Normal Modes of a Biased Resonator

Silicon micromechanical resonators have long been used for designing sensors and are now becoming increasingly important as oscillators in the consumer electronics market. In this sequence of models, a surface micromachined MEMS resonator, designed as part of a micromechanical filter, is analyzed in detail. The resonator is based on that developed in Ref. 1.

This model performs a modal analysis on the resonator, with and without an applied DC bias. The analysis begins from the stationary analysis performed in the accompanying model Stationary Analysis of a Biased Resonator — 3D; please review this model first.

Model Definition

The geometry, fabrication, and operation of the device are discussed for the Stationary Analysis of a Biased Resonator — 3D model. In this example it is no longer possible to model half of the geometry using symmetry boundary conditions, because doing so excludes all the antisymmetric vibrational modes. The geometry is therefore mirrored prior to performing the analyses, as shown in Figure 1. Note that the model could still be solved with the original geometry and symmetry boundary conditions, however the antisymmetric modes would be excluded from the solutions.

This model performs a modal analysis on the structure, with and without applied DC voltage biases of different magnitudes. The bias already exists as a parameter in the model so the prestressed eigenfrequency solver needs no adjustment to the physics settings. To compute the unbiased eigenfrequency, the solver settings are adjusted to solve only the structural mechanics problem.

Figure 1: Model geometry. In order to capture the anti-symmetric vibrational modes, it is necessary to mirror the symmetric geometry prior to solving the model. The original symmetry plane is in the center of the geoemtry.The resonator itself is shown highlighted.

Results and Discussion

Figure 2 shows the normal modes of the device, together with the eigenfrequency, in the unbiased state. The lowest three normal modes are symmetric and anti-symmetric bending modes and a torsional mode.

The symmetric bending mode is employed during the operation of the device, and its shape does not change significantly with applied bias. However, the frequency of the mode is reduced significantly by the applied bias, an effect known as spring softening. The spring softening effect can be seen in detail in Figure 2. A clear decrease in the resonant frequency is evident with increasing bias voltage. This figure should be compared with Figure 16 of Ref. 1 which shows measured experimental data for the same device. Data extracted from Ref. 1 is shown in Figure 3 along with the simulation results. The agreement between the model and the data is excellent.

Figure 2: Normal modes of the unbiased device, together with the frequency of the mode.

Figure 3: Resonant frequency of the first normal mode (a symmetric bending mode) as a function of applied DC bias. Both the COMSOL simulation data and the experimental data from Ref. 1 are shown in the plot.

Reference

1. F.D. Bannon III, J.R. Clark and C.T.-C. Nguyen, “High-Q HF Microelectromechanical Filters,” IEEE Journal of Solid State Circuits, vol. 35, no. 4, pp. 512–526, 2000.

MEMS_Module/Actuators/biased_resonator_3d_modes

Modeling Instructions

Open the existing stationary study (filename: biased_resonator_3d_basic.mph).

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file biased_resonator_3d_basic.mph.

Mirror the geometry so that asymmetric eigenmodes can be modeled.

Geometry 1

Mirror 1 (mir1)

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.

In the text field, type 1.

Click in the window and then press Ctrl+A to select both objects.

Locate the section. Select the check box.

Click

Click the

button in the toolbar.

Import experimental data into the model for comparison with the simulation.

Definitions

Interpolation 1 (int1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click .

Browse to the model’s Application Libraries folder and double-click the file biased_resonator_3d_modes_experiment.txt.

Click .

Locate the section. In the text field, type Hz.

In the text field, type V.

Locate the section. From the list, choose .

In the text field, type NaN.

Disable the symmetry node to allow anti-symmetric nodes.

Solid Mechanics (solid)

Symmetry 1

In the window, expand the node.

Right-click and choose .

Mesh 1

Size

In the window, expand the node, then click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type 1.

Click

Root

Add a study to compute the unbiased vibrational modes.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study 2

Solve for the first three modes.

Step 1: Eigenfrequency

In the window, under click .

In the window for , locate the section.

Select the check box.

In the associated text field, type 3.

Disable the electric potential and mesh displacement degrees of freedom to solve only the structural problem. This will give the vibrational modes in the absence of an electric field.

Solution 2 (sol2)

In the toolbar, click

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

In the window, expand the node, then click .

In the window for , locate the section.

Clear the check box.

In the window, click .

In the window for , locate the section.

Clear the check box.

In the window, right-click and choose .

In the dialog box, type Unbiased Eigenfrequency in the text field.

Click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Change the dataset frame to show results in the material frame. This allows the use of the deformation plot attribute.

Results

Unbiased Eigenfrequency/Solution 2 (sol2)

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Create a plot that shows the unbiased modes.

3D Plot Group 4

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Volume 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Clear the check box.

Deformation 1

Right-click and choose .

Unbiased Modes

In the window, right-click and choose .

In the dialog box, type Unbiased Modes in the text field.

Click .

Compare the mode shapes with those shown in Figure 2 for all the modes computed. To switch between the modes click and choose a different value from the list.

Add a study.

Add Study

In the toolbar, click