Modeling a Tunable Cavity Filter with a Piezoelectric Actuator

Tunable devices can be realized using varactors, phase shifters, or switches with which the reactance, phase, or path of the signal can be tweaked and the frequency response of the device is changeable. Here, a tunable bandpass filter model is designed with a piezoelectric actuator that controls the reactance of the device and results in a variable resonance frequency of the filter.

The filter design is based on a rectangular cavity filter whose resonance frequencies are given by:

where a and b are the waveguide aperture dimensions and d is the length of the waveguide cavity.

The cavity width, height, and length are a = 100 mm, b = 50 mm, and d = 50 mm, respectively, to generate 3.354 GHz of the resonance frequency for the TE101 dominant mode.

Inside the cavity, a metallic post is added and configured to create a gap between the top surface of the post and ceiling of the cavity, so the height of the post is slightly smaller than b. When the cavity is resonant at the dominant mode, the energy is confined in the center of the cavity and the response of the gap that is located in the middle becomes capacitive. The extra capacitance lowers the resonance frequency by keeping the same structure size, so the device size is effectively reduced as well.

Two shorted 50-Ω microstrip lines, which are terminated by lumped ports, are coupled into the cavity through slots on the top of the cavity. Input matching (S₁₁) and insertion loss (S₂₁) can be improved by adjusting the dimensions and locations of the slots. A circular aperture at the top of the cavity is closed with a piezoelectric actuator, and the bottom surface of the disk is finished with a layer of a material where the conductivity is high enough to have a very small skin depth.

Left: The cavity filter with a piezoelectric actuator shaped like a circular disk. The feed scheme is built with slot-coupled microstrip lines. Right: The gap size between the piezoelectric actuator and metallic post controls the resonance frequency.

All metal parts — e.g., the cavity walls, post, substrate ground planes, microstrip lines, and bottom surface of the piezoelectric device — are set as perfect electric conductors (PECs). Lead zirconate titanate (PZT-5H) is used for the piezoelectric actuator. The actuator is z-polarized, resulting in mainly z-directional deflection of the device.

When a positive DC bias is applied across the piezoelectric actuator, it will deflect toward the bottom of the cavity. This deflection makes the capacitance stronger and shifts the resonance frequency lower than the case without any deformation. The animation below plots the electric field norm at the resonance frequency. At the center of the cavity, as well as in the gap between the top of the post and the bottom of the piezo device, strong electric fields are observed.

Accurately Analyzing an RF Filter with Multiphysics Simulation

The conventional analysis method for this type of device is using a parametric sweep of the height of the metallic post geometry (instead of warping the piezoelectric actuator parabolically) to see the change of the capacitance of the filter. However, in reality, the metallic post is fixed and the actual deformation of the piezoelectric actuator is not geometrically uniform. For this reason, the parametric sweep does not address the capacitance change precisely; thus, the evaluated resonance frequency is not accurate.

To describe the real-world phenomena, the elastic deformation of the actuator and the resulting capacitance change have to be modeled with a multiphysics approach that combines a high-frequency electromagnetic and piezostructural analysis. Using this approach is seamless and intuitive in the COMSOL Multiphysics® software, as it offers you a single simulation platform.

The multiphysics and moving mesh settings in a single simulation platform to model the deformation of the piezoelectric actuator.

The deformation of the piezoelectric actuator is addressed via the combination of a few physics interfaces: the Solid Mechanics (solid), Electrostatics (es), and Moving Mesh (ale) interfaces. When the piezoelectric device deforms due to a positive and negative DC bias, the Moving Mesh interface is used to reconfigure the mesh for the Electromagnetic Waves, Frequency Domain interface, which calculates the wave propagation and resonance behavior in the microstrip lines and cavity.

By changing the resonance frequency, the electric field around 3 GHz is evanescent. The deformation of the piezoelectric actuator is exaggerated for visualization purposes. A surface plot of the electric field norm and an arrow plot of the electric field are also shown in the animation.

With an electric potential of +300 V across the piezoelectric actuator, a deflection of ~90 μm is observed, making the gap smaller and the capacitance in the gap stronger. Thus, the shift of the resonance frequency is lower than the shift at 0 bias as well as at negative bias.

The S-parameters of the tunable cavity filter. The mode uses a DC bias of ±300 V.

The S-parameter plot shows the effect of the piezoelectric device deflection on the filter’s resonance frequency. The tunable frequency range of this example is around 40 MHz. This range can be adjusted by different choices of the piezoelectric disk size and the input bias voltage.

Concluding Thoughts on Simulating Real-World Devices

The RF Module, an add-on product to COMSOL Multiphysics, helps you design, build, and optimize RF, microwave, millimeter-wave, and passive THz devices. You can model traditional devices and extend models to include other physics phenomena that are not easily measured in the lab, such as heat effects on material properties as well as structural deformation. All of the physics that you want to include can be efficiently simulated using the same simulation environment and workflow.

Next Steps

To try the tutorial model featured in this blog post, click the button below. Clicking this button will take you to the Application Gallery, where you can log into your COMSOL Access account and then download the MPH-file.

Further Resources

• Learn more about modeling RF filters in this blog post: Methods That Accelerate the Modeling of Bandpass-Filter Type Devices
• Check out a tutorial on modeling a tunable MEMS capacitor

Ultrasound focusing is widely used in various industrial applications, such as nondestructive testing (NDT) and medical imaging. For clinical applications, high-intensity focused ultrasound (HIFU) is a specific aspect of this technology where most of the power provided by the probe is carried to a targeted zone to coagulate biological tissues. This blog post discusses ultrasound focusing simulation.

Designing Transducers for Noninvasive Ultrasounds

Ultrasounds have a great advantage: They can reach a volume inside a piece of metal, a human organ, or biological tissue without the need to cut through the path of the transmitted signal to the target at which it is directed. Unlike the scalpel of a surgeon in a medical treatment, ultrasound will not leave any scarring on the skin of a patient and can still treat the targeted zone with good accuracy, while limiting the risk of damage to the surrounding healthy tissues. Focused ultrasound is used or has the potential to be used to treat diseases like prostate and breast cancer, hypertension, and even glaucoma.

There are several ways to focus ultrasound using different transducer designs, and the COMSOL Multiphysics® software is a very good tool to simulate and optimize them. Designing a transducer that will effectively produce an ultrasound field that reaches a targeted zone can be a difficult task. It depends on the frequency and power of the emitted signal; the attenuation and absorption of the medium in which the ultrasound propagates; and, of course, the position and dimensions of the transducer itself.

Figure 1: Schematic of the acoustic field generated by an ultrasound transducer.

Here are a few important aspects of an ultrasound transducer used for clinical applications (see figure above):

• The near field distance N, calculated as: (1)
  • D is the transducer diameter
  • f is the frequency
  • c is the speed of sound in the medium
• The focal distance F, which is the distance between the transducer and the focal point that is the targeted zone
• The field depth or focal zone, which defines the -6-dB signal amplitude drop from the maximum amplitude, calculated as: (2)

Two alternatives can be used to focus the signal from the transducer:

1. Modify the radius of curvature of the transducer element to a value corresponding to the focal distance (see the above schematic)
2. Introduce a phase delay when applying voltage in an array of several flat-faced transducers (see schematic below)

Figure 2: Schematic of an ultrasonic probe with an array of piezo transducers (phased array) used to focus the acoustic signal. The transducer consists of a backing material, the piezo elements, and a matching layer to the tested sample (here tissue).

COMSOL Multiphysics has been used to look at these two alternatives. Besides the ability to model ultrasound propagation, it is also very interesting to couple this simulation with a heat transfer simulation and even a damage law of biological tissues. In this way, we can quickly visualize whether the focusing effect can treat the right amount of tissues and check the location and volume of coagulation, all within the same modeling interface.

Simulating a Geometrically Focused Probe

Ultrasound can be focused directly by the way the emitting transducer is shaped. A tutorial available with the Acoustics Module provides a very good example of this phenomenon coupled with heat transfer. A few assumptions are made to the acoustics simulation, such as neglecting the nonlinear effects and shear waves, but it still provides very valuable information about the sensitivity of the focal zone to the probe parameters.

This tutorial can be adapted to most device configurations and used as a starting point for simulations. For instance, before running the heat transfer part of the simulation, we can check how the frequency modifies the size of the focal zone, hence the energy that is delivered to this zone. In the example below, three frequencies are computed at 0.5 MHz, 0.7 MHz, and 1 MHz. Figures 3–5 show the shape of the ultrasound pressure wave, the size of the focal zone with the criteria max(SPL) – 6dB, and the corresponding energy that is used to heat and coagulate tissues, respectively.

Figure 3: Simulated ultrasounds (represented in blue and red wave-type signals) are emitted and focused by a curved transducer (the surface with orange arrows at the bottom). They travel in tissues and provide a peak of intensity in the focal zone. This results in an elevation of temperature due to the absorbed energy.

When the transducer diameter and curvature are kept constant, increasing the frequency will reduce the size of the focal zone. We clearly see the smaller wavelength at a higher frequency, as well as its effect on focusing.

Figure 4: The size of the focal zone is visualized with the max(SPL) – 6dB criteria. It confirms what can be seen from the pressure plot above. The dB scale is not the same for the three visualized frequencies.

Figure 5: The acoustics intensity is plotted for all three frequencies on the same color scale in W/cm². The maximum intensity deposited is more than 10 times higher at 1 MHz than at 0.5 MHz, with all other parameters kept equal. Although the focal zone is decreased when increasing the frequency, it also means that more energy is transmitted to this zone, hence allowing for higher temperatures in the tissues.

Phase Delay Focused Probes

Another way to focus ultrasounds is to use several transducers in an array of piezoelectric elements and then control the voltage input with a phase delay for each element. This phase delay must be calculated for each array configuration, since it depends on frequency, piezoelectric elements, size, position, and, of course, on the focal distance.

For a linear array of elements, a quick method is to consider the distance d_i between the center of each element i and the focal point and to apply the phase as:

(3)

To illustrate this, let us make a geometry of a 16-element array probe and use the Acoustics Module and Heat Transfer Module with COMSOL Multiphysics to couple several interfaces:

• Pressure Acoustics, Frequency Domain
• Solid Mechanics
• Electrostatics
• Bioheat Transfer

The geometry is shown in a 2D cross section in Figure 6 below, with matching and backing layers in front of and behind the piezoelectric elements, respectively. The backing layer is used to prevent excessive vibrations. The matching layer is an intermediate material between the piezoelectric material and the biological tissue that is necessary for ultrasonic waves to efficiently enter the tissue. It has the same function as the gel used by a doctor between the probe and the skin for an echography.

Figure 6 also shows the phase delay that has been calculated based on (3) as a color and deformation plot on each element, going from 0 on the side elements to 434° in the centered elements.

When the voltage is applied on these elements, the piezoelectric material vibrates and creates an ultrasound wave that will focus at the desired focal distance due to the phase delay.

As for the geometrically focused probe, this simulation can then be coupled to the heat transfer and damage law simulation to assess the temperature elevation and the coagulated volumes in the biological tissues. The heat source from the acoustics signal, given in the plane-wave limit, is calculated as:

(4)

where α_abs is the acoustic absorption coefficient of the tissue and I_ac is the acoustic intensity magnitude.

The absorption of energy, represented by α_abs, varies significantly with the different tissues. As a result, it is also important to check if the calculated focused signal damages other tissues between the array probe and the focal zone. If these tissues are not supposed to be damaged, then the focus should be modified. In this case, the simulation allows us to quickly modify the design and operation parameters of the array probe and to validate or discard an array configuration.

Figures 7 and 8 show the shape of the ultrasound pressure wave and the corresponding energy that is focused, respectively.

Figure 6: The delay, which is calculated as a function of the frequency, the focal distance, and the size and position of the transducer elements.

Figure 7: The wave pattern is seen at a frequency of 1.5 MHz. One can decide to modify the geometric design, phase delay, and even the device frequency if the ultrasounds are not focused enough.

Figure 8: The acoustics intensity is plotted in W/cm². Here, the 16 piezoelectric transducer elements provide a low intensity that is spread out over a few millimeters on the focal zone. At this stage, the heat transfer and damage simulation could be run to decide if the temperature elevation due to nonnegligible intensity between the focal zone and the transducers (a few W/cm²) is too high or if it could be handled during the medical treatment.

Related Resource

• Read a related paper from the COMSOL Conference 2013 Rotterdam:
  • FEM Simulation for ‘Pulse-Echo’ Performances of an Ultrasound Imaging Linear Probe

References

1. Zhenya Yang, Jean-Louis Dillenseger. Phase estimation for a phased array therapeutic interstitial ultrasound probe. Conference proceedings: Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference, Institute of Electrical and Electronics Engineers (IEEE), 2012, 2012, pp. 472-5.
2. “Phased Array Tutorial”, Olympus, http://www.olympus-ims.com/en/ndt-tutorials/phased-array/.
3. “Basic principle of medical ultrasonic probes (transducer)”, Nihon Dempa Kogyo Co., Ltd., http://www.ndk.com/en/sensor/ultrasonic/basic02.html.

About the Guest Author

Thomas Clavet is a mechanical engineer from Arts et Métiers Paris Tech and KTH University in Stockholm. He has previously worked as a stress engineer in the nuclear industry and as an application engineer for COMSOL Ltd. in the U.K. and Ireland, where he met and trained several COMSOL Multiphysics users in the fields of fluid flow, heat transfer, acoustics, and structural mechanics simulations.

Thomas founded EMC3 Consulting in 2014, in the south of France, to provide his expertise in the use of COMSOL Multiphysics as a COMSOL Certified Consultant and in the fields of CFD, heat transfer, acoustics, and structural mechanics simulations.

Learn more about how EMC3 Consulting is helping companies to design better products with COMSOL Multiphysics by visiting www.emc3-consulting.com.

Interfaces Based on the Boundary Element Method in COMSOL Multiphysics®

The boundary element method (BEM) is complementary to the finite element method (FEM) and is generally available in the COMSOL Multiphysics® software as of version 5.3. There are three different types of interfaces that are based on BEM, summarized in the table below:

These interfaces are quite similar. Although this blog post focuses on the interface for electrostatics, some of the techniques shown here are applicable if you are interested in the other two interfaces.

What Is the Boundary Element Method?

In contrast to FEM, BEM doesn’t require the generation of a robust volumetric mesh throughout your computational domain, which can be difficult and resource-intensive to achieve. BEM eliminates the problem by only requiring a surface mesh, which is significantly easier to generate. However, this advantage comes at a price. The COMSOL Multiphysics implementation of BEM cannot be used to model, for example, nonlinear or general inhomogeneous materials. The table below summarizes the pros and cons of BEM and FEM in the COMSOL Multiphysics implementation.

By combining domains modeled with FEM and regions modeled with BEM, you can get the best of both worlds. For example, you can have one domain with an anisotropic material modeled with the traditional Electrostatics interface in the AC/DC Module and a surrounding isotropic domain modeled with the new Electrostatics, Boundary Elements interface.

Example: Electrostatic Precipitation Filter

To illustrate using the Electrostatics, Boundary Elements interface, let’s create a simplified model of an electrostatic precipitation filter. This type of filter is used in various industrial settings to filter particles from, for example, exhaust gases from coal power plants. An array of high-voltage wires creates a corona discharge region surrounding them, which in turn charges the unwanted particles. The charged particles then migrate in the electric field toward grounded metal plates (the collecting electrodes) and are periodically scraped off when the layer of particles becomes so thick that it deteriorates the performance of the filter.

Simulating the entire physical process of corona discharge, ionization, and charged particle migration is complicated and beyond the scope of this blog post. Instead, let’s look at the filter from a purely electrostatics perspective. This keeps the model simple, yet quite general, and illustrates a modeling approach that is applicable to a wide range of other electrical devices. If you want to know more about the details of modeling an electrostatic precipitation filter, see page 21 of COMSOL News 2012.

The filter in this example consists of 6 ground plates and 60 wires, as shown in the figure below. The wires are modeled as parametric curves and held at 50 kV.

The electrostatic precipitation filter example.

Assigning Material Properties

In the real case, this filter would be held in a frame, which we have neglected here to keep things simple. We assume that the space between and outside the plates is filled with air, which is the only material property in this model. In this example, we study this component as “hanging in midair” to get its idealized electrostatics properties. To assign air to the model, notice that there is no domain to click. Instead, you select the air region surrounding the model by selecting All voids from the selection list in the Settings window for the Air material. The only available void in this example is called the Infinite void and represents the region between the plates all the way “out to infinity”, as shown below.

For more information on the difference between solid domains and voids, see the Release Highlights page.

The settings for the Air material.

Selecting the Infinite void in this way is all that has to be done to model an infinite region when using boundary elements. Had we modeled this with FEM, then we would have needed to enclose the geometry model in a finite-sized box (or some other shape). To increase the accuracy of the computation, we would have also needed to add extra layers with Infinite Element domains surrounding the box.

Applying Boundary Conditions

The boundary conditions are set at two levels: for boundary surfaces and for edges. The figure below shows the Ground boundary condition assigned to the ground plates.

The settings for the Ground boundary condition.

There is a more interesting condition on the wires. They are assigned a Terminal edge condition with the Terminal type set to Voltage at 50 kV. In addition, the Edge radius is set to 1 mm, as shown in the figure below.

The settings for the Terminal edge condition.

Notice how the radii of the wires are entered into the model on the physics side and not on the CAD side. The CAD model of the wires consists of parameterized curves that have no radial extent but are (mathematically speaking) one-dimensional objects. This modeling approach shows a major benefit of BEM. If the model had been set up with the finite-element-based interface for electrostatics, then the wires would have to be modeled as thin spiral-shaped tubes with a finite-sized radius, thereby generating a mesh with many elements. Although possible, BEM is much more convenient.

Solvers for the Boundary Element Method

FEM produces large sparse matrices, whereas BEM generates large filled matrices. This calls for solvers specialized in manipulating such. As a matter of fact, the system matrix produced by BEM is so heavy to handle that it cannot even be formed in its entirety. Instead, only those parts of the matrix that are needed by the solver for the moment are generated. More specifically, only the matrix-vector multiplications needed for the moment are performed. The method implemented in COMSOL Multiphysics for fast matrix-vector multiplications is called the adaptive cross approximation method and is used automatically when you are using one of the BEM interfaces. If you are interested, you can read more about the related solver options for version 5.3.

On one hand, BEM requires fewer degrees of freedom in order to produce accurate results as compared with FEM. On the other hand, BEM is more computationally demanding, so in the end, the methods are comparable with regard to computational demand versus accuracy.

Postprocessing and Visualization

For finite-element-based models, the computed fields in the modeled volume are visualized using slice plots, isosurface plots, arrow plots, flux lines, etc., by means of the volumetric finite element mesh. When using BEM, there is no volumetric mesh available, so in order to visualize spatially varying fields, a regular grid is used as a substitute. The regular grid is defined as a Grid 3D data set and lets you define a rectangular box with maximum and minimum values of its extents in the x, y, and z directions. In addition, the x-, y-, and z-resolution settings correspond to the element size and determine the granularity of the visualization. In the figure below, the resolution is set to 100 by 200 by 200, which corresponds to 4 million hexahedral grid elements.

The settings for the Grid 3D data set.

Boundary element fields can be quite heavy to postprocess and visualize and it can be a good idea to turn off Automatic update of plots. The corresponding check box is available in the Settings window for the Results node, as shown below.

The check box for Automatic update of plots in the Result node settings.

The visualization below shows the electric potential field around the wires, between the plates, and surrounding the plates. By increasing the size of the Grid 3D box, you can extend the visualization to a larger volume without having to recompute the solution. This is another benefit of BEM, since with FEM, you would need to enlarge the truncation domain and recompute.

The electric potential field for the electrostatic precipitation filter example.

Models with Multiple Dielectric Materials

A limitation with BEM is that each modeling domain is required to have a constant and isotropic material property. In the case of electrostatics, each domain must have a constant permittivity. You can create models with several domains of different permittivity values. The figure below shows a MEMS capacitor model with two permittivity values:

1. Air in the exterior infinite void
2. Dielectric material between the two electrode plates

To make this type of model possible, each distinct dielectric domain needs to have its own Charge Conservation node added under the Electrostatics, Boundary Elements interface. Within each Charge Conservation domain, or group of domains, the permittivity is a constant. The capacitance of this type of device is computed using the predefined variable for capacitance under Derived Values, just like the corresponding finite-element-based model would.

A MEMS capacitor model example with multiple permittivity values.

Hybrid Finite Element and Boundary Element Modeling

COMSOL Multiphysics version 5.3 comes with a predefined multiphysics coupling that combines finite-element-based and boundary-element-based electrostatics. The figures below show another version of the MEMS capacitor model, where the dielectric material is replaced by an anisotropic piezoelectric material (PZT-5H). Since the Electrostatics, Boundary Element interface doesn’t allow for simulating anisotropic materials, the traditional finite-element-based Electrostatics interface is used in that region.

In addition, a small box surrounding the capacitor is also modeled using the finite-element-based interface. In this example, the Electrostatics, Boundary Elements interface is only active in the exterior infinite void. The coupling between the finite element region and boundary element region is defined under the Multiphysics node in the settings for Boundary Electric Potential Coupling.

The settings to combine finite-element-based and boundary-element-based electrostatics.

The figure below shows the electric potential visualized in both the finite element and boundary element regions. Some numerical artifacts due to interpolation can be seen in the junction between the finite element and boundary element domains. Such artifacts vanish when computing using a finer mesh and visualizing using a higher resolution for the Grid 3D data set.

The electric potential field for the MEMS capacitor model example.

Try It Yourself

You can download the example models highlighted in this blog post by clicking the button below.

Two other boundary-element-based electrostatics tutorials are available in the Application Gallery:

1. MEMS capacitor with a single dielectric domain
2. Capacitive position sensor

The capacitive position sensor demonstrates the use of the Electrostatics, Boundary Elements interface in combination with a Deformed Geometry interface for modeling large geometrical displacements. In addition, the same model demonstrates using the accelerated capacitance-matrix-calculation option Stationary Source Sweep, which is new in version 5.3 of COMSOL Multiphysics. The Stationary Source Sweep study type can also be used with the finite-element-based Electrostatics interface.

Using Plane Strain, Plane Stress, and Generalized Plane Strain Conditions

Under plane strain conditions, no expansion in the out-of-plane direction is allowed. There are usually stresses in that direction caused by the coupling to the in-plane strain through a nonzero Poisson’s ratio. On the other hand, when a thin sheet is studied, the plane stress assumption is more useful. In this case, the material is free to contract or expand in the out-of-plane direction and the transverse stress is zero.

If the structure is long in the transverse direction when compared with its in-plane size, but is still not restrained in the transverse direction, then neither of these assumptions is good. This is where a generalized plane strain condition becomes useful.

About the Generalized Plane Strain State Formulation

A possible generalization of the plane strain formulation is to assume that the strains are independent of the out-of-plane coordinate. In the COMSOL Multiphysics® software, this generalization can be implemented using a 2D geometry of the cross section and the Solid Mechanics interface, in which the plane strain formulation is a default option.

The strain tensor components are assumed to be functions of only the in-plane coordinates x and y (and possibly time):

(1)

Under the small-strain assumption, the strain tensor components are related to the displacement field as:

(2)

The above equations have the following 3D solution:

(3)

where a, b, and c are constant coefficients.

The corresponding out-of-plane strains are:

(4)

This strain state differs from the standard plane strain assumption only by the fact that the normal out-of-plane strain is nonzero and can vary linearly over the cross section. At the cross section z = 0, the deformation is in-plane and fully characterized by the in-plane displacement components u(x,y) and v(x,y).

The coefficients a, b, and c in the expression for the normal out-of-plane strain can be introduced as extra degrees of freedom (DOF) that are constant throughout the model (global variables). The extra strain contribution can be incorporated using the External Strain feature available in the Solid Mechanics interface.

A generalized strain formulation is important is when analyzing stress-optical effects, such as birefringence in waveguides composed of several layers of different materials (e.g., silicon-on-insulator waveguides). This stress-optical effects tutorial model shows such a case.

Example of Using the Generalized Plane Strain Formulation

To illustrate the efficiency of this approach, let’s consider a simple beam-like structure composed of two layers with square cross sections of 1 cm. The layers are made of materials with significantly different elastic and thermal properties: aluminum and nylon. The data is taken from the built-in Material Library in COMSOL Multiphysics. The length in the out-of-plane z direction is L = 20 cm. The structure is assumed to be manufactured at an elevated temperature. Due to the mismatch in the thermal expansion properties of the materials, a residual thermal stress builds up in the structure when it has cooled down to the operating temperature. This makes the structure bend slightly in the out-of-plane direction.

The following figure shows plots of the total displacement together with the deformation for a full 3D model and a 2D generalized plane strain condition:

The 2D solution computed for u(x,y) and v(x,y) within the cross section has been extruded in the out-of-plane z direction using the analytical solution for the corresponding 3D displacement field given above.

The 3D solution requires around 32,000 DOF, while the 2D solution only needs around 250 DOF.

The following plots show the variation of the out-of-plane strain and stress along one of the edges.

Strain (left) and stress (right) along the z-axis.

Around 80% of the true 3D structure has stress and strain fields similar to those predicted by the generalized plane strain theory. Only near the free ends, where the stress tends toward zero, does the strain field start to deviate from the linear distribution within the cross section.

Implementing the Generalized Plane Strain Condition in COMSOL Multiphysics®

One way to incorporate the changes needed for the generalized plane strain approximation is to start with a 2D component and the Solid Mechanics interface and then add the following nodes in the Model Builder tree:

The Model Builder tree, showing the nodes needed to implement a generalized plane strain condition.

In addition to the standard settings for a 2D problem, you must perform the following steps. First, in the Global Equations node, add the a, b, and c coefficients as DOF. Note that you do not set up any equations for those variables here. Thus, all input fields other than the variable names are kept at their default values.

The Global Equations node, showing the a, b, and c coefficients.

In the Variables node, define the out-of-plane normal strain component eZ in terms of a, b, and c.

The Variables node, showing the expression for the variable eZ.

Next, incorporate the extra strain component into the stress-strain relation in the External Strain node. Note that any expression you enter in this node is subtracted from the total strains before the elastic stresses are computed from strains using Hooke’s law. Usually, this node can be used to incorporate inelastic effects; for example, strains caused by various electromechanical multiphysics effects. Here, we use it simply as a mechanism to inject an extra strain component that is zero by default in the plane strain formulation.

The External Strain node, showing the extra strain component.

Lastly, in the Weak Contribution node, include the extra virtual work done by the out-of-plane stress. This sets up equations (in the weak form) to determine a, b, and c. Here, solid.d is the thickness in the z direction, as defined in the Solid Mechanics interface.

The Weak Contribution node, showing the weak expression.

Alternative Implementation

You can also skip the third and fourth steps above and insert the strain variable eZ directly into the equations of the Linear Elastic Material node. To do this, make sure Equation View is enabled.

The settings for Equation View.

This way, the new strain in the z direction is directly part of the material model and goes into the weak expression that is already generated in the Linear Elastic Material node.

Concluding Remarks

We have shown how to use the functionality available in COMSOL Multiphysics to model elongated structures that are free to extend in the out-of-plane direction. The use of the 2D generalized plane strain approximation allows us to reduce the computation effort significantly while reproducing the possible out-of-plane bending of the structure — a 3D effect that can be important in applications such as piezoelectric devices and optical waveguides. It is also possible to incorporate out-of-plane shearing of the structure, which can be important in some piezoelectric applications.

Further Resources

• Learn more about structural mechanics modeling on the COMSOL Blog:
  • Introduction to Modeling Stress Linearization in COMSOL Multiphysics®
  • Modeling Linear Elastic Materials – How Difficult Can It Be?

The Inkjet Printhead Nozzle: A Key Component in a Variety of Applications

Inkjet printers are widely used to provide high-resolution 2D printouts of digital images and text, where the printhead ejects small droplets of liquid from a nozzle onto a sheet of paper in a specific pattern. In addition to printing images onto paper, the inkjet technique is also common in 3D printing processes. The printhead moves over a certain type of powdered printing material and deposits a liquid through the nozzle onto the powder to effectively bind it into a predetermined 3D shape. (Tip: Check out the video on 3dprinting.com to see this process in action.) Inkjet printheads are also prevalent in life science applications for diagnosis, analysis, and drug discovery. The nozzles are used as part of a larger instrument to deposit microdroplets in a very precise fashion.

An inkjet nozzle deposits an ink droplet, which travels through the air before reaching its target. The model was created using the COMSOL Multiphysics® software.

No matter what device or machine relies on the inkjet printhead to deposit material, precision is crucial. Therefore, the quality of the end product hinges on the nozzle design.

Studying the Fluid Flow in an Inkjet Nozzle with Simulation

The droplet size for an inkjet nozzle is a key design parameter. In order to produce the desired size, you need to optimize the design of the nozzle and the inkjet’s operating conditions. Rather than build nozzle prototypes and test them in a lab, you can use simulation software to understand the physics of the fluid ejection and determine the optimal design. COMSOL Multiphysics® is one such software package.

When you expand COMSOL Multiphysics with either the CFD or Microfluidics add-on module, you can create models that help you understand how the ink properties and nozzle pressure profile affect the droplet velocity and volume as well as the presence of satellite droplets.

Model created using the level set method to track the interface between air and ink. The color plot around the droplet signifies the velocity magnitude in the air.

What happens inside the inkjet nozzle when the liquid is emitted? First, the nozzle fills with fluid. Next, as more fluid enters the nozzle, the existing fluid is forced out of the nozzle. Finally, the injection is halted, which ultimately causes a droplet of liquid to “snap off”. Thanks to the force transmitted to the droplet by the fluid in the nozzle, it travels through the air until it reaches its target. In terms of physics, inside the nozzle, there is a single-phase fluid flow. When the liquid moves through the air, the flow becomes a two-phase flow.

We won’t go into the details of how to build this model here, because you can download the step-by-step instructions in the Application Gallery.

As the simulation specialist in your organization, you are a member of a small and rather exclusive group of people tasked with serving a larger pool of colleagues and customers who rely on your models to make important business and design decisions. Wouldn’t it be nice if these stakeholders could take on some of the work that goes into rerunning simulations for different parameter changes?

Save Time by Building Apps Based on Your Model

The COMSOL Multiphysics software comes with the built-in Application Builder, which enables you to wrap your sophisticated models in custom user interfaces. By building your own apps, you can give your colleagues or customers access to certain aspects of your models, while hiding other aspects that may be unnecessary to change and too complicated to expose. For example, suppose that your colleagues in design or manufacturing want to test the performance of an inkjet nozzle for different geometries and liquid properties. Instead of coming back to you each time they want a minor change to the underlying model, they can input different values in simple fields and click on a button to plot new simulation results in the app you provide them. Since they can run their own analyses, your time can be spent on new projects, models, and apps.

To show you what we mean — and to inspire you to make your own apps — we have made a demo app based on our inkjet tutorial model. In this example, the app user can analyze various nozzle designs to see which version produces the ideal droplet size. Contact angle, surface tension, viscosity, and liquid density are all taken into account in the app. As you can see in the screenshot below, an app user can adapt the nozzle shape and operation by changing different input parameters.

An example of what an inkjet printhead design app might look like. In this demo app, users can modify liquid properties, the model geometry, and simulation time intervals.

When you build apps, you can empower other stakeholders to make better decisions faster without actually giving them access to your full underlying model. The model simply powers the app and you, as the app designer, decide what inputs the users can modify. Your original model file stays safely untouched in your care, but a variety of results are accessible by those who rely on them most.

Try It Yourself

Get started by downloading the .mph file and accompanying documentation for the tutorial model and demo app from the Application Gallery.

All you need to download the documentation is a COMSOL Access account. To get the .mph file, you will also need a valid COMSOL Multiphysics® software license or trial. Note that you can access these files directly within the product as well, via the Application Libraries.

Other Related Resources

• Watch a keynote video on industrial inkjet printheads from the COMSOL Conference 2014
• Get an introduction to modeling separated three-phase flow with COMSOL Multiphysics
• Learn about modeling piezoelectric actuators

Piezoresistive Pressure Sensors “Measure Up” to the Competition

One of the first types of commercialized MEMS devices was the piezoresistive pressure sensor. This device, which continues to dominate the pressure sensor market, is valuable in a range of industries and applications. Measuring blood pressure as well as gauging oil and gas levels in vehicle engines are just two examples.

Piezoresistive pressure sensors have applications in the biomedical field as well as the automotive industry. Left: A blood pressure measurement device. Image by Andrew Butko. Licensed under CC BY-SA 3.0, via Wikimedia Commons. Right: A vehicle’s oil gauge. Image by Marcus Yeagley. Licensed under CC BY-SA 2.0, via Flickr Creative Commons.

While piezoresistive pressure sensors require additional power to operate and feature higher noise limits, they offer many advantages over their capacitive counterparts. For one, they are easier to integrate with electronics. They also have a more linear response in relation to the applied pressure and are shielded from RF noise.

But like other MEMS devices, piezoresistive pressure sensors include multiple physics within their design. And in order to accurately assess a sensor’s performance, you need to have tools that enable you to couple these different physics and describe their interactions. The features and functionality of COMSOL Multiphysics enable you to do just that. From your simulation results, you can get an accurate overview of how your device will perform before it reaches the manufacturing stage.

To illustrate this, let’s take a look at an example from our Application Gallery.

Evaluate the Performance of a Piezoresistive Pressure Sensor with COMSOL Multiphysics®

The design of our Piezoresistive Pressure Sensor, Shell tutorial model is based on a pressure sensor that was previously manufactured by a division of Motorola that later became Freescale Semiconductor, Inc. While the production of the sensor has stopped, there is a detailed analysis provided in Ref. 1 and an archived data sheet available from the manufacturers in Ref. 2.

Our model geometry is comprised of a square membrane that is 20 µm thick, with sides that are 1 mm in length. A supporting region that is 0.1 mm wide is included around the edges of the membrane. This region is fixed on its underside, indicating a connection to the thicker handle of the device’s semiconducting material. Near one of the membrane’s edges, you can see an X-shaped piezoresistor (Xducer™) as well as some of its associated interconnects. Only some interconnects are included, as their conductivity is high enough that they don’t contribute to the device’s output.

Geometry of the sensor model (left) and a detailed view of the piezoresistor geometry (right).

A voltage is applied across the [100] oriented arm of the X, generating a current down this arm. When pressure induces deformations in the diaphragm in which the sensor is implanted, it results in shear stresses in the device. From these stresses, an electric field or potential gradient that is transverse to the direction of the current flow occurs in the [010] arm of the X — a result of the piezoresistance effect. Across the width of the transducer, this potential gradient adds up, eventually producing a voltage difference between the [010] arms of the X.

For this case, we assume that the piezoresistor is 400 nm thick and features a uniform p-type density of 1.31 x 10¹⁹ cm^-3. While the interconnects are said to have the same thickness, their dopant density is assumed to be 1.45 x 10²⁰ cm^-3.

With regards to orientation, the semiconducting material’s edges are aligned with the x- and y-axes of the model as well as the [110] directions of the silicon. The piezoresistor, meanwhile, is oriented at a 45º angle to the material’s edge, meaning that it lies in the [100] direction of the crystal. To define the orientation of the crystal, a coordinate system is rotated 45º about the z-axis in the model. This is easy to do with the Rotated System feature provided by the COMSOL software.

In this example, we use the Piezoresistance, Boundary Currents interface to model the structural equations for the domain as well as the electrical equations on a thin layer that is coincident with a boundary in the geometry. Using this kind of 2D “shell” formulation significantly reduces the computational resources required to simulate thin structures. Note that both the MEMS Module and the Structural Mechanics Module are used to perform this analysis.

Comparing the Results

To begin, let’s look at the displacement of the diaphragm after a 100 kPa pressure is applied. As the simulation plot below shows, the displacement at the center of the diaphragm is 1.2 µm. In Ref. 1, a simple isotropic model predicts a displacement of 4 µm at this point. Considering that the analytic model is derived from a crude variational guess, these results show reasonable agreement with one another.

The displacement of the diaphragm following a 100 kPa applied pressure.

When using a more accurate value for shear stress in local coordinates at the diaphragm edge’s midpoint, the local shear stress is said to be 35 MPa in Ref. 1. This is in good agreement with the minimum value from our simulation study (38 MPa). In theory, the shear stress should be the greatest at the diaphragm edge’s midpoint.

Shear stress in the piezoresistor’s local coordinate system.

The following graph shows the shear stress along the edges of the diaphragm. The maximum local shear stress of 38 MPa is at the center of each of the edges.

Local shear stress along two of the diaphragm’s edges.

Given that the dimensions of the device and the doping levels are estimates, the model’s output during normal operation is in good agreement with the information presented in the manufacturer’s data sheet. For instance, in the model, an operating current of 5.9 mA is obtained with an applied bias of 3 V. The data sheet notes a similar current of 6 mA. Further, the model generates a voltage output of 54 mV. As indicated by the data sheet, the actual device produces a potential difference of 60 mV.

Lastly, we look at the detailed current and voltage distribution inside the Xducer™ sensor. As noted by Ref. 3, a “short-circuit effect” may occur when voltage-sensing elements increase the current-carrying silicon wire’s width locally. This effect essentially means that the current spreads out into the sense arms of the X. The short-circuit effect is illustrated in the plot below. Also highlighted is the asymmetry of the potential, which is a result of the piezoresistive effect.

Current density and electric potential for a device with a 3 V bias and an applied pressure of 100 kPa.

Further Resources

• Get a demonstration of the Piezoresistive Pressure Sensor, Shell tutorial in this archived webinar
• Browse the MEMS category on the COMSOL Blog to learn more about simulation applications for MEMS devices

References

1. S.D. Senturia, “A Piezoresistive Pressure Sensor”, Microsystem Design, chapter 18, Springer, 2000.

2. Motorola Semiconductor MPX100 series technical data, document: MPX100/D, 1998.

3. M. Bao, Analysis and Design Principles of MEMS Devices, Elsevier B. V., 2005.

Xducer™ is believed to be a trademark of Freescale Semiconductor, Inc. f/k/a Motorola, Inc. Neither Freescale Semiconductor Inc. nor Motorola, Inc. has in any way provided any sponsorship or endorsement of, nor do they have any connection or involvement with, COMSOL Multiphysics® software or this model.

Piezoelectricity Powers Innovative Technology

Imagine a smart flooring technology that generates power from people’s movements. As their footsteps apply stress to the floor, a certain degree of energy is produced that helps to power lighting and other electrical needs throughout a particular building or environment. At the root of this technology, and many other innovative designs, is piezoelectricity.

Since the discovery of piezoelectricity in 1880 by the French physicists Jacques and Pierre Curie, this technology has been utilized in a variety of applications, from generating and detecting sounds to producing high voltages. You can even see the piezoelectric effect at work in the use of push-start propane barbecues, time reference sources within quartz watches, and musical instruments.

A piezoelectric violin bridge pickup. Image by Just plain Bill — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

Optimizing the design of these and other piezoelectric devices requires the use of computational tools that deliver accurate results. COMSOL Multiphysics provides such reliability, giving you greater assurance of the validity of your simulation findings.

To illustrate this, we’ve created a benchmark tutorial of a composite piezoelectric transducer. While the tutorial is a particularly useful resource for those performing ultrasonic transducer simulations, it also serves as a helpful foundation in the simulation of surface and bulk acoustic wave filters.

Analyzing a Composite Piezoelectric Transducer in COMSOL Multiphysics®

The example model of a piezoelectric transducer presented here consists of a 3D cylindrical geometry, which features a piezoceramic layer, two aluminum layers, and two adhesive layers. The layers are organized in such a way that the aluminum layers are at each end, connected to the piezoceramic layer by the two adhesive layers. In an effort to reduce memory requirements, we make use of the model’s symmetry when creating the geometry. This involves making a cut along a midplane that is perpendicular to the central axis and then cutting a 10-degree wedge.

The system operates with an AC potential applied on the electrode surfaces of each side of the piezoceramic layer. For this specific example, the potential has a peak value of 1 V within the frequency range of 20 kHz to 106 kHz. The goal of the simulation study is to calculate the admittance for a frequency range that is close to the structure’s four lowest eigenfrequencies.

We begin our analysis by identifying the eigenmodes and then running a frequency sweep across an interval that includes those first four eigenfrequencies. With its built-in functionality, COMSOL Multiphysics is able to assemble and solve the mechanical and electrical parts of this problem at the same time. This not only fosters greater efficiency in the simulation workflow, but also helps ensure that your results are accurate.

Left: A simulation plot of the lowest vibration mode. Right: A graph comparing susceptance and frequency.

Let’s take a look at the simulation results. The left plot above shows the lowest vibration eigenmode of the piezoelectric transducer, while the plot on the right highlights the input susceptance (the imaginary part of admittance) as a function of excitation frequency. These results agree with the findings presented in the paper “Finite Element Simulation of a Composite Piezoelectric Ultrasonic Transducer” (see Ref 1.). Note that because we did not use damping in this particular simulation, there is a small discrepancy near the eigenfrequencies. However, you can also simulate damping with COMSOL Multiphysics.

Advance the Design of Piezoelectric Devices with COMSOL Multiphysics® and the MEMS Module

Designing reliable piezoelectric devices is possible with tools like COMSOL Multiphysics. Its flexibility and functionality provides you with accurate results that will leave you feeling confident and pave the way for the continued advancement of your piezoelectric devices. To learn more about these capabilities, browse the resources below.

Learn More About Modeling Piezoelectric Devices

• Interested in using COMSOL Multiphysics to model your piezoelectric devices? Contact us today so we can help you evaluate the software
• Download our benchmark tutorial: Composite Piezoelectric Transducer
• Browse further blog posts relating to piezoelectricity:
  • Simulation Improves the Range of Motion in Piezoelectric Actuators
  • Optimizing the Power of a Piezoelectric Energy Harvester
  • Modeling a Stacked Piezoelectric Actuator in a Valve

References

1. Y. Kagawa and T. Yamabuchi, “Finite Element Simulation of a Composite Piezoelectric Ultrasonic Transducer”, IEEE Transactions on Sonics and Ultrasonics, vol. SU-26, no. 2, pp. 81-88, 1979.

The Mathematical Conventions of Piezoelectric Theory

Before we start discussing piezoelectric physics, let’s first review a couple of mathematical conventions. Piezoelectric materials have properties that allow strain to produce electric polarization and vice versa. The former is known as the direct piezoelectric effect, while the latter is referred to as the inverse piezoelectric effect. In mathematics, two forms of a set of matrix equations are conventionally used to describe these effects: the strain-charge form (also known as the d-form) and the stress-charge form (or the e-form). The two forms can be derived from each other by a transformation.

Note: The two forms are convertible, thus you can choose whichever option is preferred. In COMSOL Multiphysics, the strain-charge form is converted to the stress-charge form internally.

Here, we’ll choose the d-form, in which the relations are written as

where the field quantities , , , and are the strain, electric displacement, stress, and electric field, respectively.

The material parameters , , and are the compliance, coupling matrix, and relative permittivity, respectively. The superscript stands for transpose. The first equation expresses the relation that the stress field and electric field produce strain. The second equation, meanwhile, shows the electric polarization generated by the stress field and electric field. The second term of the first equation accounts for the electric contribution to the strain.

In solid mechanics, we typically consider an imaginary, small differential volume to understand the force that is exerted on a volume. Let’s consider a volume where the surfaces are aligned to each axis of the global Cartesian coordinate system (xyz-axes).

There are six distinct quantities that describe the forces per unit area on the surface of the imaginary volume. Three of them are normal to each surface and are usually denoted by , , and . The other three are parallel to each surface and are denoted by , , and . (, , and are omitted because of symmetry.)

The first index represents the normal vector of the surface under consideration. The second index indicates the direction of the force. For example, is a force along the x-axis that is exerted on a yz-plane (a plane perpendicular to the x-axis); is a force along the z-axis on an xz-plane; and so on.

The force components in a small differential cubic volume.

These six quantities are often expressed as a column vector. In COMSOL Multiphysics, the order of the quantities has two conventions: one is the standard notation and the other is the Voigt notation. In COMSOL Multiphysics, it is important to make sure that the Voigt notation is used in the Piezoelectric Devices interface. (The standard notation is used by default in the Solid Mechanics interface.)

In the Voigt notation, the stress is written as a 6 x 1 column vector (special attention should be paid to the ordering of yz-, xz-, and xy- components):

Similarly, the strain can be written as

In COMSOL Multiphysics, the coupling matrix is defined by

Here, the notation reads as the coupling coefficient for the strain component caused by the electric field component . Now, in the Voigt notation, all of the subscripts are numeric with the rule:

As such, the coupling coefficient matrix is

Thus, the electric contribution can be rewritten as

In the view of the material property settings in COMSOL Multiphysics, the matrix is flattened into a 1 x 18 row vector to be concise. It then looks like the following:

We always have the option to view the expression in the matrix form. We can do so by clicking on the Edit button in the “Output properties” section, as shown below:

A screenshot of the COMSOL Desktop shows the d coefficients in a matrix form.

Basic Piezoelectric Simulations

Now that we’ve reviewed some of the basics of piezoelectric theory, let’s turn our attention to performing simulations. Here, we’ll enter a nonzero number only in . This means that the piezoelectric material makes a normal force along the z-axis by the z-component of the electric field . We also assume that the bottom surface is mechanically fixed.

The results shown below are obtained, with the surface color representing the total displacement and the arrows indicating the electric field. The contribution of the electric field to the strain is given by . The black outline indicates the initially undeformed shape. As indicated by the plot, the volume is stretched in the z-axis direction.

An example of the d coefficients with the only nonzero in . Under an electric field along the z-axis, the volume stretches along the z-axis.

Keeping the same electric field, we’ll now enter a negative coefficient only in . With the yz-plane mechanically fixed, the volume shrinks in the x-axis direction.

An example of the d coefficients with the only nonzero negative in . Under an electric field along the z-axis, the volume shrinks in the x-axis.

The last preliminary example shows a shear force, with a nonzero number entered only in . The xz-plane is mechanically fixed and an electric field is applied along the y-axis.

An example of the d coefficients with the only nonzero in . Under an electric field along the y-axis, the volume experiences a shear along the yz-plane.

Building on these basic elements, we’ll now introduce you to a helpful trick for designing piezoelectric actuators — a piezoelec”trick”, if you will.

Improving the Range of Motion in Piezoelectric Actuators: A Novel Design

Simulating a Piezoelectric Cantilever, a Simple MEMS Device

On its own, a piece of piezoelectric material is not a MEMS device. To become such a device, it must be attached to other elastic materials. Cantilevers are one of the simplest MEMS examples, and there are two different piezoelectric types. The first is the unimorph type, which is typically a sheet made by attaching a single layer of piezoelectric material to an elastic material. The other is the bimorph type, which is composed of two piezoelectric layers and other elastic materials. The materials used for attachment can be almost anything, as long as it is possible to put an electrode layer on their surface. Here, for simplicity, we will consider a unimorph type of MEMS device.

It is worth mentioning that all of the piezoelectric material properties provided by COMSOL Multiphysics are assumed to be poled in the z-axis of the local coordinate system. If the material is poled along another direction, you need to define a coordinate system so that its third direction is aligned with the poling direction.

COMSOL Multiphysics provides convenient functionalities to set up local coordinate systems. To learn more, you can refer to page 106 in the Structural Mechanics Module Users Guide. You can also consult the Piezoelectric Shear-Actuated Beam and Thickness Shear Mode Quartz Oscillator models, found in the Application Libraries under the Piezoelectric Devices interface in the MEMS Module and also on our website.

Under the assumption referenced above, we model a unimorph piezoelectric cantilever that is fixed at one end. From the Material Libraries, we select barium titanate (BaTiO3) as a piezoelectric material and silica (SiO2) as the substrate.

The principle of the MEMS device is simple. An electric field is applied in the z-direction, causing shrinkage in the x-direction due to (=-7.8e-11 [C/N]). Note that the elastic properties of the electrode are not modeled here. To achieve the z-directional electric fields, an electric potential is applied on one side of the barium titanate, while the other side is grounded. With the left end mechanically fixed, the MEMS cantilever moves about 9 um at the free end. As the top material (BaTiO3) shrinks, it pulls the bottom material (SiO2), causing the entire device to bend in the upward direction.

A unimorph cantilever with a fixed end shows the total displacement.

But what if we have to fix both ends of the device for some reason — perhaps mechanical ruggedness? The result is obvious: There is almost no displacement, as the top material cannot shrink along the x-axis at all. More accurately, the only displacement is due to the effect, which is not intended.

A unimorph cantilever with both ends fixed shows almost no displacement.

It is, however, possible to overturn such a situation. We can do so by modifying the piezoelectric device with a little “trick”. As the resulting plot below shows, the displacement is now back to the same order of magnitude. If you look at the electric field directions, what we have done to obtain this result is apparent. The electric field is reversed in the center part of the beam. You can perform the trick by patterning the electrode into three parts: a center and two sides, grounding them alternately and applying a voltage alternately in the opposite order as in the first of the next two figures.

Material and voltage configurations in a unimorph cantilever with both ends fixed under alternate electric fields.

A unimorph cantilever with both ends fixed. Applying alternate electric fields improves the total displacement.

Let’s look at another example. It may seem a bit mysterious at first sight, as the entire electric field is aligned in the same direction. Still, we have the same performance as the previous result without alternating the electric field. We instead alternate the piezoelectric material, splitting it into three parts (again, a center and two sides) and flipping the center part.

With regards to simulation, this can be done either by using different local coordinate systems or, in a simple case like this, by changing the sign of the coupling coefficient, . In practice, we can really split the piezoelectric material up and put the flipped part in the center. An easier and more practical approach is to pattern the electrode in the same manner as the previous example, but pole the center part in the opposite direction. There is no need to split the material in such a situation.

For the unimorph cantilever with both ends fixed, alternate material and voltage configurations.

For the unimorph cantilever with both ends fixed, alternating the poling direction also improves the total displacement.

Simulating a Piezoelectric Membrane MEMS Device

Our last example is particularly interesting. Without the use of a trick, we would really have no displacement, since the entire perimeter is fixed and, as such, there is no room for any component to play a role. Applying the same trick from the previous examples, the entire-perimeter-fixed membrane can have a tremendous displacement. The simulation shown here can be used as a MEMS device that compensates for, to name an example, optical aberration (particularly spherical and astigmatism).

Alternate material and voltage configurations in a membrane MEMS device with the entire perimeter fixed.

A membrane MEMS device with the entire perimeter fixed. Applying an alternate electric field or poling direction improves the total displacement.

From the Basics to a Helpful Design Trick

We have reviewed some of the fundamental mathematics and conventions that are necessary when simulating piezoelectric materials. Further, we have walked you through the steps of performing basic simulations, then diving into more interesting applications using our piezoelec”trick”.

When it comes to the trick, the secret to significantly improving the displacement is the “inflection zone”, which is created by alternating the electric field or the material property. (Refer to US Patent 7,369,482 for more details and other examples.) The local coordinate system functionalities of COMSOL Multiphysics make it very easy to set up a simulation for such systems with alternating material orientations.

If you enjoyed today’s blog post, be sure to check out some other related blog posts that might spark your interest:

• Piezoelectric Materials: Crystal Orientation and Poling Direction
• Piezoelectric Materials: Understanding the Standards
• Piezoelectric Materials: Applying the Standards

Setting Up the Orientation of a Crystal in Two Standards

To set up the orientation of a crystal within COMSOL Multiphysics, it is necessary to specify the orientation of the crystallographic axes with respect to the global coordinate axes used to define the geometry. This is different than the manner in which the standards define the crystal orientation. Thus, some care is needed when defining the orientation of the geometry. For example, the orientation of the crystal axes will change if the orientation of the plate is changed. Here, we will show how to set up an AT cut quartz plate in different orientations in the physical geometry.

In a previous blog post, we discussed in detail the system that is used in both the IEEE 1978 standard and the IRE 1949 standard. Due to differences in the orientation of the crystallographic axes specified by each standard, the definition of the AT cut differs between them. The table below shows both definitions of the AT cut:

The difference between the standards can be understood by recalling that the plate cut from the crystal has an orientation defined by the l-w-t axes set (l-w-t stands for length, width, and thickness). The first two letters given in parentheses in the cut definition — Y and X — define the crystal axes with which the l and t axes are originally aligned. A rotation of 35.25° is then performed about the l-axis. The sense of the rotation differs between the standards, since the material properties are defined with respect to different sets of axes within the standards. This is illustrated in the figure below, which shows that the rotation about the l-axis is in a positive sense for the 1978 standard, but a negative sense for the 1949 standard.

The AT cut of quartz (mauve cuboid) shown together with a right-handed quartz crystal. The axes sets adopted by the IRE 1949 standard and the IEEE 1978 standard are shown, as well as the orientations of the l-w-t axes set in the plate.

There is another subtle difference between the two standards. As the AT cut is defined in the two standards, the thickness and length directions are reversed between them (shown in the figure above). From the figure, it is clear that to obtain exactly the same plate orientation as the 1949 standard, the 1978 standard would require an additional rotation of 180° about the w-direction. In this case, the AT cut in the 1978 standard would be defined as: (YXlw) -35.25° 180°. We need to carefully account for these differences between the standards when setting up a model in COMSOL Multiphysics.

Global Coordinate System

One way to set up a model is to keep the global coordinate system aligned with the crystal axes and simply rotate the plate to correspond with the first figure. As we will see, this method is perfectly valid, although it results in a rather inconvenient specification of the geometry.

Instead, we will consider how to define the material orientation for an AT cut quartz disc. In this COMSOL Multiphysics model, the crystal orientation is determined by the coordinate system selection in the Piezoelectric Material settings window. The crystal orientation is specified via a user-defined axis system that is selected in the coordinate system combo box, depicted below. This example is based on a simplified version of the Thickness Shear Mode Quartz Oscillator tutorial, available in our Application Gallery.

Changing the coordinate system for a piezoelectric material in COMSOL Multiphysics.

In the example above, the left-handed quartz defined in the 1978 standard is used for the material. If we wish to use the global coordinate system for the crystal orientation, then the quartz disc must be orientated in the manner shown in the first figure, with the axes set up for the 1978 standard. This can be achieved by rotating the cylinder about the x-axis.

A rotation operation is applied to the quartz cylinder.

The images below show the response of the device when it is set up in the selected orientation. The crystal is vibrating in the thickness shear mode. To obtain this response, use Study 1 in the COMSOL Multiphysics Application Gallery file and solve for a single frequency of 5.095 MHz.

IRE 1949 Standard	IEEE 1978 Standard

Thickness shear mode of an AT cut crystal for the same plate set up with the IRE 1949 (left) and the IEEE 1978 (right) standards. The driving frequency is 5.095 MHz. In both cases, the global coordinate axes in COMSOL Multiphysics correspond to the crystal axes.

Setting up the model within the IRE 1949 standard is straightforward, as COMSOL Multiphysics includes the material properties for both left- and right-handed quartz in each standard. To use the alternative standard, simply add the Quartz LH (1949) material to the model and select the quartz disc. This will override the previously added material. Then, change the rotation angle of the disc to -54.75º to orientate the disc equivalent to the plate shown in the first figure. The figure above shows that when these steps are followed, the 1949 standard gives the same result as the 1978 standard. Although the two figures appear identical, the global axes have been rotated so that they correspond to the two axes sets in the first figure.

As this example shows, it is possible to use the global coordinate system for the crystal axes. However, for a cut such as the AT cut, this results in an unusual orientation of the plate within the geometry. In a real world application, one might have several piezoelectric elements in different orientations and then this approach could not be used for all of the crystals. Therefore, it is often more convenient to specify the crystal orientation by means of a rotated coordinate system.

Rotated Coordinate System

In the COMSOL Multiphysics environment, the most convenient way to specify a rotated coordinate system is through a set of Euler angles. The Euler angles required for a given crystal cut will vary for different orientations of the plate with respect to the model global coordinates. Now we will consider how to specify the Euler angles for two different plate orientations in both of the available standards.

The best way to determine the Euler angles required within a given standard is to carefully draw a diagram that specifies the orientation of the l-w-t axes with respect to the crystal axes. Note that in some of the figures for the 1978 standard, l, w, and t are labeled as dimensions of the plate rather than as a set of right-handed axes. It is best to ensure that they are drawn as a set of right-handed axes to avoid potential confusion when determining the Euler angles for a plate in a COMSOL Multiphysics model. The Euler angles determine the orientation of the crystallographic axes (X_cr-Y_cr-Z_cr) with respect to the global coordinate system (X_g-Y_g-Z_g). Consequently, both the orientation of the plate with respect to the global system and the crystal cut determine the Euler angles.

As an example, we will consider the case where the global axes X_g-Y_g-Z_g align with the l-w-t axes (corresponding to the plate, with its thickness in the Zg direction). This is often the most convenient way to orientate the plate within a larger geometry. The figure below shows what happens when we take the first figure and rotate the plate such that the l, w, and t axes correspond to the global axes X_g-Y_g-Z_g within the two standards. For ease of comparison with the initial figure, the global axes are not in the same orientation for the two standards.

Rotated versions such that the l, w, and t axes correspond to the global axes X_g-Y_g-Z_g within the 1949 standard (left) and the 1978 standard (right). The Y and Z axes lie in a single plane.

The next figure shows the unrotated and rotated axes as seen from a side view of the first figure. This diagram represents an easier “paper and pencil” approach for determining the Euler angles.

	IRE 1949 Standard	IEEE 1978 Standard
Orientation in unrotated axes
Orientation in rotated axes

End on views of the axes orientation when cutting the crystal (top) and when the plate axes are oriented parallel to the global axes (bottom).

The following figure shows how the Euler angles are specified for a rotated system within COMSOL Multiphysics. An arbitrary rotated system can be specified by rotating first about the Z-axis, then about the rotated X-axis (marked as N in the figure below), and finally once again about the rotated Z-axis. This is known as a Z-X-Z scheme.

It is important to note that for cuts specified by means of multiple rotations, the rotations usually need to be applied in reverse when specifying the Euler angles. This is because COMSOL Multiphysics software specifies the orientation of the crystal with respect to the plate, whilst the standards used for cutting the plates from a crystal specify the orientation of the plate with respect to the crystal. It is straightforward to obtain equivalent Euler angles from the figure above.

Euler angles for the AT cut within the two standards. Both angles are positive for a right-handed rotation about the Z-axis.

Specifying a coordinate system using Euler angles through the rotated system feature.

If we use the Euler angles specified in the table above to set up the thickness shear mode for a quartz disc, then we obtain the results shown below for two plates with identical excitation and orientation. What went wrong? The problem is that the thickness direction for the AT cut is defined in opposite directions within the two standards. To obtain identical results from a model using the two standards, we could either switch the polarity of the driving electrodes or try using the alternative 1979 AT cut definition proposed above: (YXlw) -35.25° 180°. As a final exercise, let’s consider how to set up the Euler angles for this doubly rotated cut.

IRE 1949 Standard: (YXl) 35.25°	IEEE 1978 Standard: (YXl) -35.25°

Thickness shear mode of an AT cut crystal for the same plate set up with the IRE 1949 and the IEEE 1978 standards with a driving frequency of 5.095 MHz. In each image, the global axis orientation is shown on the left and the crystal axis orientation is shown on the right. The top images are aligned with the global coordinates and the lower images are shown with the crystal coordinates in the same orientation as in the first figure.

Below, we have the sequence of rotations involved in defining the cut (YXlw) -35.25° 180° and the sequence of Z-X-Z Euler rotations required to rotate the global axes onto the crystal axes. The corresponding Euler angles are provided in the table below. Note that the order of the rotations for the Euler angles is the reverse of that specified in the cut definition.

IEEE 1978 Standard: (YXlw) -35.25° 180°
1.Orientate the thickness direction (Zg) along the Y-axis of the crystal (Ycr) and the width direction (Xg) along the X-axis of the crystal (Xcr).
2. Rotate the cut by 35.35° about the l- (Xg) axis.
3. Rotate the cut by 180° about the w- (Yg) axis.
4. Reorientate the above figure so that the global axes are in a convenient orientation.

Sequence of rotations that correspond to the cut (YXlw) -35.25° 180° in the IEEE 1978 standard.

Equivalent Z-X-Z Euler Angles
1. Start with the crystal and the global axes aligned.
2. Rotate the crystal axes 180° about their Z-axis (Zcr).
3. Rotate the crystal axes -54.75° about the new crystal X-axis (Xcr).

Corresponding rotations that determine the Euler angles of the crystal axes with respect to the global axes.

Euler angles for the cut (YXlw) -35.25° 180° in the IEEE 1978 standard. This cut corresponds to exactly the same orientation of the plate in the IRE 1949 standard AT cut definition.

Finally, the figure below shows the frequency-domain response of the cut (YXlw) -35.25° 180° in comparison to the IRE 1949 standard AT cut. As expected, the responses of the two devices are now identical.

IRE 1949 Standard: (YXl) 35.25°	IEEE 1978 Standard: (YXlw) -35.25° 180°

Thickness shear mode of an AT cut crystal set up with the IRE 1949 standard compared to the cut (YXlw) -35.25° 180° in the IEEE 1978 standard with a driving frequency of 5.095 MHz. In each image, the global axis orientation is shown on the left and the crystal axis orientation is shown on the right. The top images are aligned with the global coordinates and the bottom images are shown with the crystal coordinates in the same orientation as in the first figure.

Further Reading on Piezoelectric Materials

• Download this tutorial: Thickness Shear Mode Quartz Oscillator
• Read a related blog post: Piezoelectric Materials: Understanding the Standards

Advancing the Power of Wireless Sensor Networks with Energy Harvesting

Wireless sensor networks are used today in a variety of low-power applications, from wearable healthcare to water quality monitoring. These networks include a series of sensors that measure and record physical conditions — often intermittently, over a period of time — at various locations. Through a wireless link, each sensor communicates the information it obtains to other sensors in the network as well as a base location that records the readings from all of the sensors.

One of the biggest challenges in developing wireless sensor networks is balancing energy consumption with efficiency. Battery life in sensors is often limited, making these devices more expensive to deploy, while reducing their number of applications. That’s where energy harvesting comes into play. In this process, energy is gathered from an external source (e.g., solar or thermal power) and converted into usable energy. Energy harvesting is beneficial, as it makes use of energy that would have otherwise been lost, helping to optimize the power of devices while extending their operational lifetime.

As previously referenced, energy can be harvested from the environment in a number of ways. One example is when mechanical strain is converted into electrical energy, also known as piezoelectric energy harvesting. A potential source of mechanical strain is local variations in acceleration. This is the case when a wireless sensor is mounted on a piece of machinery that is vibrating.

Let’s analyze such an energy harvester configuration in COMSOL Multiphysics.

Using Simulation to Analyze a Piezoelectric Energy Harvester

The Piezoelectric Energy Harvester tutorial model is designed to represent a simple “seismic” energy harvester. The device features a piezoelectric biomorph clamped at one end of the vibrating machinery and a proof mass mounted on the other end. A ground electrode is embedded within the biomorph, with two electrodes on the cantilever beam’s exterior surfaces. This design scheme ensures that an equal voltage is induced on the exterior electrodes.

Geometry of a piezoelectric energy harvester.

Our series of simulation analyses begins with addressing the power output as a function of vibration frequency. The plot below shows the input mechanical power and the power harvested, along with the voltage that is induced across the piezoelectric biomorph when acceleration occurs. In this case, the fixed electrical load is 12 kΩ and the acceleration magnitude is 1 g. From the results, we can identify a peak voltage at 76 Hz. This calculation is close to the resonant frequency computed for the cantilever in a separate eigenfrequency analysis (73 Hz).

Power output as a function of vibration frequency.

Let’s now measure the power output as a function of the electrical load resistance. In this scenario, we apply an acceleration of 1 g vibrating at 75.5 Hz. The results, shown in the following graph, indicate that the peak of power harvested correlates to an electrical load of 6 kΩ.

Power output as a function of electrical load resistance.

Lastly, we analyze the voltage and mechanical/electrical power as a function of mechanical acceleration. Here, the acceleration is set at 75.5 Hz, with a load impedance of 12 kΩ. As the plot below illustrates, there is a linear relationship between the voltage and the load, while the harvested energy increases quadratically.

Power output as a function of acceleration.

These results show good qualitative agreement when compared with experimental findings.

Concluding Thoughts

Simulation offers a simplified approach to studying and optimizing energy harvesting devices. The ability to easily test different device configurations accelerates the design process, while helping to produce more efficient energy harvesters. As the efficiency of these devices continues to grow, a greater number of technologies will have the chance to benefit from energy harvesting.

Next Steps

• Download the tutorial model: Piezoelectric Energy Harvester
• Read a related user story: Modeling Optimizes a Piezoelectric Energy Harvester Used in Car Tires

Piezoelectric Valves and Stacked Actuators

Piezoelectric valves are common in medical and laboratory applications because they offer many advantages, such as energy efficiency, durability, and fast response times. To open and close the valve featured in this tutorial, there is a hyperelastic material with a piezoelectric actuator sitting on top of it. When a voltage is applied to the stacked piezoelectric actuator, it deforms in a way that either pushes the hyperelastic material against the opening of the valve to seal it or moves it away from the valve to open it.

Valve, piezoelectric actuator, and seal.

Stacked piezoelectric actuators consist of two actuators stacked on top of each other. Each of the two actuators is made up of alternating layers of piezoelectric material, PZT, and very thin metal conducting layers between them. Every second metal layer is grounded, while every other layer receives an applied voltage. Similarly, the stacked PZT layers have alternating polarization directions.

Close-ups of the actuator and seal with alternating layers of PZT and metal highlighted. The top images show the PZT layers of alternating polarization directions. The bottom images show the metal substrate with an applied voltage to every other layer and the others set to a ground.

The bimorph actuator under consideration can be thought of as two stacked actuators placed one on top of the other. For a positive applied voltage, the upper and lower actuators are designed to expand laterally and contract laterally, respectively. This results in a bending of the structure (in this case, a disc), such that the center of the disc arches downwards. This forces the hyperelastic seal into contact with the valve seat — closing the valve. In the surface plot below, the stress is indicated by the color scale.

The von Mises stresses in a piezoelectric valve with a bimorph disc actuator.

Tutorial for Modeling a Stacked Piezoelectric Actuator in a Valve

The Piezoelectric Valve tutorial model, a new addition to the Application Gallery with COMSOL Multiphysics 5.1, demonstrates how to model a stacked piezoelectric bimorph disc actuator in a pneumatic valve. The MEMS Module and Nonlinear Structural Materials Module are used for this simulation.

The valve model consists of a multilayer stacked piezoelectric actuator, which in itself is a complex structure of stacked layers and electrodes. The model also includes a stainless steel substrate and a seal of hyperelastic material over the through hole of the valve.

For the simulation, we apply 50 volts to the layers. The contact pressure is determined here at the two contact pressure points of the seal. We can see that deformation of the disc is greatest at the center, which compresses the hyperelastic seal against the valve’s opening and closes the valve.

Left: The strain at the two contact surfaces of the valve’s seal. Here, we can see that the deformation of the disc is greatest at the center, which closes the valve. Right: The contact pressure at the two surface points of the valve’s seal.

Modeling a piezoelectric valve allows us to analyze the operation of the stacked piezoelectric actuator and evaluate the stress and strain in the seal and the surrounding materials. The analysis could be extended to estimate the performance of the seal with different pressure differentials applied across the valve in the closed state.

Try It Yourself

• Tutorial Download: Piezoelectric Valve