods is unclear. Also, the paper did not attempt to precisely calculate the dimple down or catastrophic pull-in voltages. These values were only calculated to ensure that the DC bias voltage was sufficient to seat the dimple against the ground plane, but not large enough to cause catastrophic pull-in. The applied DC bias voltage (86 Volts) is within the acceptable range using either the estimates in the paper or the values calculated here.
The shape of the measured transmission gain versus frequency plot is similar to the shape of the ANSYS Multiphysics calculated transfer function versus frequency plot. Also, the absolute values of the measured transmission gain and the absolute value of the calculated transfer function are similar.
This detailed analysis work was performed by Daniel Shaw of the ANSYS, Inc. technical support group. Please contact the ANSYS, Inc. Technical Support Group for more details.
Clark T. C. Nguyen's website has other related publications in PDF format.