Modern optical communication systems commonly use EO routers and macroscale MO devices. Each device, however, has its own drawbacks. EO routers require an electric field and often have operation voltages in the kilovolt range, while macroscale MO devices don’t allow for scalable solutions. The quest for small, low-power optic routing alternatives is thus an important focus for specialized researchers within this field. This search is complicated, as various specializations within physics and engineering are required to address the study and interaction of magnetism, magnetic materials, and light.
Scanning electron micrograph of an unpolished silicon-on-insulator rib waveguide used for optical routers on a chip. Image credit: J. Tioh, “Interferometric switches for transparent networks: development and integration,” 2012, Graduate Theses and Dissertations. Paper 12487.
One potential solution involves integrating optical components onto a silicon substrate to create an MO routing solution on a chip. This option reduces both the size and operation power of the device and can potentially enable new technologies like light processors. But before monolithically integrated MO routers become commonplace, there are still a few hurdles that this technology needs to overcome.
Standard industry practices, for instance, present a challenge when it comes to manufacturing monolithically integrated MO routers. To introduce new technology with minimal industry disruptions, standard practices must be used. In this case, silicon should be used as a base substrate and the combining materials must be compatible with silicon for successful monolithic integration. But bonding silicon and the suitable magneto-optic materials can be quite difficult using standard industry practices due to their crystal structures. As a result, the materials tend to become brittle and crack, significantly increasing optical losses.
Multiphysics simulation offers potential solutions to such challenges. These tools can help the research community identify optimal designs and manufacturing techniques for monolithically integrated MO routers. For his doctoral dissertation at Iowa State University, John Pritchard, an engineer who works within this field, turned to the COMSOL Multiphysics® software to provide new insight into the design and future of on-chip MO routers.
When analyzing an on-chip MO system, Pritchard chose to focus on a few key design elements. One point of focus was analyzing a codirectional coupler, a device that is commonly found in interferometer designs. The power coupling coefficients of a codirectional coupler vary based on the distance between the coupling section length and coupled waveguides. Through his simulation work, Pritchard determined how to generate an ideal power coupler coefficient by choosing a specific coupler length.
3D simulation results for the codirectional coupler. Copyright © John Pritchard.
Another point of analysis was an on-chip optical waveguide. The goal here was to design a rib waveguide that minimizes energy loss and maintains a sufficient beam profile throughout the device. To achieve this, Pritchard used silicon as the rib waveguide’s transmission medium, since it is suitably transparent to infrared light and useful for integration with electronic devices. Further, a low-index cladding model was placed between the substrate and waveguide to stop the optical mode from leaking out.
The optical mode of an SOI rib waveguide. Copyright © John Pritchard.
As for the waveguide’s silicon-on-insulator (SOI) platform, Pritchard used a buried oxide insulator on a silicon substrate. This waveguide configuration enabled him to confine relatively large optical modes in the waveguide and avoid harming the single-mode operation. Subsequent simulations of the configuration revealed that the optical mode is well confined within the waveguide and that this geometry can be used to design an interferometer. Pritchard also performed a frequency analysis of the top view of the design, as seen in the animation below and to the left. This waveguide design was deemed a success and is a significant step toward fully realizing MO routers on a chip.
Left: Wave propagation at the top of a dual waveguide and coupler at 1550 nm. Right: Mode profile of a coupler and dual waveguide. Copyright © John Pritchard.
The coupler and waveguide designs we’ve discussed thus far are ideal for the single-mode confinement of light at 1550 nm. Now, let’s see how adding a top layer of MO material to the SOI rib waveguide affects the device. Specifically, the goal is to find out the amount of light that is exposed to the Faraday rotation. This indicates when light with a rotated state of polarization interferes with nonrotated light.
Mode analysis of an SOI waveguide with a top layer of MO material. Copyright © John Pritchard.
The results, highlighted above, show that the MO material contains 3.9% of the light. Despite being a small percentage, previous research suggests that this creates sufficient Faraday rotation to observe interference at the electrical output. But for this to happen, the material needs to be magnetized with a permanent magnet or controlled magnetic field generator. Finding appropriate monolithically integrated magnetic field generators was therefore a final point of consideration.
While magnetic field generation techniques are important for creating on-chip MO modulators, the process itself is complex. The small size of MO systems makes it difficult to develop monolithically integrated magnetic field generators. To address this, Pritchard used simulation to validate the design of a novel dynamic magnetic field generator: a four-turn integrated coil.
Left: Geometry of an integrated magnetic field generator. Right: Geometry of the integrated magnetic field generator with the MO material highlighted in pink and the silicon waveguide shown in purple. Copyright © John Pritchard.
The results show that the coil generated 260 G near the center of the optical waveguide when energized with a 35 mA current. Within the tested waveguide dimensions, this magnetic field strength can magnetize Ce:YIG film on silicon.
It is possible to expand such research by investigating the field at the center of the MO material, which is part of the core of the four-turn integrated coil. Here, the simulation studies indicate that with a current of 35 mA, the magnetic field at the center of the MO material layer has a maximum field of about 210 G, a reduction possibly explained by the difference in properties of the material. Such findings speak to the potential of on-chip MO routing solutions and can be used as a resource in improving their design and manufacturing processes.
Simulation results for the magnetic field generator at the center of the coil. Copyright © John Pritchard.
Future on-chip optical network architects will have a variety of active switching and routing options, allowing them to make their networks more robust by using both EO and MO devices. While it’s important to note that the results mentioned here are preliminary and more research is needed, the simulations and design methodologies act as a proof of concept for on-chip magnetic field generators and silicon rib waveguides. They can serve as a useful foundation for continued studies on such devices, creating a path for furthering their optimization.
As John Pritchard notes: “Some of the most beautiful connections between light, magnetism, and quantum theory have led to breathtaking technologies ranging from superconductor imaging to gigawatt laser pulses. These have enabled revolutionary inventions in transportation; measurement instruments used to understand the birth of the universe; and, in the near future, optical integrated circuits.” Looking to the future, we are eager to see the continued role of multiphysics simulation in advancing optical research, a field with wide-reaching applications.
A typical underground three-phase electrical cable is made up of a bundle of three conductive cables. Each individual cable is stranded, meaning it is composed of many wires that are twisted together and compressed so that the strands are in good electrical contact. The cable can also have shielding such as a metal foil. A polymer material between the cable and shielding provides electrical insulation. Wound paper, fluids, and even pressurized gases are also used as electric insulators. The entire insulated cable bundle is then encapsulated within another dielectric and a metal sheath as well as an outer polymer coating, which protects the cable from the environment.
Left: An underground three-phase electrical cable. Right: Cross-sectional schematic of a buried three-phase cable.
The alternating current passing through the cable results in a time-varying magnetic field, which causes induced currents in the cable as well as in the surrounding metal sheaths and foil. The currents lead to a combination of Joule heating and induction heating. The cable bundle then begins to heat up, possibly causing it to fail, hence our interest in building a predictive computational model.
The electrical analysis of the cable is fairly straightforward. We usually know all of the relevant material properties (electric conductivity, permeability, and permittivity) in the cable bundle as well as exactly how much current flows through the cable and at what frequency. However, we only have a rough understanding of the electrical properties of the surrounding soil.
Thermally speaking, there are even more unknowns. The thermal properties of the surrounding soil vary based on its composition and moisture content. Even within the cable, although we know the material properties, there can be thin layers of material and small air gaps that significantly change the peak temperature.
Let’s find out how we can model these types of cables using COMSOL Multiphysics.
We can reasonably assume that underground cables are long and the surrounding environment is relatively uniform. These assumptions allow us to simplify our model by considering a 2D cross-sectional slice, similar to the one shown in the schematic above. We know that the three-phase current in the cables varies at a fixed frequency. We also know the maximum current.
We assume that the stranded bundle of copper wires is compressed together with good electrical contact, so we treat each of the three copper cables as one uniform domain across which the current can redistribute itself. Thus, we use three different Coil features to excite the three copper cables, as shown in the screenshot below. The applied excitation is of the form: 1[kA]*exp(-i*120[deg])
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That is, a 1kA peak current flows through the three cables, but the relative phases are shifted by 120° between each.
The Coil feature sets up the current flowing through one of the cables. The other two cables carry the same current, but with a 120° phase shift.
Next, we consider the thin layer of metal shielding. If the thickness of this layer is small compared to the other dimensions, then we model this metal layer via the Transition boundary condition, as shown below. This boundary condition allows us to enter a thickness and specify a set of material properties at an internal boundary of the model. The advantage of this condition is that we don’t need to explicitly model the geometry and thus don’t need to mesh this thin layer of material.
The Transition boundary condition in which the layer thickness and material properties can be entered.
The magnetic fields can extend some distance outside of the cable. Since we want to know how quickly the fields drop off, we model a region of soil around the cable. We choose this region’s size by studying progressively larger domains until the field solution shows minimal variation with an increasing domain size, a procedure described in an earlier blog post on choosing boundary conditions for coil modeling. The results of such an analysis, seen in the image below, show the magnetic field and cycle-averaged losses. It is these losses that lead to a rise in temperature.
The losses in the cable bundle and shielding with arrows representing the magnetic field. The arrow lengths are logarithmically scaled relative to the magnetic field strength.
Modeling the temperature rise of the cable seems relatively straightforward — we simply take the computed losses and include them in a thermal model. We add the Heat Transfer in Solids physics interface to our model and, within the Multiphysics branch, use the predefined features to set up a bidirectional coupling between the electromagnetic and thermal problem. Either the Frequency-Stationary or Frequency-Transient study type can be used to solve the electromagnetic problem in the frequency domain while solving the thermal problem in the steady state or time domain.
The Frequency-Stationary solver and multiphysics settings for creating the bidirectionally coupled electromagnetic heating model.
Now, even though the thermal model appears straightforward, there are a number of things that we need to keep in mind as well as features of the software that we want to be aware of. For one, the cable bundle has several thin layers of material, such as the shielding and coatings, that we might not want to model explicitly. For these, we use the Thin Layer boundary condition, which has the option of modeling the thin layer as a Thermally thin approximation, Thermally thick approximation, or General layer, as shown in the screenshot below.
The Thermally thin approximation is appropriate when the material layers have a relatively much-higher thermal conductance than their surroundings, whereas using the Thermally thick approximation is better for material layers with a relatively much-lower conductance. The General type should be used for any intermediate cases, where there are significant thermal gradients both normal to and tangentially along the layer of material. All of these options allow you to specify the layer thickness and properties, and the General type additionally allows you to specify a composite of up to five different layers.
The Thin Layer boundary condition.
The Thin Layer boundary condition is appropriate for well-defined layers of material, with known thicknesses and properties. We also need to consider the thermal resistance that arises when two materials are in contact. Heat transfer between rough surfaces in contact occurs when there is:
These effects can all be modeled via the Thermal Contact feature, seen in the following screenshot.
The Thermal Contact feature and equations.
The conductive heat transfer through the solids is strongly affected by the contact pressure. This pressure can be computed from (and coupled with) a structural analysis, as exemplified by these tutorial models:
Next, we need to consider the thermal environment, which has a great deal of variability that directly affects the cable temperature. The surrounding soil, concrete, and rocks have thermal conductivities that range from 0.1 to 5 W/m/K and densities that range from just over 1000 kg/m^{3} for very loosely packed soil to over 3000 kg/m^{3} for solid rocks. Their material-specific temperatures also vary from ~500 to 1500 J/kg/K. Furthermore, these values do not remain constant. For example, the thermal conductivity of dry and wet sand can differ by over an order of magnitude: from ~0.2 to 4 W/m/K. It is also helpful to introduce the thermal diffusivity, which is defined as and ranges from roughly for these materials.
In addition to the huge variability of the soil’s thermal properties, the thermal boundary conditions at the surface are rarely well defined. There is both convective cooling to the air and radiative cooling to the sky. The magnitude of this cooling is greatly affected by the local and temporary features at the surface. For example, dead leaves or loosely packed snow can act as a very good layer of thermal insulation that is difficult to quantify with any precision.
Lucky for us, the cables are buried deep enough that these temporary variations at the surface can often be neglected. Thus, it is reasonable to approximate the heat balance at the surface with a combination of three boundary conditions:
The solar heat load and ambient air temperature can be entered approximately or from the American Society of Heating, Refrigerating, and Air-Conditioning Engineers database of weather station data, as we describe in a previous blog post. The effective sky temperature ranges from about 230 K to 285 K (-45°C to 10°C), depending on the air temperature and cloud cover, with a typical ground surface emissivity of 0.8–0.95.
We must also consider the width and depth of our thermal domain. We need to model a sufficiently large domain of soil such that the boundary conditions don’t affect the results. For thermal loads that vary sinusoidally in time with cycle period , the distance D from the boundary at which the temperature oscillation is reduced by approximately 90% relative to the oscillation at the surface is given by: .
Assuming that the thermal boundary conditions vary sinusoidally over the year, and assuming a very high thermal diffusivity, a good rule of thumb is to model a domain that extends at least eight meters beneath the surface and is at least three times the burial depth on either side, with thermal insulation boundary conditions on the vertical boundaries and a fixed temperature boundary condition at the bottom boundary. A Temperature boundary condition is used to fix the temperature at the bottom to the average of the surface temperature over an entire year. This is a good approximation of the large thermal mass of the ground.
We can also investigate a larger soil domain to see if peak temperatures are noticeably affected. Of course, if there are known subsurface features, such as nearby water mains or building foundations, these should be included in the model.
We can solve the model using either a Frequency-Stationary or Frequency-Transient study type. Both solve for the frequency-domain form of Maxwell’s equations, but solve the thermal model as either a steady-state or time-dependent problem. Solving for the steady-state temperature requires a bit of care in interpreting the results. A steady-state analysis will assume that all thermal transients have died out, a rather severe assumption. Such results must be interpreted with care. Solving the transient problem, on the other hand, can consider all of the changing environmental conditions and loads and will give not just the peak temperatures but also the duration at which different materials are at different temperatures.
The screenshot below shows a typical model setup and sample results.
The thermal model and representative results of the cable’s temperature. The magnetic fields are only solved for the smaller circular domain around the cable, since the magnetic fields drop off rapidly in intensity.
Here, we have shown the appropriate COMSOL® software features and modeling approaches for computing the temperature rise in underground power cables. When solving such problems, keep in mind the variability in the solution that can be introduced due to the changing thermal environment, imprecisely known soil properties, and even small air gaps or standoffs within the cable itself. Of course, COMSOL Multiphysics (along with the AC/DC and Heat Transfer modules) is a great tool for modeling these situations and for considering all of the variability in the model inputs.
Interested in using COMSOL Multiphysics to model electromagnetic heating?
Electrical installations, such as substations and capacitor banks, play an important role in delivering the right amount of power to homes and businesses. But just as important as the ability of this equipment to supply such power is the overall safety of the system. For instance, when electric fields are too high in the area surrounding an electrical installation, they can pose potential health risks for the operators of the equipment as well as the general public. To this end, there are requirements that exist with regards to the maximum electric fields allowed outside the installation.
Electric fields outside electrical installations, like the capacitor bank shown above, are important considerations in the design of these systems. Image by Western Area Power. Licensed under CC BY 2.0, via Flickr Creative Commons.
Take the ICNIRP guidelines, for example. The focus of these requirements is to limit exposure to low-frequency electric fields, which are typically 50 to 60 Hz. Within these guidelines, as is typically the case, the maximum electric field is specified in two levels: one for authorized personnel and another for the general public. At 50 Hz, the general public can be exposed to a maximum of 5 kV/m, while personnel are allowed a higher exposure of 10 kV/m.
Simulation is the primary method used to verify that electrical installations meet these low-frequency requirements as well as those specified by customers. For simulation experts, performing such studies is a straightforward process, but these experts may not always be available to address simulation requests right away. This delay can result in the need for significant redesigns at a much later stage in the process.
By creating an easy-to-use app with the Application Builder, a built-in tool in the COMSOL Multiphysics® software, simulation experts can enable design engineers to verify electric field levels outside electrical installations on their own. Nils Lavesson, a principal scientist at ABB, built a simulation app in an effort to do just that.
When describing the geometry of an installation, one of the most practical problems that simulation engineers face is balancing the level of detail needed for accuracy with the level of simplification for efficiency. Designs like the ABB capacitor bank, shown below, are typically available as CAD drawings that can be imported into COMSOL Multiphysics. But these drawings are often quite detailed and require extensive simplification before they can be used in a simulation.
An ABB capacitor bank. Image by ABB.
As Lavesson was interested in the electric field significantly far away from the active parts, he opted to take the most common approach: creating a 3D CAD geometry based on the CAD drawing. In this geometry, simplified geometric shapes represent the electrical components. This usually provides a good degree of accuracy with regards to the calculated fields of interest. A typical geometry includes multiple components that are connected to different phases, with one or two fences surrounding it. This approach is useful for running simulations, but there’s a great deal of manual work that’s needed to set up the geometry. To automate this process, further simplifications are needed.
Lavesson chose to use a 2D axisymmetric geometry, which only considers one of the platforms connected to one phase. While this method may sound restrictive, it is sufficient to obtain a reasonable approximation for the final value. Most real cases involve three phases, but one of these phases is often dominant. Additional phases cause screening of the field, which makes approximating the problem with one phase a conservative approach. (We describe this in more detail in a previous blog post.)
A simplified 2D geometry for an electrical installation. The electrical components (1), first fence (2), second fence (3), ground (4), and air (5) are all highlighted. Image by Nils Lavesson and taken from his COMSOL Conference 2016 Munich paper.
Note that when approximating a 3D platform in 2D, some accuracy is lost. The platform or tower, which typically has a rectangular shape, is represented by a cylinder in 2D. Selecting the largest dimension of the platform as the cylinder’s diameter (often the diagonal distance) helps account for this and ensures that there’s no risk of the calculated field value becoming too low.
The same argument can be applied to the fences. The fence is typically built along straight lines in a real-world setting; however, in the simulation, it is represented as circular. Selecting the shortest distance between the fence and the platform within the electrical installation and using it as the distance between the two elements in the 2D geometry helps ensure that the approximation leads to a simulation of only a slightly higher field.
Another important step in designing the 2D geometry is parameterizing the voltage distribution. The analysis here considers two main cases:
Using the simplified geometry described above, Lavesson designed an easy-to-use simulation app. The app prompts users to enter various geometry parameters as well as the allowed electric field levels. Note that an air box is automatically generated with a sufficient size so as to not limit the accuracy of the simulation.
The computation — a step that often takes under a minute — begins when the user clicks the Compute button. The results from this analysis indicate to the user whether or not the design is viable. Both the calculated electric field and the geometry are displayed in various plots within the app. In the screenshot below, for instance, you can see a plot that indicates where the specified limits lie between the allowed and unallowable electric field levels. This enables the user to better visualize, and thus understand, the electric field’s distribution.
The electric field calculation app. Image by Nils Lavesson and taken from his COMSOL Conference 2016 Munich paper.
Along with access to help, users also have the option to generate a report that includes the calculation’s input parameters, plots, and results — all with just a simple click of the Report button. This makes it possible to efficiently document individual simulation studies throughout the entire design process and easily communicate each set of results to a larger group.
After completing the design of the app, Lavesson created a 3D model in COMSOL Multiphysics based on the same analysis and compared the results between the two studies. For this purpose, he used a common example geometry. While it wasn’t necessary for this analysis, it’s possible to design an app using this 3D model.
This geometry features a 2×2 square tower that is 3 m tall, with voltage increasing linearly from 100 to 300 kV. Surrounding the tower are two fences, one that is 3 m tall and another that is 2.5 m tall. These fences are 2 m and 5 m away from the tower, respectively. The evaluation of the electric field is performed at 2 m above the ground.
The 3D geometry. Image by Nils Lavesson and taken from his COMSOL Conference 2016 Munich paper.
A cut is made at the center of the platform of the full 3D geometry. The cut includes the maximum field outside the fences because of symmetry. Outside the first fence, the maximum field is evaluated as 4.6 kV/m. Outside the second fence, the maximum field is evaluated as 2.5 kV/m.
Electric field calculations outside the electrical installation tower along the cut plane in the 3D geometry. Image by Nils Lavesson and taken from his COMSOL Conference 2016 Munich paper.
This same analysis is repeated using the app. Using the same simplified geometry described above, a cylinder is used to approximate the tower, with the radius defined as half the diagonal distance of the tower. The distances to the fences are defined as the respective minimum distances. When plotting the electric field along the 2D plane, the results show that the maximum field is 5.5 kV/m outside the first fence and 3.1 kV/m outside the second fence.
Electric field calculations outside the electrical installation tower for the 2D geometry included within the app. Image by Nils Lavesson and taken from his COMSOL Conference 2016 Munich paper.
For the 2D approximation, you’ll notice that the values are slightly larger than they are for the 3D geometry. This is intentional with the use of the platform’s largest dimensions and minimum distances to the fence. Such an approach ensures that any design that falls within the electric fields for the 2D geometry also meets these requirements when a 3D simulation is performed.
As preliminary 2D simulations help to ensure that designs are not far from the optimal state, the electric field app has the potential to be quite useful to design engineers down the road. If greater accuracy in optimization is desired, a full 3D simulation can be performed to complement the app’s simulation results. As highlighted in the simulation research above, good agreement is found between the results from the app and those from the sample 3D geometry.
While this specific app is limited to a class of particular geometries, the flexibility of the Application Builder makes it possible to build similar apps that can address other geometrical configurations. Extending the scope of these simulation capabilities provides a way of optimizing design processes for a wide range of electrical installations in the years ahead.
In Part 2 of the blog series, we used the Electromagnetic Waves, Frequency Domain interface, which we call a Full-Wave simulation, and a Far-Field Domain node to determine the electric field in the far field. We then coupled a Full-Wave simulation to the Electromagnetic Waves, Beam Envelopes interface (or a Beam-Envelopes simulation) in order to precisely calculate fields in any region, regardless of the distance from the source.
The Far-Field Domain and Beam-Envelopes solutions that we looked at in the previous blog post are effective, but they share one noteworthy restriction. In each case, we assumed that a homogeneous domain surrounded the antenna in all directions. For many situations, this information is sufficient. In other simulations, you may not have a homogeneous domain surrounding your antenna and you need to account for issues like atmospheric refraction or reflection off of nearby buildings. These simulations require a different approach.
A model of several hotels in Las Vegas. A directional antenna emits rays toward the ARIA® Resort & Casino.
The Geometrical Optics interface in the Ray Optics Module, an add-on product to the COMSOL Multiphysics® software, regards EM waves as rays. This interface can account for spatially varying refractive indices, reflection and refraction from complicated geometries, and long propagation distances. However, these features come with a tradeoff. Since waves are treated as rays, this approach neglects diffraction. In other words, we are assuming that the wavelength of light is much smaller than any geometric features in our environment. You can read a more thorough description of ray optics in a previous blog post.
As you may recall, we introduced an approach to coupling a radiating and receiving antenna in Part 3 of this series. When incorporating ray optics into our multiscale modeling, we are required to use a similar but more generalized approach. Before we show you how to set up a geometrical optics simulation in COMSOL Multiphysics, let’s first review this alternate method.
As a quick refresher, we are interested in calculating the fields at the location of the receiving antenna using the following equation:
We previously used an integration operator on a single point to calculate this along the line directly between the two antennas. We now wish to retain the angular dependence, so we need to recalculate this equation for each point in the receiving antenna’s domain. Since it is impractical to add numerous points and integration operators, we need to establish a more general technique.
To do so, we replace the integration operator with a General Extrusion operator. As before, we create a variable for the magnitude of r. We then use the General Extrusion operator to evaluate the scattering amplitude at a point in the geometry that shares the same angular coordinates, , as the point in which we are actually interested.
To demonstrate this concept, we use a figure that is slightly more involved than that from the previous post. Note that the subscripts 1, 2, and r in represent a vector in component 1, a vector in component 2, and the offset between the antennas, respectively.
Image showing where the scattering amplitude should be calculated and how the coordinates of that point can be determined.
As we previously outlined, the primary complication is determining where to calculate the scattering amplitude. We want the fields at the point , which requires calculating the scattering amplitude at . The complication, of course, is that each point in the domain around the receiving antenna (each vector ) will have its own evaluation location . We evaluate this by again rescaling the Cartesian coordinates, but instead of doing it for a single point, we define it inside of the general operator so that it can be called from any location. From the above figure, we know that this point is , with corresponding equations for y and z. The operator is defined in component 1, so the source will be defined in that component. It will be called from component 2, so the x, y, z in the following expressions refer to x_{2}, y_{2}, z_{2} in the above figure.
The General Extrusion operator used for the scattering amplitude calculation. Note that this is defined in component 1.
As a bookkeeping step, we store the calculated fields in a “dummy” variable. By a dummy variable, we mean that we add in an extra dependent variable that takes the value of a calculation determined elsewhere. We do this for two reasons.
The first reason is that most variables in COMSOL Multiphysics are calculated on demand from the dependent variables. In an RF simulation, for example, the dependent variables are the three Cartesian components of the electric field: Ex, Ey, and Ez. These are determined when computing the solution. In postprocessing, every other value (electric current, magnetic field, etc.) is calculated from the electric field when required. In most cases, this is a fast and seamless process. In our case, each field evaluation point requires a general extrusion of a scattering amplitude, and each scattering amplitude point requires a surface integration as defined in the Far-Field Domain node. This can take a while and we want to ensure that we perform this calculation only once.
The second reason why we do this has to do with the element order. The Scattered Field formulation requires a background electric field. COMSOL Multiphysics then calculates the magnetic field using the differential form of Faraday’s law (also known as the Maxwell-Faraday equation). This requires taking spatial derivatives of the electric field. There are no issues when taking the spatial derivatives of an analytical function like a plane wave or Gaussian beam, but it can cause a discretization issue when applied to a solved-for variable. This is a rather advanced topic, which you can find out more about in an archived webinar on equation-based modeling.
By using a cubic dummy variable to store the electric field, we can take a spatial derivative of the electric field and still obtain a well-resolved magnetic field for use in the Scattered Field formulation. Without the increased order of the dummy variable, the magnetic field used would be underresolved. Below, you can see what it looks like to put the General Extrusion operator together with the dummy variable setup. The variable r is identical to the one used in Part 3 of this blog series and is defined in component 2.
The dummy variable implementation. Notice that the dummy variable components are called Ebx, Eby, and Ebz.
The only remaining step is to use the dummy variables — Ebx, Eby, and Ebz — in a background field simulation of the half-wavelength dipole discussed in Part 1 and Part 3.
This technique isn’t actually very good for this particular problem. There may be situations where it is useful, but the technique from Part 3 is preferred in the vast majority of cases. The received power from the two simulations is extremely close, but this method takes much longer to calculate and the file size increases drastically. In the demo examples for this post, this method took several times longer than the previous simulation method. While you may conclude that this is not a terribly useful step overall, it is useful when we incorporate ray optics into our multiscale modeling, as discussed in the next section.
A geometrical optics simulation implicitly assumes that every ray is already in the far field. Earlier in the blog series, we saw that the Far-Field Domain feature correctly calculates the electric field at arbitrary points in the far field. Here, we use that information as the input for rays in a geometrical optics simulation. The simulation geometry, symmetry, and electric dipole point source used are the same as in Part 2.
The domain assignments for the simulation. The Full-Wave simulation is performed over the entire domain, with the outer region set as a perfectly matched layer (PML). The geometrical optics simulation is only performed in this outer region. Note that this image is not to scale.
With the domains assigned, we select the Geometrical Optics interface, change the Intensity computation to Compute intensity, and select the Compute phase check box. These steps are required to properly compute the amplitude and phase of the electric field along the ray trajectory.
Settings for the Geometrical Optics interface. The Intensity computation is set to Compute intensity and the Compute phase check box is selected.
We also apply an Inlet boundary condition to the boundary between the Full-Wave simulation domain and Geometrical Optics domain. The inlet settings can be seen in the image below, but let’s walk through them one at a time. First, the Ray Direction Vector section is configured. This will launch the rays normal to the curved surface we’ve selected for the inlet — in other words, radially outwards. The variables Etheta and Ephi are calculated from the scattering amplitude according to
with a similar assignment for Ephi.
This equation comes from our previous blog post about using the Far-Field Domain node to calculate the fields at an arbitrary location. These variables are used to specify the initial phase and polarization of the rays. The variable specifies the correct spatial intensity distribution for the rays (as antennas generally do not emit uniformly) and is calculated according to , where Z is the impedance of the medium.
The initial radius of curvature has two factors. The parameter is the radius of the spherical boundary that we are launching the rays from and will correctly initialize the curvature of the ray wavefront.
Finally, we use the Cartesian components of our spherical unit vector to specify the initial principal curvature direction. This ensures that the correct polarization orientation is imparted to the rays. The wavefront shape here must be set to Ellipsoid — even though the surface is technically a sphere — because we need to be able to specify a preferred direction for polarization. If we choose Spherical, then each orientation is degenerate and we cannot make that specification.
The settings for the Inlet boundary condition in the Geometrical Optics interface. Note that you can click the image to expand it.
Beyond setting the correct frequency, the only other setting here is the placement of a Freeze Wall condition on the exterior boundary to stop the rays. Let’s take a look at the results vs. theory. As before, we express the full solution for a point dipole as a sum of two contributions, which we have labeled near field (NF) and far field (FF).
The electric fields from a geometrical optics simulation compared against theory. Geometrical optics is always in the far field, so we see excellent agreement as the distance from the source increases. For reference, the far-field domain results from the previous post would overlap exactly with the ray optics and FF theory lines.
As mentioned before, the Geometrical Optics interface is necessarily in the far field, so we do not expect to be able to correctly capture the near-field information as we did in the Beam-Envelopes solution in Part 2. This can also be seen because we seeded the ray tracing simulation with data from the Far-Field Domain node calculation. It is therefore unsurprising that there is disagreement near the source, but we can clearly see that the results match with theory as the distance from the source increases.
From looking solely at the above plot, we have to ask ourselves: “What have we actually gained here?”
This is a fair question, because the plot shown above could have been constructed directly from any of the techniques covered in the series so far. To make this clear, let’s review each of them.
Multiscale Technique | Regime of Validity | Modules Used | Notes |
---|---|---|---|
Far-Field Domain node | Far field | RF or Wave Optics | Requires the antenna to be completely surrounded by a homogeneous domain. |
Beam-Envelopes | Any field | Wave Optics | Requires specification of the phase function or wave vector. |
Geometrical Optics | Far field | Ray Optics | Can account for a spatially varying index as well as reflection and refraction from complex geometries. Diffraction is neglected. |
A summary of the multiscale modeling techniques we have covered in this blog series.
Note that any of these techniques will require a Full-Wave simulation of the radiation source. This generally requires the RF Module, although there is a subset of radiation sources that can be modeled using the Wave Optics Module instead. The Far-Field Domain node is available in both the RF and Wave Optics modules.
We originally motivated this discussion by talking about signal transmission from one antenna to another, and solved that simulation using the Far-Field Domain node in the last post. In the next blog post in this series, we’ll redo that simulation using the Geometrical Optics interface introduced here.
Access the model discussed in this blog post and any of the model examples highlighted throughout this blog series by clicking on the button above.
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Iron, one of the world’s most commonly used metals, is found in everything from automobiles and skyscrapers to furniture and cookware. Due to its popularity, iron ore is an in-demand commodity and global producers need an efficient way of harvesting it. As such, engineers are left with an interesting task: coming up with a method that finds as much iron ore as possible while minimizing exploration costs.
Iron ore that has been washed and graded. Image by Peter Craven — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
One solution is magnetic prospecting, a geological exploration method that estimates the center-of-mass coordinates and spatial extent of iron-rich layers underneath the earth’s surface. This process works because deposits of iron ore, such as those comprised of magnetite and hematite, alter the earth’s magnetic field and cause magnetic anomalies. In magnetic prospecting, probes are used to measure these local anomalies and find iron ore deposits. This saves on exploration costs.
With simulation, you can analyze and improve the magnetic prospecting process. Creating custom simulations tailored for each situation, however, requires a great deal of time. You may need to create new models or import a geometry and apply the correct magnetic simulation for each new site. As an alternative, you can use the Application Builder in the COMSOL Multiphysics® software to design a simulation app based on an underlying model. With an app, such as the demo app highlighted here, you can import customized magnetic prospecting data and apply magnetic simulations quickly and easily. In addition, colleagues can use the app to run their own analyses and get customized results without the assistance of a simulation engineer.
Today, we discuss a magnetic prospecting app that enables users to import digital elevation model (DEM) files and heightmap images as well as add magnetic simulations to these files.
Imported heightmap © OpenStreetMap contributors. See http://www.openstreetmap.org/copyright. Data available from the U.S. Geological Survey.
The Magnetic Prospecting of Ore Deposits app has two main uses:
The demo app achieves these goals by solving a magnetostatic problem with the Magnetic Fields, No Currents interface. The underlying model includes a geometry that extends both above and below a reference elevation level and thus includes an aboveground and underground region. Completely contained in the underground simulation area is an iron ore deposit, which is modeled as a uniform ellipsoid. The model computes a uniform magnetic background field from a specific geomagnetic field.
For more information about the underlying model, you can download the application documentation.
The app’s user interface consists of three tabs. To begin, let’s discuss the Geological Properties tab, pictured below. The Geological Properties tab enables users to customize the study size area, ore deposit geometry, and how the earth’s surface is represented. Here, we focus on one useful feature in this section: importing a surface geometry.
The Geological Properties tab in the Magnetic Prospecting demo app.
In the screenshot above, Default Heightmap is selected in the Surface list box of the Site Properties section. In our example, this default heightmap represents an actual area in California. While this is the app’s default location, users can easily investigate other locations by using the Surface list box to change how the surface of the earth is specified.
One option is to import an image that represents a desired elevation profile by selecting Heightmap (image file) from the drop-down menu instead of Default Heightmap. The imported image needs to be grayscale, with brighter and darker tones representing higher and lower elevations, respectively. After importing an image, users enter a value for scaling and the app automatically scales the image to the area’s dimensions.
App users can also use the Digital Elevation Model (DEM) option to import elevation data in the DEM file format, which is defined by the United States Geological Survey (USGS). This file type includes elevation data for points on a regular grid. When working with an imported DEM file, users can enter a reference elevation and the app generates an overlay of an elevation grid on the simulation area.
Example of imported topographical data showing a magnetite deposit.
After changing the application inputs in any tab, users need to recompute the solution. Similarly, after changing the geometric parameters, the geometry needs to be rebuilt. With this demo app, you only have to click the Compute and Build Site buttons in the upper-left corner to accomplish these tasks.
Moving on, let’s investigate the Magnetic Properties tab, which defines the local geomagnetic field and the ore’s magnetic properties.
Here, the app provides a few different options for adding magnetic simulations to the preexisting or imported data we discussed above. First, users can customize the geomagnetic field by defining individual components and downloading current geomagnetic field data from the National Centers for Environmental Information website. To download this data, click the Download button next to Download geomagnetic data.
Second, users can customize the ore’s magnetic properties. To accomplish this, they can use a mixture model that accounts for the ore’s magnetic permeability and the remanent magnetization of its magnetic portion.
In the Results tab, app users can add anomaly probes and customize their locations on the geometry surface. They can also receive numerical values for the anomaly at these points.
The process of adding a probe is easy. Users simply enter the location that they want to study in the Marker section and click the Update Marker button. After this, a blue marker appears at the designated location on the 3D plot. The 3D plot also shows the computed geometric field anomaly and the total perturbed field, which are represented by streamlines.
If the user is happy with the marker’s location on the 3D plot, they can click the Add Probe button to place a probe at the marker’s location. This action produces a red arrow, like the two we see on the 3D plot above. At the same time, numerical data indicating the predicted anomaly at the probe’s locations within the region is also added to the Anomaly Measurement Probes table. Users don’t even have to worry about choosing the wrong location for a probe, since at any time, they can click the Clear Probes button to start over.
When the app user is satisfied with their results, they can click the Write Report button, located in the app’s upper-left corner, to create a report that includes all of their customized simulation results. With apps like these, colleagues can generate customized results without relying on the assistance of simulation experts like you. Click the button below to try this app and get inspired to create apps of your own.
In the simulation of our receiving antenna, we will use the Scattered Field formulation. This formulation is extremely useful when you have an object in the presence of a known field, such as in radar cross section (RCS) simulations. Since there are a number of scattered field simulations in the Application Gallery, and it has been discussed in a previous blog post, we will assume a familiarity with this technique and encourage you to review those resources if the Scattered Field formulation is new to you.
The Scattered Field formulation is useful for computing a radar cross section.
When comparing the implementation we will use here with the scattering examples in the Application Gallery, there are two differences that need to be referenced explicitly. The first is that, unlike the scattering examples, we will use a receiving antenna with a Lumped Port. With the Lumped Port excitation set to Off, it will receive power from the background field. This is automatically calculated in a predefined variable, and since the power is going into the lumped power, the value will be negative. The second difference, which we will spend more time discussing, is that the receiving antenna will be in a separate component than the emitting antenna and we will have to reference the results of one component in the other to link them.
What does it mean when we have two or more components in a model? The defining feature of a component is that it has its own geometry and spatial dimension. If you would like to have a 2D axisymmetric geometry and a 3D geometry in the same simulation, then they would each require their own component. If you would like to do two 3D simulations in the same model, you only need one component, although in some situations it can be beneficial to separate them anyways.
Let’s say, for example, that you have two devices with relatively complicated geometries. If they are in the same component, then anytime you make a geometric change to one, they both need to be rebuilt (and remeshed). In separate components this would not be the case. Another common use of multiple components is submodeling, where the macroscopic structure is analyzed first and then a more detailed analysis is performed on a smaller region of the model. When we split into components, however, we then need to link the results between the simulations.
In our case, we have two antennas at a distance of 1000 λ. Separating them into distinct components is not strictly required, but we are going to do it anyways to keep things general. We will add in ray tracing later in this series and some users may find this multiple component method useful with an arbitrarily complex ray tracing geometry.
While we go through the details, it’s important that we have a clear image of the big picture. The main idea that we are pursuing in this post is that we first simulate an emitting antenna and calculate the radiated fields in a specific direction. Specifically, this is the direction of the receiving antenna. We then account for the distance between the antennas and use the calculated fields as the background field in a Scattered Field formulation for the receiving antenna. The emitting antenna is centered at the origin in component 1 and the receiving antenna is centered at the origin in component 2. Everything we will discuss here is simply the technical details of determining the emitted fields from the first simulation and using them as a background field in a second simulation.
Note: The overwhelming majority of the COMSOL Multiphysics® software models only have one component and only should have one component. Ensure that you have a sufficient need for multiple components in your model before implementing them, as there is a very real possibility of causing yourself extra work without benefit.
There are a number of coupling operators, also known as component couplings, available in COMSOL Multiphysics. Generally speaking, these operators map the results from one spatial location to another. Said in another way, you can call for results in one location (the destination), but have the results evaluated at a separate location (the source). While this may seem trivial at first glance, it is an incredibly powerful and general technique. Let’s look at a few specific examples:
As mentioned above, we want to simulate the emitting antenna (just like we did in Part 2 of the series) and calculate the radiated fields at a distance of 1000 λ. We then use a component coupling to map the fields to being centered about the origin in component 2.
If we look at the far-field evaluation discussed in Part 2, we know that the x-component of the far field at a specific location is
The only complication is determining where to calculate the scattering amplitude. This is because component couplings need the source and destination to be locations that exist in the geometry. We don’t want to define a sphere in component 1 at the actual location of the receiving antenna, since that defeats the entire purpose of splitting the two antennas into two components. What we will do instead is create a variable for the magnitude of r, and then evaluate the scattering amplitude at a point in the geometry that shares the same angular coordinates, , as the point we are actually interested in. In the image below, we show the point where we would like to evaluate the scattering amplitude.
Image showing where the scattering amplitude should be calculated and how the coordinates of that point can be determined.
We add a point to the geometry using the rescaling of the Cartesian coordinates shown in the above figure. Only x is shown in the figure, but the same scaling is also applied to y and z. For the COMSOL Multiphysics implementation, shown below, we have assumed that the receiving antenna is centered at a location of (1000 λ, 0, 0), and the two parameters used are ant_dist = and sim_r = .
The required point for the correct scattering amplitude evaluation.
Note that we create a selection group from this point. This is so that it can be referenced without ambiguity. We then use this selection for an integration operator. Since we are integrating only over a single point, we simply return the value of the integrand at that point similar to using a Dirac delta function.
The integration operator is defined using the selection group for the evaluation point.
The above discussion was all about how to evaluate the scattering amplitude at the correct location. The only remaining step is to use this in a background field simulation of the half-wavelength dipole discussed in Part 1. When we add in the known distance between the antennas, we get the following:
The variable definition for r. Note that this is defined in component 2.
The background field settings.
In the settings, we see that the expression used for the background field in x is comp1.intop1(emw.Efarx)*exp(-j*k*r)/(r/1[m]), which matches the equation cited above. Also note that r is defined in component 2, while intop1() is defined in component 1. Since we are calling this from within component 2, we need to include the correct scope for the coupling operator, comp1.intop1(). The remainder of the receiving antenna simulation is functionally equivalent to other Scattered Field simulations in the Application Gallery, so we will not delve into the specifics here.
It is interesting to note that running either the emission or background field simulations by themselves is quite straightforward. All of the complication in this procedure is in correctly calculating the fields from component 1 and using them in component 2. All of this heavy lifting has paid off in that we can now fully simulate the received power in an antenna-to-antenna simulation, and the agreement between the simulated power and the Friis transmission equation is excellent. We can also obtain much more information from our simulation than we can purely from the Friis equation, since we have full knowledge of the electromagnetic fields at every point in space.
It is worth mentioning one final point before we conclude. We have only evaluated the far field at an individual point, so there is no angular dependence in the field at the receiving antenna. Because we are interested in antennas that are generally far apart, this is a valid approximation, although we will discuss a more general implementation in Part 4.
We have now reached a major benchmark in this blog series. After discussing terminology in Part 1 and emission in Part 2, we can now link a radiating antenna to a receiving antenna and verify our results against a known reference. The method we have implemented here can also be more useful than the Friis equation, as we have fully solved for the electromagnetic fields and any polarization mismatch is automatically accounted for.
There is one remaining issue, however, that we have not discussed. The method used here is only applicable to line-of-sight transmission through a homogeneous medium. If we had an inhomogeneous medium between the antennas or multipath transmission, that would not be appropriately accounted for either by this technique or the Friis equation. To solve that issue, we will need to use ray tracing to link the emitting and receiving antennas. In Part 5 of this blog series, we will show you how we can link a radiating source to a ray optics simulation.
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 Full-Wave simulation.
We then extract information about the antenna’s emission pattern using a Far-Field Domain node, which performs a near-to-far-field 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 Far-Field 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 Far-Field 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 Far-Field 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 x-component of the electric field in the far field is:
and similar expressions apply to the y- and z-component, 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 far-field 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 Far-Field Domain node for a dipole at the origin with . For comparison, we have included the Far-Field Domain node, the full theory, as well as the near- and far-field terms individually. The fields are evaluated along an arbitrary cut line. As you can see, there is overlap between the Far-Field Domain node and the far-field theory plots, and they agree with the full theory as the distance from the antenna increases. This is because the Far-Field 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 Far-Field Domain node is correct in the far field, which you probably would have guessed from the name.
A comparison of the Far-Field Domain node vs. theory for a point dipole source.
For most simulations, the near-field and far-field 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 Beam-Envelopes. As discussed in a previous blog post, Beam-Envelopes 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 Full-Wave simulation of the fields near the source, as before, and then use Beam-Envelopes 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 Full-Wave simulation is performed everywhere. It is also possible to solve the inner region using a Full-Wave simulation and the outer region using Beam-Envelopes, 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 Beam-Envelopes simulation to our Full-Wave simulation of the dipole? This can be done in two steps involving the boundary conditions at the interface between the Full-Wave and Beam-Envelopes domains. First, we set the exterior boundary of the Full-Wave 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 Beam-Envelopes. We then specify the field values in the Beam-Envelopes Electric Field boundary condition according to the fields computed from the Full-Wave simulation, as shown here.
The Electric Field boundary condition in Beam-Envelopes. Note that the image in the top right is not to scale.
A Matched Boundary Condition is applied to the exterior boundary of the Beam-Envelopes 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 Full-Wave and Beam-Envelopes 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 Far-Field Domain node from a Full-Wave simulation and linking a Full-Wave simulation to a Beam-Envelopes simulation. In both cases, the fields near the source and in the far field are correctly computed. The coupled approach using Beam-Envelopes 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 far-field 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 long-distance 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 high-frequency electromagnetics model.
The Friis transmission equation calculates the received power for line-of-sight 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 so-called free-space 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 far-field 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 E-field 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 far-field 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 xy-plane 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 half-wavelength dipole as our receiving antenna.
A visual representation of radiation incident on a half-wavelength 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 length-to-diameter 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 half-wavelength 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 half-wave 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 xy-plane, 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 BY-SA 3.0, via Wikimedia Commons. Right: A vehicle’s oil gauge. Image by Marcus Yeagley. Licensed under CC BY-SA 2.0, via Flickr Creative Commons.
While piezoresistive pressure sensors require additional power to operate and feature higher noise limits, they offer many advantages over their capacitive counterparts. For one, they are easier to integrate with electronics. They also have a more linear response in relation to the applied pressure and are shielded from RF noise.
But like other MEMS devices, piezoresistive pressure sensors include multiple physics within their design. And in order to accurately assess a sensor’s performance, you need to have tools that enable you to couple these different physics and describe their interactions. The features and functionality of COMSOL Multiphysics enable you to do just that. From your simulation results, you can get an accurate overview of how your device will perform before it reaches the manufacturing stage.
To illustrate this, let’s take a look at an example from our Application Gallery.
The design of our Piezoresistive Pressure Sensor, Shell tutorial model is based on a pressure sensor that was previously manufactured by a division of Motorola that later became Freescale Semiconductor, Inc. While the production of the sensor has stopped, there is a detailed analysis provided in Ref. 1 and an archived data sheet available from the manufacturers in Ref. 2.
Our model geometry is comprised of a square membrane that is 20 µm thick, with sides that are 1 mm in length. A supporting region that is 0.1 mm wide is included around the edges of the membrane. This region is fixed on its underside, indicating a connection to the thicker handle of the device’s semiconducting material. Near one of the membrane’s edges, you can see an X-shaped piezoresistor (Xducer™) as well as some of its associated interconnects. Only some interconnects are included, as their conductivity is high enough that they don’t contribute to the device’s output.
Geometry of the sensor model (left) and a detailed view of the piezoresistor geometry (right).
A voltage is applied across the [100] oriented arm of the X, generating a current down this arm. When pressure induces deformations in the diaphragm in which the sensor is implanted, it results in shear stresses in the device. From these stresses, an electric field or potential gradient that is transverse to the direction of the current flow occurs in the [010] arm of the X — a result of the piezoresistance effect. Across the width of the transducer, this potential gradient adds up, eventually producing a voltage difference between the [010] arms of the X.
For this case, we assume that the piezoresistor is 400 nm thick and features a uniform p-type density of 1.31 x 10^{19} cm^{-3}. While the interconnects are said to have the same thickness, their dopant density is assumed to be 1.45 x 10^{20} cm^{-3}.
With regards to orientation, the semiconducting material’s edges are aligned with the x- and y-axes of the model as well as the [110] directions of the silicon. The piezoresistor, meanwhile, is oriented at a 45º angle to the material’s edge, meaning that it lies in the [100] direction of the crystal. To define the orientation of the crystal, a coordinate system is rotated 45º about the z-axis in the model. This is easy to do with the Rotated System feature provided by the COMSOL software.
In this example, we use the Piezoresistance, Boundary Currents interface to model the structural equations for the domain as well as the electrical equations on a thin layer that is coincident with a boundary in the geometry. Using this kind of 2D “shell” formulation significantly reduces the computational resources required to simulate thin structures. Note that both the MEMS Module and the Structural Mechanics Module are used to perform this analysis.
To begin, let’s look at the displacement of the diaphragm after a 100 kPa pressure is applied. As the simulation plot below shows, the displacement at the center of the diaphragm is 1.2 µm. In Ref. 1, a simple isotropic model predicts a displacement of 4 µm at this point. Considering that the analytic model is derived from a crude variational guess, these results show reasonable agreement with one another.
The displacement of the diaphragm following a 100 kPa applied pressure.
When using a more accurate value for shear stress in local coordinates at the diaphragm edge’s midpoint, the local shear stress is said to be 35 MPa in Ref. 1. This is in good agreement with the minimum value from our simulation study (38 MPa). In theory, the shear stress should be the greatest at the diaphragm edge’s midpoint.
Shear stress in the piezoresistor’s local coordinate system.
The following graph shows the shear stress along the edges of the diaphragm. The maximum local shear stress of 38 MPa is at the center of each of the edges.
Local shear stress along two of the diaphragm’s edges.
Given that the dimensions of the device and the doping levels are estimates, the model’s output during normal operation is in good agreement with the information presented in the manufacturer’s data sheet. For instance, in the model, an operating current of 5.9 mA is obtained with an applied bias of 3 V. The data sheet notes a similar current of 6 mA. Further, the model generates a voltage output of 54 mV. As indicated by the data sheet, the actual device produces a potential difference of 60 mV.
Lastly, we look at the detailed current and voltage distribution inside the Xducer™ sensor. As noted by Ref. 3, a “short-circuit effect” may occur when voltage-sensing elements increase the current-carrying silicon wire’s width locally. This effect essentially means that the current spreads out into the sense arms of the X. The short-circuit effect is illustrated in the plot below. Also highlighted is the asymmetry of the potential, which is a result of the piezoresistive effect.
Current density and electric potential for a device with a 3 V bias and an applied pressure of 100 kPa.
S.D. Senturia, “A Piezoresistive Pressure Sensor”, Microsystem Design, chapter 18, Springer, 2000.
Motorola Semiconductor MPX100 series technical data, document: MPX100/D, 1998.
M. Bao, Analysis and Design Principles of MEMS Devices, Elsevier B. V., 2005.
Xducer™ is believed to be a trademark of Freescale Semiconductor, Inc. f/k/a Motorola, Inc. Neither Freescale Semiconductor Inc. nor Motorola, Inc. has in any way provided any sponsorship or endorsement of, nor do they have any connection or involvement with, COMSOL Multiphysics® software or this model.
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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 high-intensity 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 double-sided 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 single-sided heating apparatus.
The model geometry for the RTA configuration.
In the above figure, the components are stored in a chamber featuring temperature-controlled 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 start-up 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 current-carrying coil generates a time-varying magnetic field in a nearby conductor. The time-varying 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 BY-SA 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 aluminum-conducting disc placed above two cylindrical, concentric coils carrying sinusoidal currents in opposite directions. The cross-sectional cut of the setup and the dimensions are shown in the image below.
A cross-sectional 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 time-varying currents in opposite directions.
To model this device in the COMSOL Multiphysics® software, we use a 2D axisymmetric geometry. Due to the time-varying 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 current-carrying coils are modeled using separate Coil features with the Homogenized Multi-Turn 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 first-order 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 high-speed 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 four-pole-pair 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 add-on 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.