Some devices require a very high degree of frequency stability with respect to changes in the environment. The most common parameter is temperature, but the same type of phenomena could, for example, be caused by hygroscopic swelling due to changes in humidity. In very high precision applications, the frequency stability requirements might specify a precision at the ppb (parts-per-billion, 10^{-9}) level. Setting up simulations that accurately capture such small effects can be a challenging task, since several phenomena can interact.
Consider a rectangular beam with the following data:
Property | Symbol | Value |
---|---|---|
Length | L | 10 mm |
Width | a | 1 mm |
Height | b | 0.5 mm |
Young’s modulus | E | 100 GPa |
Poisson’s ratio | ν | 0 |
Mass density | ρ | 1000 kg/m^{3} |
Coefficient of thermal expansion, x direction | α_{x} | 1·10^{-5} 1/K |
Coefficient of thermal expansion, y direction | α_{y} | 2·10^{-5} 1/K |
Coefficient of thermal expansion, z direction | α_{z} | 3·10^{-5} 1/K |
Temperature shift | ΔT | 10 K |
The beam geometry and mesh used in the example.
The material parameters have values that are of the same order of magnitude as those for many other engineering materials. To better separate the various effects, Poisson’s ratio is set to zero, but this assumption does not change the results in any fundamental way. Orthotropic thermal expansion coefficients are used to highlight some properties of the solution.
To analyze the effect of thermal expansion, add a Prestressed Analysis, Eigenfrequency study.
Adding the Prestressed Analysis, Eigenfrequency study.
This study consists of two study steps:
The two study steps shown in the Model Builder tree.
To compute the reference solution, you either add a separate Eigenfrequency study or run the same study sequence, but without thermal expansion.
The eigenfrequencies of the beam have been calculated for two different types of boundary conditions:
The doubly clamped beam results are shown below.
Mode Type | Eigenfrequency, Hz |
Eigenfrequency, Hz ΔT = 10 K |
Ratio |
---|---|---|---|
First bending, z direction | 50713.9 | 50425.1 | 0.9943 |
First bending, y direction | 97659.6 | 97526.2 | 0.9986 |
First twisting | 266902 | 266917 | 1.00006 |
First axial | 500000 | 500025 | 1.00005 |
Mode shapes for the doubly clamped beam.
The following table shows the cantilever beam results.
Mode Type | Eigenfrequency, Hz |
Eigenfrequency, Hz ΔT = 10 K |
Ratio |
---|---|---|---|
First bending, z direction | 8063.79 | 8066.92 | 1.00039 |
First bending, y direction | 16049.1 | 16053.7 | 1.00028 |
First twisting | 132233 | 132265 | 1.00025 |
First axial | 250000 | 250050 | 1.0002 |
Mode shapes for the cantilever beam.
The first thing to note is that the bending eigenmodes for the doubly clamped beam stand out and have a strong temperature dependence. The change is 0.6% in the first mode. For all other modes, the relative shift in frequency is significantly smaller. If you make the beam thinner, this difference would be even more pronounced. The reason for this behavior is discussed in the following sections.
In the case of the doubly clamped beam, the thermal expansion causes a compressive axial stress. With the given data, the stress is -10 MPa (computed as Eα_{x}ΔT). This stress causes a significant reduction in the stiffness of the beam — an effect often called stress stiffening, since it typically occurs in structures with tensile stresses. However, compressive stresses soften the structure.
Another way of looking at this is by performing a linear buckling analysis. You can do so by adding a Linear Buckling study to the model and using the thermal expansion caused by ΔT = 10 K as a unit load. You will then find that the critical load factor is 80.
The first buckling mode.
With a linear assumption, the beam becomes unstable at an 800 K temperature increase. At the buckling load, the stiffness has reached 0. Assuming that the stiffness decreases linearly with the compressive stress, the stiffness at ΔT = 10 K should be reduced by a factor of
Since a natural frequency is proportional to the square root of the stiffness, you can estimate the decrease to , which matches the computed value of 0.9943 well.
Stress softening also affects the twisting and axial modes, but the effect is not as obvious as it is in the bending modes.
In the cantilever beam, no stresses develop when it is heated, as it simply expands. In this case, the frequency shift is due solely to the change in geometry — an effect that is much smaller than the stress-softening effect.
The natural frequencies for the bending, torsional, and axial vibration of a beam have the following dependencies on the physical properties:
Here, the following variables have been introduced:
It is assumed that the initial dimensions of the beam are L_{0} x a_{0} x b_{0}, where a_{0} > b_{0}. After thermal expansion, the size is L x a x b.
The expansions (strains) in the three orthogonal directions are called ε_{x}, ε_{y}, and ε_{z}; respectively. In this case, they are linearly related to the thermal expansion by ε_{x} = α_{x}ΔT, ε_{y} = α_{y}ΔT, and ε_{z} = α_{z}ΔT; but in principle, it could be any type of inelastic strain.
The geometric properties scale as:
The mass density also changes. Since the same mass is now confined in a larger volume,
By introducing these expressions into the formulas for the natural frequencies, you arrive at the following expected eigenfrequency shifts:
Since the thermal expansions are very small, the approximate first-order series expansions can be expected to be accurate.
For the torsional vibrations, the situation is slightly more complicated, since the powers of a and b are mixed in the expression for the polar moment J. But if you make use of the fact that a = 2b for this geometry, then it is possible to derive a similar expression.
Now, compare the computed frequency shifts with the analytical predictions for the cantilever beam. The results are shown in the table below and the agreement is very good.
Mode Type | Computed | Predicted |
---|---|---|
First bending, z direction | 1.00039 | 1.00040 |
First bending, y direction | 1.00028 | 1.00030 |
First twisting | 1.00025 | 1.00025 |
First axial | 1.00020 | 1.00020 |
The fixed constraints at the ends of the beam cause local stress concentrations when the temperature is increased, as the transverse displacement is constrained.
The axial stress in the doubly clamped beam caused by a 10 K temperature increase.
This can have two effects:
To determine what effects the constraints should have, you must rely on your engineering judgment. Usually, the component and its surroundings are subject to temperature changes. In this situation, the possibility to add a thermal expansion to constraints in COMSOL Multiphysics comes in handy. Let’s see how the solution is affected.
Thermal expansion added to the fixed constraints for the doubly clamped beam.
For the cantilever beam, the results now change so that they perfectly match the analytical values.
Mode Type | Fixed Constraints | Stress-Free Constraints | Predicted |
---|---|---|---|
First bending, z direction | 1.00039 | 1.00040 | 1.00040 |
First bending, y direction | 1.00028 | 1.00030 | 1.00030 |
First twisting | 1.00025 | 1.00026 | 1.00025 |
First axial | 1.00020 | 1.00020 | 1.00020 |
In the analysis above, it is assumed that the material data does not depend on temperature. When looking at constrained structures (dominated by the stress-softening effect), this might be an acceptable approximation. However, with the small frequency shifts caused by geometric changes, the temperature dependence of the material must also be taken into account.
In this guide, you can see the temperature dependence of Young’s modulus for a number of metals. The stiffness decreases with temperature. For many materials, the relative change in stiffness is of an order of 10^{-4} K^{-1}. This means that for a temperature change of 10 K, you can expect a relative change in material stiffness that is of the order of 0.1%. This effect might actually be larger than the geometric effect computed above.
A small note of warning: When measuring the temperature dependence of Young’s modulus, it is important to know whether or not the geometric change caused by thermal expansion has been taken into account. In other words, you must know whether the Young’s modulus is measured with respect to the original dimensions or the heated dimensions.
When it comes to mass density, the situation is easier. When performing structural mechanics analyses in COMSOL Multiphysics, the equations are formed in the material frame. Thus, the mass density should never be given an explicit temperature dependence, since that violates mass conservation.
The coefficient of thermal expansion (CTE) usually increases with temperature. The relative sensitivity is often of the order of 10^{-3} K^{-1}. This sounds large, but it isn’t usually important when looking at the way the CTE enters the equations.
Most materials in the Material Library in COMSOL Multiphysics come with temperature-dependent material properties. In this example, you manually add a linear temperature dependence to the Young’s modulus with the following steps:
Alternatively, you can create a function and call it, with T as the argument.
Adding a linear temperature dependence to the material.
In the settings for the Linear Elastic Material, the Model Input section is now active. You then provide a temperature to be used by the material.
Adding the temperature to the material using Model Input.
After including a reduction of Young’s modulus by 1·10^{-4} K^{-1}, the resulting frequency shift turns out to be negative, rather than the positive shift observed with a constant Young’s modulus (shown in the table below).
Mode Type |
Stress-Free Constraints Constant E |
Stress-Free Constraints Temperature-Dependent E |
Difference |
---|---|---|---|
First bending, z direction | 1.00040 | 0.99990 | -0.00050 |
First bending, y direction | 1.00030 | 0.99980 | -0.00050 |
First twisting | 1.00026 | 0.99976 | -0.00050 |
First axial | 1.00020 | 0.99970 | -0.00050 |
The shift is exactly as expected for all modes — Young’s modulus is reduced by a factor 1·10^{-3} and the natural frequencies are proportional to its square root. Actually, you can include the change in Young’s modulus in the linearized expressions for the frequency shifts as:
Here, it is assumed that . The value of the coefficient β is usually negative; In this case, β = -10^{-4} K^{-1}.
For the common case of isotropic thermal expansion, these expressions simplify to:
We are looking for frequency changes that are at the ppm (parts-per-million) level. This means that it is important to avoid spurious rounding errors. There are some actions that you can take to ensure optimal accuracy.
In the settings for the Eigenfrequency node, set Search for eigenfrequencies around to a value of the correct order of magnitude.
The updated settings in the Eigenfrequency node.
Then, decrease the Relative tolerance in the settings for the Eigenvalue Solver node.
The decreased Relative tolerance in the settings for the Eigenvalue Solver node.
Change only the parameters necessary for capturing the physics. For example, use the same mesh for all studies.
If you have reason to believe that the problem is ill-conditioned, as can be the case for a slender structure, select Iterative refinement in the settings for the Direct solver.
The settings for the Direct solver, showing the option for Iterative refinement.
In version 5.3 of COMSOL Multiphysics®, the method for how inelastic strains are handled under geometric nonlinearity has been changed. A multiplicative decomposition of deformation gradients is the current default, rather than the subtraction of strains that was used in previous versions. This is one key concept to understand why it is now possible to perform this type of analysis with a very high accuracy. Let’s look at a (somewhat artificial) case where the temperature increase is 3·10^{4} K and there are no temperature dependencies in the material properties. This means that the stretches are
resulting in the volume changing by a factor of 3.952.
You can then compare the results from the prestressed eigenfrequency analysis with a standard eigenfrequency analysis on a bigger beam with L = 13 mm; a = 1.6 mm; b = 0.95 mm; and lower density, scaled by a volume factor of 3.952, ρ = 253.036 kg/m^{3}. This of course leads to large increases in the natural frequencies, as the heated object is much larger but with a lower density. The relative changes in frequency for the two approaches are shown in the following table.
Mode Type |
Thermal Expansion and Prestressed Eigenfrequency |
Larger Geometry and Lower Density |
---|---|---|
First bending, z direction | 2.2309 | 2.2308 |
First bending, y direction | 1.8759 | 1.8759 |
First twisting | 1.6702 | 1.6695 |
First axial | 1.5292 | 1.5292 |
As can be seen above, the correspondence is in excellent agreement. There is a slight difference in the twisting mode, but that disappears with a refined mesh. Actually, refining the mesh shows that the best prediction is from the prestressed eigenfrequency analysis.
We have discussed how to accurately determine changes in eigenfrequencies caused by temperature changes with COMSOL Multiphysics, as well as the effects of stress softening, geometric changes, and the temperature dependence of material properties.
For many years, landmarks like the Hoover Dam and the Pantheon have attracted the attention of tourists. While these structures are most recognized for their famous history, they have something else in common: Both were built using concrete.
Concrete was used to build some of the world’s most famous landmarks, including the Hoover Dam and the Pantheon. Left: Image by Mobilus In Mobili. Licensed under CC BY-SA 2.0, via Flickr Creative Commons. Right: Image by Roberta Dragan — Own work. Licensed under CC BY-SA 2.5, via Wikimedia Commons.
This manmade material — the most widely used in the world — is the foundation for many modern buildings and structures. In recent years, there has been a growing trend of embedding sensors within these structures as a means for measuring the concrete’s condition. This provides an efficient way of monitoring how important parameters like temperature, humidity, and pressure impact the strength and stability of the material.
To design reliable sensors for this purpose, engineers must have an understanding of the properties of concrete and their influence on sensor performance. But due to its unique properties, concrete can be a rather complex material to analyze. For instance, because concrete is viscoelastic, it experiences a time-dependent strain increase when a constant load is applied. This effect, known as viscoelastic creep, is particularly prominent when concrete is subjected to forces for a long period of time. Even when the temperature conditions are steady, concrete structures may undergo a volumetric change. When this occurs without environmental thermal exchange, it is called shrinkage.
The unique properties of concrete make it a complex material to study. Image by Les Chatfield. Licensed under CC BY 2.0, via Flickr Creative Commons.
With the flexibility of COMSOL Multiphysics, a research team from STMicroelectronics in Italy was able to successfully model the complex phenomena in concrete and predict their influence on the performance of an embedded sensor design. This shows an example of a technology group that is versed in one type of application — sensors and the optimization of their designs — using COMSOL Multiphysics to study another application that directly affects their application — the deformation of concrete.
The researchers began by first addressing concrete viscoelasticity. To realistically describe the concrete’s viscoelastic behavior, the team implemented their own Kelvin chain model in COMSOL Multiphysics and solved strains within each of the eight Kelvin branches.
To validate the model, the researchers used a theoretical model of a simple system featuring a concrete cylinder under uniaxial pressure. The cylinder, which is 7 cm in height and diameter, has a compressive strength of 48 MPa. The analysis includes the assumption that the environment is comprised of 50% relative humidity.
The researchers built a model in COMSOL Multiphysics of this same system, comparing their simulation results for the creep with trends observed in the theoretical model. As highlighted below, the findings show perfect agreement with one another.
Comparison of the results for the creep experiment. A constant load of 1 MPa is applied in the simulation study. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
The team then shifted gears to modeling the time-dependent concrete shrinkage. This involves using a strategy based on thermal phenomena. After calculating the shrinkage profile via equations, a unitary thermal coefficient is applied to the concrete material. For the theoretical model, a temperature profile in agreement with the computed shrinkage is used. The resulting thermal strain is meant to imitate the actual shrinkage.
Using the same benchmark model from the creep test, the team calculated the shrinkage strain. As before, the two results show perfect agreement.
Comparison of the results for shrinkage strain. In the simulation study, the time frame is around 1500 days. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
After validating their COMSOL Multiphysics model for both viscoelastic creep and shrinkage, the researchers sought to address a more complex scenario — when a silicon sensor is embedded within the concrete. While concrete creep and shrinkage can affect all sensor types, analyzing their effect is especially relevant for pressure sensors.
To represent a simplified sensor structure for measuring pressure, a cylindrical sensor with a height of 600 μm and diameter of 2 mm is used. The membrane (the sensing portion of the configuration) is 10 μm thick with a radius of 700 μm and an internal cavity depth of 50 μm. At the top of the cylinder, a constant load of 10 MPa is applied.
The geometry of the simulated sensor structure (a), with a zoomed-in version depicting the sensor’s axial displacement (b). Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
Initially, with only creep equations applied, the creep greatly affects membrane displacement over time. Its impact is particularly noticeable at the center of the membrane and close to one of its edges.
The difference between the vertical displacement at the center of the membrane and near one of its edges. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
The next aspect evaluated is the potential relationship between creep-induced changes and stress inside the membrane. The results show that stress does increase over time. The example below highlights this with respect to radial stress distribution.
The radial stress distribution on the sensor when the time span begins (a) and when it ends (b). Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
The following plots provide even more detail, showing the radial and angular stress components along the membrane’s radius when the analysis first starts and when it ends. Due to the creep effect, these components vary over time. Assuming that stress-sensing piezoresistive elements have been fabricated on the membrane, it is possible to observe the impact of a time-dependent creep-induced variation on sensor performance. Note that the positioning of the piezoresistive elements impacts these results.
Radial stress (a) and angular stress (b) along the radius. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
The behavior of the membrane is also influenced by shrinkage. When adding shrinkage to the model that only previously accounted for creep, a small change in deformation occurs near the center of the membrane. On the other hand, shrinkage has a large effect on stress distribution.
The two stress component distributions at the end of the time frame are then compared, with one analysis that includes shrinkage and one that doesn’t. When only creep is considered, there are changes in stress distribution on the membrane of both components. It is therefore assumed that there is also a shrinkage-induced effect on sensor performance. Once again, the location of the piezoresistor elements impacts the results. Note that the shrinkage effect has a greater influence on the angular stress than on the radial stress.
Comparison of the radial stress (a) and angular stress (b) along the radius — both with and without shrinkage. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
As the results indicate, creep and shrinkage — two of concrete’s unique properties — change the deformation and stress within a sensor’s membrane. This in turn affects sensor output, specifically the output voltage of the piezoresistors that are implemented on the membrane. Such findings are critical in the design of reliable pressure sensors for monitoring the condition of concrete.
There are a number of industries that rely on accelerometers. Take automotive designers, for instance, who often use these electromechanical devices to analyze shock and vibrations in safety testing. In addition, the developers of consumer electronics use these devices as a means of detecting orientation in digital cameras and tablet computers.
Automotive safety testing is just one application of accelerometers.
To detect the size and direction of an object’s acceleration, accelerometers feature sensor packages. Together, the components of these packages determine the frequencies for which the accelerometer provides accurate measurements. For example, modern commercial products typically feature a bandwidth of 10 to 20 kHz. As technologies continue to evolve and higher frequencies need to be measured, sensor packages must be able to handle higher bandwidths.
Recognizing this, a team of researchers from the Fraunhofer Institute for High-Speed Dynamics, Ernst-Mach-Institut, EMI, and the Albert-Ludwigs-Universität Freiburg used the COMSOL Multiphysics® software to design and analyze a sensor package for a high-g accelerometer. At the heart of the design is a novel piezoresistive sensor chip, one that can measure transient accelerations up to 100,000 g. Compared to today’s state-of-the-art sensors, this piezoresistive sensor’s figure of merit (sensitivity multiplied by resonance frequency) is around an order of magnitude greater.
To begin, let’s look at the design of the piezoresistive sensor chip. It includes:
The sensor chip. Image by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
In COMSOL Multiphysics, this configuration is fully modeled as a silicon MEMS device.
For the package itself, three of these chips are integrated onto a single ceramic plate. Oriented at right angles to one another, the chips are sensitive in the x-, y-, and z-directions.
The sensor — its package included — acts as a complex mass spring system. Bending in the plate as a result of acceleration causes the piezoresistive elements to stretch and compress, which in turn produces changes in electrical resistance.
The complete sensor package. Image by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
While the researchers tested multiple sensor package designs, we focus on one specific example here. The geometry for this sensor package, shown below, was imported into COMSOL Multiphysics® via LiveLink™ for Inventor®.
Each color represents the following:
The example sensor package design with its cap lifted (a) and an expanded view (b). Image by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
To analyze the sensor’s behavior over a specific frequency range, the researchers used two approaches:
The first approach provides the value and shape of the resonance frequencies, while the latter depicts the stress and displacement of components within the sensor package. This information is used to determine the relative resistance of the piezoresistive bridges as well as compute the output signals of the sensor chips.
The simulation results shown here correspond to the following sensor package design parameters:
Parameter | Setting |
---|---|
Wall thickness | 1 mm |
Cap thickness | 200 µm |
Package material | Titanium |
Adhesive layer thickness | 20 µm |
Adhesive Young’s modulus | 2.5 GPa |
Sensor chip | Type M (0.65 µ V/V/g) |
Let’s look at the sensor’s output signal for the frequency range of 0 to 250 kHz. Note that this signal is computed for 100,000 g and a supply voltage of 1 V. Further, a limit of 5% is defined with regards to the sensor’s maximum change in sensitivity.
Output signal for the sensor at various excitation frequencies. The plot on the left shows the whole frequency spectrum, while the plot on the right shows a close-up view of 0 to 100 kHz. Images by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
From the plot on the left, it initially appears that the curve’s behavior is flat until about 130 kHz. But with a closer view, shown in the plot on the right, the sensitivity changes are also visible at lower frequencies. With the 5% limit in place, the potential bandwidth of the sensor package is 47 kHz.
The sensor package modes are also analyzed, specifically those shown as peaks in the frequency spectrum. As highlighted in the previous plot, the first mode, or “cap mode”, occurs at 39 kHz and has minimal influence on sensitivity. Aside from the cap oscillations, the first mode, or “package mode”, occurs at 128 kHz. This mode, which is also highlighted above, has a significant effect on the output signal.
Left: The first cap mode, deflecting in the z-direction. Right: The first package mode, oscillating in the y-direction. Images by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
From the modal analysis, an additional oscillation is observed at 287 kHz. As the main source of displacement for the sensor element, this mode is expected to have the most influence on the sensor’s signal. To test this assumption, the researchers turned to experimental tests.
Main displacement mode inside the sensor element. Image by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
When adapting the simulation study to the experimental phase, the researchers used the following parameters. For practical reasons, these parameters differ slightly from those used in the model.
Parameter | Setting |
---|---|
Wall thickness | 1 mm |
Cap thickness | 200 µm |
Package material | Titanium |
Adhesive layer thickness | 20 to 70 µm |
Adhesive Young’s modulus | 0.56 GPa |
Sensor chip | Type L (1.3 µ V/V/g) |
An additional simulation study helped to predetermine the ranges of expected frequencies:
In the experiment, a small glass hammer stimulates the sensor oscillation in order for the researchers to measure the eigenfrequencies. A sampling rate of 10 MHz is used to record the sensor’s impulse answer.
Impulse answer of the sensor to an oscillation. Image by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
To examine the impulse answer for relevant frequencies, the signal is converted into the frequency domain. As indicated below, multiple peaks appear at higher frequencies. The highest peak at 930 kHz represents the first eigenfrequency of the actual sensor chip. The lower frequencies, up to about 70 kHz, are a portion of the excitation impulse.
Left: Impulse answer of the sensor from 0 to 2 MHz. Right: Impulse answer of the sensor from 0 to 350 kHz. Images by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
Something interesting to note is the peak that appears at 153 kHz. This represents the sensor element’s oscillation that is expected between 129 and 200 kHz. This finding supports the theory that this oscillation has the biggest influence on the sensing element.
In the sensitivity analysis, an acceleration of 8600 g is applied to each axis. A titanium Hopkinson bar is used to produce the shock load, with an attachment included to ensure equal distributions of the load on each sensor axis.
The attachment for the Hopkinson bar. Image by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
The measured output signals, shown in the plot below, are used to compute the sensitivities of the different axes. The expected sensitivity is 1.3 µ V/V/g, with a potential maximum deviation of 30%. The greatest deviation in sensitivity occurs at the x-axis (around 23%), while the other axes have much lower percentages. Note that all of the sensor chips fall within this deviation range.
The measured output signals under a load of 8600 g. Image by R. Langkemper, R. Külls, J. Wilde, S. Schopferer, and S. Nau and taken from their COMSOL Conference 2016 Munich paper.
These findings show good agreement with the expected values from the simulation results, further highlighting the suitability of the sensor package design for a high-g accelerometer.
Autodesk, the Autodesk logo, and Inventor are registered trademarks or trademarks of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries.
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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.
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 10^{19} cm^{-3}. While the interconnects are said to have the same thickness, their dopant density is assumed to be 1.45 x 10^{20} 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.
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.
S.D. Senturia, “A Piezoresistive Pressure Sensor”, Microsystem Design, chapter 18, Springer, 2000.
Motorola Semiconductor MPX100 series technical data, document: MPX100/D, 1998.
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.
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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.
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.
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.
The food that we consume on a day-to-day basis, particularly carbohydrates, is a powerful source of energy for the human body. For the body to utilize energy from carbohydrates and store glucose for later use, it is crucial that its cells properly absorb the sugar. The key to this process is insulin, a hormone the body signals to the pancreas cells to release into the bloodstream, allowing sugars to enter the cells and be used for energy.
But what happens when the body fails to produce enough insulin or if it doesn’t work in the way that it should? In this case, the glucose fails to be absorbed by the cells and will instead remain in the bloodstream, resulting in rather high blood glucose levels. Referred to as diabetes, this metabolic disease relates to cases where the body produces little or no insulin (Type 1) or does not properly process blood sugar or glucose (Type 2). Note that in the latter type, a lack of insulin can develop as the disease progresses.
A device for injecting insulin. Image by Sarah G. Licensed under CC BY 2.0, via Flickr Creative Commons.
In both Type 1 and Type 2 diabetes, insulin injections serve as a viable treatment option. These injections, however, can cause pain when applied by a heavy single-needle mechanical pump. To minimize patients’ discomfort, researchers have investigated the potential of using a microneedle-based MEMS drug delivery device to administer insulin dosages. Not only would the stackable structure be minimal in size and easy to apply to the skin, but it would also provide a safer and less painful approach to applying injections.
Here’s a look at how a research team from the University of Ontario Institute of Technology used simulation to evaluate such a device…
Let’s begin with the design of the micropump model. The researchers developed a MEMS-based insulin micropump, placing a piezoelectric actuator on top of a diaphragm membrane comprised of silicone, with a viscous Newtonian fluid flowing through it. Note that the design itself is based on the minimum dosage requirement for diabetes patients — this typically ranges from 0.01 to 0.015 units per kg per hour.
Vibrations from the actuator create a positive/negative volume in the main chamber of the pump, which then pulls the fluid from the inlet gate and pushes it toward the outlet gate. Two flapper check valves control the direction of the fluid from the inlet to the outlet leading to the microneedle array, with a distributor connecting the outlet gate to the microneedle substrate. The established discharge pressure then pushes the fluid out of the microneedles to the skin’s outer layer.
The following set of images show the dimensions and cross section of the micropump as well as a more detailed layout of the model setup, respectively.
The MEMS-based piezoelectric micropump design. Image by F. Meshkinfam and G. Rizvi and taken from their COMSOL Conference 2015 Boston paper.
A 2D layout of the micropump model setup. Image by F. Meshkinfam and G. Rizvi and taken from their COMSOL Conference 2015 Boston paper.
To accurately study the performance of the micropump, the researchers utilized three different physics interfaces in COMSOL Multiphysics: the Solid Mechanics, Piezoelectric Devices, and Fluid-Structure Interaction (FSI) interfaces. The fluid flow that occurs from the inlet to the outlet via the action of the flapper check valve is described by the Navier-Stokes equations. Upon a wave signal exciting the piezoelectric actuator, the diaphragm disk and piezoelectric actuator move together, with an FSI moving mesh presenting the deformed solid boundary to the fluid domain as a moving wall boundary condition. Within the solid wall of the pump, this moving mesh follows the structural deformation. The FSI interface also accounts for the fluid force acting on the solid boundary, making the coupling between the fluid and solid domains fully bidirectional.
For their simulation analyses, the research team applied different input voltages and input exciting frequencies to the micropump design, studying various elements of the device’s behavior. The range of the voltages spanned from 10 to 110 V, while the exciting frequencies ranged from 1 to 3 Hz.
Let’s look at the results for an input voltage of 110 V and an input exciting frequency of 1 Hz. The plot on the left depicts the inflow and outflow rates, showing very little leakage for both. The plot on the right shows the established discharge and suction pressures at the inlet and outlet. At the inlet gate, a negative pressure denotes suction pressure, while a negative pressure at the outlet gate represents discharge pressure.
Left: Inflow and outflow rates. Right: Discharge and suction pressures. Images by F. Meshkinfam and G. Rizvi and taken from their COMSOL Conference 2015 Boston paper.
As part of their analyses, the researchers measured the stress and deflection of the flapper check valves as well as the velocity field of the fluid. You can see their results in the following set of plots.
Left: Von Mises stress in the flapper check valves and velocity field of the fluid. Right: Deflection of the flapper check valves. Images by F. Meshkinfam and G. Rizvi and taken from their COMSOL Conference 2015 Boston paper.
The studies shown here, as well as those conducted with alternative inputs, indicate that the micropump design performs properly from the minimum to maximum spectrum of pressure and flow rates. Such a configuration can therefore serve as a viable alternative for applying insulin injections, providing a safer and less painful method of treatment for diabetes patients. The researchers hope to use their simulation findings as a foundation for creating more durable and dynamic insulin micropump designs in the future.
In any simulation study, there are complex theories and physics that must be considered in order to obtain accurate, realistic results. What is unique about simulation apps is that they are able to incorporate these elements into the underlying model while hiding such complexities behind a user-friendly interface. Such capabilities give those without simulation expertise the ability to set up their own numerical tests, all while enhancing their understanding of various scientific methods along the way.
Within the university setting, simulation apps are evolving as a powerful tool for introducing students to challenging concepts and advancing their modeling skills. What’s more is that as students continue with their studies, professors can take advantage of the Application Builder’s flexibility and incorporate greater complexity into an app’s design to further advance their learning.
The Tubular Reactor app, available in our Application Gallery, offers just one example of how teachers can simplify the process of teaching mathematical modeling concepts to students.
So how, you might wonder, are professors incorporating apps into their curriculum? Let’s take a look at two cases.
In 2015, Mumbai University in India introduced MEMS technology as a new course within their curriculum, combining theory with hands-on experience in the lab. Like many educators around the world, professors at this university recognize the importance of ensuring that students understand the basic principles of scientific methods at an early stage. The students, however, are often not yet ready to set up numerical problems on their own right away. They need to first invest a lot of time in learning the simulation software and mastering its use before they can carry out their own tests. To address this challenge, the professors turned to the Application Builder in COMSOL Multiphysics, having a set of apps developed that could better illustrate complex ideas.
With the easy-to-use apps — all of which are designed to be self-explanatory — students have been able to more quickly grasp the theory behind various scientific concepts. Each app includes notes that provide a clearer overview of its underlying theory as well as access to numerous design parameters that depict how device performance changes when parameters are modified.
“Since the students do not need to learn the software to use apps, this approach offers a quick way to understand the underlying principles,” notes Darshana Sankhe, a professor at the university’s DJ Sanghvi College of Engineering. “It frees up time for students to focus on other aspects of the topic at hand.”
The apps themselves cover a range of MEMS technology. Take the pressure sensor moisture absorption app, for instance, which shows students how MEMS devices react to a moist environment. The thermal expansion app, meanwhile, offers an overview of MEMS materials and their properties, while demonstrating the working principles behind a thermal actuator. The screenshots below highlight some of the additional apps created for the course.
MEMS-based apps utilized by professors at Mumbai University. Left: Piezoelectric shear-actuated beam app. Right: Piezoresistive pressure sensor app.
The benefits of incorporating such apps into the course’s layout extend not just to the students, but to the professors as well. Chetana Sangar, a professor and head of the Department of Electronics and Telecommunication at the Bharat College of Engineering, offers insight: “Not only have these apps helped students, but they have also helped us professors prepare for this new course. The apps that we have developed are very useful and I have recommended them to my colleagues in other colleges as well.”
At the Eindhoven University of Technology, professors also recognize the importance of teaching students the principles of scientific methods as early on as possible. The challenge here is that students don’t yet possess the knowledge and skills that are required to solve realistic heat transfer problems on their own. Recognizing this factor, professors at the university sought to simplify the learning process — and offer hands-on experience — in a first-year scientific methods course on building physics. Their answer came in the form of building an app.
The app itself is centered around a 3D heating experiment that involves analyzing three different methods for determining the thermal conductivity of polymethylmethacrylat (PMMA), with the goal of finding out which one is most accurate. The simulation study focuses on a box composed of expanded polystyrene, with a PMMA sheet attached at the top. For the case of Fourier’s law of conduction, one of the studied methods, a heat flux sensor is installed to determine the sheet’s thermal conductivity.
By using a simulation app, students are able to perform data analysis at an early stage, simulating output sensor data, from temperatures to heat fluxes. Such data is then used to calculate the sample’s heat conduction coefficient (the objective parameter), information that can then be compared to the input value. In this way, students successfully learn how to verify both their simulation and data analysis methods. Further, they are able to bring the skills and knowledge that they obtain through virtual testing, including how to improve the expected responses of the sensor, and apply them when conducting the actual experiments.
The 3D heating simulation app designed by professors at the Eindhoven University of Technology.
Along with the advantage of helping students understand key concepts faster, Jos van Schijndel, an assistant professor at the university, notes how the hands-on approach is also generating greater student engagement and interest: “The students are much more enthusiastic to start with the simulation because they can interact very intuitively with the simulation results from the beginning. This stimulates their curiosity to find out more about what’s behind the underlying model.”
Simulation apps, by design, hide complex theories and physics behind a user-friendly, customizable interface. These same advantages that are helping to optimize design workflows across a range of industries are also allowing professors to more efficiently teach students. When developing an app, professors can tailor the design in a way that best suits the needs of their specific course, from the included physics to the overall degree of complexity. Such flexibility enables the creation of an interactive learning tool that correlates with the needs of a particular set of students and can evolve with them as their skill set grows.
When using either the AC/DC Module, the MEMS Module, or the Plasma Module, the Terminal condition can be applied to the boundaries of any domains through which conduction or displacement currents can flow. With this boundary condition, it is possible to apply a Current, Voltage, or Power excitation as well as a connection to an externally defined Circuit or a Terminated connection with known impedance.
Regardless of the type of excitation or the physics interface being used, the Terminal condition always specifies the voltage, but optionally adds more equations to the model. For example, when using a Terminal condition with a specified current, the software automatically adds an Integration Component Coupling feature to integrate the total current through the specified boundary. The software also adds a Global Equation that introduces one additional degree of freedom to the model for the terminal voltage, such that the current through the terminal equals the user-specified current.
This combination of Global Equations with an Integration Component Coupling is quite flexible and you may already be familiar with its usage for structural mechanics and heat transfer modeling. Let’s now see how we can easily switch between different terminal types.
A schematic of a block of material with a ground and a Terminal condition on opposite sides. The Terminal boundary condition will be switched between a voltage or a current source.
We will look at a very simple electric current model involving just a block of material with a grounded boundary on one side and a current-type Terminal boundary condition on the other. We will start by considering the steady-state case and address how to apply a current or voltage excitation by adding a Global Equation. The Global Equation itself is added to the Electric Currents interface (To add Global Equations to a physics interface, make sure to toggle on Advanced Physics Options underneath the Show menu in the Model Builder.)
First, we take a look at the Terminal settings. As we can see from the screenshot below, the Terminal type is Current, and the applied current is the variable Current
, which will be solved for via the Global Equation.
The current-type Terminal condition with an applied current, which will be controlled by the Global Equation.
The Global Equations settings control the applied current for the Terminal condition.
The settings for the Global Equation are shown in the screenshot above. There is a single equation for the variable Current
, and the equation that must be satisfied is
(Current-1[A])/1[A]
Since this equation, by definition, must equal zero, the applied current equals 1 Amp. This is a straightforward equation; it does not include any feedback from the model, but rather sets the value of Current
. The Global Equation itself is nondimensionalized, since we will also want to satisfy an equation for voltage. Switching to a voltage excitation can be done by simply changing this equation to
(ec.V0_1-3[V])/3[V]
where the variable ec.V0_1
is automatically defined by the Terminal boundary condition.
We are thus applying a current such that the terminal voltage is equal to 3 Volts. This equation does introduce a feedback from the model, but the model is still linear. It will still solve in a single iteration, but it does require using a direct solver. If you try this out yourself, you can see that you can now switch between a voltage and a current excitation simply by changing the Global Equation for a stationary problem. Next, we will look at how you can dynamically switch between these excitations during a transient simulation.
Let’s suppose that we have a power source driving a system that exhibits variable resistance. For example, resistance changes with temperature due to Joule heating and induction heating. Let’s also suppose that, as the resistance changes, our power source can supply a constant current up to some peak voltage, or a constant voltage up to some peak current.
To model this type of switch, we will use the Events interface. We have previously written about the Events interface for implementing a thermostat for a thermal problem and we recommend reviewing that blog post for technical details and relevant solver settings.
The Events interface contains four features: a Discrete States feature, an Indicator States feature, and two Implicit Events features. First, the Discrete States feature defines a single-state variable, CC
, which acts as a flag indicating if the power supply is in constant-current mode, CC=1
, or constant-voltage mode, CC=0
. Initially, our power supply will be in constant-current mode. Next, there is an Indicator States feature defining two indicator-state variables, PeakV
and PeakI
, which should vary smoothly over time. Lastly, there are two Implicit Events features, which will track these two indicator-state variables and change the discrete-state variable, CC
, to zero or one if the logical conditions are met. These settings are all shown in the screenshots below.
The Discrete States feature defines a flag that signals the Terminal state.
The Indicator States feature defines two different possible events indicators.
The Implicit Events toggle the Discrete States variable.
There is only one task left to do: modify the global equation for the variable Current
to be
CC*((Current-1[A])/1[A])+(1-CC)*(ec.V0_1-3[V])/3[V]
You can see that this is a sum of the two expressions developed earlier for either current control or voltage control using the CC
flag to switch between them. Once this is done, it is only a matter of solving in the time domain with the study settings described in our earlier blog post and using a direct solver for the Electric Currents voltage field, the Terminal voltage, and the Global Equation for the current, as shown in the screenshot below. The Events variables can be solved in a separate segregated step.
Solver settings showing how the variables are segregated and that a direct solver is being used.
With these features, we have now implemented the following power supply behavior:
To generate some representative results, we will explicitly make the total resistance of our domain vary over time, as shown in the plot below. We can see from the subsequent plots of current and voltage that the supplied current is initially constant but the voltage rises due to increased resistance. The power supply then switches over to constant-voltage mode and the current varies. As the resistance goes back down, the current rises to its peak and then the power supply switches back to fixed-current mode.
As the device resistance varies over time, the source switches between constant current and constant voltage to ensure that the maximum current and voltage are never exceeded.
We have demonstrated a control scheme that enforces both a current and voltage maximum, implemented with Global Equations and the Terminal boundary condition. The functionality of the Terminal boundary condition used here is not limited to the Electric Currents interfaces. The Terminal condition is also available within the Magnetic and Electric Fields interface as well as the Magnetic Fields interface, where it is called the Boundary Feed or Gap Feed condition. The Magnetic Fields interface also includes a Multi-Turn Coil domain feature that can be used equivalently.
It is also possible to use this type of control scheme during a Frequency-Transient study, which often occurs in Joule or induction heating, wherein the electromagnetic problem is solved in the frequency domain and the thermal problem, which solves for the temperature variations that lead to impedance changes, is solved in the time domain. This can be particularly useful in the modeling of, for example, RF heating and ablation.
The Batteries & Fuel Cells Module, Corrosion Module, Electrochemistry Module, and Electrodeposition Module all contain the Electrode Current and Electrolyte Current boundary conditions that can be used equivalently to the Terminal condition demonstrated here. In particular, the Capacity Fade of a Lithium-Ion Battery tutorial model shows how to model the charging and discharging of a battery.
We hope you can see that with a little bit of imagination, it is possible to implement some quite complex control schemes with the Terminal boundary condition and the Events interface.
If you have a particular application that you’d like to model with COMSOL Multiphysics, please don’t hesitate to contact us.
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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.
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.
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.
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.
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:
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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:
Standard | AT Cut Definition |
---|---|
IRE 1949 | (YXl) 35.25° |
IEEE 1978 | (YXl) -35.25° |
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.
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 |
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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.
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 | |
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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.
Z | X | Z | |
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IRE 1949 Standard | 0° | 54.75° | 0° |
IEEE 1978 Standard | 0° | 125.25° | 0° |
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° |
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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° | |
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1.Orientate the thickness direction (Z_{g}) along the Y-axis of the crystal (Y_{cr}) and the width direction (X_{g}) along the X-axis of the crystal (X_{cr}). | |
2. Rotate the cut by 35.35° about the l- (X_{g}) axis. | |
3. Rotate the cut by 180° about the w- (Y_{g}) 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 | |
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1. Start with the crystal and the global axes aligned. | |
2. Rotate the crystal axes 180° about their Z-axis (Z_{cr}). | |
3. Rotate the crystal axes -54.75° about the new crystal X-axis (X_{cr}). |
Corresponding rotations that determine the Euler angles of the crystal axes with respect to the global axes.
X | Z | X | |
---|---|---|---|
IEEE 1978 Standard: (YXlw) -35.25° 180° | 180° | -54.75° | 0° |
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° |
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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.
Electrostatic actuators offer certain advantages, including the use of lithographic techniques for ease of manufacture, as well as the ability to use any conductive material as a resonator.
Let’s consider the schematic of an electrostatic actuator consisting of a parallel plate capacitor connected to a mechanical spring, as shown in the diagram below. The electrostatic force acting on the capacitor plate is given by
where is the total stored electrical energy in the system composed of the capacitor with capacitance and an ideal battery (not shown) supplying the voltage to the capacitor. Note that this simplistic force expression only applies for a parallel plate capacitor in which one of the plates is rigidly moving up and down while remaining parallel to the fixed plate. It also ignores any edge effects. We will use this simplification to study the behavior of an actuator as it captures the basic concepts of equilibrium and pull-in phenomenon.
Schematic of an electrostatic actuator.
The capacitance can be calculated as , where is the permittivity of the dielectric between the parallel plates and is the area of the plate. The spring represents the elasticity of the moving electrode.
For this 1D analysis, the restoring force in a linear spring is . However, in a general scenario with a large enough displacement, , the relationship between restoring force and displacement is no longer linear. This is conceptually referred to as large deformation or geometric nonlinearity. We will therefore define the restoring mechanical force in the nonlinear spring as . You can learn more about geometric nonlinearity in these previous blog posts: “Modeling Linear Elastic Materials — How Difficult Can It Be?” and “What Is Geometric Nonlinearity?”
The net force on the capacitor plate is given by
(1)
At equilibrium, the electrostatic force is balanced by the mechanical force so that . This leads us to the equilibrium equation
(2)
where is the equilibrium displacement.
We can then solve Equation (1) for the equilibrium displacement .
A pull-in point is defined as the equilibrium point beyond which the system becomes unstable. Practically, this means that for values of voltage greater than the voltage corresponding to the pull-in point (pull-in voltage), the plates will snap together.
Estimating this pull-in voltage is an important aspect of designing microresonators. To estimate the pull-in voltage, we will need to analyze the stability of an equilibrium point. For simplicity, we will take the case of a linear spring for estimating the pull-in point. As per the theory of stability of an equilibrium point,
From Equation (2), for the case of a linear spring (small deformation analysis), the voltage at the equilibrium point can be expressed as
(3)
where .
Evaluating at and using Equation (3), we find
Solving the above equation for gives
We note that for , and for , . This indicates that is our pull-in displacement and for values of , the system becomes unstable. Using Equation (3), we also find that voltage has a maxima at . The plot below shows the relationship between and .
The pull-in voltage is given by
In the above derivations for pull-in displacement and pull-in voltage, we assume a linear spring. Inclusion of geometric nonlinearity will result in some changes in these values.
Let’s take a look at the forces on the capacitor plate about an equilibrium point . For a small displacement, , around the equilibrium point , the total force acting on the parallel plate capacitor is
For a small enough displacement, , we can perform a series expansion about . Keeping terms to only a second order in , we get
(4)
Using Equation (2) to cancel out equilibrium terms, we can rewrite the force as
is the stiffness corresponding to the spring force and is the stiffness corresponding to the electrostatic force. acts in a way to reduce the mechanical stiffness. This is known as the spring softening phenomenon. The value of , in the case shown here, increases with an increase in . This is known as stress stiffening. Such an effect is a result of geometric nonlinearity and is discussed in this previous blog post.
In the time-harmonic response analysis of an electrostatic actuator, the actuation voltage can be considered a sum of a DC bias voltage and an AC signal, so that , where and is the perturbation amplitude. The electrostatic force is given by
where is the equilibrium point given by solving Equation (2) with . We will consider a linear frequency response analysis. For a linear harmonic analysis, the displacements will also be of the form . Expanding the above equation, we get
We can also expand the term about the equilibrium point to give
Keeping only the terms to a first order in perturbation amplitude and frequency , we get
The electrostatic force is thus a sum of a DC force and a time-harmonic force at the excitation frequency. Note that in this derivation, we are ignoring the small DC component proportional to and a force component at twice the excitation frequency. We can similarly derive the expression for the mechanical force for linear time-harmonic analysis with a DC bias.
We have discussed the theory behind the principles of electrostatic actuation using the case of a spring connected to a parallel plate capacitor to analyze the equilibrium, pull-in phenomenon, and time-harmonic response in an electrostatic actuator. We have also discussed the principles of electrostatic spring softening and stress stiffening that occur in an electrostatic actuator.
While the principles discussed in this blog post are important for understanding the fundamentals of an actuator, the expressions derived above cannot be used for an accurate analysis of an actual 3D microresonator. These are based on simplifications and approximations enabling a lumped analysis. Finite element analysis would give us a more accurate picture of the equilibrium, pull-in, and time-harmonic analysis of a microresonator.
In an upcoming blog post, we will take a look at how we can model microresonators using the dedicated Electromechanics physics interface in COMSOL Multiphysics.
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