Nonequilibrium cold plasmas are characterized by an electron temperature that is much higher than the gas temperature. During plasma modeling, the ion temperature is often set to equal the gas temperature. This is an acceptable approximation, as long as the ions undergo sufficient collisions with neutral gas molecules and then thermalize with the background gas. This is especially true in inductively coupled plasmas (ICP), where the pressure is low and the ions’ mean free path length comes closer to the plasma reactor’s length scale. Moreover, the number of collisions are low, therefore, the ion temperature is somewhere in between the gas and electron temperatures.
While COMSOL Multiphysics does not solve for the ion temperature, there are some options available for you to do so.
You can choose to set the ion temperature to equal the gas temperature or use a userdefined value or expression. Moreover, you can also elect to define a correlation between the electric field and the ion mobility and employ an Einstein relation to calculate it, using the Local Field Approximation (LFA) — available in the COMSOL software.
As mentioned, your choice in ion temperature (especially for lowpressure plasmas) could significantly impact your model’s results. Below, you will find a theoretical reason that helps explains this phenomenon.
For the heavy species transport (and ion transport), a continuity equation with a drift diffusion approximation is solved for each species. The variation of the mass fraction ,w_k, for species k depends on a flux, \mathbf{j}_k, and a reaction term, R_k. In this case, convection and thermal diffusion are neglected for simplicity:
To compute the flux, \mathbf j_k, a mixture averaged diffusion coefficient, D_{k,m}, and the ion mobility, \mu_{k,m}, are required:
Based on the kinetic theory of gases, binary diffusion coefficients, D_{kj}, are calculated to get the mixture averaged diffusion coefficient, D_{k,m}. You may have already noticed that LenardJones parameters, \sigma and \epsilon / k_B, have to be specified for each plasma species:
The ion mobility is then calculated, using an Einstein relation according to:
At the reactor walls, the ion flux, \mathbf j_k, to the wall, is computed according to:
The ion temperature is needed to compute the ion mobility and flux to the reactor walls, so the choice in ion temperature especially affects the ions’ transport properties within the plasma model. If the migration part of the flux is large, in comparison to the diffusion part, then the choice in ion temperature particularly grows in importance. This is notably true in cases at very low pressures or at high electric field strengths.
To reiterate, you can also compute the ion temperature with the help of the LFA, available in COMSOL Multiphysics.
The LFA assumes that the local velocity distribution of the particles is balanced with the local electric field. Therefore, quantities, like ion temperature or ion mobility, can be expressed in terms of (reduced) electric field. The LFA requires that local changes in the electric field are small in comparison to the mean free path length. However, this is not always true in the boundary layer, particularly.
The following expression, for the reduced electron mobility as a function of the reduced field, can be found in the paper “Twofluid modelling of an abnormal lowpressure glow discharge” and is used in a subsequent ICP example, below.
In the equation above, the reduced electric field,E/n, is given in Townsends (Td).
To demonstrate the impact your ion temperature choice has on an ICP model, let’s take a look at an example.
An inductively coupled plasma reactor (similar to the GEC ICP Reactor, Argon Chemistry model) was modeled three times with varying ion temperatures. Because ICPs work at particularly low pressures, the ion temperature choice has to be considered carefully.
The ion temperature was:
The other model parameters were as follows:
Model Parameters  

Gas Temperature  300 K 
Coil Power  500 W 
Pressure  0.02 torr 
Electron Mobility  4E24 (1/(m*V*s)) 
The mean ion temperature from Model 3, which was computed from D_{k,m} / \mu_{k,m}, amounts to 0.22 eV –, or 2515 K.
The following figures represent the electron density for all three models after 0.001 seconds.
Model 1: Electron density (T_ion = 300 K).
Model 2: Electron density (T_ion = 0.1 eV).
Model 3: Electron density (T_ion from LFA).
As seen above, using a higher ion temperature value significantly increases the electron density.
The modeling results are also compared in the table below. The maximum electron density, maximum electron temperature, and the absorbed power are displayed.
Max. Electron Density [1/m³]  Max. Electron Temperature [eV]  Resistive Losses [W]  

1. T_i = 300 \text K  4.3E17  4.1  387 
2. T_i = 0.1 \text {eV}  2.6E18  2.8  407 
3. Local Field Approximation  3.3E18  2.3  41 
Based on the table, we can deduce that increasing the ion temperature not only leads to a significant increase in electron density, but also the absorbed power. Additionally, the electron temperature noticeably decreases.
The example above illustrates the impact the choice in ion temperature has on the modeling results of an ICP. A comparison of the results with literature values is essential in judging which assumptions give the best outcomes.
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During a period of time in which energy efficiency and sustainability are heavily emphasized, magnetic cooling has found its way into new technologies, from industrial to household applications. Based on the magnetocaloric effect, this cooling technology involves the phenomenon in which a temperature of a magnetocaloric material is altered by exposing it to an applied magnetic field. This applied field causes the magnetic dipoles to align, resulting in an increase in temperature. The removal of this magnetic field causes the atoms to become disorganized and the material then cools.
With continued research on the optimization of this technology, the potential to reduce energy consumption in homes and offices across the world has become a more realistic goal. This left one group of researchers wondering if this same method could be used to address another source of high energy consumption — heating and air conditioning in electric vehicles.
Examples of electric cars (“Ride and Drive EVs Plug’n Drive Ontario” by Mariordo. Licensed under Creative Commons AttributionShare Alike 2.0 via Wikimedia Commons).
Using COMSOL Multiphysics, a team of researchers from the National Institute of Applied Science designed a magnetocaloric HVAC system for an electric vehicle.
These vehicles rely on energy from batteries for heating and air conditioning, just as they do for operation. The level of energy required is furthered by the vehicle’s lack of available heat waste from the thermal engine, which makes it easier to heat the internal space of conventional vehicles. The additional need for cooling to prevent overheating in the vehicle’s battery further contributes to the energy usage, while highlighting the importance of adequate cooling systems.
In their design, the researchers used a 2D model to analyze an active magnetic regenerator refrigeration cycle for a magnetic refrigeration system. In this case, the magnetocaloric regenerator was comprised of thin parallel plates, with microchannels featuring heat transfer fluid alternating in between.
The geometry of an active magnetic regenerator. Image by A. Noume, C. Vasile, and M. Risser and is taken from the presentation titled “Modeling of a Magnetocaloric System for Electric Vehicles“.
As a means to optimize the efficiency of the system, the team simulated the behavior of the magnetocaloric regenerator coupled with the circulating fluid. They particularly focused on the convective heat transfer coefficient connected to the heat transfer between magnetocaloric material and coolant — an especially important parameter in the overall design.
During the refrigeration cycle simulation, researchers analyzed the hot and coldend temperature variation. The temperature span — the difference between the maximum and minimum temperature — was found to be around 8 K. Adding new materials and alloys was recognized as a potential method of optimizing thermal properties in future designs of these systems.
The results of this study provide a valuable foundation for the use of magnetic cooling technology in electric vehicles. Both rooted in the quest for lower energy consumption, the combination of these two innovative technologies could greatly enhance the autonomy of electric vehicles and make magnetic cooling more mobile.
Bell Labs, the research arm of AlcatelLucent, is committed to designing and implementing new technologies for significantly improving energy management for the next generation of telecommunications products. Working to meet this goal, Bell Labs founded the GreenTouch consortium, a leading organization of researchers dedicated to reducing the carbon footprint of information and communications technology. It is the goal of GreenTouch and Bell Labs to demonstrate the key components needed to increase network energy efficiency by a factor of 1,000 compared to 2010 levels.
The Thermal Management and Energy Harvesting & Storage Research Group within the Efficient Energy Transfer (ηET) Department at Bell Labs (led by Dr. Domhnaill Hernon) is one such group working towards this goal. The department focuses on two main areas. One focus of the thermal research group is to deliver gamechanging thermal technology into AlcatelLucent products across all scales and across multiple disciplines ranging from reliable active air cooling to single and multiphase liquid cooling. One way that this is done is through research for improving the thermal management of laser light transmission in photonic devices by developing an approach to achieve a 50 to 70 percent reduction in energy usage per bit. This approach is explored here.
In addition, the department performs research into Alternative Energy and Storage solutions to enable power autonomous deployment of wireless sensors and small cell technology. In this blog post, we will focus on the production of an energy harvesting device used to power wireless sensors that can produce up to 11 times more energy than current approaches.
In order to improve energy efficiency in photonic devices, the Thermal Management department is using multiphysics simulation to model new designs for cooling photonic devices, which rely on the thermoelectric effect for cooling. Photonic devices used for communications contain a thermoelectric material that is used to cool the device.
In these materials, a temperature difference is created when an electric current is applied to the material, resulting in one side of the material heating up and the other side cooling down. When thermoelectric materials are used to control the temperature of photonics devices, the system is known as a thermally integrated photonics system (TIPS). Currently, a large thermoelectric (TEC) cooler is used to cool off the entire system within the photonic device. While TECs can be used for precise temperature control, they are highly inefficient. The group’s new approach improves thermal management by using an individual micro TEC (μTEC) to cool down each laser in the device.
Schematic of the thermally integrated photonics system (TIPS) architecture, which includes microthermoelectric and microfluidic components.
Using COMSOL Multiphysics, the team simulated a TIPS architecture to be used in new laser devices, including the electrical, optical, and thermal performance of the device. In addition to cooling, these devices are used by telecommunication laser devices in order to maintain the correct output wavelength, output optical power, and data transmission rates.
The team investigated temperature control and heat flux management in the integrated TIPS and μTEC architecture using simulation. In particular, they investigated how temperature control can be archived in these systems through the integration of μTECs with the semiconductor laser architectures. The simulation of the integrated laser and μTECs can be seen in the image below, on the right.
Multiphysics simulation of a laser with an integrated μTEC where temperature (surface plot), current density (streamlines), and heat flux (surface arrows) are shown.
Another project currently being conducted by Bell Labs is the design of an energy harvesting device that can convert ambient vibrations from motors, AC, and HVAC into usable energy. This would prevent the need for the replacement of batteries used in wireless sensors frequently utilized across the network. Applications for this new design include the monitoring of energy usage in large facilities, and in sensors for the future Internet of Things (IoT).
The team’s design used the principles of the conservation of momentum and velocity amplification to convert vibrations into electricity using electromagnetic induction. The device uses a novel approach with multiple degrees of freedom to amplify the velocity of the smallest mass in the system. Simulation played a big part in the design, as parametric sweeps allowed the team to determine how different structural, electrical, and magnetic parameters would affect one another and the design as a whole. The figure below shows the novel design (left) along with the simulation of the device (right).
Left: Prototype of novel machinedspring energy harvester. Right: Simulation of the energy harvester, showing von Mises stress.
Although these new designs are not yet available on the market, researchers at Bell Labs believe that because of the accuracy achieved through their simulations, the devices should be ready for commercial production in as little as five years. Whereas previously these designs would have taken years of timeconsuming physical testing, the Bell Labs team anticipates that these devices will be available with a much faster timetomarket, thanks to the use of multiphysics simulation.
For more detail, read the full story “Meeting HighSpeed Communications Energy Demands Through Simulation“, which appeared in Multiphysics Simulation magazine.
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Piezoelectric materials become electrically polarized when strained. From a microscopic perspective, the displacement of charged atoms within the crystal unit cell (when the solid is deformed) produces a net electric dipole moment within the medium. In certain crystal structures, this combines to give an average macroscopic dipole moment and a corresponding net electric polarization. This effect, known as the direct piezoelectric effect, is always accompanied by the inverse piezoelectric effect, in which the solid becomes strained when placed in an electric field.
Several material properties must be defined in order to fully characterize the piezoelectric effect within a given material. The relationship between the material polarization and its deformation can be defined in two ways: the straincharge or the stresscharge form. Different sets of material properties are required for each of these equation forms.
To complicate things further, there are two standards used in the literature: the IEEE 1978 Standard and the IRE 1949 standard, and the material properties take different forms within the two standards. IEEE actually revised the 1978 standard in 1987, but this version of the standard contained a number of errors and was subsequently withdrawn. Confused yet? I was when I first started reading the literature!
Today’s blog post describes in detail the different equation forms and standards, with a focus on the particular case of quartz — the material that causes the most confusion. In both academia and industry, the quartz material properties are commonly defined within the older 1949 IRE standard. Meanwhile, other materials are now almost always defined using the 1978 IEEE standard. To make matters worse, it is not common to indicate which standard is being employed when specifying the material properties.
The coupling between the structural and electrical domains can be expressed in the form of a connection between the material stress and its permittivity at constant stress or as a coupling between the material strain and its permittivity at constant strain. The two forms are given below.
The straincharge form is written as:
where S is the strain, T is the stress, E is the electric field, and D is the electric displacement field. The material parameters s_{E}, d, and ε_{rT} correspond to the material compliance, coupling properties, and relative permittivity at constant stress. ε_{0} is the permittivity of free space. These quantities are tensors of rank 4, 3, and 2, respectively. The tensors, however, are highly symmetric for physical reasons. They can be represented as matrices within an abbreviated subscript notation, which is usually more convenient. In literature, the Voigt notation is almost always used.
Within this notation, the above two equations can be written as:
\left(
\begin{array}{l}
S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}s_{E11} & s_{E12} &s_{E13} &s_{E14} &s_{E15} &s_{E16}\\
s_{E21} & s_{E22} &s_{E23} &s_{E24} &s_{E25} &s_{E26}\\
s_{E31} & s_{E32} &s_{E33} &s_{E34} &s_{E35} &s_{E36}\\
s_{E41} & s_{E42} &s_{E43} &s_{E44} &s_{E45} &s_{E46}\\s_{E51} & s_{E52} &s_{E53} &s_{E54} &s_{E55} &s_{E56}\\s_{E61} & s_{E62} &s_{E63} &s_{E64} &s_{E65} &s_{E66}\\\end{array}
\right)\left(
\begin{array}{l}T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
+
\left(
\begin{array}{lll}
d_{11} & d_{21} & d_{31} \\
d_{12} & d_{22} & d_{32} \\
d_{13} & d_{23} & d_{33} \\
d_{14} & d_{24} & d_{34} \\
d_{15} & d_{25} & d_{35} \\
d_{16} & d_{26} & d_{36} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\left(
\begin{array}{l}
D_{x} \\
D_{y} \\
D_{z} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}
d_{11} & d_{12} &d_{13} & d_{14} & d_{15} & d_{16}\\
d_{21} & d_{22} &d_{23} & d_{24} & d_{25} & d_{26}\\
d_{31} & d_{32} &d_{33} & d_{34} & d_{35} & d_{36}\\
\end{array}
\right)\left(
\begin{array}{l}
T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
+
\epsilon_0 \left(
\begin{array}{lll}
\epsilon_{rT11} & \epsilon_{rT12} & \epsilon_{rT13} \\
\epsilon_{rT21} & \epsilon_{rT22} & \epsilon_{rT23} \\
\epsilon_{rT31} & \epsilon_{rT32} & \epsilon_{rT33} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\end{array}
The stresscharge form is as follows:
The material parameters c_{E}, e, and ε_{rS} correspond to the material stiffness, coupling properties, and relative permittivity at constant strain. ε_{0} is the permittivity of free space. Once again, these quantities are tensors of rank 4, 3, and 2 respectively, but can be represented using the abbreviated subscript notation.
Using the Voigt notation and writing out the components gives:
\left(
\begin{array}{l}
T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}c_{E11} & c_{E12} &c_{E13} &c_{E14} &c_{E15} &c_{E16}\\
c_{E21} & c_{E22} &c_{E23} &c_{E24} &c_{E25} &c_{E26}\\
c_{E31} & c_{E32} &c_{E33} &c_{E34} &c_{E35} &c_{E36}\\
c_{E41} & c_{E42} &c_{E43} &c_{E44} &c_{E45} &c_{E46}\\c_{E51} & c_{E52} &c_{E53} &c_{E54} &c_{E55} &c_{E56}\\c_{E61} & c_{E62} &c_{E63} &c_{E64} &c_{E65} &c_{E66}\\\end{array}
\right)\left(
\begin{array}{l}S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
+
\left(
\begin{array}{lll}
e_{11} & e_{21} & e_{31} \\
e_{12} & e_{22} & e_{32} \\
e_{13} & e_{23} & e_{33} \\
e_{14} & e_{24} & e_{34} \\
e_{15} & e_{25} & e_{35} \\
e_{16} & e_{26} & e_{36} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\left(
\begin{array}{l}
D_{x} \\
D_{y} \\
D_{z} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}
e_{11} & e_{12} &e_{13} & e_{14} & e_{15} & e_{16}\\
e_{21} & e_{22} &e_{23} & e_{24} & e_{25} & e_{26}\\
e_{31} & e_{32} &e_{33} & e_{34} & e_{35} & e_{36}\\
\end{array}
\right)\left(
\begin{array}{l}
S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
+
\epsilon_0 \left(
\begin{array}{lll}
\epsilon_{rS11} & \epsilon_{rS12} & \epsilon_{rS13} \\
\epsilon_{rS21} & \epsilon_{rS22} & \epsilon_{rS23} \\
\epsilon_{rS31} & \epsilon_{rS32} & \epsilon_{rS33} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\end{array}
The matrices defined in the above equations are the key material properties that need to be defined for a piezoelectric material. Note that for many materials, a number of the elements in each of the matrices are zero and several others are related, as a result of the crystal symmetry.
Using the international notation for describing crystal symmetry, the symmetry group of quartz is Trigonal 32. The nonzero matrix elements take different values within different standards, which can result in confusion when specifying the material properties for a simulation, especially for quartz, where two different standards are commonly employed.
Finally, there is another complication in the case of quartz. Quartz crystals do not have symmetry planes parallel to the vertical axis. Correspondingly, they occur in two types: left or righthanded (this is known as enantiomorphism). Each one of these enantiomorphic forms results in different signs for particular elements in the material property matrices.
The material property matrices appropriate for quartz and other Trigonal 32 materials are shown below. Note that the symmetry relationships between elements in the matrix hold irrespective of the standard used or whether the material is right or lefthanded.
\left(
\begin{array}{cccccc}
c_{E11} & c_{E12} &c_{E13} & c_{E14} & 0 & 0\\
c_{E12} & c_{E11} &c_{E13} & c_{E14} &0 & 0\\
c_{E13} & c_{E13} &c_{E33} & 0 & 0 & 0\\
c_{E14} & c_{E14} & 0 & c_{E44} & 0 & 0 \\
0 & 0 & 0 & 0 & c_{E44} & c_{E14}\\
0 & 0 & 0 & 0 & c_{E14} & \frac{1}{2}\left(c_{E11}c_{E12}\right)\\
\end{array}
\right)
&
\left(
\begin{array}{cccccc}
s_{E11} & s_{E12} &s_{E13} & s_{E14} & 0 & 0\\
s_{E12} & s_{E11} &s_{E13} & s_{E14} &0 & 0\\
s_{E13} & s_{E13} &s_{E33} & 0 & 0 & 0\\
s_{E14} & s_{E14} & 0 & s_{E44} & 0 & 0 \\
0 & 0 & 0 & 0 & s_{E44} & 2 s_{E14}\\
0 & 0 & 0 & 0 & 2 s_{E14} & 2\left(s_{E11}s_{E12}\right)\\
\end{array}
\right)
\\
\left(
\begin{array}{cccccc}
e_{11} &e_{11} & 0 & e_{14} & 0 & 0 \\
0 & 0 & 0 & 0 & e_{14} & e_{11}\\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right)
&
\left(
\begin{array}{cccccc}
d_{11} & d_{11} & 0 & d_{14} & 0 & 0 \\
0 & 0 & 0 & 0 & d_{14} & 2d_{11} \\
0 & 0 & 0 & 0 & 0 & 0\\
\end{array}
\right)
\\
\left(
\begin{array}{ccc}
\epsilon_{rS11} & 0 & 0 \\
0 & \epsilon_{rS11} & 0 \\
0 & 0 & \epsilon_{rS33} \\
\end{array}
\right)
&
\left(
\begin{array}{ccc}
\epsilon_{rT11} & 0 & 0 \\
0 & \epsilon_{rT11} & 0 \\
0 & 0 & \epsilon_{rT33} \\
\end{array}
\right)
\\
\end{array}
Having defined a set of material properties in terms of matrices that operate on the different components of the stress or the strain in the x,y,z axes system, all that remains is to define a consistent set of axes to use when writing down the material properties.
Correspondingly, all of the standards define a consistent set of axes for each of the relevant crystal classes. Unfortunately, in the particular case of quartz, subsequent standards have not used the same sets of axes, and the adoption of the most recent standard has not been widespread. Therefore, it is important to understand exactly which standard a given set of material properties is defined in.
The two relevant standards are:
The orientation of the axes set with the crystal can be determined by specifying the orientation with respect to the atoms in the unit cell of the crystal (which is not that helpful in practice) or by specifying the orientation with respect to the crystal forms. A crystal form is a set of crystal faces or planes that are related by symmetry. Particular forms commonly appear in crystal specimens found in rocks and are used to identify different minerals.
The Quartz Page has a series of helpful figures for identifying the common crystal forms, termed m, r, s, x, and z, as well as a further page specifying the Miller indices of the corresponding planes. Since the standards typically use crystal forms to orientate the axes, this approach is adopted in the figure below, which shows the two axes sets that relate to the 1978 and 1949 standards. Note that both left and righthanded quartz are shown in the figure.
Crystallographic axes defined for quartz within the 1978 IEEE standard (solid lines) and the 1949 standard (dashed lines). Click on the image to view a larger version.
As a result of the different crystal axes, the signs of the material properties for both right and lefthanded quartz can change depending on the particular standard employed. The table below summarizes the different signs that occur for the quartz material properties:
IRE 1949 Standard 
IEEE 1978 Standard 


Material Property 
RightHanded Quartz 
LeftHanded Quartz 
RightHanded Quartz 
LeftHanded Quartz 
s_{E14} 
+ 
+ 
– 
– 
c_{E14} 
– 
– 
+ 
+ 
d_{11} 
– 
+ 
+ 
– 
d_{14} 
– 
+ 
– 
+ 
e_{11} 
– 
+ 
+ 
– 
e_{14} 
+ 
– 
+ 
– 
Usually, piezoelectrics, such as quartz, are supplied in thin wafers that have been cut at a particular angle, with respect to the crystallographic axes. The orientation of a piezoelectric crystal cut is frequently defined by the system used in both the 1949 and 1978 standards. The orientation of the cut, with respect to the crystal axes, is specified by a series of rotations, using notation that takes the form illustrated below:
Diagram showing how a GT cut plate of quartz is defined in the IEEE 1978 standard. The crystal shown is righthanded quartz.
The first two letters of the notation given in the brackets describe the orientation of the thickness and length of the plate that is being cut from the crystal. From the figure on the left, it is clear that the thickness direction (t) is aligned with the Yaxis and the length direction (l) is aligned with the Xaxis. The plate also has a third dimension, its width (w). After the first two letters, a series of rotations are defined about the edges of the plate.
In the example above, the first rotation is about the laxis, with an angle of 51°. The negative angle means that the rotation takes place in the opposite direction to a righthanded rotation about the axis. Finally, an additional rotation about the resulting taxis is defined, with an angle (in a righthanded sense) of 45°.
Most practical cuts use one or two rotations, but it is possible to have up to three rotations within the standard, allowing for completely arbitrary plate orientations.
Note that since the crystallographic axes are defined differently in the 1949 and the 1978 standards, the crystal cut definitions differ between the two. A common cut for quartz plates is the AT cut, which is defined in the two standards in the following manner:
Standard 
AT Cut Definition 

1949 IRE 
(YXl) 35.25° 
1978 IEEE 
(YXl) 35.25° 
The figure below shows how the two alternative definitions of the AT cut correspond to the two alternative definitions of the axes employed in the standards.
The AT cut of quartz is defined as (YXl) 35.25° in the IRE 1949 standard and (YXl) 35.25° in the IEEE 1978 standard. The figure shows the cut defined in a righthanded crystal of quartz. The reason for the difference between the standards is related to the different conventions for the orientation of the crystallographic axes. In the IRE 1949 standard, the rotation occurs in a positive or righthanded sense about the laxis (which in this case is aligned with the Xaxis). As a result of the different axes set employed in the IEEE 1978 standard, the rotation corresponds to a negative angle in this standard.
We have now seen how the two different standards result in different definitions of the material properties and different definitions of the crystal cuts.
In a followup blog post, we will explore how to set up a COMSOL Multiphysics model using the two standards. COMSOL Multiphysics provides material properties for quartz using both of the available standards, so it is possible to set up a model using whichever standard you are most familiar with. Stay tuned for that.
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Although it is possible to set up and solve a 3D model of a conical horn antenna, such a model would require a relatively large amount of computational resources to solve. We can solve for the electromagnetic fields much more quickly by exploiting the symmetry of the structure. Because we are dealing with a cone, our model is structurally symmetric around its axis, i.e., it’s axisymmetric.
Now, although the structure is axisymmetric, the electromagnetic fields will have some variation around the azimuth of the axis, that is, the fields have an azimuthal variation. The RF Module and the Wave Optics Module allow you to model axisymmetric structures with different azimuthal mode numbers.
We can exploit this feature; by building a 2D axisymmetric model and solving for several different azimuthal mode numbers, we can build a model that solves much quicker and uses less memory than a full 3D model. I like the sound of that. But first, a quick note on horn antennas.
There are various types of horn antennas in terms of both overall shape and pattern of the inside. These attributes determine the antenna’s beam profile, bandwidth, and crosspolarization.
Crosspolarization means that the electromagnetic fields are polarized opposite to what is intended. For example, we want the fields to be polarized vertically, but they are instead polarized horizontally.
The funnel part of the antenna is connected to a waveguide, which feeds electromagnetic waves into the antenna. The shape of the horn will dictate what application it’s suited for. For example, sectoral horns (labeled b and c in the image below) are typically used for wide search radar antennas.
Various horn antenna shapes: a) pyramidal; b) sectoral, Eplane; c) sectoral, Hplane; d) conical; e) exponential. “Horn antenna types” by Chetvorno — Own work. Licensed under Creative Commons Zero, Public Domain Dedication via Wikimedia Commons.
The antenna in our case is both shaped like a cone (labeled d in the image above) and has a corrugated surface inside; it’s a corrugated conical horn antenna fed by a circular waveguide. The waveguide passes the excited TE mode through the corrugated funnel, which, in turn, generates a TM mode. Due to the corrugated surface throughout the cone, the modes are mixed, leading to a lower crosspolarization at the aperture than the original excited TE mode.
Conical horn antenna: A visualization in 3D based on a 2D axisymmetric model. The waveguide feeds the antenna with the TE_{1m} mode (m = ±1), which mixes with the TM_{1m} mode as it propagates through the antenna.
Above, I mentioned what crosspolarization is, but why would we want to reduce it? Well, if we have a lot of crosspolarization, our signal may interfere with other channels nearby, if they have alternating vertical and horizontal polarization. We wouldn’t want that.
To investigate the crosspolarization, we can use COMSOL Multiphysics and the RF Module to set up a model. As we learned earlier, we can save time by solving this as a 2D axisymmetric model instead of in 3D. We can do that by using the Electromagnetic Waves, Frequency Domain interface.
I will skip over the stepbystep model setup and head straight to the fun stuff — the results. If you want to reproduce the plots shown here, feel free to download the model documentation and MPHfile from the Model Gallery.
First, we can see what the directive beam pattern of the antenna is:
Farfield plot: The directive beam pattern of the antenna.
Next, we can analyze the electric field at the antenna’s entrance and exit. By solving the model for both m = +1 and m = 1, we can compare the linear polarization in the x and ydirection at the exit.
Electric field at the entrance and exit of the antenna for the linear superposition of m = +1 and m = 1.
At the waveguide feed, the field is mostly in the xdirection, but not linearly polarized. At the aperture, the field is very nearly linearly polarized. To quantify the polarization in both directions, we can evaluate the integral of the absolute value of each field component over the conical horn antenna’s entrance and exit. Doing so, we’ll find that the ratio is roughly 5:1 at the entrance and about 40:1 at the exit. In other words, we have reduced the crosspolarization by approximately a factor of 8.
The Electrical showcase, called Designing and Modeling Electrical Systems and Devices, is a resource created by COMSOL applications engineers and developers alike to demonstrate the modeling capabilities of COMSOL Multiphysics in a comprehensive and resourceoriented guide.
In the showcase, you’ll find information about the six modules offered by COMSOL specifically designed for simulating such diverse applications as transformers, electronic packaging, the propagation of waves in and around structures, analyzing microwave devices and antennas, and much more. The showcase introduces the philosophy of the COMSOL software by demonstrating the various ways in which you can use it to perform detailed simulations with realworld accuracy. The showcase is divided into sections to display this functionality most effectively; you’ll find sections on Joule and Induction Heating, Optics and Photonics, and Plasma physics, just to name a few. Explore these and other areas by selecting from the available categories.
When exploring the showcase, and the COMSOL Product Suite in general, the philosophy of the multiphysics approach to modeling will become apparent. We developed the Product Suite to allow you to conduct fullycoupled analyses of applications involving multiple physics in one and the same simulation environment. For electrical engineers, an example of this modeling approach is the design of a power transmission line, where heat transfer, structural mechanics, and electromagnetics all come into play.
In a power transmission line, operating temperatures can affect loadcarrying capabilities and even the protective coating that surrounds electrical cables. As a result of the heat produced as current flows through the conductor, the system temperature rises and the electrical and thermal material properties of the conductor change. The interaction between these physics can change their expected behavior, altering, for example, the currentcarrying capabilities of the conductor as well as the durability of the cable’s protective coating.
In this application, the electrical and thermal effects are interdependent and strongly coupled. Therefore, the simulation must couple the physics the same way they are in the real world. The objective is to find a selfconsistent solution that satisfies all physics. This is the key strength of COMSOL Multiphysics, where accurate analyses are accomplished through the use of a truly multiphysics code that allows engineers to apply an unlimited number of physics analyses in a single simulation.
The Designing and Modeling Electrical Systems and Devices resource shows you more about this approach to multiphysics modeling and how you can employ it in your own simulations.
Once you have finished exploring the different application areas, you can contact COMSOL experts to ask any additional questions you might have. A short form is provided at the end of the showcase that will put you directly in contact with our application engineers.
Explore all the free resources mentioned above and learn how multiphysics modeling can help improve your R&D process at www.comsol.com/showcase/electrical.
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Frequency selective surfaces (FSS) are periodic structures that function as filters for plane waves, such as microwave frequency waves. These structures can transmit, absorb, or reflect different amounts of radiation at varied frequencies. Typically, they have a bandstop or bandpass frequency response.
Frequency selective surfaces are used in a variety of applications. For example, the article “Picking the Pattern for a Stealth Antenna” from COMSOL News 2013 describes how engineers at Altran used FSS as RF filters to reduce the radar cross sections (RCS) of stealth antennas. In the article, designers employed FSS to reduce antenna gain in order to lower the RCS. There, the FSS were designed to absorb incident radiation, rather than scatter it. FSS surfaces are typically constructed with metallic patterns that are arranged periodically. Complementary split ring resonators can be used to build such structures.
As a type of planar resonator, complementary split ring resonators are primarily used to simulate metamaterial elements. When designing a bandpass structure, for example, they can be arranged periodically. Modeling these resonators in a periodic configuration can become quite complex and timeconsuming. However, you can overcome these design challenges by implementing periodic boundary conditions into your model.
COMSOL Multiphysics, together with the RF Module, enables you to model a periodic complementary split ring resonator with ease by utilizing perfectly matched layers and periodic boundary conditions. As an example, we can refer to the Frequency Selective Surface, Periodic Complementary Split Ring Resonator model, which is available in our Model Gallery.
In this model, a split ring slot is patterned on a thin copper layer (which is modeled as a perfect electric conductor) that rests on a PTFE substrate that is 2 mm in thickness.
A single unit cell of the complementary split ring resonator. The model is created with periodic boundary conditions.
To simulate a 2D infinite array, as shown above, you can model just one unit cell of the complementary split ring resonator. This is done using Floquetperiodic boundary conditions on each of the four sides of the unit cell.
To learn how to do this, check out the model documentation, where stepbystep instructions are provided.
While this post mostly focuses on how you can save time modeling using periodic boundary conditions, this particular model’s documentation goes into further detail regarding the periodic structure’s bandpass frequency response in terms of Sparameters, as shown below.
Sparameter plot showing the periodic structure functions as a bandpass filter near 4.6 GHz.
The electron energy distribution function (EEDF) is essential in plasma modeling because it is needed to compute reaction rates for electron collision reactions. Because electron transport properties can also be derived from the EEDF, the choice of the EEDF you use influences the results of the plasma model. If the plasma is in thermodynamic equilibrium, the EEDF has a Maxwellian shape. In most plasmas, for technical purposes, deviations from the Maxwellian form occur.
To describe the EEDF, several possibilities are available, such as a Maxwell or Druyvesteyn function. In addition, a generalized form is available, which is an intermediate between the Maxwell and the Druyvesteyn function.
Maxwell 
f(\epsilon)=\varphi^{3/2}\beta_1\exp\left(\frac{\epsilon\beta_2}{\varphi}\right)
\beta_1=\Gamma(5/2)^{3/2}\Gamma(3/2)^{5/2},\ \beta_2=\Gamma(5/2)\Gamma(3/2)^{1}

Druyvesteyn 
f(\epsilon)=\varphi^{3/2}\beta_1\exp\left(\left(\frac{\epsilon\beta_2}{\varphi}\right)^2\right)
\beta_1=\Gamma(5/4)^{3/2}\Gamma(3/4)^{5/2},\ \beta_2=\Gamma(5/4)\Gamma(3/4)^{1}

Generalized 
f(\epsilon)=\varphi^{3/2}\beta_1\exp\left(\left(\frac{\epsilon\beta_2}{\varphi}\right)^g\right)
\beta_1=\Gamma(5/2g)^{3/2}\Gamma(3/2g)^{5/2},\ \beta_2=\Gamma(5/2g)\Gamma(3/2g)^{1}

Here, ϵ is the electron energy, (eV); \varphi is the mean electron energy, (eV); and g is a factor between 1 and 2. For a Maxwell distribution function, g is equal to 1, while g equals 2 for a Druyvesteyn distribution. Lastly, \Gamma is the incomplete Gamma function.
As the Druyvesteyn EEDF is based on a constant (electron energy independent) cross section, the Maxwellian EEDF is based on constant collision frequency. The distribution functions assume that elastic collisions dominate, thus the effect of inelastic collisions (e.g., excitation or ionization) on the distribution function is insignificant. In such a case, the distribution function becomes spherically symmetric. In elastic collisions with neutral atoms, the electrons’ direction of motion is changed, but not their energies (due to the large mass difference).
If the electrons are in thermodynamic equilibrium among each other, the distribution function is Maxwellian. However, this is only true if the ionization degree is high. Here, electronelectron collisions drive the distribution towards a Maxwellian shape. Inelastic collisions of electrons with heavy particles lead to a drop of the EEDF at higher electron energies. Therefore, a Druyvesteyn distribution function often gives more accurate results for a lower ionization degree.
Furthermore, the EEDF can be computed by solving the Boltzmann equation. The Boltzmann equation describes the evolution of the distribution function, f, in a sixdimensional phase space:
To solve the Boltzmann equation and, therefore, compute the EEDF, drastic simplifications are necessary. A common approach is to expand the distribution function in spherical harmonics. The EEDF is assumed to be almost spherically symmetric, so the series can be truncated after the second term (a socalled twoterm approximation). This approach is the most accurate way to compute the EEDF because an anisotropic perturbation, due to inelastic collisions, is taken into account. However, this is also the most computationally expensive approach.
Maxwellian EEDF in eV^{–1} for mean electron energies from 2 — 10 eV.
Normally, the distribution function is divided by \sqrt{\epsilon} for illustration purposes. This kind of distribution function is also known as an electron energy probability function (EEPF). For a Maxwellian function, this results in a straight line with a slope of (1/k_B T), as shown below.
Maxwellian EEPF in eV^{–3/2} for mean electron energies from 2 — 10 eV.
A Druyvesteyn distribution has the maximum and mean energy shifted to higher values. The highenergy tail drops much faster in comparison to a Maxwellian distribution. As the electrons reach enough energy for excitation or ionization, elastic collisions occur. This leads to the fall of the Boltzmann distribution function, which is observed below.
Comparison of a Maxwell, Druyvesteyn, and Boltzmann distribution function.
Mean electron energy 5 eV, electron density 1\cdot10^{16}\ \mathrm{m}^{3}, ionization degree 1\cdot10^{9}
In the plasma model, the EEDF is needed to compute the rate coefficients, k_k, for the electron collisions reactions, according to the equation:
In the above equation, \gamma = \sqrt{2q/m_e}, the electron energy is ϵ, and \sigma_k is the cross section for reaction, k.
The rate coefficients for excitation and ionization highly depend on the shape of the EEDF. This is due to the exponential dropoff in the population of electrons at energies exceeding the activation threshold. Using a Maxwellian EEDF can lead to an overestimation of the ionization rate, which is shown below.
Rate coefficients for argon ionization computed with different kinds of EEDFs.
Furthermore, the electron transport properties can be computed by means of the EEDF, using the Boltzmann Equation, TwoTerm Approximation interface. The computed transport coefficients have less dependence on the type of EEDF.
Reduced electron mobility computed with different kinds of EEDFs.
As the rate coefficients can differ by orders of magnitude, we must be aware that the discharge characteristics also change drastically when changing the EEDF. In the Plasma Module Model Library, there is a model of dielectric barrier discharge (DBD). This model simulates electrical breakdown in atmospheric pressure argon. The model was recomputed with the three different kinds of EEDFs, and we compared the results. The next two figures show the total current at the grounded electrode and the instantaneously absorbed power in the plasma. The plasma is driven with a sinusoidal voltage with a frequency of 50 kHz. The figures show the behavior over two periods.
Total discharge current in the DBD vs. time.
Absorbed power of the DBD vs. time.
The results look quite similar. So, the choice of the EEDF influences the modeling results, not orders of magnitude, but in this case much less than a factor of two. This, of course, depends on the model and the specific results you wish to extract.
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Spinal cord stimulation (SCS) is used to relieve chronic back and leg pain by surgically implanting electrodes that apply an electrical potential directly to the spine. The treatment works by sending electrical pulses through the spinal cord that communicate with and shut down cells that are sending pain signals to the brain. The pulses can be increased or decreased, or the patterns of stimulation can be changed by an external programmer and accessed by the patient.
Patients receiving SCS treatment are implanted with the electrodes in the epidural space overlying the dorsal column. However, a few weeks after the device has been implanted, fibrotic scar forms under the electrode contacts, changing the electrical conducting characteristics of the tissue surrounding the electrode. Often, this means that the stimulation must be reprogrammed in order to deliver the correct stimulus to the intended target cells, a process that is generally done through trial and error.
Stimulator electrode array on the surface of outer dural skin of the spinal cord. Scar formation is shown in green. Image courtesy of Kris Carlson.
Although SCS has been in use since the 1960s and around 30,000 such procedures are done each year, there is still not a precise understanding of how the technique works or why stimulation patterns change after implantation. Using COMSOL Multiphysics in combination with other software, the team at Beth Israel Deaconess modeled how scar formation can affect current distribution in the spinal cord (formerly, research was conducted at Lahey Clinic, Burlington, MA, where researcher Jay Shils is still affiliated).
Their research is one of the first to examine the effects of scar tissue on SCS treatments, and the first to use computational modeling to do so. In their study, the team created a 3D finite element model of the spinal cord, scar tissue, and electrodes to determine if they could use simulation to accurately predict and define the necessary modifications to the stimulus pattern after scar tissue has formed.
The June 2014 volume of Neuromodulation features the paper “Modeling Effects of Scar on Patterns of Dorsal Column Stimulation” by Jeffrey Arle et al., Beth Israel Deaconess. The cover image COMSOL Multiphysics slice plots are an excerpt from their paper, showing the field potential of a central cathode flanked by two anodes during spinal column stimulation with and without scarring. Image used with permission from John Wiley & Sons, Inc.
The simulations created by the team were used to determine which axons in the dorsal columns would become activated when exposed to stimulation, and how particular stimulation patterns would change in response to scar tissue. The model looked at how a scar would change the penetration depth and pattern of active fibers in the spinal column. The team’s expectation was that resistive scar formation would reduce the energy penetrating to spinal cord fibers and therefore reduce the pain relief felt by patients. Surprisingly, they found that when a scar forms under either of the anodes that flank a central cathode, the field shape is distorted, resulting in greater energy reaching the spinal cord with unpredictable effects.
Volume plots looking through the back of a hidden spinal cord toward the stimulator electrode array (i.e., a ventral view). The plots show the distribution of electric potential on the surface of the white matter (axons or nerve fibers) of the spinal cord. A: Normal stimulation from a central cathode flanked by anode “guards” that focus the field toward the center of the white matter. B: After scar formation under the right anode, the desired electric field is distorted, stimulating large numbers of nerve fibers in untargeted areas. The result is lack of pain relief and uncomfortable sensations. C: Reprogramming the electrode array to attempt to compensate for scar formation. Without finite element modeling to indicate the resulting field redistribution, such reprogramming is like “flying blind”. Here, the right electrode amplitude has been increased. D: A second attempt to reprogram the electrode array to compensate for scar. The left electrode amplitude has been decreased and the two electrodes above and below the electrode blocked by scar have been turned on. Image courtesy of Kris Carlson.
Their results showed that, with a scar, conductivity in the tissue would be reduced to 0.15 S/m compared to the initial conductivity of 0.6 S/m. The model also took into account two different compensation patterns — attempts that a stimulus programmer might make in order to recapture the original baseline of the stimulation pattern before it was disrupted by the scar.
Topdown slice plots of electric potential distribution from the stimulator electrode array. Compare to previous figure. A: Normal stimulation pattern. Finite element modeling has shown that, in spinal cord stimulation, the nerve fibers resulting in pain relief lie in a thin 0.3 mm strip on the very outer layer of the white matter (inner cylinder), which is roughly the area touched by the yellow region of electric potential between the two blue regions. Even a small shift of electric field changing its magnitude or symmetry results in loss of pain relief and possible uncomfortable sensations. B: Distorted field distribution pattern after scar formation under the left anode. C and D: Patterns after two attempts to reprogram the electrode array to compensate for scar formation. Image courtesy of Kris Carlson.
The model they created with COMSOL Multiphysics showed how, with scar tissue, there is more activation with deeper penetration than in the baseline, noscar condition, as well as how compensation efforts failed to recapture the baseline stimulation pattern. In the model, the activation energies beneath the scar itself are greatly reduced, but the total energy from the stimulation remains the same. The researchers related this finding to Kirchoff’s current laws about the conservation of overall energy. What this means for the patient is that, in many cases, higher than desired levels of current will reach deeper tissues, potentially stimulating too many fibers and causing discomfort.
So what do these results mean for the future of spinal stimulation treatments? The research conducted by Arle et al. provides a deeper understanding of how scar tissue can affect SCS and demonstrates a need for stimulation programmers to better define and incorporate information about scar tissue on a patientbypatient basis. Not only can this research be applied to SCS, but brain stimulation techniques could also benefit from a deeper understanding of how scar tissue can affect stimulation. In their paper, the researchers state that, “further analyses and the knowledge from FEM modeling, especially with systematic parameter variations, may guide programmers toward effective compensation for scar formation [...] future iterations of programming devices may be able to allow programmers to make appropriate changes more efficiently.”
First, let us look at a rectangular wound multiturn coil, as shown in the figure below. A spherical modeling domain contains a rectangular coil. The coil domain represents several hundred turns of wire wrapped around a rectangular profile. The lead wires that excite the coil are neglected from the model, and we treat the coil as a closed loop of current. The Multiturn coil feature is used to compute and apply a uniform current distribution around the profile of the coil, and the steadystate magnetic field is plotted. Note that the coil domain has a constant crosssectional area as we follow the path of the current around it.
A rectangular coil with current flowing around the winding direction. Current flow (blue arrows) and magnetic field are plotted.
This modeling domain has three planes of symmetry, i.e. planes about which the geometry is exactly mirrored. Let us now see how we can use this geometric symmetry as well as our knowledge of the magnetic field and what direction the current is flowing to reduce the size of our modeling domain.
The Magnetic Insulation boundary condition represents a mirror symmetry plane for the magnetic field. The magnetic field will be exactly mirrored as you cross the plane.
The Magnetic Insulation boundary condition — “cut perpendicular to J and parallel to B“.
This boundary condition also means that the magnetic field is zero in the normal direction to the boundary. That is, the magnetic field must be tangential to this boundary. As a consequence, this boundary condition has the physical interpretation of a boundary through which current can only flow in the normal direction. The modeling rule can be summarized: “Use Magnetic Insulation to cut perpendicular to J and parallel to B.”
The Perfect Magnetic Conductor boundary condition, on the other hand, represents a mirror symmetry plane for the current. From a mathematical point of view, it can be thought of as the “opposite” of the Magnetic Insulation boundary condition.
The Perfect Magnetic Conductor boundary condition — “cut perpendicular to B and parallel to J“.
The current vector will be exactly mirrored as you cross the plane and can have no normal component, so the current must flow tangentially. This boundary condition enforces that the magnetic field can have no tangential component as you approach the boundary, so the magnetic field can only point in the normal direction and cannot change sign as you cross the boundary. The modeling rule can be summarized: “Use Perfect Magnetic Conductor to cut perpendicular to B and parallel to J.”
The original geometry can be reduced in size to a oneeighth model representing the original geometry. Orthogonal planes through the center of the coil are used to partition the domains as shown below.
A oneeighth symmetry model of a rectangular coil with current flowing around the winding direction. The Magnetic Insulation (magenta) and Perfect Magnetic Conductor (cyan) boundary conditions are applied along the appropriate symmetry planes for this problem.
The Magnetic Insulation boundary condition is applied at the two boundaries representing the planes through which the current will flow normally. If the coil is excited with a voltage boundary condition, it is important to reduce the voltage by a factor of two for each Magnetic Insulation symmetry condition applied. If the coil is excited with current, the applied current does not need to be changed, but the postprocessed coil voltage should be scaled by a factor of two for each Magnetic Insulation symmetry condition.
The Perfect Magnetic Conductor condition is used at the plane along which the current will flow tangentially. Since the Perfect Magnetic Conductor condition cuts the coil in half, it is important to divide the applied current in half when a current excitation is used. On the other hand, if a voltage excitation is used, then the postprocessed coil current must be scaled up by a factor of two for each Perfect Magnetic Conductor symmetry plane.
In almost all cases, the Magnetic Insulation and Perfect Magnetic Conductor boundary conditions are sufficient to significantly reduce the size of your model. As we saw earlier, these conditions enforce the current and magnetic fields to be either normal or tangential to the boundary. But what if we have a geometric symmetry plane where the fields do not have such a symmetry? In such cases, the Periodic (boundary) Condition may be appropriate.
The Periodic Condition is used when all we know is that the solution must be periodic.
The Periodic (boundary) Condition allows for more general symmetry where both the current and the magnetic field vector can be at an angle to the boundary. The usage of this condition is limited to cases where the magnetic sources as well as the structure are periodic in space. Typically, the full geometry can be reduced to the smallest repetitive element, a unit cell, limited by periodic conditions.
Consider the structure of a toroidal inductor wound with a single strand of wire, shown below. The wire can be modeled fairly accurately as a single continuous spiral around the toroid, as long as we again neglect the asymmetry due to the lead wires. We can model the wire as an edge current, flowing tangentially to the wires.
A spirally wound toroidal inductor. The arrows (blue) indicate the direction of current flow. The magnetic field in the core is shown.
To exploit as much symmetry as possible here, we can consider the unit cell that is just a small slice of the original model containing a single turn of the winding. The Periodic Condition is used along the sides of the slice. When using this boundary condition, the mesh must be identical on the periodic faces, so the Copy Face functionality should be used to ensure identical meshes. As we can see from the image below, the size of the model can be reduced by the number of windings, greatly reducing the problem size.
Periodic Conditions can greatly reduce the model size for certain geometries.
The generality of the Periodic Condition comes at a price compared to the more basic Magnetic Insulation and Perfect Magnetic Conductor conditions. As it links the unknown fields on one side of the geometry to those on the opposite side, it makes the system matrix more dense and expensive to solve. Therefore, do not use it if the more basic conditions apply.
By reducing the model size, we also reduce the computational requirements significantly. In fact, computational requirements grow exponentially with problem size, so the more symmetries that we can use, the better. Even if you don’t have symmetry in the full problem that you want to solve, it is often advisable to work with a smaller model that does have symmetry in the initial developmental stages of your modeling.
]]>Anechoic chambers are noisecanceling rooms that are designed to absorb sound or electromagnetic waves. Acoustic anechoic chambers, which typically have noise levels of around 1020 dBA, can be used to test loudspeakers and the directivity of noise radiation, the sound quality of certain products (such as a HarleyDavidson), as well as to decrease the noise level produced by certain products (washing machines, computer fans, etc.).
First developed during World War II as a way to test aircraft that absorbed or scattered radar signals, RF anechoic chambers are still used today for a variety of different purposes. Like acoustic anechoic chambers, an RF anechoic chamber provides a space where no incident energy waves are present, allowing for devices to be tested without interference. Examples include satellites and the antenna performance of devices such as cell phones, RFID tags, and GPS. When testing onboard aircraft systems, these chambers can be large enough to house the entire aircraft itself.
The image below shows the cones, or pyramidal absorbers, that line the walls, ceiling, and floor of an RF anechoic chamber at the Surface Sensors and Combat Systems Facility at Naval Surface Warfare Center (NSWC) Dahlgren.
Testing a maritime antenna in an anechoic chamber.
You can easily model an RF anechoic chamber using COMSOL Multiphysics and the RF Module with perfectly matched layers and periodic boundary conditions. Let’s explore the one of our models from the Model Gallery: Using the Modeling of Pyramidal Absorbers for an Anechoic Chamber.
In our model example, pyramidal lossy structures (absorbers) are placed in an infinite array. When an incident wave strikes one of the pyramidal structures, many small reflections are created as the electromagnetic wave passes into the pyramid and is reflected into a second pyramid. The absorber is made of radiantabsorbent (RAM) material, which means that as the wave strikes the pyramid, the incident field is partially reflected and partially transmitted into the nearby absorber. Therefore, after many reflections and partial transmissions, the wave’s amplitude is drastically reduced by the time it reaches the base of the pyramid. In the model, we imitate the microwave absorption of a conductive carbon loadedfoam with a conductive material at σ = 0.5 S/m.
Below, you can see the infinite 2D array of the pyramidal structures. The structure can be modeled using Floquetperiodic boundary conditions on four sides, where one unit of the model contains the pyramidal structure as well as the block beneath it made of the same material. Above the pyramid is a perfectly matched layer (PML) and the space above and between the pyramid and PML is filled with air. At the bottom of the pyramid and block is a layer of highly conductive material that is used to block any noise from outside the chamber (shown in orange in the image above). This layer is modeled as a perfect electric conductor (PEC) in our model.
In the plot below on the left, the norm of the electric field and power flow in the model is shown for the case where the angle of incidence on the pyramid is 30°. As we can see, the strength of the incident wave is strong near the tip of the absorber and deceases toward the base of the pyramid. On the right, the graph displays the scattering parameter (Sparameter) for yaxis polarized waves as a function of incident angle.