Let’s begin by discussing a traditional antenna simulation using COMSOL Multiphysics and the RF Module. When we simulate a radiating antenna, we have a local source and are interested in the subsequent electromagnetic fields, both nearby and outgoing from the antenna. This is fundamentally what an antenna does. It converts local information (e.g., voltage or current) into propagating information (e.g., outgoing radiation). A receiving antenna inverts this operation and changes incident radiation into local information. Many devices, such as a cellphone, act as both receiving and emitting antennas, which is what enables you to make a phone call or browse the web.
Antennas of the Atacama Large Millimeter Array (ALMA) in Chile. ALMA detects signals from space to help scientists study the formation of stars, planets, and galaxies. Needless to say, the distance these signals travel is much greater than the size of an antenna. Image licensed under CC BY 4.0, via ESO/C. Malin.
In order to keep the required computational resources reasonable, we model only a small region of space around the antenna. We then truncate this small simulation domain with an absorbing boundary, such as a perfectly matched layer (PML), which absorbs the outgoing radiation. Since this will solve for the complex electric field everywhere in our simulation domain, we will refer to this as a FullWave simulation.
We then extract information about the antenna’s emission pattern using a FarField Domain node, which performs a neartofarfield transformation. This approach gives us information about the electromagnetic field in two regions: the fields in the immediate vicinity of the antenna, which are computed directly, and the fields far away, which are calculated using the FarField Domain node. This is demonstrated in a number of RF models in the Application Gallery, such as the Dipole Antenna tutorial model, so we will not comment further on the practical implementation here.
One question that occasionally comes up in technical support is: “How do I use the FarField Domain node to calculate the radiated field at a specific location?” This is an excellent question. As stated in the RF Module User’s Guide, the FarField Domain node calculates the scattering amplitude, and so determining the complex field at a specific location requires a modification for distance and phase. The expression for the xcomponent of the electric field in the far field is:
and similar expressions apply to the y and zcomponent, where r is the radial distance in spherical coordinates, k is the wave vector for the medium, and emw.Efarx is the scattering amplitude. It is worth pointing out that emw.Efarx is the scattering amplitude in a particular direction, and so it depends on angular position , but not radial position. The decrease in field strength is solely governed by the 1/r term. There are also variables emw.Efarphi and emw.Efartheta, which are for the scattering amplitude in spherical coordinates.
To verify this result, we simulate a perfect electric dipole and compare the simulation results with the analytical solution, which we covered in the previous blog post. As we stated in that post, we split the full results into two terms, which we call the near and farfield terms. We briefly restate those results here.
where is the dipole moment of the radiation source and is the unit vector in spherical coordinates.
Below, we can see the electric fields vs. distance calculated using the FarField Domain node for a dipole at the origin with . For comparison, we have included the FarField Domain node, the full theory, as well as the near and farfield terms individually. The fields are evaluated along an arbitrary cut line. As you can see, there is overlap between the FarField Domain node and the farfield theory plots, and they agree with the full theory as the distance from the antenna increases. This is because the FarField Domain node will only account for radiation that goes like 1/r, and so the agreement improves with increasing distance as the contribution of the 1/r^{2} and 1/r^{3} terms go to zero. In other words, the FarField Domain node is correct in the far field, which you probably would have guessed from the name.
A comparison of the FarField Domain node vs. theory for a point dipole source.
For most simulations, the nearfield and farfield information is sufficient and no further work is necessary. In some cases, however, we also want to know the fields in the intermediate region, also known as the induction or transition zone. One option is to simply increase the simulation size until you explicitly calculate this information as part of the simulation. The drawback of this technique is that the increased simulation size requires more computational resources. We recommend a maximum mesh element size of for 3D electromagnetic simulations. As the simulation size increases, the number of mesh elements increases, and so do the computational requirements.
Another option is to use the Electromagnetic Waves, Beam Envelopes interface, which here we will simply refer to as BeamEnvelopes. As discussed in a previous blog post, BeamEnvelopes is an excellent choice when the simulation solution will have either one or two directions of propagation, and will allow us to use a much coarser mesh. Since the phase of the emission from an antenna will look like an outgoing spherical wave, this is a perfect solution for determining these fields. We perform a FullWave simulation of the fields near the source, as before, and then use BeamEnvelopes to simulate the fields out to an arbitrary distance, as required.
The simulation domain assignments. If the outer region is assigned to PML, then a FullWave simulation is performed everywhere. It is also possible to solve the inner region using a FullWave simulation and the outer region using BeamEnvelopes, as we will discuss below. Note that this image is not to scale, and we have only modeled 1/8 of the spherical domain due to symmetry.
How do we couple the BeamEnvelopes simulation to our FullWave simulation of the dipole? This can be done in two steps involving the boundary conditions at the interface between the FullWave and BeamEnvelopes domains. First, we set the exterior boundary of the FullWave simulation to PMC, which is the natural boundary condition for that simulation. The second step is to set that same boundary to an Electric Field boundary condition for BeamEnvelopes. We then specify the field values in the BeamEnvelopes Electric Field boundary condition according to the fields computed from the FullWave simulation, as shown here.
The Electric Field boundary condition in BeamEnvelopes. Note that the image in the top right is not to scale.
A Matched Boundary Condition is applied to the exterior boundary of the BeamEnvelopes domain to absorb the outgoing spherical wave. The remaining boundaries are set to PEC and PMC according to symmetry. We must also set the solver to Fully Coupled, which is described in more detail in two blog posts on solving multiphysics models and improving convergence from a previous blog series on solvers.
If we again examine the comparison between simulation and theory, we see excellent agreement over the entire simulation range. This shows that the PMC and Electric Field boundary conditions have enforced continuity between the two interfaces and they have fully reproduced the analytical solution. You can download the model file in the Application Gallery.
A comparison of the electric field of the FullWave and BeamEnvelopes simulations vs. the full theory.
In today’s blog post, we examined two ways of computing the electric field at points far away from the source antenna and verified the results using the analytical solution for an electric point dipole. These two techniques are using the FarField Domain node from a FullWave simulation and linking a FullWave simulation to a BeamEnvelopes simulation. In both cases, the fields near the source and in the far field are correctly computed. The coupled approach using BeamEnvelopes has the additional advantage in that it also computes fields in the intermediate region. In the next post in the series, we will combine the calculated farfield radiation with a simulation of a receiving antenna and determine the received power. Stay tuned!
Multiscale modeling is a challenging issue in modern simulation that occurs when there are vastly different scales in the same model. For example, your cellphone is approximately 15 cm, yet it receives GPS information from satellites 20,000 km away. Handling both of these lengths in the same simulation is not always straightforward. Similar issues show up in applications such as weather simulations, chemistry, and many other areas.
While multiscale modeling can be a general topic, we will focus our attention on the practical example of antennas and wireless communication. When we wirelessly transmit data via antennas, we can break the operation down into three main stages:
Modern communications require longdistance wireless data transfer via antennas.
The two length scales that we will consider for this process are the wavelength of the radiation and the distance between the antennas. To use a specific example, FM radio has a wavelength of approximately three meters. When you listen to the radio in your car, you are often ten km or more away from the radio tower. Because many antennas, such as dipole antennas, are similar in size to a wavelength, we will not consider this to be another distinct length scale. As a result, we have one length scale for the emitting antenna, a different length scale for the signal propagation from source to destination, and then the original length scale again for the receiving antenna.
Let’s go over some of the most important equations, terms, and considerations when working with multiple scales in the same highfrequency electromagnetics model.
The Friis transmission equation calculates the received power for lineofsight communication between two antennas separated by a lossless medium. The equation is
where the subscripts r and t discriminate between the transmission antenna and the receiving antenna, G is the antenna gain, P is the power, is the reflection coefficient for impedance mismatch between antenna and transmission line, p is the polarization mismatch factor, λ is the wavelength, r is the distance between the antennas and is associated with the socalled freespace path loss, and and are the angular spherical coordinates for the two antennas.
Note that we have explicitly included two impedance mismatch terms, and so:
The Friis transmission equation is derived in many texts, so we will not do so again here.
A visualization of the gain for a transmitting and receiving antenna. When using the Friis transmission equation, we require the orientation of each antenna for correct gain specification. The distance between the antennas is r.
Let’s now discuss spherical coordinates , since they are incredibly useful for antenna radiation and we will use them repeatedly. Starting from the Cartesian coordinates (x, y, z), we can easily express these as follows.
For convenience, we have used the actual COMSOL Multiphysics commands — sqrt(), acos(), and atan2(,) — instead of their mathematical symbols. In our simulation setup, we will also make use of the Cartesian components of the spherical unit vector .
Similar assignments can be made for the Cartesian components of and , but is the most important for our purposes. This will be discussed later in this blog series when we cover ray optics.
A given point shown in both Cartesian (x, y, z) and spherical coordinates. The unit vectors for the spherical coordinates are also included. Note that the spherical unit vectors are functions of location.
We are generally interested in the radiated power from antennas. The power flux in W/m^{2} is represented by the complex Poynting vector .
Many antenna texts also use radiation intensity, which is defined as the power radiated per solid angle and measured in W/steradian. Mathematically speaking, this is . For clarity, we have included two conventions here, as it is common to use in electrical engineering, while physicists will generally be more familiar with . We can then calculate the radiated power by integrating this quantity over all angles.
Gain and directivity are similar in that they both quantify the radiated power in a given direction. The difference is that gain relates this radiated power to the input power, whereas directivity relates this to the overall radiated power. Put more simply, gain accounts for dielectric and conductive losses and directivity does not. Mathematically, this reads as and for gain and directivity, respectively. P_{in} is the power accepted by the antenna and P_{rad} is the total radiated power. While both quantities can be of interest, gain tends to be the more practical of these two as it accounts for material loss in the antenna. Because of its prevalence and usefulness, we also include the definition of gain (in a given direction) from “IEEE Standard Definitions of Terms for Antennas”, which is: “The ratio of the radiation intensity, in a given direction, to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically.”
IEEE also includes three notes about gain in their definition:
In practice, an actual antenna will be connected to a transmission line. Because the antenna and the transmission line may not have the same impedance, there can be a loss factor due to impedance mismatch. The realized gain is simply the gain when accounting for impedance mismatch. Mathematically, this is , where is the reflection coefficient from transmission line theory, Z_{c} is the characteristic impedance of the transmission line, and Z is the impedance of the antenna.
When using a lumped port with a characteristic impedance in COMSOL Multiphysics, the farfield gain that is calculated corresponds to the IEEE realized gain. This is important to mention explicitly, since various definitions of gain have changed over the last few decades. Starting with COMSOL Multiphysics version 5.3, which will be released in 2017, the variable names in the COMSOL software will be changed to match the IEEE definitions.
The realized gain and electric field from a Vivaldi antenna, simulated using COMSOL Multiphysics and the RF Module. You can find the Vivaldi Antenna tutorial model in the Application Gallery.
The terms we have discussed so far have referred to antennas emitting radiation, but they are also generally applicable to receiving antennas. The reason we have put more emphasis on emission thus far is because antennas generally obey reciprocity (the Lorentz reciprocity theorem is a fixture in most antenna textbooks). Reciprocity means that an antenna’s gain in a specific direction is the same regardless of whether it is emitting in that direction or receiving a signal from that direction. Practically speaking, you can calculate the gain in any direction from a single simulation of an emitting antenna, which is easier than simulating the inverse process for each desired direction.
When we talk about receiving antennas, we are often interested in calculating the received power for an incoming signal. This can be done by multiplying the effective area, , of the antenna by the incident power flux and accounting for impedance mismatch in the line, yielding . As you may expect, this bears a striking similarity to several terms of the Friis transmission equation.
Today, we will talk about one type of emitter: the perfect electric point dipole. Depending on the literature, you may have seen this referred to as a perfect, ideal, or infinitesimal dipole. This emitter is a common representation of radiation for electrically small antennas. The solution for the field is
where is the dipole moment of the radiation source (not to be confused with the polarization mismatch) and k is the wave vector for the medium.
One breakdown of the various regions for the electromagnetic field generated from an electrically small antenna.
In this equation, there are three factors of 1/r^{n}. The 1/r^{2} and 1/r^{3} terms will be more significant near the source, while the 1/r term will dominate at large distances. While the electromagnetic field will be continuous, it is common to refer to different regions of the field based on the distance from the source. One such distribution for an electrically small antenna is shown above, although there are other conventions that refer to the magnitude of kr.
Later, we will see how to calculate the fields at any distance from a given source, but the most important region for antenna communications is the far field or radiation zone, which is the region farthest away from the source. In this region, the fields take the form of spherical waves, , a fact that we will take advantage of.
We will now split up the Efield equation above into two terms. For simplicity, we will call the 1/r term the far field (FF) and the 1/r^{2} and 1/r^{3} terms the near field (NF).
As mentioned before, we can calculate the radiated power in watts by integrating over all angles. Note that only the farfield term will contribute to this integral, which is a primary reason why the far field is of practical interest to antenna engineers. The total power radiated from a point dipole is , where Z_{0} is the impedance of free space and c is the speed of light. The maximum gain is 1.5 and is isotropic in the plane normal to the dipole moment (e.g., the xyplane for a dipole in ).
A note on units: The equations above are given with the traditional definition of the dipole moment in Coulomb*meters (Cm). In antenna and engineering texts, it is common to specify an infinitesimal current dipole in Ampere*meters (Am). COMSOL Multiphysics follows the engineering convention. The two definitions are related by a time derivative, so for a COMSOL software implementation, the dipole moment should be multiplied by a factor of to obtain the infinitesimal current dipole.
We will use a perfectly conducting halfwavelength dipole as our receiving antenna.
A visual representation of radiation incident on a halfwavelength dipole antenna.
Many texts cover an infinitely thin wire, which has an impedance of and a directivity of . It is worth mentioning that the antenna impedance will change from these values for an antenna of finite radius. The receiving antenna we use here has a length of 0.47 λ and a lengthtodiameter ratio of 100. With these values, we simulate an impedance of , which is close to the infinitely thin value and also agrees reasonably well with experimental values. Regrettably, there is no theoretical value to compare to this number, but this highlights the need for numerical simulation in antenna design.
The comparison between the directivity of the infinitely thin dipole and our simulated dipole antenna is shown below. Because the antenna is lossless, this is equivalent to the antenna gain. You can download the dipole antenna model here.
A comparison of the directivity for two halfwavelength antennas (oriented in z) as a function of theta. The COMSOL Multiphysics® simulation is of a finite radius cylinder and the theory is for an infinitely thin antenna.
We can now use the Friis transmission equation to calculate the power that is emitted from a perfect point dipole and received by a halfwave dipole antenna. To use this equation, we simply need to know the gain and impedance mismatch (or realized gain), wavelength, distance between the antennas, and input power. Since we are using a point electric dipole, we have a dipole moment instead of input power and impedance mismatch. We can account for this by removing the impedance mismatch term and replacing the input power by the radiated power of the perfect electric dipole from above — effectively saying that power in equals power out.
If we assume that our emitter and detector are both located in the xyplane, are polarization matched, and are separated by 1000 λ, as well as that the dipole moment of the emitter is 1 Am in , the Friis equation yields a received power of 380 μW. We will simulate this value in part 3 of this series for verification of our simulation technique. We can then use our simulation to confidently extract results and introduce complexity that the Friis equation cannot account for.
In this blog post, we have introduced the idea of multiscale modeling and discussed all of the relevant terms, definitions, and theory that we will need moving forward. For those of you with a strong background in electromagnetics and antenna design, this has likely been a quick review. If the concepts presented here are new to you, we strongly recommend further reading in a book on classical electromagnetics or antenna theory.
In the following blog posts, we will focus primarily on practical implementation of multiscale modeling in COMSOL Multiphysics and we will repeatedly refer to concepts discussed today.
Stay tuned for more installments in our multiscale modeling blog series:
One of the first types of commercialized MEMS devices was the piezoresistive pressure sensor. This device, which continues to dominate the pressure sensor market, is valuable in a range of industries and applications. Measuring blood pressure as well as gauging oil and gas levels in vehicle engines are just two examples.
Piezoresistive pressure sensors have applications in the biomedical field as well as the automotive industry. Left: A blood pressure measurement device. Image by Andrew Butko. Licensed under CC BYSA 3.0, via Wikimedia Commons. Right: A vehicle’s oil gauge. Image by Marcus Yeagley. Licensed under CC BYSA 2.0, via Flickr Creative Commons.
While piezoresistive pressure sensors require additional power to operate and feature higher noise limits, they offer many advantages over their capacitive counterparts. For one, they are easier to integrate with electronics. They also have a more linear response in relation to the applied pressure and are shielded from RF noise.
But like other MEMS devices, piezoresistive pressure sensors include multiple physics within their design. And in order to accurately assess a sensor’s performance, you need to have tools that enable you to couple these different physics and describe their interactions. The features and functionality of COMSOL Multiphysics enable you to do just that. From your simulation results, you can get an accurate overview of how your device will perform before it reaches the manufacturing stage.
To illustrate this, let’s take a look at an example from our Application Gallery.
The design of our Piezoresistive Pressure Sensor, Shell tutorial model is based on a pressure sensor that was previously manufactured by a division of Motorola that later became Freescale Semiconductor, Inc. While the production of the sensor has stopped, there is a detailed analysis provided in Ref. 1 and an archived data sheet available from the manufacturers in Ref. 2.
Our model geometry is comprised of a square membrane that is 20 µm thick, with sides that are 1 mm in length. A supporting region that is 0.1 mm wide is included around the edges of the membrane. This region is fixed on its underside, indicating a connection to the thicker handle of the device’s semiconducting material. Near one of the membrane’s edges, you can see an Xshaped piezoresistor (Xducer™) as well as some of its associated interconnects. Only some interconnects are included, as their conductivity is high enough that they don’t contribute to the device’s output.
Geometry of the sensor model (left) and a detailed view of the piezoresistor geometry (right).
A voltage is applied across the [100] oriented arm of the X, generating a current down this arm. When pressure induces deformations in the diaphragm in which the sensor is implanted, it results in shear stresses in the device. From these stresses, an electric field or potential gradient that is transverse to the direction of the current flow occurs in the [010] arm of the X — a result of the piezoresistance effect. Across the width of the transducer, this potential gradient adds up, eventually producing a voltage difference between the [010] arms of the X.
For this case, we assume that the piezoresistor is 400 nm thick and features a uniform ptype density of 1.31 x 10^{19} cm^{3}. While the interconnects are said to have the same thickness, their dopant density is assumed to be 1.45 x 10^{20} cm^{3}.
With regards to orientation, the semiconducting material’s edges are aligned with the x and yaxes of the model as well as the [110] directions of the silicon. The piezoresistor, meanwhile, is oriented at a 45º angle to the material’s edge, meaning that it lies in the [100] direction of the crystal. To define the orientation of the crystal, a coordinate system is rotated 45º about the zaxis in the model. This is easy to do with the Rotated System feature provided by the COMSOL software.
In this example, we use the Piezoresistance, Boundary Currents interface to model the structural equations for the domain as well as the electrical equations on a thin layer that is coincident with a boundary in the geometry. Using this kind of 2D “shell” formulation significantly reduces the computational resources required to simulate thin structures. Note that both the MEMS Module and the Structural Mechanics Module are used to perform this analysis.
To begin, let’s look at the displacement of the diaphragm after a 100 kPa pressure is applied. As the simulation plot below shows, the displacement at the center of the diaphragm is 1.2 µm. In Ref. 1, a simple isotropic model predicts a displacement of 4 µm at this point. Considering that the analytic model is derived from a crude variational guess, these results show reasonable agreement with one another.
The displacement of the diaphragm following a 100 kPa applied pressure.
When using a more accurate value for shear stress in local coordinates at the diaphragm edge’s midpoint, the local shear stress is said to be 35 MPa in Ref. 1. This is in good agreement with the minimum value from our simulation study (38 MPa). In theory, the shear stress should be the greatest at the diaphragm edge’s midpoint.
Shear stress in the piezoresistor’s local coordinate system.
The following graph shows the shear stress along the edges of the diaphragm. The maximum local shear stress of 38 MPa is at the center of each of the edges.
Local shear stress along two of the diaphragm’s edges.
Given that the dimensions of the device and the doping levels are estimates, the model’s output during normal operation is in good agreement with the information presented in the manufacturer’s data sheet. For instance, in the model, an operating current of 5.9 mA is obtained with an applied bias of 3 V. The data sheet notes a similar current of 6 mA. Further, the model generates a voltage output of 54 mV. As indicated by the data sheet, the actual device produces a potential difference of 60 mV.
Lastly, we look at the detailed current and voltage distribution inside the Xducer™ sensor. As noted by Ref. 3, a “shortcircuit effect” may occur when voltagesensing elements increase the currentcarrying silicon wire’s width locally. This effect essentially means that the current spreads out into the sense arms of the X. The shortcircuit effect is illustrated in the plot below. Also highlighted is the asymmetry of the potential, which is a result of the piezoresistive effect.
Current density and electric potential for a device with a 3 V bias and an applied pressure of 100 kPa.
S.D. Senturia, “A Piezoresistive Pressure Sensor”, Microsystem Design, chapter 18, Springer, 2000.
Motorola Semiconductor MPX100 series technical data, document: MPX100/D, 1998.
M. Bao, Analysis and Design Principles of MEMS Devices, Elsevier B. V., 2005.
Xducer™ is believed to be a trademark of Freescale Semiconductor, Inc. f/k/a Motorola, Inc. Neither Freescale Semiconductor Inc. nor Motorola, Inc. has in any way provided any sponsorship or endorsement of, nor do they have any connection or involvement with, COMSOL Multiphysics® software or this model.
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For those in the semiconductor industry, rapid thermal processing (RTP) is recognized as an important step in producing semiconductors. In this manufacturing process, silicon wafers are heated to temperatures greater than 1000°C in a few seconds or less. This is often achieved by using highintensity lasers or lamps as heat sources. The temperature of the wafer is then slowly reduced in order to prevent any dislocations or water breakage that could occur as a result of thermal shock. The applications for RTP range from activating dopants to chemical vapor deposition, a topic that we’ve discussed previously on the blog.
Rapid thermal annealing (RTA) is a subset of RTP. This process involves rapidly heating an individual wafer from ambient temperature to somewhere in the range of 1000 to 1500 K. For RTA to be effective, there are a few considerations that need to be made. For one, the step must occur quickly; otherwise, the dopants can be diffused too much. Also important to the step’s success is preventing overheating and nonuniform temperature distributions. This facilitates the need for accurate measurements of the wafer’s temperature during RTA, which are typically achieved using either thermocouples or IR sensors.
A schematic of a common RTA apparatus.
An IR sensor, when ideally positioned, only receives radiation that is reflected and emitted by the silicon wafer. This is otherwise known as secondary radiation. Other desirable characteristics of sensors include short response times and high levels of accuracy. To design an optimal IR sensor, you could perform a parameter optimization in COMSOL Multiphysics. But before that step, and what we’ll focus on here, is using simulation to determine if an IR sensor is the more appropriate choice for an RTA configuration when compared to the inexpensive thermocouple.
As highlighted in the diagram above, RTA often makes use of doublesided heating in many applications. In such a setup, IR lamps are placed above and below the silicon wafer. For our Rapid Thermal Annealing tutorial, we chose to model a singlesided heating apparatus.
The model geometry for the RTA configuration.
In the above figure, the components are stored in a chamber featuring temperaturecontrolled walls with a set point of 400 K. The geometry of the chamber walls are thus omitted, as this results in a closed cavity. The model further assumes that radiation and convection cooling dominate the physical system. A heat transfer coefficient is used to model the convective cooling of the wafer and sensor to the gas.
The lamp, meanwhile, is treated as a solid object that has a volume heat source of 25 kW. Insulation is included on all sides of the object, except for the top surface. It is through this top surface, which faces the wafer, that the heat leaves the lamp as radiation. The model uses a low heat capacity for the solid to capture its transient startup time. The other thermal properties of the lamp are the same as those of copper metal.
Let’s begin by looking at the temperature distribution in the lamp, wafer, and sensor after 10 seconds of heating. As the simulation plot shows, there is a significant difference between the temperature of the wafer, which is around 1800 K, and the temperature of the sensor, which is around 1100 K. You may also notice that there is an uneven temperature distribution in the wafer. While not included in our example model, reconfiguring the heat source could help address this issue.
The transient temperature field after ten seconds of heating.
We also want to see how well the temperature of the sensor reflects that of the wafer’s surface. For this purpose, it is helpful to plot the temperature transient of the centerpoint on the wafer’s surface facing the lamp along with the temperature at a certain point on the top surface of the sensor. In the plot below, these two measurements are denoted by T_{wafer} and T_{sensor}, respectively. The temperature transient of the lamp (T_{lamp}) and the irradiation power at the surface of the sensor I_{sensor} are also shown.
Comparison of the temperature transients for the individual components of the RTA configuration along with the irradiation power at the sensor’s surface.
As the results indicate, the temperature of the sensor poorly reflects that of the wafer. A thermocouple’s signal would therefore not be very useful in regulating this process. But the IR detector does show good agreement with the characteristics of the wafer temperature. Such accurate measurements of the wafer’s temperature are achieved with a scalar amplification.
There are some drawbacks to IR sensors that are important to mention as well. For instance, the IR sensor has far less inertia than the wafer. While the wafer needs some time to heat up, the sensor detects the radiation as soon as it starts. Further, an IR signal depends on the wafer’s emissivity. Because the emissivity varies with temperature, the response is nonlinear. The signal is also rather sensitive to changes within the geometry. With tools like COMSOL Multiphysics, you can fully study such phenomena and gain a better understanding of how to optimize your RTA configuration for successful semiconductor manufacturing.
Electrodynamic magnetic levitation takes place when a rotating and/or moving permanent magnet or a currentcarrying coil generates a timevarying magnetic field in a nearby conductor. The timevarying magnetic fields induce eddy currents in the conductor, which create an opposing field. This in turn gives rise to a repulsive force between the conductive material and the magnetic source. This process is the fundamental operating principle of all magnetic levitation systems.
A magnet levitating above a superconductor. Image by Julien Bobroff — Own work. Licensed under CC BYSA 3.0, via Wikimedia Commons.
Let’s take a look at a benchmark model based on Transient Electromagnetic Analysis Methods (TEAM) problem 28: An Electromagnetic Levitation Device. This problem consists of a circular aluminumconducting disc placed above two cylindrical, concentric coils carrying sinusoidal currents in opposite directions. The crosssectional cut of the setup and the dimensions are shown in the image below.
A crosssectional view of concentric coils and an aluminum disc. All of the dimensions are in millimeters.
The model is shown in 3D below.
A 3D model of the electrodynamic levitation device, showing the levitated disc and two concentric coils carrying timevarying currents in opposite directions.
To model this device in the COMSOL Multiphysics® software, we use a 2D axisymmetric geometry. Due to the timevarying currents as well as the induced eddy currents, we model the magnetic fields using the Magnetic Fields interface in the AC/DC Module. The opposite currentcarrying coils are modeled using separate Coil features with the Homogenized MultiTurn Coil model. The electrodynamic force is computed in the aluminum plate by using the Force Calculation feature, which calculates the Maxwell stress tensor in it.
The rigid body dynamics of the plate are solved as a system of ordinary differential equations (ODE) via the Global ODEs and DAEs interface. The firstorder ODEs for position and velocity are given by:
Since the electromagnetic force dynamically varies with distance between the plate and the coil, it is necessary to solve the Magnetic Field interface for the dynamical change in plate position. Therefore, we model the movement of the plate using the Moving Mesh interface. We compare the simulation results and the TEAM benchmark data for the corresponding position of the oscillating disc in the following plot.
Comparing simulation results and TEAM data of the vertical plate oscillation as a function of time.
Animation depicting the oscillation of the conductive disc above the two concentric coils for 0.6 seconds.
The mechanical rotation of a magnetic source, such as a radially magnetized Halbach rotor, induces eddy currents above a passive conductive guideway (e.g., aluminum). This creates an opposing magnetic field that interacts with the source magnetic field to produce both lift and thrust forces simultaneously. This device is called an electrodynamic wheel (EDW).
The concept of EDW levitation for highspeed transportation is illustrated in the following figure. The production of the thrust or braking force depends on the relative slip speed, s_{l}, defined as the difference between the circumferential speed, v_{c}, and the translational speed, v_{x}. For instance, s_{l} = v_{c} — v_{x}, where v_{c} = ω_{m}r_{o} and ω_{m} = ω_{e}P. Here, ω_{m} is the mechanical angular speed, ω_{e} is the electrical angular speed, and P is the number of pole pairs in the Halbach rotor.
The conceptual design of a fourpolepair EDW maglev, depicting the conductive track and rotating and/or traveling Halbach rotor.
If the circumferential speed is greater than the translational speed (for positive slip), a thrust force is generated. If the opposite is true, there will be a breaking force.
Using the Rotating Machinery in 2D and 3D, Magnetic interface, we can incorporate both the translational and rotational motion in the same model. The rotational motion is specified using a Prescribed Rotational Velocity feature. The translational motion of the maglev (Halbach rotor) is accounted for in the conducting track using the Velocity (Lorentz) term with the opposite sign. The permanent magnets are modeled using the default Ampère’s Law features with remanent flux density B_{r} = 1.42[T]. Since the magnetization direction is radial or azimuthal, we choose the cylindrical coordinate system for convenience.
The transient simulation is performed for a step change in the mechanical angular speed of the rotor. The results for the lift and thrust forces as a function of time are shown below. These forces are calculated by two different methods: the Maxwell stress tensor (using the Force Calculation feature) and the Lorentz method.
The comparison of lift and thrust forces on the maglev as a function of time. Results from both the Maxwell stress tensor and Lorentz method are shown.
In the second step, the stationary simulation is performed for a range of translational speeds. A drag force is generated if there is no rotation or the circumferential speed is less than the translational speed. The simulation results for the lift and drag forces for various speeds are shown below.
The comparison of lift and drag forces on the maglev as a function of time. Results from both the Maxwell stress tensor and Lorentz method are shown.
Animation depicting the surface plot of the magnetic flux density in the air and magnets; current density in the guideway; and the contour plot of magnetic vector potential, A_{z}. The clockwise rotation of the Halbach rotor and the field interactions are shown.
In today’s blog post, we demonstrated how to model two electrodynamic magnetic levitation devices using COMSOL Multiphysics and the addon AD/DC Module. We featured TEAM problem 28: Electrodynamic Levitation Device and compared the results with experimental data from literature. We also highlighted the working principle of an electrodynamic wheel magnetic levitation system. Our simulation results show the lift and drag/thrust forces generated by this system for a step change in the angular speed and various translational velocities.
At Reelight, we are developing an affordable bicycle safety light that is extremely easy for the end user to install. Along with a stronger and more flexible mounting system, we needed to develop a new power generation platform. Using simulationbased design, we created a power platform that is easy to use and quick to install.
If you have been to Copenhagen in Denmark, you have probably noticed the hubmounted flashing lights that are on the majority of bikes. Urban commuters prefer these lights because they’re already attached to the bike, so they can’t be forgotten at home. The bicycle lights are usually sold together with the bikes, thus mounted by the local bike store.
With our research, we aimed to create a light tailored for the consumer by making it much easier to install. To us, ease of installation means that the light fits any bike, regardless of size or accessories. Also, the installation should take less than five minutes and shouldn’t require any additional tools, besides the small Allen key that is included. The concept for the Reelight safety bike light is shown below. The light on each wheel only needs two parts, both of which are simple to install.
The Reelight safety light with a small builtin generator. The circular spoke unit accelerates the rotor in the light’s generator with each pass. The light itself is mounted on the frame with a coated steel wire rope.
The bike light is always on, yet it doesn’t require any batteries to operate. It also does not have a mechanical part connected to the wheel that often produces an irritating sound. To achieve this design goal, we used simulation to create a power generation system that relies on magnetic induction.
The classic power platform for our bicycle safety light is a pure inductionbased system. To create sufficient power with this technology, the module mounted on the spokes must span more than one spoke. For our new light, we wanted to keep the spoke unit small and installed on a single spoke to give the user maximum flexibility. In order to accomplish this, we chose to use a simple type of synchronous machine.
Inside the light, a permanent magnet rotor is aligned with a laminated core, on which a coil is wound. The rotor is spun into action by the permanent magnet mounted on the spoke, also called the exciter. This transfers the mechanical energy from the bike to the rotor and lights up the LEDs. An exploded view can be seen below, showing the relevant naming conventions for the generator parts of the light.
Exploded view of the safety light. The rotor, coil, iron core, and exciter are the subject of further analysis in this blog post.
This concept seems easy to understand, but actually implementing the production is a different story. The bike safety light itself has several design challenges, most of which could be identified by building prototypes and simply playing around with ideas. Since the bike light is a small product, with a limited number of parts, we can use 3D printing to save time and easily test different designs.
When it comes to designing the power generator of the bike light, on the other hand, prototyping has major drawbacks. For instance, the lead time on magnets can stall the iterations on the product. By using the COMSOL Multiphysics® software to model the power generation, we increase the iteration speed in our development cycles. Furthermore, simulation gives us insight into what works, or does not work, in the bike light design. We can use this knowledge to improve the product faster than with only prototyping.
Even though this power platform is designed for bicycles, it’s still an electrical machine and exhibits some wellknown phenomena. Starting the rotor is much like starting an inertia load with a synchronous machine. At some point, depending on the inertia of the rotor and the torque from the magnet, the rotor will be unable to start and the net mechanical transferred power approaches zero.
Since bicycles have a somewhat limited range of operating speeds, it’s possible to find an optimal solution for this range. One of the aims of the model is to investigate the startup performance of the machine, hence the power generation at different speeds.
The geometry of the machine is simplified to an iron core, rotor with permanent magnets, and the exciter, as depicted below. We use the Rotating Machinery interface and create two identity pairs, each of which has a matching rotation. The exciter rotates proportionally to the speed of the bike, i.e., prescribed rotational velocity, while the rotation of the rotor is calculated with its dynamic equation, i.e., prescribed rotation.
Geometry of the electrical machine, showing the iron core for the coil winding, arc segment permanent magnets, and exciter magnet (the rightmost magnet). The remaining geometry is used either for identity pairs in the Rotating Machinery interface or for reasons of mesh control.
To determine the angular velocity and angle of the rotor, per Newton’s second law, we need to know the torque acting on the system and the inertia. We find the inertia of the rotor by setting up an integration for the rotor volume, then using this integration to integrate over the density in the Variables section. An additional 10% is added to the inertia as an experience figure for the shaft, which is not modeled explicitly.
Integration of the rotor’s permanent magnet, used to determine the inertia of the rotor in the Variables section.
We determine the magnetic torque on the rotor by setting up a force calculation, thus accounting for the force between the magnets and the reluctance force. We also add the torque contribution from the damping in the bearing as well as the electrical power. This is set up in the Global Equations node, where the torque summation and the inertia are used to calculate the angle, and angular velocity.
To keep the model simple and save computational power, the induced current is computed by differentiating the flux in the iron core and multiplying it with the number of windings. This approximation of a 3D coil allows us to neglect the coil itself, and its accompanying vector potential formulation, by sacrificing some accuracy. The electrical circuit is implemented in an equation in the Variables section to find the power from the circuit, thus the torque for the rotational equation.
When running a simulation with moving parts, you can create an animation of the simulation results in COMSOL Multiphysics, as we did for our bike safety light model. The animation seems to fascinate everyone I show it to, most likely because it is easy to visualize and understand the results.
The magnetic field in the coil of the rotor, shown at four different times: t = 0.0 s (top left), t = 0.025 s (top right), t = 0.050 s (bottom left), and t = 0.075 s (bottom right).
The model is sliced through the center of the rotor, perpendicular to the rotational axis. The arrows represent the B vector in the spatial (rotated) frame.
We can also analyze the induced voltage in the coil directly in the model and use this information to estimate the luminous flux from the flash that the light will emit. Several parameters are highly influential on the induced voltage. Since the purpose of the generator is to induce voltage, this parameter is subject to optimization. The voltage output from the coil for this geometry is shown below.
Induced voltage in the stator coil of the rotor.
The torque contributions are the most interesting parameters that this model shows. This is where we can analyze the startup behavior. In general, the transferred mechanical power is the prerequisite for generating electrical power and more light for our cyclist.
Torque contributions from the exciter, bearing damping, and electrical circuit.
This dynamic model takes approximately 30 minutes to solve and is supported by a series of static sweep studies, where the geometry of the different components can be investigated further and faster. Using simulation, we were able to investigate magnetic induction and optimize a power generation source for our bicycle safety light design.
Rune Ryberg Thygesen is a mechanical engineer with a master’s degree in electromechanical system design from Aalborg University. He is a research and development manager at Reelight ApS, a Danish firm that designs batteryfree bike lights for the aftermarket.
]]>The Hyperloop is a proposed mode of transportation in which a vehicle, or pod, travels at the speed of sound through a lowpressure tube. At this speed, a magnetic suspension offers several advantages over systems such as air bearings or wheels. To test this, Delft’s Hyperloop team modeled their pod’s magnetic suspension in the COMSOL Multiphysics® software.
The original idea for the Hyperloop came from Elon Musk, the man behind SpaceX, who promoted it as a faster, cleaner, and cheaper alternative to a proposed highspeed rail track between San Francisco and LA. His idea was to place a small pod in a tube with a mild vacuum and have the pod float on air bearings instead of wheels. The reduced pressure in the tube eliminates most of the air drag. The remaining air would then be compressed for an even bigger reduction in drag and to enable the use of air bearings.
While Musk himself does not intend to build a Hyperloop, SpaceX decided to host a competition to accelerate its realization. They built a 1.5 km halfscale test track and invited teams from around the world to design a pod for the tube.
Rendering of Delft’s Hyperloop competition vehicle.
Delft’s Hyperloop competition vehicle at the reveal event in July 2016.
The Delft Hyperloop team, who is competing in the SpaceX Hyperloop competition final in January 2017, consists of approximately 30 students from the Delft University of Technology in the Netherlands. By carefully reviewing the concept of the complete system, the team, which includes me, found that using a magnetic suspension would be much more beneficial than the originally proposed air bearings, reducing the complexity and cost.
While there are different types of magnetic suspensions, the main distinction is between an actively controlled attractive system, known as electromagnetic suspension, and an inherently stable and passive repulsive electrodynamic system. We used the latter in the Delft Hyperloop competition vehicle, and the concept is quite simple. Moving a permanent magnet over a conductive surface, such as a sheet of aluminum, induces eddy currents in the surface, which in turn cause a Lorentz force on the magnets moving over them, lifting them up. Since the competition tube will be fitted with an aluminum track, this concept of levitation can be used in Delft’s pod.
Demonstration of the electrodynamic suspension. The handle is used to spin an aluminum disk (below the plastic cover), where a small neodymium magnet with a Hyperloop enclosure is being visibly lifted from the disk.
Left: Visualization of the brake magnets, which are positioned on both sides of the Ibeam on the track. Right: Eddy currents in the track as a magnet moves over it.
For linear motion, the Lorentz force on a moving magnet has two components: one lifts the magnet up and the other opposes the magnet in the direction of motion. These are referred to as the lift and drag component, respectively, with the latter known for its use in eddy current brakes. These forces depend on factors like speed, magnet dimensions, field strength, conductivity of the track, and gap height. Choosing a magnet configuration therefore requires research on how its parameters affect the lift and drag components.
For the Hyperloop, it’s best that there is a low drag component and high lifting force. This is due to how a large air gap decreases the track’s accuracy demands, a major cost factor in a Hyperloop system.
Left: Rendering of the front bogie, including the “skis”, or lift bogies, with the lift magnets on the sides and the brakes in the middle. Right: The rear bogie with lift bogies, brakes, and suspension. The Ibeam in the track is visible here.
Our Hyperloop pod uses four magnetic “skis” filled with Halbach arrays, a configuration of permanent magnets in which the field is strong toward the track and weak toward the passengers. We used COMSOL Multiphysics and the AC/DC Module to determine the relations between the magnet parameters and the lift and drag forces with the Magnetic and Electric Fields interface. We then performed parametric studies to plot the forces for different speeds and gap heights for a certain magnet configuration.
Typical lift and drag curves for a constant gap height can be seen in the figure below. The drag component peaks and then diminishes for higher speeds, contrasting with, for instance, air drag. This makes it attractive for highspeed travel. Increasing the size of the individual magnets also has a positive effect on reducing the drag.
Typical lift and drag force curves for an electrodynamic suspension.
We used similar simulations to design our brake system. Naturally, we chose to use eddy current brakes in the system design. When the pod needs to stop, the brakes can be moved toward the track. As this needs a large drag component to be effective, we used COMSOL Multiphysics to quickly come up with magnet configurations that satisfy this and other requirements.
To validate our models, we built test setups to evaluate the force versus speed as well as gap curves for the final magnets, finding good agreement between the simulation and experimental results. Knowing that the simulation is accurate is very important as these curves are essential for the fullvehicle model and control systems. The simulations are at the basis of the vehicle, and with the COMSOL® software, we designed magnet arrays that will lift the vehicle when it moves at low speeds. The magnet arrays will also be able to maintain a comfortable cruising gap height of more than 20 mm.
Bauke Kooger is a master’s student in applied physics at the Delft University of Technology, and his passion lies in applying physics to technology. Having had previous experience in designing a battery for an electric racing car, he knows how to apply his knowledge to the simulation, design, and testing of the magnetic suspension system for the Delft Hyperloop team.
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Since antennas radiate electromagnetic waves, it is important to ensure that their radiated fields don’t return to the radiating source. The absorbers attached to the chamber’s walls play a major role in this process by absorbing incident waves on their surfaces. This makes the anechoic chamber a key aspect in an antenna simulation. However, including the anechoic chamber is challenging because it increases the simulation’s computational requirements.
A biconical antenna excited in an anechoic chamber, used to test for electromagnetic interference (EMI) and electromagnetic compatibility (EMC).
The anechoic chamber shown above is smaller than the typical anechoic chamber that meets CISPR specifications, but its computational cost is still high. This model requires over 16 GB of RAM. To improve computational efficiency, we should simplify the model, while maintaining the accuracy of the computation. As discussed in a previous blog post, we can accomplish this by using a perfectly matched layer (PML). This reduces the memory use to less than 2 GB without sacrificing simulation accuracy.
To efficiently mimic the real world in our antenna simulation, we need to choose the correct boundary conditions and physics features. Accurately reflecting realworld conditions in your simulation environment, while still keeping your model memory and time efficient, can be challenging. In the table below, we outline some realworld antenna scenarios and the optimal modeling feature to choose.
Real World  Simulation Environment  

Basic  Advanced  
Anechoic chamber absorbing electromagnetic waves  Scattering boundary condition  Perfectly matched layer 
Metallic antenna body and surface  Perfect electric conductor 

Network analyzer measuring Sparameters for antenna input matching properties  Port or lumped port  Numeric TEM port 

Farfield domain and calculation 
When setting up an antenna model, you do not need many complicated boundary conditions. You can actually build an antenna in COMSOL Multiphysics by deploying only four features. Let’s see how to do this with a printed dipole antenna example.
The geometry of a printed dipole antenna.
A printed dipole antenna’s geometry consists of four objects:
Geometry  Purpose 

Block  Polystyrene foam board 
Rectangle  Printed metallic layer 
Rectangle  Port location 
Sphere  Air domain 
The geometry is configured with only two materials: a userdefined polystyrene foam board and the air that encloses the simulation domain. Use the following table to choose the correct physics features:
Physics Feature  Purpose 

Perfect Electric Conductor boundary condition  Mimics metallic surfaces with a high conductivity 
Lumped port  Excites the antenna and measures Sparameters 
Scattering boundary condition  Absorbs the incident wave to minimize any reflection 
Farfield domain and calculation  Calculates the farfield radiation pattern, directivity, and gain 
A Perfect Electric Conductor boundary condition imposed on a rectangular strip.
For the intended operating frequency, the simulation may only take a few seconds. The RF Module provides the default Sparameter evaluation, electric field distribution plot, and polar farfield plot. It also gives you the 3D farfield radiation pattern plot, which shows the computed directivity and gain.
The farfield radiation pattern of a printed dipole antenna. The computed directivity is 2.15 dB, which is close to that of an ideal halfwave dipole antenna.
Although simulating an antenna is a straightforward process, it is a good idea to start with a simple structure, whether you are a beginner or an expert. This way, you can ensure that the basic modeling process is correct for the simple geometry before adding complex design elements.
The RF Module also enables you to combine electromagnetics with any other type of physics. You can see and change all of the physics features in the modeling environment and clearly define every physics property. Taking into account multiple physical effects, as well as knowing the underlying physics involved, is useful when validating your antenna design.
To capture the details of the physics, such as the loss on metallic surfaces, the Perfect Electric Conductor boundary condition can be replaced by the Transition boundary condition for a geometrically very thin lossy layer or an Impedance boundary condition for surfaces of a lossy volume. You can also use a PML instead of a Scattering boundary condition, which assumes that the incident wave is normal to the surface.
After setting up these physics features, you can begin to design your antenna, whether its shape is traditional, wideband, multiband, or an array.
You can access many antenna examples for a wide range of applications in the Application Library in the RF Module. The tutorial models range from conventional antennas, such as halfwave dipole and microstrip patch antennas, to wideband and multiband antennas, including Vivaldi, fractal, spiral, and helical antennas. There is also an antenna array example, which can be useful when designing devices for the 5G mobile network.
Traditional antennas, like the halfwave dipole and microstrip patch antenna shown below, are good examples to start with when learning how to model antennas in the COMSOL Multiphysics® software. Their geometries are relatively easy to build and it’s simple to validate the results with wellknown analytical solutions.
For example, you can simulate a halfwave dipole antenna to find its omnidirectional radiation. Or, you can model a microstrip patch antenna to see if the electric field is confined to the radiating edges.
Conventional antennas: halfwave dipole antenna with a quarterwave coaxial balun (left) and microstrip patch antenna (right).
Sometimes, we need to cover many different frequency ranges with a single antenna. By tweaking the radiating structure and using the multiple resonance behaviors of a certain part of the metallic body or slot, we can meet the system’s specifications without deploying multiple antennas. One such example is the popular Vivaldi antenna, also called a tapered slot antenna.
Using fractal algorithms, such as those from Sierpinski, Koch, and Hilbert, you can generate interesting results for antenna applications. For instance, unlike the halfwave dipole antenna, which can only be used for a single frequency resonance, the Sierpinski fractal antenna doesn’t require additional matching networks to adjust the antenna input impedance to the reference characteristic impedance of 50 ohms for higherorder resonances.
Wideband and multiband antennas: Vivaldi antenna (left) and Sierpinski fractal monopole antenna (right).
When the input power to your antenna is regulated due to the path loss in a communication link budget, your initial design may not fulfill its needed specifications. This is a potential issue that the 5G mobile network faces and it can be resolved using an antenna array.
While antenna arrays sound cool, they can greatly increase the simulation size and computational cost if modeled explictly. When we only want to check the antenna array’s performance feasibility, we can simplify the simulation using the antenna array factors, which are mathematical properties from antenna theory. This way, we can maintain the computational efficiency of the simulation.
In COMSOL Multiphysics, we can access all of the equations that are being solved and modify them with an alternative formulation. We can also create an antenna array simulation app that allows us to control the implemented equations and the parameters that we’d like to change.
Antenna arrays: monopole antenna array (top) and slotcoupled microstrip patch antenna array synthesizer (bottom).
In order to characterize your antenna system and evaluate its EMI/EMC, it has to be measured in an anechoic or reverberation chamber using one of three wellknown test antennas: the logperiodic, biconical, or doubleridged horn antenna. You can examine these test antennas in your simulation and enhance their performance. These antennas are also a type of wideband antennas and usually cover 20 MHz ~ 200 MHz, 200 MHz ~ 2 GHz, and 2 GHz ~ 20 GHz, respectively. When you simulate them, you need to set the mesh either based on the highest frequency or parametrically updated for each analysis frequency during the frequency sweep.
Test and measurement antennas: a logperiod (left) and biconical antenna (right). These antennas are popularly used in an anechoic chamber.
You test your antenna design in an anechoic chamber, but it will eventually be deployed in real systems. You also have to evaluate the antenna’s performance when it is implemented on different types of platforms. The effect of one radiating device on another device’s system can result in problems, such as radio frequency interference. You can address these interference problems through simulation. For instance, you can use simulation to see how a car windshield’s antenna affects a cable harness and evaluate antenna crosstalk on an airplane’s fuselage.
Simulation showing a car windshield antenna’s effect on a cable harness (left) and antenna crosstalk on an airplane’s fuselage (right).
With the RF Module, you can also expedite antenna modeling with the body of revolution approach, which doesn’t demand a lot of computational resources. With this approach, you can quickly model antennas like the corrugated horn antenna and conical horn lens antenna.
Fast numerical modeling of a corrugated horn antenna (left) and a conical horn lens antenna (right).
In this blog post, we briefly reviewed efficient antenna modeling techniques and several different types of antennas that can be designed using the RF Module. Each antenna model begins with a simple geometry configuration and boundary conditions. You then gradually add more complex parts and elements to your simulation. This way, you can easily debug and tune your antenna model. Using this information and the featured examples, you can start designing antennas in COMSOL Multiphysics with optimized computational efficiency and speed.
The operating modes and types of electrical machines are defined by the way that their windings are connected. Their fundamental principle of operation is based on the voltages and currents flowing through these windings. Independent of the type of machine, the windings can be categorized as concentrated or distributed, with further subcategories such as fractional and integral also applied.
For concentrated windings, as the name suggests, each pole in the machine will have a set of conductors passing through the same slot. On the other hand, for distributed windings, the number of slots will be greater than the number of poles, so the conductors of each pole will be distributed among the number of slots. The benefits and differences of using concentrated windings versus distributed windings are outside of the scope of this blog post.
Our induction motor model was built using a pole pitch of 60°. This means that there is 60° of separation from the bottom of one stator slot to another. To create some uniform flux density lines and ensure induction on the steel rotor to produce motion, we need to make sure that there is some separation between each stator slot. In many rotor topologies, this gap is filled with the stator tooth. However, in this example, we have selected 15° of air gap, which means that each stator slot will cover 45° of the full 60° needed.
Threephase induction motor schematic depicting the dimensions and phase configurations of the original model.
At this point, what we have described might sound complicated, but it is actually quite simple. As we mentioned earlier, the design of windings in electrical machines depends on their voltage phasor diagrams. In the figure below, the blue lines represent the phases, while the orange lines represent their negative counterparts.
Threephase phasor diagram.
The induction motor used for this example is a threephase, twopole machine, and it can be described exactly by the phasor diagram, where 60° defines the pole pitch in the stator. The only challenge here is that these angles of separation between each stator only work in the twopole case. We need to create a relationship between the electrical angles shown in the phasor diagram, which describe the circular motion of the rotor, and the actual mechanical angle, which describes the physical location of each stator slot. The electrical angle is represented by the following equation:
Under the Definitions node, we can create selections, that is, group geometrical entities such as domains, boundaries, edges, and points. For the twopole machine, there are two points of interest: identifying the stator slot corresponding to each phase and identifying the direction of the current (inward or outward). Note that while there are multiple ways to perform the same operation, this specific case serves as an example of how selections can be used in a model.
In the figure below, we have parameterized the ball selection in such a way that it is possible for us to always select the midpoint of each stator slot. As mentioned earlier, coil pitch is 45º and since we are working with a circular geometry, it is easy to parameterize the x and ycoordinates to follow each coil location around the geometry:
Creating a parameterized ball selection in COMSOL Multiphysics.
Take a look at the screenshot shown above. Ball 1 represents phase –A and Ball 2 represents phase A. These elements have been collected under a single selection via the Union selection feature. This allows us to easily call them from the physics interface under each Coil feature.
Linking Coil 1: Phase A to the Union 1 selection and the Reversed Current Direction to the Ball 1 selection.
The highlighted boxes in the above screenshot show that the physics node can be linked to the domain selection. Here, the green box represents the Union node and the blue box represents Ball 1. Such linking is necessary to indicate where the current direction is reversed.
The order of the phases will determine the direction in which the rotor moves. We will therefore use the same sequence as the original example, starting from phase –A. Now let’s consider a fourpole machine and describe the electrical angle and mechanical angle as well as the phase and direction of the current. The following table provides an overview of these elements.
Phase  A  B  C  A  B  C  A  B  C  A  B  C 

Current Direction  
θ_{electrical}  0  60  120  180  240  300  360  420  480  540  600  660 
θ_{mechanical}  0  30  60  90  120  150  180  210  240  270  300  330 
Relationship between the phase, current direction, electrical angle, and mechanical angle for a fourpole machine.
If done manually, this task can become rather tedious. To perform the same operation, we would need to create twelve ball selections, group them into three unions, and then identify every reversed current direction. Now imagine doing this for an eight or a tenpole machine and having to manually change each selection for every winding design. As we’ll demonstrate next, the Application Builder becomes quite handy in such situations.
With Application Builder, you can develop a customized user interface that is tailored to your specific needs. Here, we have parameterized our model in order to create a function that depends on the stator pole pitch and the mechanical angle and can be used to model an induction threephase machine for multiple poles. As noted earlier, there are many ways to perform such an operation, but our goal is to showcase the capabilities and advantages of using selections and the Application Builder to do so.
The functions entered as x and y positions within the ball selections can be described by the following expressions, respectively:
where θ_{s} is the coil pitch and n is just an integer ranging from zero to the total number of coils in the model.
Since we are working with three phases and two current directions, we can easily use the mod function (%) to identify the two interesting attributes of each coil: the phase and current direction. If we start counting coils from phase –A until all of the mechanical angles are covered for each coil slot, we would end up with something similar to the table below for the fourpole case.
Phase  A  B  C  A  B  C  A  B  C  A  B  C 

Coil Number (i)  0  1  2  3  4  5  6  7  8  9  10  11 
i%3  0  1  2  0  1  2  0  1  2  0  1  2 
i%2  0  1  0  1  0  1  0  1  0  1  0  1 
Description of the algorithm used to identify each phase and when to reverse the current direction.
Here, a coil number (i) of 0 represents phase –A, a coil number of 1 represents phase B, and so on. Using i%3, we can easily identify phase A as 0, phase B as 1, and phase C as 2. Further, for each phase starting from phase –A, we can see that the negative current is identified as 1, whereas the positive side of the current is identified as 0.
To see how this works as described in the previous table, let’s look at a section of the code created in the Method Editor of the Application Builder.
Generating automated ball selections using the Method Editor for each coil.
The above image only shows how to create the selections for phase A. Here, we use three counters: i counts from 0 until the last coil (11 in the fourpole case). If i%3 is true, then we can say it is phase A, and we will store those selections in array phaseA. We’ll then need to check for the current direction. If i%2 is 0, then the direction of the current is considered negative, and we will store those selections in another array called phaseArev.
So far we have created ball selections, but we still need to create the unions. This operation is rather easy to perform as we will always have six unions in our case: phase A, phase B, and phase C, along with the three unions corresponding to the reversed current direction.
Creating union selections using the Method Editor for each phase.
In the same fashion, we’ll need to make sure that each union is assigned to its corresponding physics interface.
Linking the coils to the union selections and the reversed current direction to the ball selections using the Method Editor.
We have now finalized the creation of selections that will depend on the number of poles entered by users of the app. With more time and additional creativity, you could further extend the functionality of this app via the Form Editor. This tool gives you the ability to design an app that enables users to, for instance, assign material properties, account for time dependency of the solution (transient or harmonic), and even automate postprocessing. Such features are highlighted in the screenshot below.
A simulation app based on the original induction motor model. This app creates and displays selections based on the relationship between the electrical and mechanical angles.
As noted in this earlier blog post, a tokamak is an experimental device meant to produce thermonuclear fusion power. The tokamak heats the hydrogen fuel to more than 150,000,000°C, which forces electrons to separate from their nuclei, thus creating plasma. Magnetic fields are used to confine the plasma within a vacuum vessel. These magnetic fields keep the hot plasma away from the walls of the vessel and stronger fields can enhance the plasma’s performance.
Plasma visible in the window of a tokamak. Image by Bobmumgaard — Own work. Licensed under CC BYSA 4.0, via Wikimedia Commons.
The magnetic fields’ ability to affect the plasma’s performance is one reason why researchers at MIT’s PSFC are focusing on a high magnetic field approach to fusion. Because fusion power is proportional to the magnetic field strength to the fourth power, any increase in the magnetic field strength greatly increases the device’s power and provides better confinement for the plasma.
The Advanced Divertor eXperiment (ADX), a proposal created by PSFC and collaborators, intends to develop a compact tokamak with high magnetic fields. This tokamak would have densities, heat fluxes, and temperatures typical of a reactor, but would use short plasma discharges. A key innovation that makes the ADX different from other tokamaks is that rather than being made of a single cylinder, the vacuum vessel is modular and consists of five axisymmetric shells. These separate shells provide a powerful advantage as they enable engineers to change parts of the design, such as the divertor, as further research is done and new discoveries are made.
The proposed ADX tokamak design.
A major challenge for tokamaks is handling the high heat and particle exhaust from the plasma. These fluxes are taken care of by the divertor, and the ADX will be able to perform tests with a number of different divertor configurations and evaluate how each performs based on measured data.
Another issue for tokamaks is plasma disruptions. During normal operation, the plasma carries a large electric current. For example, the plasma in the ADX is designed to carry 2 MA. In a disruption, the plasma equilibrium is lost when the plasma moves from its equilibrium position and then loses all of its current on a short timescale i.e., 1 millisecond. This type of disruption, where the plasma moves and then loses its current, is called a vertical displacement event (VDE).
During a VDE, changes in the plasma result in rapidly changing magnetic fields, which drive eddy currents in the surrounding conductive structures such as the vacuum vessel. When these eddy currents cross the poloidal and toroidal magnetic fields in the tokamak, they create large Lorentz forces that the structure must be able to withstand.
Left: PSFC’s vacuum vessel design. Right: Eddy currents in the vacuum vessel’s walls.
In a VDE, the plasma can move closer to the vessel walls, and this proximity leads to larger loads on the vessel during the disruption. With this in mind, the PSFC engineers used a VDE as a test case to see if their vessel could support ADX operation.
To reduce eddy currents, the vessel itself consists of Inconel 625. The researchers at PSFC chose Inconel 625 because of its high electrical resistivity and high strength. The electrical resistivity of this nickelbased alloy helped to lessen the magnitude of the eddy currents in the vessel. When the team tested their initial design with numerical simulation, they found large stresses and displacements within the vessel, as shown below.
PSFC’s structural model of the vacuum vessel with a purple boundary showing where it is fixed. The simulations show stress (a) and displacement (b) during a VDE.
The engineers then added a reinforcement block on one of the vessel’s boundaries. This stabilized the vacuum vessel, greatly reducing stress and wall displacement. With the addition of the support block, the simulation results indicate that the vessel can handle a plasma disruption and function properly in the ADX.
PSFC’s structural model of the vacuum vessel after adding another support. The simulations show stress (a) and displacement (b) in the optimized model during a VDE.
When it comes to designing fusion devices, simulation serves as a powerful tool for balancing efficiency with accuracy. For the research team at PSFC, the COMSOL Multiphysics® software proved to offer reliable predictions of the magnetic fields in their tokamak design, matching well with recorded data. The flexibility of the platform also made it easy for the engineers to shift from one type of physics to the next, creating a smooth transition between different studies.
With their simulation findings, the engineers at PSFC can ensure that their vacuum vessel, once manufactured, performs well within the ADX. Such advancements offer new potential in making nuclear fusion, and the power that it will provide, a closer reality.
Because they can be focused to the smallest spot size of all electromagnetic beams, Gaussian beams can deliver the highest resolution for imaging, as well as the highest power density for a fixed incident power, which can be important in fields such as material processing. These qualities are why lasers are such attractive light sources. To obtain the tightest possible focus, most commercial lasers are designed to operate in the lowest transverse mode, called the Gaussian beam.
As such, it would be reasonable to want to simulate a Gaussian beam with the smallest spot size. There is a formula that predicts real Gaussian beams in experiments very well and is convenient to apply in simulation studies. However, there is a limitation attributed to using this formula. The limitation appears when you are trying to describe a Gaussian beam with a spot size near its wavelength. In other words, the formula becomes less accurate when trying to observe the most beneficial feature of the Gaussian beam in simulation. In a future blog post, we will discuss ways to simulate Gaussian beams more accurately; for the remainder of this post, we will focus exclusively on the paraxial Gaussian beam.
A schematic illustrating the converging, focusing, and diverging of a Gaussian beam.
Note: The term “Gaussian beam” can sometimes be used to describe a beam with a “Gaussian profile” or “Gaussian distribution”. When we use the term “Gaussian beam” here, it always means a “focusing” or “propagating” Gaussian beam, which includes the amplitude and the phase.
The paraxial Gaussian beam formula is an approximation to the Helmholtz equation derived from Maxwell’s equations. This is the first important element to note, while the other portions of our discussion will focus on how the formula is derived and what types of assumptions are made from it.
Because the laser beam is an electromagnetic beam, it satisfies the Maxwell equations. The timeharmonic assumption (the wave oscillates at a single frequency in time) changes the Maxwell equations to the frequency domain from the time domain, resulting in the monochromatic (single wavelength) Helmholtz equation. Assuming a certain polarization, it further reduces to a scalar Helmholtz equation, which is written in 2D for the outofplane electric field for simplicity:
where for wavelength in vacuum.
The original idea of the paraxial Gaussian beam starts with approximating the scalar Helmholtz equation by factoring out the propagating factor and leaving the slowly varying function, i.e., , where the propagation axis is in and is the slowly varying function. This will yield an identity
This factorization is reasonable for a wave in a laser cavity propagating along the optical axis. The next assumption is that , which means that the envelope of the propagating wave is slow along the optical axis, and , which means that the variation of the wave in the optical axis is slower than that in the transverse axis. These assumptions derive an approximation to the Helmholtz equation, which is called the paraxial Helmholtz equation, i.e.,
The special solution to this paraxial Helmholtz equation gives the paraxial Gaussian beam formula. For a given waist radius at the focus point, the slowly varying function is given by
where , , and are the beam radius as a function of , the radius of curvature of the wavefront, and the Gouy phase, respectively. The following definitions apply: , , , and .
Here, is referred to as the Rayleigh range. Outside of the Rayleigh range, the Gaussian beam size becomes proportional to the distance from the focal point and the intensity position diverges at an approximate divergence angle of .
Definition of the paraxial Gaussian beam.
Note: It is important to be clear about which quantities are given and which ones are being calculated. To specify a paraxial Gaussian beam, either the waist radius or the farfield divergence angle must be given. These two quantities are dependent on each other through the approximate divergence angle equation. All other quantities and functions are derived from and defined by these quantities.
In COMSOL Multiphysics, the paraxial Gaussian beam formula is included as a builtin background field in the Electromagnetic Waves, Frequency Domain interface in the RF and Wave Optics modules. The interface features a formulation option for solving electromagnetic scattering problems, which are the Full field and the Scattered field formulations.
The paraxial Gaussian beam option will be available if the scattered field formulation is chosen, as illustrated in the screenshot below. By using this feature, you can use the paraxial Gaussian beam formula in COMSOL Multiphysics without having to type out the relatively complicated formula. Instead, you simply need to specify the waist radius, focus position, polarization, and the wave number.
Screenshot of the settings for the Gaussian beam background field.
Plots showing the electric field norm of paraxial Gaussian beams with different waist radii. Note that the variable name for the background field is ewfd.Ebz
.
In the scattered field formulation, the total field is linearly decomposed into the background field and the scattered field as . Since the total field must satisfy the Helmholtz equation, it follows that , where is the Laplace operator. This is the full field formulation, where COMSOL Multiphysics solves for the total field. On the other hand, this formulation can be rewritten in the form of an inhomogeneous Helmholtz equation as
The above equation is the scattered field formulation, where COMSOL Multiphysics solves for the scattered field. This formulation can be viewed as a scattering problem with a scattering potential, which appears in the righthand side. It is easy to understand that the scattered field will be zero if the background field satisfies the Helmholtz equation (under an approximate Sommerfeld radiation condition, such as an absorbing boundary condition) because the righthand side is zero, aside from the numerical errors. If the background field doesn’t satisfy the Helmholtz equation, the righthand side may leave some nonzero value, in which case the scattered field may be nonzero. This field can be regarded as an error of the background field. In other words, under certain conditions, you can qualify and quantify exactly how and by how much your background field satisfies the Helmholtz equation. Let’s now take a look at the scattered field for the example shown in the previous simulations.
Plots showing the electric field norm of the scattered field. Note that the variable name for the scattered field is ewfd.relEz
. Also note that the numerical error is contained in this error field as well as the formula’s error.
The results shown above clearly indicate that the paraxial Gaussian beam formula starts failing to be consistent with the Helmholtz equation as it’s focused more tightly. Quantitatively, the plot below may illustrate the trend more clearly. Here, the relative L2 error is defined by , where stands for the computational domain, which is compared to the mesh size. As this plot suggests, we can’t expect that the paraxial Gaussian beam formula for spot sizes near or smaller than the wavelength is representative of what really happens in experiments or the behavior of real electromagnetic Gaussian beams. In the settings of the paraxial Gaussian beam formula in COMSOL Multiphysics, the default waist radius is ten times the wavelength, which is safe enough to be consistent with the Helmholtz equation. It is, however, not a “cutoff” number, as the approximation assumption is continuous. It’s up to you to decide when you need to be cautious in your use of this approximate formula.
Semilog plot comparing the relative L2 error of the scattered field with the waist size in the units of wavelength.
In the above plot, we saw the relationship between the waist size and the accuracy of the paraxial approximation. Now we can check the assumptions that were discussed earlier. One of the assumptions to derive the paraxial Helmholtz equation is that the envelope function varies relatively slowly in the propagation axis, i.e., . Let’s check this condition on the xaxis. To that end, we can calculate a quantity representing the paraxiality. As the paraxial Helmholtz equation is a complex equation, let’s take a look at the real part of this quantity, .
The following plot is the result of the calculation as a function of x normalized by the wavelength. (You can type it in the plot settings by using the derivative operand like d(d(A,x),x)
and d(A,x)
, and so on.) We can see that the paraxiality condition breaks down as the waist size gets close to the wavelength. This plot indicates that the beam envelope is no longer a slowly varying one around the focus as the beam becomes fast. A different approach for seeing the same trend is shown in our Suggested Reading section.
Real part of the paraxiality along the xaxis for paraxial Gaussian beams with different waist sizes.
Today’s blog post has covered the fundamentals related to the paraxial Gaussian beam formula. Understanding how to effectively utilize this useful formulation requires knowledge of its limitation as well as how to determine its accuracy, both of which are elements that we have highlighted here.
There are additional approaches available for simulating the Gaussian beam in a more rigorous manner, allowing you to push through the limit of the smallest spot size. We will discuss this topic in a future blog post. Stay tuned!