One of the basic tasks of any navigation system is to keep track of an object’s position and orientation, as well as their rates of change. Extreme accuracy may be required, particularly in space travel. For example, a communications satellite can be sensitive to angular velocities as small as one thousandth of a degree per hour.
While this accuracy requirement may seem daunting, this fundamental task of attitude control can be posed as a simple question: How do I determine how fast I’m spinning, and about what axis?
In principle, this task is the same for any observer in any rotating frame of reference — even a guest in the revolving restaurant shown below.
A photograph of the revolving restaurant at Ambassador Hotel, the oldest revolving restaurant in India. Image by AryaSnow — Own work. Licensed under CC BY-SA 4.0, via Wikimedia Commons.
Suppose you’re the guest in such a rotating restaurant, trying to determine its angular velocity, Ω (unit: rad/s).
The simplest approach is to look outside. Pick a stationary object, like a building or a tree, and see if its location is changing over time in your field of view.
The above image shows the position of a tree in the observer’s field of view (i.e., through the window) at an initial time t_{1} and a later time t_{2}. Let θ (unit: rad) be the angle between these two lines of sight. If the tree is very far away compared to the size of the restaurant, you could estimate the angular velocity as
Because space travel is so much more demanding than the restaurant example described above, there are a few caveats to consider. In space, the idea of a “stationary object” is a bit tricky. For example, when using a Sun sensor for attitude control of a satellite in a geostationary orbit, you must also account for the relative motion as the earth orbits around the sun. A star sensor, on the other hand, can be extremely accurate because stars other than the sun can be considered fixed in space for many purposes, and because a star more closely approximates a point of light rather than a continuous source over some finite angle.
Because of the accuracy requirements of spacecraft attitude detection and control, the finite size of the object being observed must also be considered. For a line of sight to the sun, for example, you need to know what part of the sun you’re looking at. For arbitrary rotation in 3D, at least two objects are needed, because you don’t always know how the axis of rotation is oriented.
Next, suppose we’re back at the restaurant, except that all of the windows are covered. Because your view of the outdoors is blocked, you can’t rely on any stationary object to get information about your rotating frame of reference.
There are several experiments you could perform within your rotating frame of reference to determine its angular velocity. For example, you could put a ball on the floor and see if it rolls, presumably due to a centrifugal force. (This requires that you know where the axis of rotation is — not always a given in space travel!) Another approach would be to use a mechanical gyroscope.
A third approach, explained in the following section, is to exploit the unique properties of light; namely, its uniform speed in a vacuum in all frames of reference. When light propagates in a rotating frame of reference, it reveals a phenomenon known as the Sagnac effect. A ring laser gyroscope takes advantage of this effect. Such gyroscopes have become a popular alternative to traditional mechanical gyroscopes, which use rotating masses, because the ring laser gyro has no moving parts, thus reducing the cost of maintenance.
The easiest way to visualize the Sagnac effect is to consider two counterpropagating light rays — that is, two rays going in opposite directions — that are constrained to move in a ring. The ring is rotating counterclockwise with a constant angular velocity Ω. (The SI unit is radians per second, but for inertial navigation systems, we might work in degrees per hour instead.)
The two rays are initially released at a point P_{0} along the ring. The rays go around the ring at the speed of light in opposite directions, while the release point rotates with the frame of reference. By the time the clockwise ray returns to the release position, it has moved to a new location shown by P_{1}, and the distance it has traveled is somewhat less than one full circle. By the time the counterclockwise ray returns to the release position, it has moved to a different location, P_{2}, and the distance it has traveled is greater than one full circle.
Of course, the movement shown here is greatly exaggerated. In reality, the displacement of P_{1} and P_{2} from P_{0} (and from each other) might be 10 billion times smaller! Even then, the tiny difference in the distance traveled (and similarly, the transit time) between the two rays is detectable because it’s accompanied by a phase shift, which produces an interference pattern between the rays. If we let ΔL represent the difference in the distance traveled by the two rays, then
(1)
where A is the area of the ring and c_{0} = 299,792,458 m/s is the speed of light in a vacuum.
As it turns out, Eq. (1) isn’t just true for circular paths, but for other shapes as well. The optical path difference only depends on the area enclosed by the loop and not by its shape. A more general derivation of Eq. (1) can be accomplished using the principles of general relativity. At its core, the Sagnac effect is a relativistic phenomenon, for which a classical derivation gives the same results to first order. For a more rigorous application of the theory, see Refs. 1–2.
In this section, we examine a model of a basic Sagnac interferometer. This shares the same fundamental operating principle as the ring laser gyro, but is simpler to set up because we don’t need to consider the presence of a lasing medium along the beam path. (Besides the intensity gain, such a lasing medium can introduce many other complications, such as dispersive effects, that we can ignore for illustrative purposes.) However, a Sagnac interferometer with a given geometry will introduce the same optical path difference and phase delay as a ring laser gyro with the same arrangement of mirrors, so we can still learn quite a lot from it.
The basic Sagnac interferometer geometry consists of a beam splitter, two mirrors, and an obstruction to absorb the outgoing rays. It is illustrated below.
A few geometry parameters for this model are tabulated below.
Name | Expression | Value | Description |
---|---|---|---|
λ_{0} | N/A | 632.8 nm | Vacuum wavelength |
R | N/A | 10 cm | Ring radius |
b | 17.3 cm | Triangle side length | |
P | 52.0 cm | Triangle perimeter | |
A | 130 cm^{2} | Triangle area |
The geometry is sometimes designed in a square rather than a triangle, with mirrors at three vertices and a beam splitter at the other. Rays are traced through the system with directions indicated by arrows. Because the whole apparatus is rotating counterclockwise, the rays going counterclockwise propagate a slightly longer distance than the rays going clockwise, before reaching the obstruction.
To better visualize this phenomenon, see the two animations below. (Note again that the rotation here is exaggerated by a factor of about ten billion!)
In the left animation, the observer stands in an inertial (nonaccelerating) frame of reference. Thus the rays go along straight paths but they hit the mirrors at different times. In the right animation, the observer is “riding” the spacecraft and is thus in a noninertial frame. (Strictly speaking, even in this rotating frame, the counterpropagating rays go at the same speed; the speed of light is the same in any frame of reference!)
For the geometry parameters given above, applying Eq. (1) gives the optical path difference between the counterpropagating rays as about 8 × 10^{-16}, or 0.8 femtometers. This is about the radius of a proton; clearly a difficult quantity to measure! Rather than report path length directly, Sagnac interferometers and ring laser gyros usually report the frequency difference or beat frequency Δν, given by
(2)
where ν (in Hz) is the frequency of the light, and L is the optical path length for light going around the perimeter of the triangle.
Note that L is not necessarily the perimeter of the triangle itself, since there might be a comoving medium such as a lasing medium with n ≠ 1 along the beam path. In this example, we assume the space between the mirrors and beam splitter is a vacuum. The beat frequency is on the order of 1 Hz, which is certainly much easier to measure than a distance equal to the proton radius.
This model uses the Geometrical Optics interface to trace rays through the Sagnac interferometer geometry. The two mirrors are given the dedicated Mirror boundary condition, which causes specular reflection. The beam splitter uses the Material Discontinuity boundary condition with a user-defined reflectance of 0.5, so that both of the counterpropagating beams have the same intensity.
To rotate the apparatus, use the Rotating Domain feature, as shown below.
The resulting plot shows the rays propagating in both directions through the system of mirrors, but because the mirrors move so slowly relative to the speed of light, the two paths are indistinguishable in this image. If we zoomed in by a factor of 10 billion or so, we’d be able to discern two triangles spaced a tiny distance apart.
In the following plot, the beat frequency is given as a function of the angular velocity of the interferometer. As expected from Eqs. (1)–(2), this relationship is linear. Some numerical noise is visible in the bottom-left corner of the plot. This is due to numerical precision and is explained in greater detail in the model documentation.
The Sagnac interferometer described above, along with related devices like ring laser gyros and fiber optic gyros, are examples of inertial navigation systems, which predict an object’s position and orientation by starting from a known position and then integrating the translational and angular velocity over time. In practice, inertial navigation systems are usually combined with absolute measurements of position and orientation relative to some other object in space. This absolute measurement might be done with an Earth sensor, Sun sensor, or star sensor; with RF beacons at known locations on the earth’s surface; with measurements of the earth’s magnetic field; or with any combination of these.
The uncertainty of an inertial navigation system grows over time due to small errors in the measurement of the translational and angular velocity. Periodically taking an absolute measurement using one of the sensors described above resets this uncertainty to a more reasonable value. A prediction of the uncertainty over time might look like the following graph.
We’ve successfully demonstrated the Sagnac effect in a simple interferometer using ray optics simulation. The resulting beat frequency agrees with the more rigorous theory, which is based on general relativity, as long as the velocity of all of the moving parts is much smaller than the speed of light. The magnitude of the optical path difference due in a Sagnac interferometer or ring laser gyro depends only on the area enclosed by the counterpropagating beams, not on the geometry of the loop.
Explore the Sagnac interferometer model by clicking the button below. Once in the Application Gallery, you can log into your COMSOL Access account and download the MPH-file (with a valid software license) as well as a tutorial for this model.
Tunable devices can be realized using varactors, phase shifters, or switches with which the reactance, phase, or path of the signal can be tweaked and the frequency response of the device is changeable. Here, a tunable bandpass filter model is designed with a piezoelectric actuator that controls the reactance of the device and results in a variable resonance frequency of the filter.
The filter design is based on a rectangular cavity filter whose resonance frequencies are given by:
where a and b are the waveguide aperture dimensions and d is the length of the waveguide cavity.
The cavity width, height, and length are a = 100 mm, b = 50 mm, and d = 50 mm, respectively, to generate 3.354 GHz of the resonance frequency for the TE101 dominant mode.
Inside the cavity, a metallic post is added and configured to create a gap between the top surface of the post and ceiling of the cavity, so the height of the post is slightly smaller than b. When the cavity is resonant at the dominant mode, the energy is confined in the center of the cavity and the response of the gap that is located in the middle becomes capacitive. The extra capacitance lowers the resonance frequency by keeping the same structure size, so the device size is effectively reduced as well.
Two shorted 50-Ω microstrip lines, which are terminated by lumped ports, are coupled into the cavity through slots on the top of the cavity. Input matching (S_{11}) and insertion loss (S_{21}) can be improved by adjusting the dimensions and locations of the slots. A circular aperture at the top of the cavity is closed with a piezoelectric actuator, and the bottom surface of the disk is finished with a layer of a material where the conductivity is high enough to have a very small skin depth.
Left: The cavity filter with a piezoelectric actuator shaped like a circular disk. The feed scheme is built with slot-coupled microstrip lines. Right: The gap size between the piezoelectric actuator and metallic post controls the resonance frequency.
All metal parts — e.g., the cavity walls, post, substrate ground planes, microstrip lines, and bottom surface of the piezoelectric device — are set as perfect electric conductors (PECs). Lead zirconate titanate (PZT-5H) is used for the piezoelectric actuator. The actuator is z-polarized, resulting in mainly z-directional deflection of the device.
When a positive DC bias is applied across the piezoelectric actuator, it will deflect toward the bottom of the cavity. This deflection makes the capacitance stronger and shifts the resonance frequency lower than the case without any deformation. The animation below plots the electric field norm at the resonance frequency. At the center of the cavity, as well as in the gap between the top of the post and the bottom of the piezo device, strong electric fields are observed.
The conventional analysis method for this type of device is using a parametric sweep of the height of the metallic post geometry (instead of warping the piezoelectric actuator parabolically) to see the change of the capacitance of the filter. However, in reality, the metallic post is fixed and the actual deformation of the piezoelectric actuator is not geometrically uniform. For this reason, the parametric sweep does not address the capacitance change precisely; thus, the evaluated resonance frequency is not accurate.
To describe the real-world phenomena, the elastic deformation of the actuator and the resulting capacitance change have to be modeled with a multiphysics approach that combines a high-frequency electromagnetic and piezostructural analysis. Using this approach is seamless and intuitive in the COMSOL Multiphysics® software, as it offers you a single simulation platform.
The multiphysics and moving mesh settings in a single simulation platform to model the deformation of the piezoelectric actuator.
The deformation of the piezoelectric actuator is addressed via the combination of a few physics interfaces: the Solid Mechanics (solid), Electrostatics (es), and Moving Mesh (ale) interfaces. When the piezoelectric device deforms due to a positive and negative DC bias, the Moving Mesh interface is used to reconfigure the mesh for the Electromagnetic Waves, Frequency Domain interface, which calculates the wave propagation and resonance behavior in the microstrip lines and cavity.
By changing the resonance frequency, the electric field around 3 GHz is evanescent. The deformation of the piezoelectric actuator is exaggerated for visualization purposes. A surface plot of the electric field norm and an arrow plot of the electric field are also shown in the animation.
With an electric potential of +300 V across the piezoelectric actuator, a deflection of ~90 μm is observed, making the gap smaller and the capacitance in the gap stronger. Thus, the shift of the resonance frequency is lower than the shift at 0 bias as well as at negative bias.
The S-parameters of the tunable cavity filter. The mode uses a DC bias of ±300 V.
The S-parameter plot shows the effect of the piezoelectric device deflection on the filter’s resonance frequency. The tunable frequency range of this example is around 40 MHz. This range can be adjusted by different choices of the piezoelectric disk size and the input bias voltage.
The RF Module, an add-on product to COMSOL Multiphysics, helps you design, build, and optimize RF, microwave, millimeter-wave, and passive THz devices. You can model traditional devices and extend models to include other physics phenomena that are not easily measured in the lab, such as heat effects on material properties as well as structural deformation. All of the physics that you want to include can be efficiently simulated using the same simulation environment and workflow.
To try the tutorial model featured in this blog post, click the button below. Clicking this button will take you to the Application Gallery, where you can log into your COMSOL Access account and then download the MPH-file.
Picture a micromirror as a single string on a guitar. The string is so light and thin that when you pluck it, the surrounding air dampens the string’s motion, bringing it to a standstill.
Because this damping effect is important to many MEMS devices, micromirrors have a wide variety of potential applications. For instance, these mirrors can be used to control optic elements, an ability that makes them useful in the microscopy and fiber optics fields. Micromirrors are found in scanners, heads-up displays, medical imaging, and more. Additionally, MEMS systems sometimes use integrated scanning micromirror systems for consumer and telecommunications applications.
Close-up view of an HDTV micromirror chip. Image by yellowcloud — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
When developing a micromirror actuator system, engineers need to account for its dynamic vibrating behavior and damping, both of which greatly affect the operation of the device. Simulation provides a way to analyze these factors and accurately predict system performance in a timely and cost-efficient manner.
To perform an advanced MEMS analysis, you can combine features in the Structural Mechanics Module and Acoustics Module, two add-on products to the COMSOL Multiphysics simulation platform. Let’s take a look at frequency-domain (time-harmonic) and transient analyses of a vibrating micromirror.
We model an idealized system that consists of a vibrating silicon micromirror — which is 0.5 by 0.5 mm with a thickness of 1 μm — surrounded by air. A key parameter in this model is the penetration depth; i.e., the thickness of the viscous and thermal boundary layers. In these layers, energy dissipates via viscous drag and thermal conduction. The thickness of the viscous and thermal layers is characterized by the following penetration depth scales:
where is the frequency, is the fluid density, is the dynamic viscosity, is the coefficient of thermal conduction, is the heat capacity at constant pressure, and is the nondimensional Prandtl number.
For air, when the system is excited at a frequency of 10 kHz (which is typical for this model), the viscous and thermal scales are 22 µm and 18 µm, respectively. These are comparable to the geometric scales, like the mirror thickness, meaning that thermal and viscous losses must be included. Moreover, in real systems, the mirrors may be located near surfaces or in close proximity to each other, creating narrow regions where the damping effects are accentuated.
The frequency-domain analysis provides insight into the frequency response of the system, including the location of the resonance frequencies, Q-factor of the resonance, and damping of the system.
The micromirror model geometry, showing the symmetry plane, fixed constraint, and torquing force components.
In this example, we use three separate interfaces:
By modeling the detailed thermoviscous acoustics and using the Thermoviscous Acoustics, Frequency Domain interface, we can explicitly include thermal and viscous damping while solving the full linearized Navier-Stokes, continuity, and energy equations. In doing so, we accomplish one of the main goals for this model: accurately calculating the damping experienced by the mirror.
To set up and combine the three interfaces, we use the Acoustics-Thermoviscous Acoustics Boundary and Thermoviscous-Acoustics-Structure Boundary multiphysics couplings. We then solve the model using a frequency-domain sweep and an eigenfrequency study. These analyses enable us to study the resonance frequency of the mirror under a torquing load in the frequency domain.
Let’s take a look at the displacement of the micromirror for a frequency of 10 kHz and when exposed to the torquing force. In this scenario, the displacement mainly occurs at the edges of the device. To view displacement in a different way, we also plot the response at the tip of the micromirror over a range of frequencies.
Micromirror displacement at 10 kHz for phase 0 (left) and the absolute value of the z-component of the displacement field at the micromirror tip (right).
Next, let’s view the acoustic temperature variations (left image below) and acoustic pressure distribution (right image below) in the micromirror for a frequency of 11 kHz. As we can see, the maximum and minimum temperature fluctuations occur opposite to one another and there is an antisymmetric pressure distribution. The temperature fluctuations are closely related to the pressure fluctuations through the equation of state. Note that the temperature fluctuations fall to zero at the surface of the mirror, where an isothermal condition is applied. The temperature gradient near the surface gives rise to the thermal losses.
Temperature fluctuation field within the thermoviscous acoustics domain (left) and the pressure isosurfaces (right).
The two animations below show a dynamic extension of the frequency-domain data using the time-harmonic nature of the solution. Both animations depict the mirror movement in a highly exaggerated manner, with the first one showing an instantaneous velocity magnitude in a cross section and the second showing the acoustic temperature fluctuations. These results indicate that there are high-velocity regions close to the edge of the micromirror. We determine the extent of this region into the air via the scale of the viscous boundary layer (viscous penetration depth). We can also identify the thermal boundary layer or penetration depth using the same method.
Animation of the time-harmonic variation in the local velocity.
Animation of the time-harmonic variation in the acoustic temperature fluctuations.
When the problem is formulated in the frequency domain, eigenmodes or eigenfrequencies can also be identified. From the eigenfrequency study (also performed in the model), we can determine the vibrating modes, shown in the animation below (only half the mirror is shown as symmetry applies). Our results show that the fundamental mode is around 10.5 kHz, with higher modes at 13.1 kHz and 39.5 kHz. The complex value of the eigenfrequency is related to the Q-factor of the resonance and thus the damping. (This relationship is discussed in detail in the Vibrating Micromirror model documentation.)
Animation of the first three vibrating modes of the micromirror.
As of version 5.3a of the COMSOL® software, a different take on this example solves for the transient behavior of the micromirror. Using the same geometry, we extend the frequency-domain analysis into a transient analysis. To achieve this, we swap the frequency-domain interfaces with their corresponding transient interfaces and adjust the settings of the transient solver. In the simulation, the micromirror is actuated for a short time and exhibits damped vibrations.
The resulting model includes some of the most advanced air and gas damping mechanisms that COMSOL Multiphysics has to offer. For instance, the Thermoviscous Acoustics, Transient interface generates the full details for the viscous and thermal damping of the micromirror from the surrounding air.
In addition, by coupling the transient perfectly matched layer capabilities of pressure acoustics to the thermoviscous acoustics domain, we can create efficient nonreflecting boundary conditions (NRBCs) for this model in the time domain.
Let’s start with the displacement results. The 3D results (left image below) visualize the displacement of the micromirror and the pressure distribution at a given time. We also generate a plot (right image below) to illustrate the damped vibrations caused by thermal and viscous losses. The green curve represents the undamped response of the micromirror when the surrounding air is not coupled to the mirror movement. The time-domain simulations make it possible to study transients of the system, like the decay time, and the response of the system to an anharmonic forcing.
Micromirror displacement and pressure distribution (left) and the transient evolution of the mirror displacement (right).
We can also examine the acoustic temperature variations surrounding the micromirror. The isothermal condition at the micromirror surface produces an acoustic thermal boundary layer. As with the frequency-domain example, the highest and lowest temperatures are located opposite to one another.
In addition, by calculating the acoustic velocity variations of the micromirror, we see that a no-slip condition at the micromirror surface results in a viscous boundary layer.
Acoustic temperature variations (left) as well as acoustic velocity variations for the x-component (center) and z-component (right).
These examples demonstrate that we can analyze micromirrors using advanced modeling features available in the Acoustics Module in combination with the Structural Mechanics Module. For more details on modeling micromirrors, check out the tutorials below.
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Strain gauges measure how structures — both manmade and biological — react to an applied strain. These devices are common in the mechanical and civil engineering fields for monitoring the structural health of bridges, detecting soil pressure changes near oil drilling platforms, and testing aircraft components. Strain gauges can even be used to analyze bone structure in humans and animals.
A mechanical strain gauge used to measure the growth of cracks on the Hudson-Athens Lighthouse in New York. Image by Roy Smith. Licensed under CC BY-SA 2.5, via Wikimedia Commons.
In terms of performance, MEMS-based gauges have several advantages over standard foil strain gauges. For one, MEMS sensors have higher strain sensitivity than foil sensors, resulting in more accurate measurements. MEMS-based gauges also have higher fracture strength and can withstand high operating and bonding temperatures, which makes them more durable than their foil counterparts and expands their range of applications.
DETF strain gauges have their own set of distinct advantages:
To optimize the design of a new DETF sensor, researchers from the School of Engineering Technology at Purdue University used MEMS simulation.
The research team created a 3D geometry of a strain gauge with the MEMS Module, an add-on to COMSOL Multiphysics. The model consists of a DETF — including the beams, base, and anchors — and an electrostatic comb drive. The researchers wanted to validate the simulation results, so they used the dimensions of an analytical model when setting up the model’s components.
The DETF strain gauge. Image courtesy A. Bardakas, H. Zhang, and D. Leon-Salas.
The research team used several of the built-in capabilities of the MEMS Module when modeling the strain gauge. The Thin-Film Damping feature was used to compute the forces between solid surfaces and the surrounding air. This effect accounts for the main cause of damping in the DETF.
The team also used the MEMS Module to set up a prestressed frequency analysis. This is important for many MEMS devices, as it helps determine the initial frequency of the device and how the frequency shifts once a load is applied.
Using a parametric sweep, the researchers determined how a range of applied forces affect the gauge without having to manually change the value and recompute the model each time. This enabled them to optimize their design more efficiently. Based on the results of the parametric sweep, the team modified the geometry to ensure that the device could accommodate a range of forces.
The resonance frequency of the DETF gauge was computed via two methods. Using a frequency analysis, the researchers found a resonance frequency of 84.060 kHz for the model. This is slightly higher than the frequencies found using a fundamental mode analysis (83.263 and 83.271 kHz). This difference is likely because a denser mesh was used for the mode analysis.
Resonance frequency of the DETF strain gauge for different mode shapes. Image courtesy A. Bardakas, H. Zhang, and D. Leon-Salas
The team used the model to optimize the DETF strain gauge design by balancing its sensitivity and structural integrity. Next, they plan to use a two-mask silicon-on-insulator process to fabricate the design. In addition, the researchers plan to investigate strain loss in the device via further analyses and experiments.
A Kelvin probe can determine the contact potential difference (or work function difference) of materials. This type of probe, which is based on a time-varying capacitor, works by using two electrodes:
When these electrodes are electrically connected, the Fermi levels reach equilibrium. The electrons from the material with the lower work function flow into the material with the higher work function. This flow creates a contact potential difference, charging the capacitor. Additionally, the vibrations in the movable electrode change the electrical energy stored inside the capacitor and create a flowing electrical current. By measuring this current, we can determine the work function difference. It’s also possible to establish a compensated operation and make the current zero by applying an external bias voltage. This approach results in a bias voltage equal to the work function difference between the two electrode materials.
Among other things, Kelvin probes are used for Kelvin probe force microscopy (KPFM), a noncontact technique for finding the work function of surfaces at molecular and atomic scales. This technique is often used to study the nanoscale electrical properties of metal and semiconductor surfaces and devices. KPFM can also be used to examine the electrical properties of organic materials and devices.
Kelvin probe force microscopy. Image by Inkwina — Own work. Licensed under CC BY-SA 4.0, via Wikimedia Commons.
Of course, Kelvin probes are not only variable in terms of their application but also in their design. For instance, the movable electrode has a few potential shapes, such as a circular plate and a narrow tip. Having a circular plate enables the probe to form a parallel plate capacitor. This type of capacitor produces a large capacitance change and current that can be measured without needing a complex electrical compensation scheme. When the probe has a narrow tip, it can scan an area and generate a lateral resolution. This option requires an electrical compensation circuit because the capacitance change and resulting current are low.
To analyze different probe geometries and their sensitivity, a research team from the Institute for Fast Mechatronic Systems at Heilbronn University and the Faculty of Electrical Engineering at the University of Applied Science used COMSOL Multiphysics simulations. Their goal was to study the capacitance characteristics of a Kelvin probe and determine optimal geometries.
The research team’s 2D axisymmetric Kelvin probe model includes both a vibrating and fixed electrode. The movable electrode (located above the fixed electrode) has a prescribed harmonic displacement and defined contact potential difference. In addition, its movement is set up as a sine wave. The fixed electrode is fixed mechanically and set to an electrical ground.
Simulation is ideal for analyzing Kelvin probes because models of analytical plate capacitors become more inaccurate as their radii are scaled down to the submillimeter domain.
2D Kelvin probe geometry. Image by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
The researchers used this model to perform parametric sweeps to test various lateral dimensions and lengths of the movable electrode. They also analyzed different tip geometries. Let’s take a look at the results…
The stepwise simulations show individual effects when the probe radii and lengths are tested separately. Regarding the lateral dimensions, different movable electrode radii are studied while the length is kept constant. The simulations show that, as mentioned above, the simple analytical formula for parallel plate capacitors becomes less accurate for smaller radii, making numerical analysis indispensable.
Nominal capacitance (left) and capacitance change (right) for different radii. Images by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
In the probe length simulations, different lengths are tested with a constant radius. These simulations show that the capacitance change and the nominal capacitance vary with the length in a different way. For a probe length less than about 2 mm, the capacitance change shows a significant decrease. This result indicates that the Kelvin probe needs to maintain a minimum length.
Nominal capacitance (left) and capacitance change (right) for different lengths. Images by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
So far, we’ve only looked at the simulation results for the probe radius and length separately. However, the research shows that reducing the probe’s radius and length at the same time dramatically increases the individual effects on the probe.
The nominal capacitance (left) and capacitance change (right) of the movable electrode as a function of the length and radius. Images by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
To learn more about these geometry changes, the researchers performed additional postprocessing studies. The results reveal that each geometry must be reviewed separately due to the lack of fixed proportion between length and radius for analytical simplifications.
To find the optimal tip geometry for Kelvin probes of different sizes, the researchers compared three different tip geometries:
The tip geometries lead to a variety of electric energy densities, causing them to generate different capacitance behaviors.
Flat (left), round (middle), and spiky (right) tip geometries. Images by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
In this analysis, the length is kept constant at 2 mm and the radius is swept. To generate more precise calculations for the different geometries while still allowing for sine movement and parametric sweeps, the model includes various airspace domains for ease of meshing, such as the one shown below for the round tip.
The airspace meshing for a round tip geometry. Image by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
The simulations show that the flat tip design has the highest nominal capacitance and capacitance change. Additionally, the capacitance values become closer for all three geometries when the tips have smaller lateral dimensions. Note that at these small scales, using narrow tips becomes less important.
Nominal capacitance (left) and capacitance changes (right) for different tips. Images by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
As shown below, the electric potential for all three geometries is comparable in the close gap. Stray capacitances are accounted for and found to only have a small influence on the Kelvin probe.
Electric potential for a flat (left), round (middle), and spiky (right) tip geometry. Images by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
A final point of study is determining a suitable equivalent circuit. Here, a simple load resistor is added to the model to study a complete Kelvin probe setup.
An equivalent electrical circuit of a Kelvin probe, used for a simple voltage measurement. Image by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
The capacitance characteristics are calculated for a Kelvin probe with an electrical circuit and a voltage drop over the load resistor. When compared to a SPICE model based on the analytical equations of a plate capacitor, these results show a good correlation.
Comparison of the COMSOL Multiphysics simulation and SPICE model for the characteristic transient capacitance behavior of a Kelvin probe with an electrical circuit (left) and for the transient voltage drop over the load resistor (right). Images by S. Ciba, A. Frey, and I. Kuehne and taken from their COMSOL Conference 2016 Munich paper.
The research team’s results offer insight into ways to optimize the geometry of a Kelvin probe. In the future, the researchers plan to design a highly sensitive electrical circuit to further optimize the probe’s performance.
The trend of miniaturization is one that we can see in a variety of applications, including mobile phones and computers. The same can be said for the design of satellites used in space missions. The devices used in NASA’s Space Technology 5 (ST5) mission are just one example.
Microsatellites mounted on a payload structure for the ST5 mission. Image by NASA. Licensed under the public domain, via Wikimedia Commons.
Due to the payload complexity of microsatellites — and the desire to extend their reach outside of Earth’s orbit — active thermal control is very important. Such control demands more power and also increases the mass of the satellite with added parts. The challenge is to design a thermal control system that can meet these power and mass demands while still removing excess heat in a controlled manner.
With this in mind, NASA used electrostatic comb drives for actuation in their ST5 mission. These actuation systems were paired with two different radiator designs: a louvre and a shutter configuration. The mission helped to validate the use of high-voltage MEMS technology in thermal subsystems.
Left: The shutter concept. Right: An optical microscope image of the shutter radiator design. Images by L. Pasqualetto Cassinis and taken from his COMSOL Conference 2016 Munich paper.
Looking to build upon these initial findings, a researcher from TU Delft considered an alternative to using electrostatic comb drives: thermal actuators. These devices provide relatively high displacement with little applied voltage and are less sensitive to radiation than their electrostatic counterparts. To validate their potential in such applications and further optimize their design, the researcher turned to the COMSOL Multiphysics® software.
For this analysis, two models were built in COMSOL Multiphysics. The first is a 3D structural model of the shutter array, a configuration chosen based on its robustness.
3D shutter array model. Image by L. Pasqualetto Cassinis and taken from his COMSOL Conference 2016 Munich presentation.
The second is a 3D multiphysics model of a two-arm thermal actuator made of polysilicon — a model based on the Joule Heating of a Microactuator tutorial. An applied voltage generates electric current through the two hot arms, raising the temperature of the actuator. This temperature increase leads to thermal expansion, which then causes the actuator to bend. In addition to these hot arms, the thermal actuator includes a cold arm, with a gap that separates the two types. Note that the hot arms have more electrical resistance than the cold arm, thus greater Joule heating.
Thermal actuator model geometry. This image is taken from the documentation for the Joule Heating of a Microactuator tutorial.
To validate the thermal actuator model, the researcher compared the simulation results with analytical results and checked if the output displacement was close to the requirement of 3 µm. In the model, the displacement is 2.54 µm — a value comparable to that of analytical results (2.11 µm) and also near the required displacement. Note that the theoretical model only includes one hot arm, which can account for some of the differences in displacement values. Further, the simulation shows agreement in regard to temperature distribution, with the highest temperature at the center of the actuator.
A spring-like force is added to the shutter model to account for stiffness. With varying forces applied to the device, the shutter exhibits elastic behavior. The estimated stiffness obtained via the study is incorporated into the thermal actuator model. When varying the voltage to evaluate tip displacement via actuation, high voltages are needed to produce reasonable displacement. Additionally, as expected, the maximum displacement occurs at the center of the actuator instead of the tip.
After verifying the thermal actuator model, the researcher sought to optimize its configuration. In this optimization study, the length of the actuator is varied along with the gap between the hot arms and the cold arm. Per analytical results, both variables are assumed to have a strong impact on tip displacement.
In the initial optimization study, an applied voltage of 2.7 V produces a shutter stiffness of 10^{9} N/m^{3} and a displacement of 2.98 µm. Additionally, the maximum temperature that the device reaches is significantly lower than the melting temperature of silicon.
The displacement (left) and temperature (right) of the thermal actuator with an applied voltage of 2.7 V. Images by L. Pasqualetto Cassinis and taken from his COMSOL Conference 2016 Munich presentation.
Reducing the required applied voltage was the focus of a later optimization study. Just a few volts can be crucial in, for instance, applications of CubeSats — a type of miniaturized satellite used for space research — where power demand is limited. For this study, multiple objective variables are considered and the gap between arms is included as a control variable. With this approach, the displacement comes closer to 3 µm and the applied voltage is reduced to about 2.5 V.
Advancing the design of miniaturized satellites is key to extending their use in space exploration. As we’ve highlighted with this thermal actuator example, simulation is a useful tool for testing active thermal control techniques in these systems, improving their safety and reach. We look forward to seeing how this technology will continue to advance in the future and the potential role that simulation will play.
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.
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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.