Phononic crystals are rather unique materials that can be engineered with a particular band gap. As the demand for these materials continues to grow, so does the interest in simulating them, specifically to optimize their band gaps. COMSOL Multiphysics, as we’ll show you here, can be used to perform such studies.
A phononic crystal is an artificially manufactured structure, or material, with periodic constitutive or geometric properties that are designed to influence the characteristics of mechanical wave propagation. When engineering these crystals, it is possible to isolate vibration within a certain frequency range. Vibration within this selected frequency range, referred to as the band gap, is attenuated by a mechanism of wave interferences within the periodic system. Such behavior is similar to that of a more widely known nanostructure that is used in semiconductor applications: a photonic crystal.
Optimizing the band gap of a phononic crystal can be challenging. We at Veryst Engineering have found COMSOL Multiphysics to be a valuable tool in helping to address such difficulties.
When it comes to creating a band gap in a periodic structure, one way to do so is to use a unit cell composed of a stiff inner core and a softer outer matrix material. This configuration is shown in the figure below.
A schematic of a unit cell. The cell is composed of a stiff core material and a softer outer matrix material.
Evaluating the frequency response of a phononic crystal simply requires an analysis of the periodic unit cell, with Bloch periodic boundary conditions spanning a range of wave vectors. It is sufficient to span a relatively small range of wave vectors covering the edges of the so-called irreducible Brillouin zone (IBZ). For rectangular 2D structures, the IBZ (shown below) spans from Γ to X to M and then back to Γ.
The irreducible Brillouin zone for 2D square periodic structures.
The Bloch boundary conditions (known as the Floquet boundary conditions in 1D), which constrain the boundary displacements of the periodic structure, are as follows:
where k_{F} is the wave vector.
The source and destination are applied once to the left and right edges of the unit cell and once to the top and bottom edges. This type of boundary condition is available in COMSOL Multiphysics. Due to the nature of the boundary conditions, a complex eigensolver is needed. The system of equations, however, is Hermitian. As such, the resulting eigenvalues are real, assuming that no damping is incorporated into the model. The COMSOL software makes this step rather easy, as it automatically handles the calculation.
We set up our eigensolver analysis as a parametric sweep involving one parameter, k, which varies from 0 to 3. Here, 0 to 1 defines a wave number spanning the Γ-X edge, 1 to 2 defines a wave number spanning the X-M edge, and 2 to 3 defines a wave number spanning the diagonal M-Γ edge of the IBZ. For each parameter, we solve for the lowest natural frequencies. We then plot the wave propagation frequencies at each value of k. A band gap appears in the plot as a region in which no wave propagation branches exist. Aside from very complex unit cell models, completing the analysis takes just a few minutes. We can therefore conclude that this approach is an efficient technique for optimization if you are targeting a certain band gap location or if you want to maximize band gap width.
To illustrate such an application, we model the periodic structure shown above, with a unit cell size of 1 cm × 1 cm and a core material size of 4 mm × 4 mm. The matrix material features a modulus of 2 GPa and a density of 1000 kg/m^{3}. The core material, meanwhile, has a modulus of 200 GPa and a density of 8000 kg/m^{3}. The figure below shows no wave propagation frequencies in the range of 60 to 72 kHz.
The frequency band diagram for selected unit cell parameters.
To demonstrate the use of the band gap concept for vibration isolation, we simulate a structure consisting of 11 x 11 cells from the periodic structure analyzed above. These cells are subjected to an excitation frequency of 67.5 kHz (in the band gap).
The structure used to illustrate vibration isolation for an applied frequency in the band gap.
The animation below highlights the response of the cells. From the results, we can gather how effective the periodic structure is at isolating the rest of the structure from the applied vibrations. The vibration isolation is still practically efficient, even if fewer periodic cells are used.
An animation of the vibration response at 67.5 kHz.
Note that at frequencies outside of the band gap, the periodic structure does not isolate the vibrations. These responses are depicted in the figures below.
The vibration response at frequencies outside of the band gap. Left: 27 kHz. Right: 88 kHz.
To learn more about the 2D band gap model presented here, head over to the COMSOL Exchange, where it is available for download.
Electrical cables, also called transmission lines, are used everywhere in the modern world to transmit both power and data. If you are reading this on a cell phone or tablet computer that is “wireless”, there are still transmission lines within it connecting the various electrical components together. When you return home this evening, you will likely plug your device into a power cable to charge it.
Various transmission lines range from the small, such as coplanar waveguides on a printed circuit board (PCB), to the very large, like high voltage power lines. They also need to function in a variety of situations and conditions, from transatlantic telegraph cables to wiring in spacecraft, as shown in the image below. Transmission lines must be specially designed to ensure that they function appropriately in their environments, and may also be subject to further design goals, including required mechanical strength and weight minimization.
Transmission wires in the payload bay of the OV-095 at the Shuttle Avionics Integration Laboratory (SAIL).
When designing and using cables, engineers often refer to parameters per unit length for the series resistance (R), series inductance (L), shunt capacitance (C), and shunt conductance (G). These parameters can then be used to calculate cable performance, characteristic impedance, and propagation losses. It is important to keep in mind that these parameters come from the electromagnetic field solutions to Maxwell’s equations. We can use COMSOL Multiphysics to solve for the electromagnetic fields, as well as consider multiphysics effects to see how the cable parameters and performance change under different loads and environmental conditions. This could then be converted into an easy-to-use app, like this example that calculates the parameters for commonly used transmission lines.
Here, we examine a coaxial cable — a fundamental problem that is often covered in a standard curriculum for microwave engineering or transmission lines. The coaxial cable is so fundamental that Oliver Heaviside patented it in 1880, just a few years after Maxwell published his famous equations. For the students of scientific history, this is the same Oliver Heaviside who formulated Maxwell’s equations in the vector form that we are familiar with today; first used the term “impedance”; and helped develop transmission line theory.
Let us begin by considering a coaxial cable with dimensions as shown in the cross-sectional sketch below. The dielectric core between the inner and outer conductors has a relative permittivity (\epsilon_r = \epsilon' -j\epsilon'') of 2.25 – j*0.01, a relative permeability (\mu_r) of 1, and a conductivity of zero, while the inner and outer conductors have a conductivity (\sigma) of 5.98e7 S/m.
The 2D cross section of the coaxial cable, where we have chosen a = 0.405 mm, b = 1.45 mm, and t = 0.1 mm.
A standard method for solving transmission lines is to assume that the electric fields will oscillate and attenuate in the direction of propagation, while the cross-sectional profile of the fields will remain unchanged. If we then find a valid solution, uniqueness theorems ensure that the solution we have found is correct. Mathematically, this is equivalent to solving Maxwell’s equations using an ansatz of the form \mathbf{E}\left(x,y,z\right) = \mathbf{\tilde{E}}\left(x,y\right)e^{-\gamma z}, where (\gamma = \alpha + j\beta) is the complex propagation constant and \alpha and \beta are the attenuation and propagation constants, respectively. In cylindrical coordinates for a coaxial cable, this results in the well-known field solution of
which then yields the parameters per unit length of
where R_s = 1/\sigma\delta is the sheet resistance and \delta = \sqrt{2/\mu_0\mu_r\omega\sigma} is the skin depth.
While the equations for capacitance and shunt conductance are valid at any frequency, it is extremely important to point out that the equations for the resistance and inductance depend on the skin depth and are therefore only valid at frequencies where the skin depth is much smaller than the physical thickness of the conductor. This is also why the second term in the inductance equation, called the internal inductance, may be unfamiliar to some readers, as it can be neglected when the metal is treated as a perfect conductor. The term represents inductance due to the penetration of the magnetic field into a metal of finite conductivity and is negligible at sufficiently high frequencies. (The term can also be expressed as L_{Internal} = R/\omega.)
For further comparison, we can compute the DC resistance directly from the conductivity and cross-sectional area of the metal. The analytical equation for the DC inductance is a little more involved, and so we quote it here for reference.
Now that we have values for C and G at all frequencies, DC values for R and L, and asymptotic values for their high-frequency behavior, we have excellent benchmarks for our computational results.
When setting up any numerical simulation, it is important to consider whether or not symmetry can be used to reduce the model size and increase the computational speed. As we saw earlier, the exact solution will be of the form \mathbf{E}\left(x,y,z\right) = \mathbf{\tilde{E}}\left(x,y\right)e^{-\gamma z}. Because the spatial variation of interest is primarily in the xy-plane, we just want to simulate a 2D cross section of the cable. One issue, however, is that the 2D governing equations used in the AC/DC Module assume that the fields are invariant in the out-of-plane direction. This means that we will not be able to capture the variation of the ansatz in a single 2D AC/DC simulation. We can find the variation with two simulations, though! This is because the series resistance and inductance depend on the current and energy stored in the magnetic fields, while the shunt conductance and capacitance depend on the energy in the electric field. Let’s take a closer look.
Since the shunt conductance and capacitance can be calculated from the electric fields, we begin by using the Electric Currents interface.
Boundary conditions and material properties for the Electric Currents interface simulation.
Once the geometry and material properties are assigned, we assume that the conductors are equipotential (a safe assumption, since the conductivity difference between the conductor and the dielectric will generally be near 20 orders of magnitude) and set up the physics by applying V_{0} to the inner conductor and grounding the outer conductor to solve for the electric potential in the dielectric. The above analytical equation for capacitance comes from the following more general equations
where the first equation is from electromagnetic theory and the second from circuit theory.
The first and second equations are combined to obtain the third equation. By inserting the known fields from above, we obtain the previous analytical result for C in a coaxial cable. More generally, these equations provide us with a method for obtaining the capacitance from the fields for any cable. From the simulation, we can compute the integral of the electric energy density, which gives us a capacitance of 98.142 pF/m and matches with theory. Since G and C are related by the equation
we now have two of the four parameters.
At this point, it is also worth reiterating that we have assumed that the conductivity of the dielectric region is zero. This is typically done in the textbook derivation, and we have maintained that convention here because it does not significantly impact the physics — unlike our inclusion of the internal inductance term discussed earlier. Many dielectric core materials do have a nonzero conductivity and that can be accounted for in simulation by simply updating the material properties. To ensure that proper matching with theory is maintained, the appropriate derivations would need to be updated as well.
In a similar fashion, the series resistance and inductance can be calculated through simulation using the AC/DC Module’s Magnetic Fields interface. The simulation setup is straightforward, as demonstrated in the figure below.
The conductor domains are added to a Single-Turn Coil node with the Coil Group feature, and the reversed current direction option ensures that the direction of current through the inner conductor is the opposite of the outer conductor, as indicated by the dots and crosses. The single-turn coil will account for the frequency dependence of the current distribution in the conductors, as opposed to the arbitrary distribution shown in the figure.
We refer to the following equations, which are the magnetic analog of the previous equations, to calculate the inductance.
To calculate the resistance, we use a slightly different technique. First, we integrate the resistive loss to determine the power dissipation per unit length. We can then use the familiar P = I_0^2R/2 to calculate the resistance. Since R and L vary with frequency, let’s take a look at the calculated values and the analytical solutions in the DC and high-frequency (HF) limit.
“Analytic (DC)” and “Analytic (HF)” refer to the analytical equations in the DC and high-frequency limits, respectively, which were discussed earlier. Note that these are both on log-log plots.
We can clearly see that the computed values transition smoothly from the DC solution at low frequencies to the high-frequency solution, which is valid when the skin depth is much smaller than the thickness of the conductor. We anticipate that the transition region will be approximately located where the skin depth and conductor thickness are within one order of magnitude. This range is 4.2e3 Hz to 4.2e7 Hz, which is exactly what we see in the results.
Now that we have completed the heavy lifting to calculate R, L, C, and G, there are two other significant parameters that can be determined. They are the characteristic impedance (Z_{c}) and complex propagation constant (\gamma = \alpha + j\beta), where \alpha is the attenuation constant and \beta is the propagation constant.
In the figure below, we see these values calculated using the analytical formulas for both the DC and high-frequency regime as well as the values determined from our simulation. We have also included a fourth line: the impedance calculated using COMSOL Multiphysics and the RF Module, which we will discuss shortly. As can be seen, our computations agree with the analytical solutions in their respective limits, as well as yielding the correct values through the transition region.
A comparison of the characteristic impedance, determined using the analytical equations and COMSOL Multiphysics. The analytical equations plotted are from the DC and high-frequency (HF) equations discussed earlier, while the COMSOL Multiphysics results use the AC/DC and RF Modules. For clarity, the width of the “RF Module” line has been intentionally increased.
Electromagnetic energy travels as waves, which means that the frequency of operation and wavelength are inversely proportional. As we continue to solve at higher and higher frequencies, we need to be aware of the relative size of the wavelength and electrical size of the cable. As discussed in a previous blog post, we should switch from the AC/DC to RF Module at an electrical size of approximately λ/100. If we use the cable diameter as the electrical size and the speed of light inside the dielectric core of the cable, this yields a transition frequency of approximately 690 MHz.
At these higher frequencies, the cable is more appropriately treated as a waveguide and the cable excitation as a waveguide mode. Using waveguide terminology, the mode we have been examining is a special type of mode called TEM that can propagate at any frequency. When the cross section and wavelength are comparable, we also need to account for the possibility of higher-order modes. Unlike a TEM mode, most waveguide modes can only propagate above a characteristic cut-off frequency. Due to the cylindrical symmetry in our example model, there is an equation for the cut-off frequency of the first higher-order mode, which is a TE11 mode. This cut-off frequency is f_{c} = 35.3 GHz, but even with the relatively simple geometry, the cut-off frequency comes from a transcendental equation that we will not examine further in this post.
So what does this cut-off frequency mean for our results? Above that frequency, the energy carried in the TEM mode that we are interested in has the potential to couple to the TE11 mode. In a perfect geometry, like we have simulated here, there will be no coupling. In the real world, however, any imperfections in the cable could cause mode coupling above the cut-off frequency. This could result from a number of sources, from fabrication tolerances to gradients in the material properties. Such a situation is often avoided by designing cables to operate below the cut-off frequency of higher-order modes so that only one mode can propagate. If that is of interest, you can also use COMSOL Multiphysics to simulate the coupling between higher-order modes, as with this Directional Coupler tutorial model (although beyond the scope of today’s post).
Simulation of higher-order modes is ideally suited for a Mode Analysis study using the RF or Wave Optics modules. This is because the governing equation is \mathbf{E}\left(x,y,z\right) = \mathbf{\tilde{E}}\left(x,y\right)e^{-\gamma z}, which is exactly the form that we are interested in. As a result, Mode Analysis will directly solve for the spatial field and complex propagation constant for a predefined number of modes. We can use the same geometry as before, except that we only need to simulate the dielectric core and can use an Impedance boundary condition for the metal conductor.
The results for the attenuation constant and effective mode index from a Mode Analysis. The analytic line in the left plot, “Attenuation Constant vs Frequency”, is computed using the same equations as the high-frequency (HF) lines used for comparison with the results of the AC/DC Module simulations. The analytic line in the right plot, “Effective Refractive Index vs Frequency”, is simply n = \sqrt{\epsilon_r\mu_r}. For clarity, the size of the “COMSOL — TEM” lines has been intentionally increased in both plots.
We can clearly see that the Mode Analysis results of the TEM mode match the analytic theory, and that the computed higher-order mode has its onset at the previously determined cut-off frequency. It is also incredibly convenient that the complex propagation constant is a direct output of this simulation and does not require calculations of R, L, C, and G. This is because \gamma is explicitly included and solved for in the Mode Analysis governing equation. These other parameters can be calculated for the TEM mode, if desired, and more information can be found in this demonstration in the Application Gallery. It is also worth pointing out that this same Mode Analysis technique can be used for dielectric waveguides, like fiber optics.
At this point, we have thoroughly analyzed a coaxial cable. We have calculated the distributed parameters from the DC to high-frequency limit and examined the first higher-order mode. Importantly, the Mode Analysis results only depend on the geometry and material properties of the cable. The AC/DC results require the additional knowledge of how the cable is excited, but hopefully you know what you’re attaching your cable to! We used analytic theory solely to compare our simulation results against a well-known benchmark model. This means that the analysis could be extended to other cables, as well as coupled to multiphysics simulations that include temperature change and structural deformation.
For those of you who are interested in the fine details, here are a few extra points in the form of hypothetical questions.
The following texts were referred to during the writing of this post and are excellent sources of additional information:
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 |
---|---|
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 | |
---|---|---|
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 | |
---|---|---|---|
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° |
---|---|
Thickness shear mode of an AT cut crystal for the same plate set up with the IRE 1949 and the IEEE 1978 standards with a driving frequency of 5.095 MHz. In each image, the global axis orientation is shown on the left and the crystal axis orientation is shown on the right. The top images are aligned with the global coordinates and the lower images are shown with the crystal coordinates in the same orientation as in the first figure.
Below, we have the sequence of rotations involved in defining the cut (YXlw) -35.25° 180° and the sequence of Z-X-Z Euler rotations required to rotate the global axes onto the crystal axes. The corresponding Euler angles are provided in the table below. Note that the order of the rotations for the Euler angles is the reverse of that specified in the cut definition.
IEEE 1978 Standard: (YXlw) -35.25° 180° | |
---|---|
1.Orientate the thickness direction (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 | |
---|---|
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° |
---|---|
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.
Whenever an alternating electric current (or a direct current, for that matter) is applied to living tissue, there will be heat generation and temperature rise due to Joule heating. The ability to target this heat to specific localized tissue areas is a key advantage of the radiofrequency tissue ablation technique.
In one of many medical applications, a cancerous tumor is a localized target. Using heat, the temperature of the area is raised to kill the cancer cells. Alternating current is used (rather than direct) to avoid stimulating nerve cells and causing pain. When alternating current is used, and the frequency is high enough, the nerve cells are not directly stimulated.
To understand how we can model this process, let’s examine the figures below, which show some of the key concepts of this technique.
A tumor within healthy tissue. Capillaries perfuse blood through the tissue and tumor.
When an undesirable tissue mass is identified, such as a tumor, a doctor can use either a monopolar or bipolar applicator to inject current into and around the tumor. The current comes from a generator and varies sinusoidally in time. Frequencies of 300 to 500 kHz are common, although the procedure can use much lower frequencies.
There are a wide variety of electrode configurations ranging from flat plates and single needles to a cluster of needles, depending on the desired shape of the heated domain and how the doctor will access the tissue. One common class of applicator is deployed through the circulatory system by using a long, flexible catheter and then extending a set of needles from the distal end into the tissue to be heated.
A monopolar applicator is made up of a needle and patch applicator, whereas a bipolar applicator consists of two needle electrodes. More than two applicators and other applicator configurations are also possible. By convention, one electrode is called the ground, or reference, electrode. The voltage applied at the other electrode is with respect to this ground.
A monopolar radiofrequency applicator and a patch electrode on the skin’s surface.
A bipolar applicator primarily heats the region between the electrodes.
An engineer designing one of these devices has a complicated problem to solve. The shape of the heated tissue depends on the shape and number of electrodes; which part is insulated and which is not; and ultimately, the thermal energy absorption distribution of the nearby tissue over time.
The sharp, pointed ends of the needle electrodes complicate the design process, since they lead to high current densities and thus uneven temperature rise along the needle. For the cancerous tumor application, the goal is to kill the undesirable tissue mass and leave the surrounding healthy tissue unharmed. For shrinking collagen, the goal is still to heat tissue, but to avoid any possibility of damaging cells. COMSOL Multiphysics simulation streamlines and shortens this process.
To properly model this procedure, we must build a model of the electric current flow through the tissue as well as the heat generation and temperature rise. Let’s explore these steps.
We begin by examining the typical material properties of both the applicator and living tissue and discuss how these materials behave at an operating frequency of 500 kHz. The table below shows the representative electrical conductivity, \sigma; relative permittivity, \epsilon_r; skin depth, \delta; and complex-valued conductivity, (\sigma+j\omega \epsilon_0 \epsilon_r) at 500 kHz.
Although there is a variation to the electrical conductivity and relative permittivity of different tissues, for the purposes of this discussion, we will approximate the human body as having the properties of a weak saline solution. The actual properties of tissue do not vary by much more than one order of magnitude from this value, while the conductivity of the electrode and insulator are over five orders of magnitude larger or smaller.
Electrical Conductivity (S/m) | Relative Permittivity | Skin Depth at 500 kHz (m) | Complex Conductivity at 500 kHz (S/m) | |
---|---|---|---|---|
Metal Electrode | 10^{6} | 1 | ~10^{-4} | 10^{6} + j 4 x 10^{-6} |
Polymer Insulator | 10^{-12} | 2 | ~10^{10} | 10^{-12} + j 9 x 10^{-5} |
“Average” Human Tissue | 0.5 | 65 | 1 | 0.5 + j 0.0003 |
We compute the skin depth to decide if we need to compute the magnetic fields and any heating due to induced currents. At 500 kHz, the electrical skin depth of the human body is on the order of one meter, while the heated regions have a typical size on the order of a centimeter. Hence, we can make the approximation that heating due to induced currents in the tissue is negligible and need not be calculated. Note that this approximation will not be valid if some small pieces of metal exist within the tissue, such as a stent within a nearby blood vessel.
We can also see from the magnitude of the complex conductivity in the above table that the electrodes are essentially perfect conductors when compared to tissue. Similarly, the polymer insulators can be well approximated as perfect insulators when compared to human tissue.
This information lets us choose the form of our governing equation. Under the assumption that magnetic fields and induction currents are negligible and operating at a constant frequency, we can solve the frequency-domain form of the electric currents equation. Further assuming that the human body itself does not generate any significant currents, the governing equation is:
which solves for the voltage field, V, throughout the modeling domain. The electric field is computed from the gradient of the voltage: \mathbf{E} = -\nabla V. The total current is \mathbf{J} = (\sigma+j\omega \epsilon_0 \epsilon_r) \mathbf{E} and the cycle-averaged Joule heating is Q = \frac{_1}{^2} \Re (\mathbf{J}^* \cdot \mathbf{E} ).
Since the conductors are essentially perfectly conducting compared to the tissue, we can omit these domains from our electrical model. That is, we can assume that all surfaces of the metal electrodes are equipotential. This is reasonable if the equivalent free-space wavelength (\lambda = c_0/f = 600m) is much larger than the model size. When using the AC/DC Module, we can use the Terminal boundary condition to fix the voltage on all surfaces of the electrode. The Terminal boundary condition can specify the applied voltage, total current, or total power fed into the boundaries.
It is reasonable to ask why the conductor is omitted, for there is indeed some finite heat loss within the electrode itself. The heating within the electrode, however, is many orders of magnitude lower than in the surrounding tissue. Although the currents in the conductor can be quite high, the electric field (the variation in the voltage along the electrode) is quite small, hence the heating is negligible.
Similarly, since the insulators are essentially perfect, these domains can also be eliminated from the electrical model. In the insulators, the electric fields may be quite high, but the current is essentially zero, which again means negligible heating. The Electric Insulation boundary condition, \mathbf{n} \cdot \mathbf{J} = 0, can be applied on the boundaries of the insulators and implies that no current (neither conduction nor displacement currents) passes through these boundaries. There is one caveat to this: If the electrodes are completely enclosed within the insulators, then there will be significant displacement currents in the insulators and these domains should be included in the model.
On the exposed surface of the skin, the Electric Insulation boundary condition is also appropriate. However, if there is an external electrode patch applied to the skin’s surface, then current can pass through the skin to the electrode. The conductivity of skin is lower than that of the underlying tissue, and this should be modeled. However, we may not want to model the skin explicitly as a separate domain. In such cases, the Distributed Impedance boundary condition applies the condition \mathbf{n} \cdot \mathbf{J} = Z_s^{-1}(V-V_0), where V_0 is the external electrode voltage and Z_s is the equivalent computed impedance of the skin.
A schematic of such a model is shown below, with representative material properties and boundary conditions. Now that the electrical model is addressed, let’s move on to the thermal model.
A schematic of an electrical model of radiofrequency tissue ablation. Representative material properties are shown on the left. The modeling domain and governing equations are shown to the right.
The objective of the thermal model is quite straightforward: to compute the rise in tissue temperature over time due to the electrical heating and predict the size of the ablated region. The governing equation for temperature, T, is the Pennes Bioheat equation:
where \rho and C_p are the density and specific heat of the tissue, while \rho_b and C_{p,b} are the density and specific heat of the blood perfusing through the tissue at a rate of \omega_b. T_b is the arterial blood temperature and Q_{met} is the metabolic heat rate of the tissue itself. This equation is implemented within the Heat Transfer Module. If the last two terms are omitted, then the above equation reduces to the standard transient heat transfer equation.
It is also necessary to specify boundary conditions on the exterior of the modeling domain. The most conservative condition would be the Thermal Insulation boundary condition, which implies that the body is perfectly insulated. This would lead to the fastest rise in temperature over time. A more physically realistic boundary condition would be the Convective Heat Flux condition:
with a heat transfer coefficient of h = 5-10 W/m^2K and an external temperature of T_{ext}=20-25 ^{\circ}C. This reasonably approximates the free convective cooling from uncovered skin to ambient conditions.
Along with the change in temperature, we also want to compute the tissue damage. The Heat Transfer Module offers two different methods for evaluating this:
Along with these predefined damage integrals, it is also possible to implement a user-defined equation for damage analysis via the equation-based modeling capabilities of COMSOL Multiphysics.
Representative radiofrequency ablation results from a 2D axisymmetric model. Two insulated applicators are inserted into a tumor within the body to heat and kill the diseased tissue. The plotted results include the voltage field (top left), resistive heating (bottom left), and the temperature and size of the completely damaged tissue at two different times (right).
We have now developed a model that is a combination of a frequency-domain electromagnetics problem and a transient thermal problem. COMSOL Multiphysics solves this coupled problem using a so-called frequency-transient study type. The frequency-domain problem is a linear stationary equation, since it is reasonable to assume that the electrical properties are linear with respect to electric field strength over one period of oscillation. Thus, COMSOL Multiphysics first solves for the voltage field using a stationary solver and then computes the resistive heating. This resistive heating term is then passed over to the transient thermal problem, which is solved with a time-dependent solver. This solver computes the change in temperature over time.
The frequency-transient study type automatically accounts for material properties that change with temperature and the tissue damage fraction. If the temperature rises or tissue damage causes the material properties to change sufficiently to alter the magnitude of the resistive heating, then the electrical problem is automatically recomputed with updated material properties. This can also be described as a segregated approach to solving a multiphysics problem.
In such thermal ablation processes, it is also common to vary the magnitude of the applied electrical heating to pulse the load on and off at known times. In such situations, the Explicit Events interface can be used, as described in our earlier blog post on modeling periodic heat loads. If you instead want to model the heat load changing as a function of the solution itself, then the Implicit Events interface can be used to implement feedback, as described in our earlier blog post on implementing a thermostatic controller.
If you are interested in studying radiofrequency tissue ablation, there are several other resources worth exploring. If your electrodes have sharp edges and you are concerned about localized heating near these edges, consider adding fillets to your model, since a sharp edge leads to a locally inaccurate result for the heating. Also keep in mind that, despite any locally inaccurate heating, the total global heating will nevertheless be quite accurate with a sharp edge. Thus, the filleting of sharp edges is not always necessary, since the local temperature field can still be quite accurate.
If there are any relatively thin layers of materials that have relatively higher or lower electrical conductivity compared to their surroundings, consider using the Electric Shielding or Contact Impedance boundary conditions for the electrical problem. There are similar boundary conditions available for thin layers in thermal models as well.
If you are interested in modeling at much higher frequencies, such as in the microwave regime, then you need to consider an electromagnetic wave propagating through the tissue. In such cases, look to the RF Module and the Conical Dielectric Probe for Skin Cancer Diagnosis example in the Application Gallery. At even higher frequencies in the optical regime, a range of modeling approaches are possible, as described in our blog post on modeling laser-material interactions.
The heat source for your problem need not even be electrical. High-intensity focused ultrasound is another ablation technique and can be modeled, as described in the Focused Ultrasound Induced Heating in Tissue Phantom tutorial in the Application Gallery.
In closing, we have shown that COMSOL Multiphysics, in conjunction with the AC/DC Module and Heat Transfer Module, gives you the capability and flexibility to model radiofrequency tissue ablation procedures.
If you are interested in using COMSOL Multiphysics for this type of modeling, or have any other questions, please contact us.
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The Philips Hue system works by varying the amount of blue, green, and red light that is output, which you can set directly from your smartphone. If you are sensitive to a specific color of light, you can simply avoid it. You can set the lighting depending on your mood to help concentrate, energize, read, or relax. For example, there is a “Concentrate” mode that preferentially outputs more blue light, which has shown to enhance the ability to concentrate. When relaxing in the evenings, I use the “Sunset” mode, which provides more red and orange tints.
Having lived with the system for a while now, I’ve also found some long-term advantages:
A comparison of some of the lighting system’s settings in my apartment. Left: Soft white. Middle: Red. Right: Blue rain.
I tried convincing my parents to buy the system, but my sales pitch didn’t sway them. I recently bought them the system as a Christmas present, since I am such a good son. The first comment I heard when demonstrating the system was: “Wow, the light feels so natural.” This prompted me to investigate why this is, and whether COMSOL Multiphysics® software can be used to investigate the underlying physics. The answer lies in the emission spectrum produced by the high-efficiency LED bulbs. By comparing the emission spectrum of natural light to that produced by incandescent, fluorescent, and LED bulbs, we can better understand this phenomenon.
The emission spectra for natural daylight, as well as incandescent, fluorescent, and LED bulbs, are plotted below. As you will see, the emission spectra are very different, and none of them can perfectly replicate natural daylight.
Let’s start with daylight arriving at the earth’s surface from the sun. There is currently no way of reproducing the emission spectrum with a manmade light source. However, light pipes (or light tubes) can be used to redirect incoming daylight into underground locations, such as subway stations. One example of this is the subterranean train station in Berlin. A light pipe extends out above the station (shown below, in the left image) and collects light, which is transmitted through a special pipe and down into the station underground (shown below, on the right).
Left: A light tube at the entrance to a train station in Berlin. Image by Dabbelju — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons. Right: The light tube transmits light into the underground terminal. Image by Till Krech — Flickr. Licensed under CC BY 2.0, via Wikimedia Commons.
The light pipe creates a more natural illumination of the train station during the day. The obvious drawback of this approach is that it won’t work during the night, creating the need for an artificial light form that mimics natural daylight.
The emission spectrum for natural light generally follows the Planck distribution in the visible part of the spectrum, as we can see below. No color is dramatically favored over another, although the intensity is highest in the light blue region, around 460 nm.
The emission spectrum of visible light arriving at the earth’s surface from the sun.
An incandescent light bulb contains a tungsten filament that is resistively heated when a current is conducted through it. At temperatures around 2000 K, the filament starts to emit visible light. To prevent the tungsten wire from burning up, the bulb is filled with a gas, usually argon. The heat generated in the filament is transported to the surroundings through radiation, convection, and conduction. An incandescent bulb emits a greater proportion of red light than natural daylight. Emission even extends into the infrared part of the electromagnetic spectrum, which wastes energy and reduces the overall efficiency of the bulb.
The emission spectrum in the visible range of a typical incandescent bulb.
A fluorescent lamp typically consists of a long, glass tube containing a low-pressure mixture of mercury and an inert gas. Inside of this tube, a nonequilibrium discharge is produced (a plasma). This means that the electron temperature is different from the temperature of the surrounding gas mixture. For example, the electron temperature can be on the order of over 20,000 K, but the gas temperature stays relatively close to room temperature, 300 K. Since the plasma is not in equilibrium, the electron impact reactions modify the chemical composition of the gas mixture in a manner governed by the collisional processes. These collisions can produce electronically excited neutrals, which can subsequently produce spontaneous emission of photons at specific wavelengths.
Visible light is produced by two mechanisms: optical emission directly from the discharge, or by exciting phosphors on the surface of the tube. Fluorescent lights often cause problems for people suffering from a visual disorder called Irlen syndrome, and anecdotally, people often complain of headaches and migraines when exposed to fluorescent lights for extended periods of time.
As you can see in the graph below, the emission spectrum in a fluorescent light source looks rather strange. The quantization is either due to direct emission from the plasma or by the phosphors, but to a human eye, the light emitted still seems white. Like incandescent bulbs, fluorescent bulbs can be inefficient because the plasma needs to be sustained and it emits radiation in the nonvisible range.
The emission spectrum of a typical fluorescent bulb.
LEDs are revolutionizing the lighting industry, as they are often much more efficient and durable than traditional incandescent light technologies. For example, typical consumer LED light bulbs operate at 10-20% of the power needed to run an incandescent bulb of comparable brightness. They also have lifetimes of over 25,000 hours, compared to only 1000 hours for incandescent bulbs.
LEDs are so much more efficient than incandescent bulbs because they function in a very different way. LEDs are semiconductor devices that emit light when electrons in the conduction band transition across the bandgap via radiative recombination with holes in the valence band. Unlike incandescent bulbs, LEDs emit light over a very narrow range of wavelengths.
Initially, red, green, and yellow LEDs were developed in the 1950s and 1960s. However, it was the invention of the blue LED that led to the creation of new, efficient white light sources. Blue light emitted from such LEDs can be used to stimulate a wider spectrum of emission from a phosphor layer around the LED casing, or can be directly combined with red and green LEDs to create white light.
As shown in the graph below, the LED spectra for a warm, white setting gets closer to that of natural daylight. There is more blue light than the incandescent bulb and nearly all of the power is emitted within the visible spectrum. This is why LED bulbs are so efficient — very little energy is wasted on emission in the ultraviolet or infrared range.
The emission spectrum of a typical LED bulb on a warm, white setting.
The different emission spectra are plotted on the same axis below. While none of the bulbs exactly reproduce natural daylight, the LED bulb is clearly the best approximation. All of the emission occurs within the visible range, making the device very efficient.
Emission spectra from daylight and typical incandescent, fluorescent, and LED bulbs.
In general, incandescent and fluorescent bulbs have a fixed optical output. LED bulbs with a fixed emission spectrum are also available. By plotting the emission spectra of the different light sources, we can infer that LED bulbs most closely replicate natural daylight.
As we’ve seen in this blog post, there are many different ways of creating artificial light. All of the methods described above can be modeled in various levels of detail using COMSOL Multiphysics with either the Semiconductor, Plasma, Heat Transfer, or Ray Optics modules.
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In the 1860s, an inventor named Henry Dircks built upon an older Italian optical illusion that manipulated visual effects using glass and light, calling it the Dircksian Phantasmagoria. His effect never gained popularity because it was complicated and expensive, requiring theaters to be completely rebuilt to incorporate the trick.
John H. Pepper, who was lecturing at the Royal Polytechnic Institute in London at the time, came up with an easy way to implement Dirck’s effect in existing theaters using just a sheet of glass. Since Pepper popularized the illusion, it became known as Pepper’s Ghost. Pepper started showing the illusion at theaters around England and Australia, puzzling audiences. One local newspaper even reported that accomplished physicist Michael Faraday returned to Pepper after seeing the illusion and demanded an explanation.
The original Pepper’s Ghost optical illusion involves placing a large piece of glass at an angle between a brightly lit “stage” room into which viewers look straight ahead and a hidden room. The glass reflects the hidden room, kept dark, that holds a “ghostly” scene. When the lights in the hidden room are slightly raised to illuminate the scene, the lights in the stage room are slightly dimmed, and the apparition appears to the audience.
Behind-the-scenes of a basic Pepper’s Ghost illusion. Image by Wapcaplet — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
Over the years, amusement parks and haunted houses began to adapt different variations of Pepper’s Ghost to achieve the effect of ghostly apparitions in their displays.
Today, through a type of extremely high-quality video projection technology that combines motion capture technology with 3D computer-generated imaging (CGI), “digital doubles” of celebrities and political world leaders can be virtually projected to large crowds by displaying the graphics through a large-scale Pepper’s Ghost contraption. This burgeoning computer-generation technology enables the illusion of moving subjects by combining animation technology with previously recorded digital footage, using a variety of advanced effects and software. This special effect is also commonly used in film.
In the case of Tupac Shakur, a transparent film was suspended at an angle in front of the stage. Footage of the musician, taken from past live performances, was displayed on this film using an off-stage projector. The footage was enhanced and combined with animation software to create a crystal clear, high-resolution video for the performance. Through animation, the video footage was manipulated to make it appear as though Tupac was interacting with the other performers onstage. Instead of the Pepper’s Ghost effect of older times, which was transparent and “ghastly”, the quality of motion graphics make onlookers mistake the projected image for the real thing.
Pepper’s Ghost can be explained using ray optics. To start, the glass, or transparent film, used in a Pepper’s Ghost illusion has a different refractive index than the air around it; that is, light in the two media propagates at different speeds. When light reaches a boundary between two materials with different refractive indices, typically some of the light is reflected and the rest is refracted, or transmitted at an angle. The amount of light that is reflected and refracted is governed by the Fresnel equations and depends on the angle of incidence and polarization of the incoming light, as well as the neighboring materials.
Consider the setup used for the classical Pepper’s Ghost illusion, consisting of a stage, an additional room out of direct sight of the audience, and a wall of glass angled between the audience and both rooms. The light from the lit-up stage is refracted as it enters and leaves the pane of glass. The audience can see the stage, but not the glass, just as if they are watching any old stage play. When the lights in the side room are turned on, light propagates from the “ghost” to the glass. Some of the light is reflected by the glass and reaches the audience. The crowd sees rays of light from both the regular stage and the hidden room. This projects the hidden image in a semitransparent, or “ghostly”, manner.
In the scene above, some of the incident rays from the lit stage are refracted through the glass toward the audience, while some of the incident rays from the hidden room are reflected by the glass. The images in the two rooms both reach the audience, creating the Pepper’s Ghost illusion.
Just before Halloween, my colleague Matt Milhomme and I built our own Pepper’s Ghost projector with a few old jewel CD cases and a cell phone.
We made a "3D hologram" to project a COMSOL model on a cell phone… #SimulationFriday #PeppersGhost pic.twitter.com/3tQSTuDlrC
— COMSOL (@COMSOL_Inc) October 30, 2015
It is pretty simple to replicate our device and make your own Pepper’s Ghost projector if you follow these steps:
*We made our own video to play an RF simulation with our DIY Pepper’s Ghost projector. You can, instead, choose from a variety of videos on YouTube that are designed to implement the Pepper’s Ghost effect. The videos are simply made up of four identical animations displayed in a symmetrical square shape, over a black background. You can find a selection of videos to play with your projector if you search for “3D Hologram Video”.
A common modeling assumption is to specify the linear magnetic permeability in the constitutive relation. It is often a good practice to assume that the material responds linearly with respect to the applied field in the initial modeling stage. In COMSOL Multiphysics, this can be done by simply applying a constant value for magnetic permeability within the constitutive equations in the magnetic interfaces.
However, many ferromagnetic materials exhibit nonlinear behavior, as the magnetization depends nonlinearly on the magnetic fields, even for a small change. These materials also exhibit hysteresis, a history dependence of the applied magnetic field on magnetization. Modeling hysteresis behavior is computationally demanding and difficult. The nonlinear magnetic materials available in COMSOL Multiphysics do not include the full hysteresis loop, but instead, the average BH curve that incorporates magnetic saturation effects in the first quadrant, as discussed in this blog post.
These magnetization curves are also called dc or normal magnetization curves, which are obtained by plotting the locus of the maximum values of B and H at the tips of the hysteresis loops. These magnetic saturation curves can be used directly in the stationary and time-dependent studies, but not for the frequency domain. To solve in the frequency domain, you would need a “cycled averaged” BH/HB curve that approximates a nonlinear material at the fundamental frequency.
The Effective Nonlinear Magnetic Curves Calculator app can be used to generate the effective BH/HB curves for frequency-domain (time-harmonic) simulations. These effective BH/HB curves can be directly used in the magnetic interfaces in the AC/DC Module in COMSOL Multiphysics version 5.2, which has built-in support for these materials.
Note: The full hysteresis loop can be implemented in COMSOL Multiphysics by adding extra partial differential equations (PDEs) to describe the material models; for example, the Jiles-Atherton model in the time domain. A 3D time-domain model demonstrating the Jiles-Atherton vector hysteresis model is available in the Application Gallery. In addition, the AC/DC magnetic interfaces in COMSOL Multiphysics version 5.2 support material models defined in external C code. This allows the app creator to have the app user define subroutines to describe material models; for example, by implementing the full hysteresis loop and using those material models in a magnetic simulation in full 3D geometry. The example demonstrating the C code material model is available here.
Magnetic interfaces in the AC/DC Module of COMSOL Multiphysics version 5.2 now support the effective HB/BH curve material model, which can be used to approximate the behavior of a nonlinear magnetic material in a frequency-domain simulation without the additional computational cost of a full transient simulation. To be able to use this new effective HB/BH curve material model, we require the effective H_{eff}(B) or B_{eff}(H) relations, defined as an interpolation function.
This utility application can be used to compute the interpolation data starting from the material’s H(B) or B(H) relations. The interpolation data for the H(B) or B(H) relations can be imported from a text file or entered in a table. The application can then compute the interpolation data for the H_{eff}(B) or B_{eff}(H) relations using two different energy methods: Simple Energy and Average Energy. When using the AC/DC Module, the output result plots of effective HB/BH curves can be exported either as a text file or as a Material Library file that can be imported into COMSOL Multiphysics for the frequency-domain simulation of magnetic materials.
The user interface consists of four different sections: Ribbon, Material Information (Input & Results), Curve Plots, and Curve Analysis, as shown in the image below. The material data for the magnetic flux density (B) and magnetization field (H) can be directly typed in the table or imported from a text file using the Import Curve Data button on the menu.
The customized user interface for the Effective Nonlinear Magnetic Curves Calculator app.
The Ribbon section contains six buttons for different operations. Click on the Use Default Curve Data button to use the default input BH curve that is already loaded in the application. If you wish to bring in your own curve, click on the Import Curve Data button to open the Import Curve dialog box (as shown below) and import a text file containing the interpolation data for the BH or HB curve. In the Import Curve dialog box, specify the file to import by clicking on the Browse button.
The text file must contain pairs of values separated by whitespace characters or commas, with one pair for each line. Choose either BH Curve or HB Curve from the Import curve as combo box. For a BH curve, the first column represents H values and the second column represents B values and vice versa for an HB curve. The default data table will be replaced with the imported data, which, if needed, can be edited in the table. Rows can be added or removed using the buttons below the table.
Dialog box for the Import Curve Data button in the Ribbon section.
In the Curve Analysis section, curve data is automatically analyzed each time the data is modified or imported from the file. Curve analysis includes three conditions that the imported data must satisfy: the curve must contain the value (0,0); the curve must be strictly monotonic; and the curve must be nonnegative. If any of these conditions are not fulfilled, modify the values in the table to correct the problem. The linearized permeability at zero field (the slope of the curve at H = 0) is also computed and displayed.
Once the data is modified or imported, click on the Compute button in the ribbon to compute the effective curves using the Simple Energy and Average Energy methods. The computed values for the effective B field for both the Simple Energy and Average Energy methods are displayed in the last two columns of the table. The plots for the original BH and HB curves as well as the H_{eff}(B) and B_{eff}(H) curves for both methods are displayed in the graphics window, as shown below.
The app’s user interface, displaying the computed data and plots for magnetic curves.
The computed interpolation data for the effective HB/BH curves can be exported for further use in other COMSOL Multiphysics applications. Click on the Export Data button in the ribbon to open the Export Material Data dialog box. You can export the data as a text file or to the Material Library by choosing the respective option in the Export as combo box.
In the Text File export option, you could choose any one of the averaging methods and curve types. This exported text file contains a pair of values in each line. This text file can be imported, for example, in an Interpolation Function node in a COMSOL Multiphysics application and used for defining the effective HB/BH curve for frequency-domain magnetic simulations.
The Export Material Data dialog box, illustrating the Text File (left) and Material Library (right) data export options.
You could also export the curve data as a COMSOL Multiphysics Material Library file by using the Material Library export option in the Export as combo box (see the image above, on the right). The material in this exported Material Library file contains the HB curve, BH curve, effective HB curve, and effective BH curve based on the selected averaging method (Simple Energy or Average Energy). You could also include the linearized relative permeability by selecting the Include linearized relative permeability at zero field check box. The exported Material Library file can be added into the Material Library, as shown in the picture below.
The Material Browser window, showing the steps for adding the exported Material Library file into the Material Library.
Note: You can now use any of the materials that are available under the Nonlinear Magnetic folder in the COMSOL Multiphysics Material Library for a frequency-domain simulation by first converting the available HB/BH curves into effective HB/BH curves using this utility app. Simply add the material in the COMSOL Multiphysics model, export the BH curve or HB curve data as a text file, import the text file into the Effective Nonlinear Magnetic Curve Calculator app, evaluate and export the effective HB/BH curves, and finally import the effective HB/BH curves into the same COMSOL Multiphysics model for frequency-domain simulations. The Soft Iron (with losses) and Soft Iron (without losses) materials under the AC/DC folder already contain the effective HB/BH curves, which can be used directly in frequency-domain simulations.
The embedded model in this app computes the effective nonlinear magnetic curves for materials using the Simple Energy and Average Energy methods. The integration expressions for calculating effective magnetic flux density strengths are given below. The integration method available in this app is selected based on the paper by Gerhard Paoli, Oszkár Biró, and Gerhard Buchgraber.
where H is the amplitude of the time-harmonic magnetic field, B(H) is the material’s nonlinear BH relation, H(t) is the time-dependent oscillating magnetic field, and T is the arbitrary period of oscillation.
For more information about the model, refer to the PDF document by clicking on the Documentation button in the ribbon of the app.
To illustrate this new effective HB/BH curve material model in COMSOL Multiphysics version 5.2, let’s take a look at the square-shaped, closed, magnetic core excited by the multiturn coil on one arm, as shown in the image below, on the left. The magnetic core is modeled in the Magnetic Fields physics interface using Ampère’s Law, with three different material types: nonlinear HB curve, nonlinear effective HB curve, and linear material.
The first nonlinear HB curve model is solved in the time domain, whereas the other two material models are solved in the frequency domain at 1 kHz. The magnetic flux density at one of the corners inside of the magnetic core is measured and compared for all three different material models; see the image below, on the right. As expected, the effective HB/BH curve model behaves much closer to that of the nonlinear HB/BH curve model in the time domain. However, the time-domain model still exhibits the higher harmonics, unlike the other two models. The linear material model is quite different compared to the other two models. Therefore, for many applications where the higher harmonics are not important, the effective HB/BH curves can be appropriate, as they are less computationally expensive. You can download this example here.
The magnetic flux density norm surface plot on the magnetic core (left). A comparison of the magnetic flux density norm at a point inside of the magnetic core for three material models (right).
In this blog post, we have discussed the various material models available for modeling nonlinear magnetic materials. We also detailed the Effective Nonlinear Magnetic Curves Calculator application and explained how to utilize this app to generate the cycled-average effective HB/BH curves for the frequency-domain simulation of magnetic devices. Finally, we demonstrated an example using three types of material models (BH/HB curves, effective HB/BH curves, and linearized material) and compared the results.
If you are interested in modeling nonlinear magnetic materials for time-harmonic or time-dependent studies, please contact us.
In many modern electronics, from laptops to smartphones, touchscreen technology is becoming the norm. The growing popularity of touchscreen devices, which enable users to directly interact with the displayed information, is driving greater competition across various industries for enhanced performance and accuracy. As such, it is important for designers and engineers to verify that a touchscreen design can perform well under a range of conditions, while maintaining fast time-to-market and low costs.
Touchscreen technology is becoming more common in electronic devices, such as laptops. Image by Intel Free Press. CC BY-SA 2.0, via Wikimedia Commons.
A simulation-based approach can help to meet these needs. With simulation platforms like COMSOL Multiphysics, you can study the effectiveness of a product within various environments, isolating specific factors and identifying areas requiring optimization. As tests are performed in a virtual setting, you remove the need for building physical prototypes, reducing the overall product development costs.
Some touchscreen designs may, of course, go through several modifications before their ideal configuration is achieved. This can translate into a rather heavy workload for you, the simulation engineer. As your colleagues rely on you to run each of these tests, they must also wait for the results before communicating feedback to the customers. But what if there was a way to bridge the gap in simulation expertise?
Simulation apps are the answer. As the creator of an app, you can control the parameters that users are able to access, ensuring accuracy in the simulation results. Now, rather than spending time running simulations for every design change, you can empower your colleagues to run their own tests. Not only does this free up time for you to focus on other projects, but it also allows for more efficient communication with customers.
Parade Technologies, a leading supplier of touchscreen technology, is already experiencing similar advantages of building and sharing simulation apps.
Researchers at Parade Technologies help design touchscreen technology for a wide range of applications, including smartphones, automotive environments, and home appliances. For every touchscreen design, simulation plays a key role in its development and optimization. In an article from Multiphysics Simulation 2015, Peter Vavaroutsos, a member of the touchscreen modeling group at Parade Technologies, discussed the value of using COMSOL Multiphysics to study touchscreen technology: “Simulation has been a very valuable tool for ensuring that our product responds effectively over a range of different environments and conditions, since we can single out certain factors and determine how to most effectively optimize performance.”
The team is now extending simulation power throughout their organization by using the Application Builder in COMSOL Multiphysics to turn their complex models into easy-to-use apps. In the past, simulation engineers were called upon to run minor parameter changes. And, when sales engineers attempted to run simulations on their own, the experts would have to look over the simulation results to make sure that they were accurate — a process that could be rather time-consuming.
Using apps, the support staff at Parade Technologies is able to confidently make adjustments to design parameters. In the capacitive touchscreen app shown below, users have the ability to modify parameters such as finger location and the thickness of various layers within a touchscreen’s capacitive sensor. Additionally, the app creates a report that offers details about the capacitance matrix, information that is key in the development of capacitive sensors.
A touchscreen app created by Parade Technologies (formerly Cypress Semiconductor).
Greater accessibility to simulation power throughout the organization has enabled simulation engineers at Parade Technologies to focus more of their time on developing new technologies, rather than running simulations for every design modification and verifying others’ results. Empowered to run their own simulation tests, the support team can now deliver accurate results faster to customers around the world, accelerating the overall design workflow.
The detection and removal of landmines and IEDs is important for both humanitarian and military purposes. While the term for the process of detecting these mines — minesweeping — is the same in both cases, the removal process is referred to as demining in times of relative peace and mine clearance during times of war. The latter case refers to when mines are removed from active combat zones for tactical reasons as well as for the safety of soldiers.
When a war ends, landmines may still be in the ground and detonate under civilians, leading to casualties. The majority of the mines are located in developing countries that are trying to recover from recent wars. Aside from being politically unstable, these countries are unable to farm viable land that is strewn with IEDs, keeping their economies in poor positions. Unfortunately, finding and removing the dangerous devices can be rather difficult.
A U.S. Army detection vehicle digs up an IED during a training exercise.
In efforts to locate and remove landmines, a mechanical approach is one option. With this method, an area with known landmines is bombed or plowed using sturdy, mine-resilient tanks to detonate them safely. For a more natural approach, dogs, rats, and even honeybees are trained to detect landmines with their sense of smell, and they are usually too light to trigger detonation. Biological detection methods offer another option, utilizing plants and bacteria that change color or become fluorescent in the presence of certain explosive materials. Once the mines are detected, they are safely removed from the area.
A trained rat searches for landmines in a field.
One method can provide more knowledge about an area that contains IEDs: electromagnetic detection. An important element within electromagnetic detection is a process called ground-penetrating radar (GPR), which uses electromagnetic waves to create an image of a subsurface, revealing the buried objects.
GPR involves sending electromagnetic waves into a subsurface (the ground) through an antenna. The transmitter of the antenna sends the waves, and the receiver collects the energy reflected off of the different objects in the subsurface, recording the patterns as real-time data.
Data from a traditional GPR scan of a historic cemetery.
With recent developments in landmine cloaking technology, identifying buried objects through traditional GPR has become more challenging. Dr. Reginald Eze and George Sivulka from the City University of New York — LaGuardia Community College and Regis High School sought to improve electromagnetic IED detection by testing the method under different variables and environmental situations. By creating an intelligent subsurface sensing template with the help of COMSOL Multiphysics, the research team was able to determine better ways to safely locate and remove landmines and IEDs.
Let’s dive a bit deeper into their simulation research, which was presented at the COMSOL Conference 2015 Boston.
When setting up their model of the mine-strewn area, the researchers needed to ensure that they were accurately portraying a real-world landmine scenario. They started with a basic 2D geometry and defined the target objects and boundaries. The different layers of the model featured:
The physical parameters in the model included relative permittivity; relative permeability; and the conductivity of the air, dry soil, wet soil, and TNT (the explosive material used in the landmine).
Using the Electromagnetic Waves, Frequency Domain interface in the RF Module, the team built a model consisting of air, soil, and the landmine. Additionally, a perfectly matched layer (PML) was used to truncate the modeling domain and act as a transparent boundary to outgoing radiation, thus allowing for a small computational domain. A transverse electric (TE) plane wave was applied to the computational domain in the downward direction. The scattering results were analyzed via LiveLink™ for MATLAB®.
The scattering effect of a wave on a landmine in wet soil (left) compared to dry soil (right).
The research team studied the radar cross section (RCS), which quantifies the scattering of the waves off of various objects. Their studies were based on five key factors:
With each adjustment to an environmental parameter, a parametric sweep was performed every 0.5 GHz from 0.5 GHz to 3.0 GHz. The parametric sweeps enabled an educated selection of the optimal frequency for IED detection in every possible environmental scenario.
A parametric sweep used to identify the optimal frequency for a landmine detection system.
The simulation results pointed out the differences in scattering patterns depending on the parameters. For example, as the depth of the target increased, the scattering effects became more negligible. The relation between how deep the mine was buried and the scattering showed a clear connection to the soil’s interference with the wave.
The results also showed that dry soil has more interference with the RF signal than wet soil. Both the size and depth of the mine were related to the amount of scattering. For instance, the more shallow the mine was buried, the more easily it was detected. The parameter sweep of the frequencies indicated that the optimal frequency to detect anomalies in the subsurface scan was 2 GHz.
The scattering amplitude for a landmine buried in an air/wet soil/dry soil layer combination (left) compared to air/dry soil/wet soil (right).
Studying the parameters and their effects on the scattering patterns of the waves offers insight into the objects that are being detected, including their chemical composition. Such knowledge makes it easier to identify an object, whether a TNT-based landmine, another type of IED, a rock, or a tree root.
Through simulation analyses, the researchers gained a more comprehensive understanding of the microphysical parameters and their impact on the scattering of waves off of different objects. This gave them a better idea of the remote sensing behavior, offering potential for increased accuracy in landmine detection and removal. Such advancements could lead to safer environments, particularly within developing areas of the world.
MATLAB is a registered trademark of The MathWorks, Inc.
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Skin cancer affects numerous people around the world and is recognized as the most common form of cancer in the United States. Despite its prominence, this disease is highly treatable when skin tumors are detected early and removed. These tumors can be identified during monthly self-examinations and with the help of medical professionals. However, noninvasive skin tumor detection tools, such as dielectric probes, are emerging as an alternative.
To identify tumors, dielectric probes can utilize a millimeter wave with frequencies of either 35 GHz or 95 GHz. This millimeter wave has a sensitive reflective response to water content, which it uses as a means of detecting skin tumors. Such tumors possess a different scattering parameter or S-parameter than that of healthy skin, and the probes locate tumors by identifying these abnormal S-parameters.
Through simulation, we can evaluate the functionality of a conical dielectric probe and ensure its safety as an alternative for detecting skin tumors.
Our 2D axisymmetric tutorial model consists of a metallic circular waveguide, a tapered PTFE dielectric rod, a skin phantom, an air domain, and perfectly matched layers (PMLs).
In this example, we model our waveguide as a perfect electric conductor (PEC) and assume that its conductivity is high enough to negate any loss. The waveguide terminates at a circular port on one end and is connected to the dielectric rod on the other end. The dielectric rod is designed for impedance matching between the waveguide and the air domain. It is symmetrically tapered and supported on the rim of the waveguide by a ring structure. The tip of the rod touches the skin phantom, and the whole device uses a low-power 35 GHz Ka-band millimeter wave when operating.
Left: Dielectric probe model. Right: The probe interacting with a skin tumor.
To analyze the validity of the probe design, we first observe the electromagnetic properties of the circular waveguide and dielectric probe without the skin phantom. From the simulation results, we can conclude that the probe is functional.
The dielectric rod’s wave propagation without the skin phantom.
Next, we increase the complexity of our model with two additions: a healthy skin phantom and a skin phantom containing a tumor. This enables us to calculate and compare the S-parameters for each of these cases. Our findings show that the S_{11} value of the healthy phantom is -9.84 dB, while the phantom containing a tumor features an S-parameter value of -8.87 dB. These values indicate that more reflection occurs when the probe touches the skin phantom with a tumor. We can expect such a result, as tumors have a higher moisture content than healthy skin.
While we found the S-parameter approach to be functional, we also want to ensure that it is safe. To do so, we study the temperature distribution over the skin phantom surface in order to find the fraction of necrotic (damaged due to heat) tissue.
Left: Temperature variation on a skin phantom with a tumor. Right: Plot of the necrotic tissue.
Our analysis of a skin phantom with a tumor shows that, after ten minutes of low-powered millimeter wave exposure, the temperature change is within 0.06°C. Even at the relatively hotter spot, the temperature remains very close to the initial temperature of 34°C. With this information, we can assume that there are no harmful temperature differences. Furthermore, our results show that the fraction of necrotic tissue is negligibly small, indicating that the temperature rise induced by the probe has a negligible effect on the tissue.
Look at any device that uses electricity and you will likely find a power electronic system at the heart of its operation. Power electronics refers to the technology that is used to control and convert power from one form to another. While the conversion process may vary depending on the application, the goal remains the same — process power efficiently, safely, and at a low cost.
Silicon devices, namely insulated-gate bipolar transistors (IGBTs), currently dominate the power electronics industry. These power electronic converters are used to process most sources of electricity at least once or, more often, multiple times. The efficiency of these converters typically ranges from 70% to 90%, which translates into large amounts of money being wasted.
In recent years, wide band gap semiconductors such as silicon carbide (SiC) have given new hope in advancing power electronics systems. These materials feature band gaps that are significantly higher than their silicon counterparts, improving the power quality of the electronic devices that rely on them while reducing their size and energy losses. As the U.S. Department of Energy noted in a 2014 article, wide band gap semiconductors are essential to developing the next generation of high-performance power electronics and will lead to more affordable products and large energy savings.
Cell phones and electric vehicles are just two examples of technologies that rely on power electronics. The use of wide band gap semiconductors within these systems could help them become more reliable and efficient. Image on left by Reinraum — Own work, via Wikimedia Commons. Image on right by NJo — Own work, via Wikimedia Commons.
When studying wide band gap semiconductors and their relationship to the electronic system as a whole, it’s important to consider the effects and interaction of a variety of physical phenomena, including thermal, electrical, and mechanical behavior. At Wolfspeed, this has prompted a simulation-based approach to such analyses using COMSOL Multiphysics, a platform that is utilized across many projects throughout the company.
“COMSOL Multiphysics simulations have been essential tools for our engineers to extract a more detailed understanding of our products, virtually assess real-world performance, and reduce the amount of prototyping needed,” Brice McPherson, a development engineer at Wolfspeed, stated in a Multiphysics Simulation 2015 article.
With simulation as a key element in their design workflow, engineers at Wolfspeed embraced the concept of creating apps to further extend the reach and benefits of simulation power. This journey began with the development of applications to analyze the wide band gap technology discussed above.
The first simulation app that Wolfspeed created was based on a simple geometry. The purpose of the app was to evaluate the fusing current and impedance of the wires designed to interconnect semiconductor devices. The results from the simulation tests helped determine how many of the tiny bond wires were necessary within a particular application.
The app was built around a complex, parameterized model, ensuring accuracy in the simulation results while presenting the necessary information in a user-friendly format. A drag-and-drop user interface (UI) and intuitive control and entry fields helped narrow the learning curve behind building the app, simplifying the overall design process.
This app was just the start. Next, the team developed an app to help configure and analyze arrangements of SiC MOSFETs and diodes in their high-performance power modules. The app, shown below, automatically arranges devices in the available area, applies relevant thermal conditions, and provides users with a realistic approximation of how hot the devices will become at their specific operating point. All of the relevant performance metrics, including thermal resistance, loop inductance, package resistance, and parasitic capacitance, are included in the app’s results.
The Wolfspeed app’s UI.
Since then, engineers at Wolfspeed have continued to develop more apps, ranging from simple design tools to more complex analyzers. Newer apps have been distributed to people throughout the company’s engineering team, helping to optimize the design workflow and provide a better prediction of how products will perform. For supervisors and those on the marketing team, simulation apps have also proven to be a powerful tool, enabling them to address customer requests more promptly and effectively.
“The Application Builder tailors the simulation experience to your needs and makes simulation more accessible to a wider audience,” McPherson stated. “It is an enormous way for engineers to save time.”
As we just saw, Wolfspeed is already experiencing many of the benefits of building and distributing simulation apps. As they look ahead to the future, some points of focus include adding complexity to the underlying models of the apps as well as increasing the accuracy of their simulation tools. Once a greater number of apps have been developed, there are plans to eventually create a suite of apps and host them on COMSOL Server™.
The team at Wolfspeed is also hoping to use apps as a way to train new employees during the onboarding process. The Application Builder is designed to hide the complexity of an actual model, instead showing only those parameters that are relevant to a specific analysis. Trying out apps thus provides a simple introduction to the COMSOL Multiphysics software platform, helping employees become familiar with the feel of the software while encouraging them to dig deeper into the full extent of how it works. And, as McPherson mentions, using apps to demonstrate the effects of changing the variables and conditions on structures specific to the company’s industry serves as a great way to introduce employees to their new line of work.
Designing apps for products is another point of interest at the company. These simulation tools could be useful to companies or people across a wide range of industries. Once deployed, the apps could serve as foundations for different groups to run their own simulation tests based on design needs for their specific product.