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Capacitive Micromachined Ultrasonic Transducer with Lumped Model
Introduction
A capacitive micromachined ultrasonic transducer (CMUT) is a microscale transmitter-receiver that converts an electrical signal to ultrasound or vice versa for high-resolution imaging applications. As a mechanical system, a CMUT can be modeled as a spring-mass-damper system with one degree of freedom (DOF). Such a model can be useful when analyzing an array of CMUTs where FEM modeling would be impractical. This tutorial demonstrates how a lumped model of a MEMS transducer can be derived from its FEM model using the Lumped Mechanical System interface and the Parameter Estimation study. The Lumped Mechanical System interface is available in the Multibody Dynamics Module. The Parameter Estimation study is available in the Optimization Module.
Model Definition
This tutorial makes use of an existing FEM model and adds a lumped model defined by the Lumped Mechanical System (LMS) interface.
FEM Model of the CMUT
The tutorial begins with opening the Capacitive Micromachined Ultrasonic Transducer model from the Application Library. You can refer to the accompanying documentation for discussions on the device geometry and operation. The CMUT is a single-DOF spring-mass-damper system similar to the one in Figure 1.
Figure 1: Schematic of a spring-mass-damper system with single-DOF defined by the mass m, the spring constant k, and the damping coefficient c.
As described in Ref. 2, the solution to the equation of motion of the spring-mass-damper system is an oscillatory motion with the characteristic natural frequency ωn in radian per second given by Equation 1 or in cycle per second give by Equation 2:
(1)
(2)
Because the CMUT has distributed mass and stiffness, the constants k and m of the FEM model are replaced by the effective spring constant keff and the effective mass meff, respectively.
To estimate the value of keff, a Stationary study is added to the model. In this study, uniform pressure is applied to the top surface of the CMUT and membrane displacement is measured. The total force is p_max*l^2, with p_max the applied pressure and l^2 the area of the CMUT. Varying p_max and then plotting the total force against displacement gives keff = 63,000 N/m.
Next, a Frequency Domain, Prestressed analysis is done to compute the frequency response between 7.2 and 7.8 MHz which serves as reference data in the subsequent Parameter Estimation study.
From Capacitive Micromachined Ultrasonic Transducer, f0 = 7.50 MHz. Recalling that ωn = 2πfn and using the obtained values of keff and f0, solving Equation 3 below (Ref. 2) gives meff = 2.837·10−11 kg. These parameters are used to define the LMS model.
(3)
LMS Model of the CMUT
The LMS interface is used to define the lumped model using the previously obtained values for keff and meff, and a best-guess value for c. Next, the Frequency Domain study is done to obtain the frequency response of the LMS model from 7.2 to 7.8 MHz. At this stage, the frequency response of the LMS model and may not match the reference data very accurately.
Parameter Estimation Study
Next, the Parameter Estimation (PE) study runs a Frequency Domain analysis on the LMS model while simultaneously varying the parameters in order to minimize the error between the responses of the LMS and the reference data. This results in a final set of LMS model parameters giving the best match to the reference data.
Results and Discussion
Figure 1 shows the total force (integrated over the surface of CMUT) versus maximum displacement (measured at the center). The slope of this plot gives keff = 63,000 N/m.
Figure 1: The plot total force versus z-displacement from the FEM simulation. The slope of the plot is the effective spring constant keff of the lumped model.
Figure 2 shows the z-displacement versus frequency from the Frequency Domain, Prestressed study for the range of 7.2 to 7.8 MHz. The maximum z-displacement at the center of the device and is 30 nm at 7.5 MHz. This frequency response is used as reference data in subsequent PE study.
Figure 2: Plot of z-displacement versus frequency from FEM simulation.
From the previous results, the initial values of the lumped model parameters are summarized in Table 1.
Table 1: initial values of lumped model parameters.
Parameter
Value
c
1E-5 N·s/m
keff
63E3 N/m
meff
2.837 E-11 kg
Figure 3 compares the frequency response of the FEM and the lumped models computed with initial values for c, keff, and meff. At this stage, the frequency response of the lumped model does not match the FEM model very accurately.
Figure 3: Plot of z-displacement versus frequency from FEM simulation and lumped model using the initial values in Table 1.
The plot in Figure 4 is automatically generated by the Parameter Estimation study and shows the closely matching frequency responses of the LMS model and the reference data. The corresponding set of model parameters are listed in Table 2.
Figure 4: Automatically generated plots of z-displacement (μm) versus frequency show very good match between the LMS model and reference data after. The LMS model is using the parameters obtained from the Parameter Estimation study.
Table 2: values of lumped parameters computed from parameter estimation study.
Parameter
Values
c
1.986E-5 N·s/m
keff
98.677E3 N/m
meff
4.441E-11 kg
References
1. C. Chou, P. Chen, H. Wu, T. Hsu, and M. Li, “Piston-Shaped CMOS-MEMS CMUT Front-End Featuring Force-Displacement Transduction Enhancement,” Proceedings of the 21st International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers), pp. 26–29, 2021.
2. M. I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, 2011.
Application Library path: MEMS_Module/Sensors/capacitive_micromachined_ultrasonic_transducer_lumped_model
Modeling Instructions
Start by opening the FEM model.
Application Libraries
1
From the File menu, choose Application Libraries.
2
In the Application Libraries window, select MEMS Module > Sensors > capacitive_micromachined_ultrasonic_transducer in the tree.
3
Click  Open.
Before setting up the Stationary study to measure the effective spring constant, add a second Boundary Load feature.
Component 1 (comp1)
In the Model Builder window, expand the Component 1 (comp1) node.
Solid Mechanics (solid)
Boundary Load 2
1
In the Model Builder window, expand the Component 1 (comp1) > Solid Mechanics (solid) node.
2
Right-click Solid Mechanics (solid) and choose Boundary Load.
3
In the Settings window for Boundary Load, locate the Force section.
4
Specify the fA vector as
-p_max
z
5
Select Boundary 22 only.
Set up a Stationary study to measure the effective spring constant. Use only the second Boundary Load by disabling the first Boundary Load.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Stationary.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 3
Stationary - Force vs. Displacement
1
In the Settings window for Stationary, type Stationary - Force vs. Displacement in the Label text field.
2
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
3
In the tree, select Component 1 (comp1) > Solid Mechanics (solid), Controls spatial frame > Boundary Load 1.
4
Click  Disable.
5
Clear the Modify model configuration for study step checkbox.
6
In the Solve for column of the table, under Component 1 (comp1), clear the checkboxes for Electrostatics (es), Electrical Circuit (cir), and Moving Mesh.
7
In the Solve for column of the table, under Component 1 (comp1) > Multiphysics, clear the checkbox for Electromechanics 1 (eme1).
8
Click to expand the Study Extensions section. Select the Auxiliary sweep checkbox.
9
Click  Add.
10
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
p_max (Maximum pressure)
range(0,2e5,1e6)
Pa
11
In the Model Builder window, click Study 3.
12
In the Settings window for Study, locate the Study Settings section.
13
Clear the Generate default plots checkbox.
14
In the Study toolbar, click  Compute.
From the results of Study 3, plot the force versus displacement.
Results
1D Plot Group 5
In the Results toolbar, click  1D Plot Group.
Global 1
Right-click 1D Plot Group 5 and choose Global.
Effective Spring Constant
1
In the Settings window for 1D Plot Group, type Effective Spring Constant in the Label text field.
2
Locate the Data section. From the Dataset list, choose Study 3/Solution 4 (sol4).
3
Locate the Plot Settings section. Select the Flip the x- and y-axes checkbox.
4
Locate the Legend section. From the Position list, choose Lower right.
Global 1
1
In the Model Builder window, click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
abs(minop1(w))
µm
 
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type p_max*l^2.
6
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
abs(minop1(w))
µm
Maximum displacement
7
In the Effective Spring Constant toolbar, click  Plot.
From this plot, the effective spring constant is approximately 63,000 N/m. This value will be used in the lumped model.
For the next Frequency Domain, Prestressed study, add the Damping feature under Linear Elastic Material.
Solid Mechanics (solid)
Linear Elastic Material 1
In the Model Builder window, under Component 1 (comp1) > Solid Mechanics (solid) click Linear Elastic Material 1.
Damping 1
1
In the Physics toolbar, click  Attributes and choose Damping.
2
In the Settings window for Damping, locate the Damping Settings section.
3
From the Damping type list, choose Isotropic loss factor.
4
From the ηs list, choose User defined. In the associated text field, type 1e-2.
Set up a Frequency Domain, Prestressed study. Use only the first Boundary Load feature by disabling the second Boundary Load.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Solid Mechanics > Frequency Domain, Prestressed.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 4
Step 1: Stationary
1
In the Settings window for Stationary, locate the Physics and Variables Selection section.
2
Select the Modify model configuration for study step checkbox.
3
In the tree, select Component 1 (comp1) > Solid Mechanics (solid), Controls spatial frame > Boundary Load 2.
4
Right-click and choose Disable.
Frequency Domain Perturbation, FEM
1
In the Model Builder window, under Study 4 click Step 2: Frequency-Domain Perturbation.
2
In the Settings window for Frequency-Domain Perturbation, type Frequency Domain Perturbation, FEM in the Label text field.
3
Locate the Study Settings section. From the Frequency unit list, choose MHz.
4
In the Frequencies text field, type range(7.2,0.01,7.8).
5
Click to expand the Study Extensions section. Select the Auxiliary sweep checkbox.
6
Click  Add.
7
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
p_max (Maximum pressure)
0.01[MPa]
Pa
8
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
9
In the tree, select Component 1 (comp1) > Solid Mechanics (solid), Controls spatial frame > Boundary Load 2.
10
Right-click and choose Disable.
11
In the Model Builder window, click Study 4.
12
In the Settings window for Study, locate the Study Settings section.
13
Clear the Generate default plots checkbox.
14
In the Study toolbar, click  Compute.
From the results of Study 4, plot the frequency response of the FEM model.
Results
Frequency Response
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Frequency Response in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 4/Solution 5 (sol5).
Global 1
1
Right-click Frequency Response and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
minop1(w)
µm
FEM
4
From the Expression evaluated for list, choose RMS for total solution.
5
Locate the x-Axis Data section. From the Axis source data list, choose freq.
6
Click to expand the Legends section. Find the Include subsection. Clear the Solution checkbox.
7
In the Frequency Response toolbar, click  Plot.
Define and enter the values for the following parameters. These are the initial values for the lumped model.
Global Definitions
Parameters 2
1
In the Home toolbar, click  Parameters and choose Add > Parameters.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
keff
63000[N/m]
63000 N/m
Effective spring constant
meff
keff/(2*pi*7.5[MHz])^2
2.837E-11 kg
Effective mass
c
1e-5[N*s/m]
1E-5 N·s/m
Damping coefficient
fapp
p_max*l^2
0.0040322 N
Applied force
Add the Lumped Mechanical System interface and set up the lumped model.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Structural Mechanics > Lumped Mechanical System (lms).
4
Click the Add to Component 1 button in the window toolbar.
5
In the Home toolbar, click  Add Physics to close the Add Physics window.
Lumped Mechanical System (lms)
Spring 1 (K1)
1
In the Physics toolbar, click  Global and choose Spring.
2
In the Settings window for Spring, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p1
1
p2
0
4
Locate the Component Parameters section. In the k text field, type keff.
Damper 1 (C1)
1
In the Physics toolbar, click  Global and choose Damper.
2
In the Settings window for Damper, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p1
1
p2
0
4
Locate the Component Parameters section. In the c text field, type c.
Mass 1 (M1)
1
In the Physics toolbar, click  Global and choose Mass.
2
In the Settings window for Mass, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p1
1
p2
2
4
Locate the Component Parameters section. In the m text field, type meff.
Force Node 1 (frc1)
1
In the Physics toolbar, click  Global and choose Force Node.
2
In the Settings window for Force Node, locate the Terminal Parameters section.
3
In the fp10 text field, type fapp.
4
Locate the Node Connections section. In the table, enter the following settings:
Label
Node name
p1
2
For the next Frequency Domain, change the values of some LMS parameter.
Global Definitions
Parameters 2
1
In the Model Builder window, under Global Definitions click Parameters 2.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
fapp
0.01[MPa]*l^2
4.0323E-5 N
Applied force
Set up a Frequency Domain study for the LMS model.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Frequency Domain.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 5
Frequency Domain, LMS
1
In the Settings window for Frequency Domain, type Frequency Domain, LMS in the Label text field.
2
Locate the Study Settings section. From the Frequency unit list, choose MHz.
3
Locate the Physics and Variables Selection section. In the Solve for column of the table, under Component 1 (comp1), clear the checkboxes for Electrostatics (es), Solid Mechanics (solid), Electrical Circuit (cir), and Moving Mesh.
4
In the Solve for column of the table, under Component 1 (comp1) > Multiphysics, clear the checkbox for Electromechanics 1 (eme1).
5
Locate the Study Settings section. In the Frequencies text field, type range(7.2,0.01,7.8).
6
In the Model Builder window, click Study 5.
7
In the Settings window for Study, locate the Study Settings section.
8
Clear the Generate default plots checkbox.
9
In the Study toolbar, click  Compute.
From the results of Study 5, plot the frequency response of the LMS model.
Results
Frequency Response
1
In the Model Builder window, under Results click Frequency Response.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the x-axis label checkbox.
4
Select the y-axis label checkbox. In the associated text field, type Displacement (um).
Global 2 - LMS
1
Right-click Frequency Response and choose Global.
2
In the Settings window for Global, type Global 2 - LMS in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 5/Solution 7 (sol7).
4
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Lumped Mechanical System > Two port components > M1 > lms.M1.uRMS - Displacement, RMS (M1) - m.
5
In the Frequency Response toolbar, click  Plot.
6
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
lms.M1.uRMS
µm
LMS
7
Locate the Legends section. Find the Include subsection. Clear the Solution checkbox.
8
In the Frequency Response toolbar, click  Plot.
Copy the result of Study 4 to a table for use as reference data in the Parameter Estimation study.
Global 1
In the Model Builder window, right-click Global 1 and choose Copy Plot Data to Table.
FEM Reference Data
1
In the Model Builder window, under Results > Tables click Table 1.
2
In the Settings window for Table, type FEM Reference Data in the Label text field.
Set up a Parameter Estimation study based on the previous Frequency Domain study for the LMS model.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Empty Study.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 5
Step 1: Frequency Domain, LMS
In the Model Builder window, under Study 5 right-click Step 1: Frequency Domain, LMS and choose Copy.
Study 6
In the Model Builder window, right-click Study 6 and choose Paste Frequency Domain.
Parameter Estimation
1
In the Study toolbar, click  Optimization and choose Parameter Estimation.
2
In the Settings window for Parameter Estimation, locate the Experimental Data section.
3
From the Data source list, choose Result table.
4
Locate the Data Column Settings section. In the table, enter the following settings:
Columns
Type
Settings
freq (MHz)
Frequency
Frequency unit=MHz
5
From the Frequency unit list, choose MHz.
6
In the table, click to select the cell at row number 2 and column number 2.
7
In the Model expression text field, type comp1.lms.M1.uRMS.
8
In the Unit text field, type um.
9
From the Scale list, choose Manual.
10
In the Scale value text field, type 1e-9.
Select the model parameters to be included in the study. Specify their initial values, scaling, and the lower and upper bounds. For this study, a default plot will be generated automatically comparing the FEM reference data and the lumped model using the final values of the lumped parameters.
11
Locate the Estimated Parameters section. Click  Add three times.
12
In the table, enter the following settings:
Parameter
Initial value
Scale
Lower bound
Upper bound
Unit
c (Damping coefficient)
1e-5[N*s/m]
3e-5[N*s/m]
 
 
N*s/m
meff (Effective mass)
2.882e-11[kg]
5e-11[kg]
 
 
kg
keff (Effective spring constant)
63000[N/m]
1e5[N/m]
 
 
N/m
13
Locate the Parameter Estimation Method section. From the Method list, choose Levenberg–Marquardt.
14
From the Least-squares time/parameter list method list, choose Use only least-squares data points.
Because in the Frequency Domain study the variables are complex, the option for Split complex variables in real and imaginary parts must be enabled.
Solution 8 (sol8)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 8 (sol8) node, then click Compile Equations: Frequency Domain, LMS.
3
In the Settings window for Compile Equations, locate the Study and Step section.
4
Select the Split complex variables in real and imaginary parts checkbox.
5
Click  Run.
Results
Parameter estimation