In Part 2 of the blog series, we used the Electromagnetic Waves, Frequency Domain interface, which we call a FullWave simulation, and a FarField Domain node to determine the electric field in the far field. We then coupled a FullWave simulation to the Electromagnetic Waves, Beam Envelopes interface (or a BeamEnvelopes simulation) in order to precisely calculate fields in any region, regardless of the distance from the source.
The FarField Domain and BeamEnvelopes 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 addon 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 FarField 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 MaxwellFaraday 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 solvedfor variable. This is a rather advanced topic, which you can find out more about in an archived webinar on equationbased 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 wellresolved 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 halfwavelength 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 FarField 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 FullWave 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 FullWave 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 FarField 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 farfield 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 nearfield information as we did in the BeamEnvelopes solution in Part 2. This can also be seen because we seeded the ray tracing simulation with data from the FarField 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 

FarField Domain node  Far field  RF or Wave Optics  Requires the antenna to be completely surrounded by a homogeneous domain. 
BeamEnvelopes  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 FullWave 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 FarField 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 FarField 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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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 farfield evaluation discussed in Part 2, we know that the xcomponent 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 halfwavelength 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 antennatoantenna 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 lineofsight 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 FullWave simulation.
We then extract information about the antenna’s emission pattern using a FarField Domain node, which performs a neartofarfield transformation. This approach gives us information about the electromagnetic field in two regions: the fields in the immediate vicinity of the antenna, which are computed directly, and the fields far away, which are calculated using the FarField Domain node. This is demonstrated in a number of RF models in the Application Gallery, such as the Dipole Antenna tutorial model, so we will not comment further on the practical implementation here.
One question that occasionally comes up in technical support is: “How do I use the FarField Domain node to calculate the radiated field at a specific location?” This is an excellent question. As stated in the RF Module User’s Guide, the FarField Domain node calculates the scattering amplitude, and so determining the complex field at a specific location requires a modification for distance and phase. The expression for the xcomponent of the electric field in the far field is:
and similar expressions apply to the y and zcomponent, where r is the radial distance in spherical coordinates, k is the wave vector for the medium, and emw.Efarx is the scattering amplitude. It is worth pointing out that emw.Efarx is the scattering amplitude in a particular direction, and so it depends on angular position , but not radial position. The decrease in field strength is solely governed by the 1/r term. There are also variables emw.Efarphi and emw.Efartheta, which are for the scattering amplitude in spherical coordinates.
To verify this result, we simulate a perfect electric dipole and compare the simulation results with the analytical solution, which we covered in the previous blog post. As we stated in that post, we split the full results into two terms, which we call the near and farfield terms. We briefly restate those results here.
where is the dipole moment of the radiation source and is the unit vector in spherical coordinates.
Below, we can see the electric fields vs. distance calculated using the FarField Domain node for a dipole at the origin with . For comparison, we have included the FarField Domain node, the full theory, as well as the near and farfield terms individually. The fields are evaluated along an arbitrary cut line. As you can see, there is overlap between the FarField Domain node and the farfield theory plots, and they agree with the full theory as the distance from the antenna increases. This is because the FarField Domain node will only account for radiation that goes like 1/r, and so the agreement improves with increasing distance as the contribution of the 1/r^{2} and 1/r^{3} terms go to zero. In other words, the FarField Domain node is correct in the far field, which you probably would have guessed from the name.
A comparison of the FarField Domain node vs. theory for a point dipole source.
For most simulations, the nearfield and farfield information is sufficient and no further work is necessary. In some cases, however, we also want to know the fields in the intermediate region, also known as the induction or transition zone. One option is to simply increase the simulation size until you explicitly calculate this information as part of the simulation. The drawback of this technique is that the increased simulation size requires more computational resources. We recommend a maximum mesh element size of for 3D electromagnetic simulations. As the simulation size increases, the number of mesh elements increases, and so do the computational requirements.
Another option is to use the Electromagnetic Waves, Beam Envelopes interface, which here we will simply refer to as BeamEnvelopes. As discussed in a previous blog post, BeamEnvelopes is an excellent choice when the simulation solution will have either one or two directions of propagation, and will allow us to use a much coarser mesh. Since the phase of the emission from an antenna will look like an outgoing spherical wave, this is a perfect solution for determining these fields. We perform a FullWave simulation of the fields near the source, as before, and then use BeamEnvelopes to simulate the fields out to an arbitrary distance, as required.
The simulation domain assignments. If the outer region is assigned to PML, then a FullWave simulation is performed everywhere. It is also possible to solve the inner region using a FullWave simulation and the outer region using BeamEnvelopes, as we will discuss below. Note that this image is not to scale, and we have only modeled 1/8 of the spherical domain due to symmetry.
How do we couple the BeamEnvelopes simulation to our FullWave simulation of the dipole? This can be done in two steps involving the boundary conditions at the interface between the FullWave and BeamEnvelopes domains. First, we set the exterior boundary of the FullWave simulation to PMC, which is the natural boundary condition for that simulation. The second step is to set that same boundary to an Electric Field boundary condition for BeamEnvelopes. We then specify the field values in the BeamEnvelopes Electric Field boundary condition according to the fields computed from the FullWave simulation, as shown here.
The Electric Field boundary condition in BeamEnvelopes. Note that the image in the top right is not to scale.
A Matched Boundary Condition is applied to the exterior boundary of the BeamEnvelopes domain to absorb the outgoing spherical wave. The remaining boundaries are set to PEC and PMC according to symmetry. We must also set the solver to Fully Coupled, which is described in more detail in two blog posts on solving multiphysics models and improving convergence from a previous blog series on solvers.
If we again examine the comparison between simulation and theory, we see excellent agreement over the entire simulation range. This shows that the PMC and Electric Field boundary conditions have enforced continuity between the two interfaces and they have fully reproduced the analytical solution. You can download the model file in the Application Gallery.
A comparison of the electric field of the FullWave and BeamEnvelopes simulations vs. the full theory.
In today’s blog post, we examined two ways of computing the electric field at points far away from the source antenna and verified the results using the analytical solution for an electric point dipole. These two techniques are using the FarField Domain node from a FullWave simulation and linking a FullWave simulation to a BeamEnvelopes simulation. In both cases, the fields near the source and in the far field are correctly computed. The coupled approach using BeamEnvelopes has the additional advantage in that it also computes fields in the intermediate region. In the next post in the series, we will combine the calculated farfield radiation with a simulation of a receiving antenna and determine the received power. Stay tuned!
Multiscale modeling is a challenging issue in modern simulation that occurs when there are vastly different scales in the same model. For example, your cellphone is approximately 15 cm, yet it receives GPS information from satellites 20,000 km away. Handling both of these lengths in the same simulation is not always straightforward. Similar issues show up in applications such as weather simulations, chemistry, and many other areas.
While multiscale modeling can be a general topic, we will focus our attention on the practical example of antennas and wireless communication. When we wirelessly transmit data via antennas, we can break the operation down into three main stages:
Modern communications require longdistance wireless data transfer via antennas.
The two length scales that we will consider for this process are the wavelength of the radiation and the distance between the antennas. To use a specific example, FM radio has a wavelength of approximately three meters. When you listen to the radio in your car, you are often ten km or more away from the radio tower. Because many antennas, such as dipole antennas, are similar in size to a wavelength, we will not consider this to be another distinct length scale. As a result, we have one length scale for the emitting antenna, a different length scale for the signal propagation from source to destination, and then the original length scale again for the receiving antenna.
Let’s go over some of the most important equations, terms, and considerations when working with multiple scales in the same highfrequency electromagnetics model.
The Friis transmission equation calculates the received power for lineofsight communication between two antennas separated by a lossless medium. The equation is
where the subscripts r and t discriminate between the transmission antenna and the receiving antenna, G is the antenna gain, P is the power, is the reflection coefficient for impedance mismatch between antenna and transmission line, p is the polarization mismatch factor, λ is the wavelength, r is the distance between the antennas and is associated with the socalled freespace path loss, and and are the angular spherical coordinates for the two antennas.
Note that we have explicitly included two impedance mismatch terms, and so:
The Friis transmission equation is derived in many texts, so we will not do so again here.
A visualization of the gain for a transmitting and receiving antenna. When using the Friis transmission equation, we require the orientation of each antenna for correct gain specification. The distance between the antennas is r.
Let’s now discuss spherical coordinates , since they are incredibly useful for antenna radiation and we will use them repeatedly. Starting from the Cartesian coordinates (x, y, z), we can easily express these as follows.
For convenience, we have used the actual COMSOL Multiphysics commands — sqrt(), acos(), and atan2(,) — instead of their mathematical symbols. In our simulation setup, we will also make use of the Cartesian components of the spherical unit vector .
Similar assignments can be made for the Cartesian components of and , but is the most important for our purposes. This will be discussed later in this blog series when we cover ray optics.
A given point shown in both Cartesian (x, y, z) and spherical coordinates. The unit vectors for the spherical coordinates are also included. Note that the spherical unit vectors are functions of location.
We are generally interested in the radiated power from antennas. The power flux in W/m^{2} is represented by the complex Poynting vector .
Many antenna texts also use radiation intensity, which is defined as the power radiated per solid angle and measured in W/steradian. Mathematically speaking, this is . For clarity, we have included two conventions here, as it is common to use in electrical engineering, while physicists will generally be more familiar with . We can then calculate the radiated power by integrating this quantity over all angles.
Gain and directivity are similar in that they both quantify the radiated power in a given direction. The difference is that gain relates this radiated power to the input power, whereas directivity relates this to the overall radiated power. Put more simply, gain accounts for dielectric and conductive losses and directivity does not. Mathematically, this reads as and for gain and directivity, respectively. P_{in} is the power accepted by the antenna and P_{rad} is the total radiated power. While both quantities can be of interest, gain tends to be the more practical of these two as it accounts for material loss in the antenna. Because of its prevalence and usefulness, we also include the definition of gain (in a given direction) from “IEEE Standard Definitions of Terms for Antennas”, which is: “The ratio of the radiation intensity, in a given direction, to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically.”
IEEE also includes three notes about gain in their definition:
In practice, an actual antenna will be connected to a transmission line. Because the antenna and the transmission line may not have the same impedance, there can be a loss factor due to impedance mismatch. The realized gain is simply the gain when accounting for impedance mismatch. Mathematically, this is , where is the reflection coefficient from transmission line theory, Z_{c} is the characteristic impedance of the transmission line, and Z is the impedance of the antenna.
When using a lumped port with a characteristic impedance in COMSOL Multiphysics, the farfield gain that is calculated corresponds to the IEEE realized gain. This is important to mention explicitly, since various definitions of gain have changed over the last few decades. Starting with COMSOL Multiphysics version 5.3, which will be released in 2017, the variable names in the COMSOL software will be changed to match the IEEE definitions.
The realized gain and electric field from a Vivaldi antenna, simulated using COMSOL Multiphysics and the RF Module. You can find the Vivaldi Antenna tutorial model in the Application Gallery.
The terms we have discussed so far have referred to antennas emitting radiation, but they are also generally applicable to receiving antennas. The reason we have put more emphasis on emission thus far is because antennas generally obey reciprocity (the Lorentz reciprocity theorem is a fixture in most antenna textbooks). Reciprocity means that an antenna’s gain in a specific direction is the same regardless of whether it is emitting in that direction or receiving a signal from that direction. Practically speaking, you can calculate the gain in any direction from a single simulation of an emitting antenna, which is easier than simulating the inverse process for each desired direction.
When we talk about receiving antennas, we are often interested in calculating the received power for an incoming signal. This can be done by multiplying the effective area, , of the antenna by the incident power flux and accounting for impedance mismatch in the line, yielding . As you may expect, this bears a striking similarity to several terms of the Friis transmission equation.
Today, we will talk about one type of emitter: the perfect electric point dipole. Depending on the literature, you may have seen this referred to as a perfect, ideal, or infinitesimal dipole. This emitter is a common representation of radiation for electrically small antennas. The solution for the field is
where is the dipole moment of the radiation source (not to be confused with the polarization mismatch) and k is the wave vector for the medium.
One breakdown of the various regions for the electromagnetic field generated from an electrically small antenna.
In this equation, there are three factors of 1/r^{n}. The 1/r^{2} and 1/r^{3} terms will be more significant near the source, while the 1/r term will dominate at large distances. While the electromagnetic field will be continuous, it is common to refer to different regions of the field based on the distance from the source. One such distribution for an electrically small antenna is shown above, although there are other conventions that refer to the magnitude of kr.
Later, we will see how to calculate the fields at any distance from a given source, but the most important region for antenna communications is the far field or radiation zone, which is the region farthest away from the source. In this region, the fields take the form of spherical waves, , a fact that we will take advantage of.
We will now split up the Efield equation above into two terms. For simplicity, we will call the 1/r term the far field (FF) and the 1/r^{2} and 1/r^{3} terms the near field (NF).
As mentioned before, we can calculate the radiated power in watts by integrating over all angles. Note that only the farfield term will contribute to this integral, which is a primary reason why the far field is of practical interest to antenna engineers. The total power radiated from a point dipole is , where Z_{0} is the impedance of free space and c is the speed of light. The maximum gain is 1.5 and is isotropic in the plane normal to the dipole moment (e.g., the xyplane for a dipole in ).
A note on units: The equations above are given with the traditional definition of the dipole moment in Coulomb*meters (Cm). In antenna and engineering texts, it is common to specify an infinitesimal current dipole in Ampere*meters (Am). COMSOL Multiphysics follows the engineering convention. The two definitions are related by a time derivative, so for a COMSOL software implementation, the dipole moment should be multiplied by a factor of to obtain the infinitesimal current dipole.
We will use a perfectly conducting halfwavelength dipole as our receiving antenna.
A visual representation of radiation incident on a halfwavelength dipole antenna.
Many texts cover an infinitely thin wire, which has an impedance of and a directivity of . It is worth mentioning that the antenna impedance will change from these values for an antenna of finite radius. The receiving antenna we use here has a length of 0.47 λ and a lengthtodiameter ratio of 100. With these values, we simulate an impedance of , which is close to the infinitely thin value and also agrees reasonably well with experimental values. Regrettably, there is no theoretical value to compare to this number, but this highlights the need for numerical simulation in antenna design.
The comparison between the directivity of the infinitely thin dipole and our simulated dipole antenna is shown below. Because the antenna is lossless, this is equivalent to the antenna gain. You can download the dipole antenna model here.
A comparison of the directivity for two halfwavelength antennas (oriented in z) as a function of theta. The COMSOL Multiphysics® simulation is of a finite radius cylinder and the theory is for an infinitely thin antenna.
We can now use the Friis transmission equation to calculate the power that is emitted from a perfect point dipole and received by a halfwave dipole antenna. To use this equation, we simply need to know the gain and impedance mismatch (or realized gain), wavelength, distance between the antennas, and input power. Since we are using a point electric dipole, we have a dipole moment instead of input power and impedance mismatch. We can account for this by removing the impedance mismatch term and replacing the input power by the radiated power of the perfect electric dipole from above — effectively saying that power in equals power out.
If we assume that our emitter and detector are both located in the xyplane, are polarization matched, and are separated by 1000 λ, as well as that the dipole moment of the emitter is 1 Am in , the Friis equation yields a received power of 380 μW. We will simulate this value in part 3 of this series for verification of our simulation technique. We can then use our simulation to confidently extract results and introduce complexity that the Friis equation cannot account for.
In this blog post, we have introduced the idea of multiscale modeling and discussed all of the relevant terms, definitions, and theory that we will need moving forward. For those of you with a strong background in electromagnetics and antenna design, this has likely been a quick review. If the concepts presented here are new to you, we strongly recommend further reading in a book on classical electromagnetics or antenna theory.
In the following blog posts, we will focus primarily on practical implementation of multiscale modeling in COMSOL Multiphysics and we will repeatedly refer to concepts discussed today.
Stay tuned for more installments in our multiscale modeling blog series:
Since antennas radiate electromagnetic waves, it is important to ensure that their radiated fields don’t return to the radiating source. The absorbers attached to the chamber’s walls play a major role in this process by absorbing incident waves on their surfaces. This makes the anechoic chamber a key aspect in an antenna simulation. However, including the anechoic chamber is challenging because it increases the simulation’s computational requirements.
A biconical antenna excited in an anechoic chamber, used to test for electromagnetic interference (EMI) and electromagnetic compatibility (EMC).
The anechoic chamber shown above is smaller than the typical anechoic chamber that meets CISPR specifications, but its computational cost is still high. This model requires over 16 GB of RAM. To improve computational efficiency, we should simplify the model, while maintaining the accuracy of the computation. As discussed in a previous blog post, we can accomplish this by using a perfectly matched layer (PML). This reduces the memory use to less than 2 GB without sacrificing simulation accuracy.
To efficiently mimic the real world in our antenna simulation, we need to choose the correct boundary conditions and physics features. Accurately reflecting realworld conditions in your simulation environment, while still keeping your model memory and time efficient, can be challenging. In the table below, we outline some realworld antenna scenarios and the optimal modeling feature to choose.
Real World  Simulation Environment  

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

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

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

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

Perfect Electric Conductor boundary condition  Mimics metallic surfaces with a high conductivity 
Lumped port  Excites the antenna and measures Sparameters 
Scattering boundary condition  Absorbs the incident wave to minimize any reflection 
Farfield domain and calculation  Calculates the farfield radiation pattern, directivity, and gain 
A Perfect Electric Conductor boundary condition imposed on a rectangular strip.
For the intended operating frequency, the simulation may only take a few seconds. The RF Module provides the default Sparameter evaluation, electric field distribution plot, and polar farfield plot. It also gives you the 3D farfield radiation pattern plot, which shows the computed directivity and gain.
The farfield radiation pattern of a printed dipole antenna. The computed directivity is 2.15 dB, which is close to that of an ideal halfwave dipole antenna.
Although simulating an antenna is a straightforward process, it is a good idea to start with a simple structure, whether you are a beginner or an expert. This way, you can ensure that the basic modeling process is correct for the simple geometry before adding complex design elements.
The RF Module also enables you to combine electromagnetics with any other type of physics. You can see and change all of the physics features in the modeling environment and clearly define every physics property. Taking into account multiple physical effects, as well as knowing the underlying physics involved, is useful when validating your antenna design.
To capture the details of the physics, such as the loss on metallic surfaces, the Perfect Electric Conductor boundary condition can be replaced by the Transition boundary condition for a geometrically very thin lossy layer or an Impedance boundary condition for surfaces of a lossy volume. You can also use a PML instead of a Scattering boundary condition, which assumes that the incident wave is normal to the surface.
After setting up these physics features, you can begin to design your antenna, whether its shape is traditional, wideband, multiband, or an array.
You can access many antenna examples for a wide range of applications in the Application Library in the RF Module. The tutorial models range from conventional antennas, such as halfwave dipole and microstrip patch antennas, to wideband and multiband antennas, including Vivaldi, fractal, spiral, and helical antennas. There is also an antenna array example, which can be useful when designing devices for the 5G mobile network.
Traditional antennas, like the halfwave dipole and microstrip patch antenna shown below, are good examples to start with when learning how to model antennas in the COMSOL Multiphysics® software. Their geometries are relatively easy to build and it’s simple to validate the results with wellknown analytical solutions.
For example, you can simulate a halfwave dipole antenna to find its omnidirectional radiation. Or, you can model a microstrip patch antenna to see if the electric field is confined to the radiating edges.
Conventional antennas: halfwave dipole antenna with a quarterwave coaxial balun (left) and microstrip patch antenna (right).
Sometimes, we need to cover many different frequency ranges with a single antenna. By tweaking the radiating structure and using the multiple resonance behaviors of a certain part of the metallic body or slot, we can meet the system’s specifications without deploying multiple antennas. One such example is the popular Vivaldi antenna, also called a tapered slot antenna.
Using fractal algorithms, such as those from Sierpinski, Koch, and Hilbert, you can generate interesting results for antenna applications. For instance, unlike the halfwave dipole antenna, which can only be used for a single frequency resonance, the Sierpinski fractal antenna doesn’t require additional matching networks to adjust the antenna input impedance to the reference characteristic impedance of 50 ohms for higherorder resonances.
Wideband and multiband antennas: Vivaldi antenna (left) and Sierpinski fractal monopole antenna (right).
When the input power to your antenna is regulated due to the path loss in a communication link budget, your initial design may not fulfill its needed specifications. This is a potential issue that the 5G mobile network faces and it can be resolved using an antenna array.
While antenna arrays sound cool, they can greatly increase the simulation size and computational cost if modeled explictly. When we only want to check the antenna array’s performance feasibility, we can simplify the simulation using the antenna array factors, which are mathematical properties from antenna theory. This way, we can maintain the computational efficiency of the simulation.
In COMSOL Multiphysics, we can access all of the equations that are being solved and modify them with an alternative formulation. We can also create an antenna array simulation app that allows us to control the implemented equations and the parameters that we’d like to change.
Antenna arrays: monopole antenna array (top) and slotcoupled microstrip patch antenna array synthesizer (bottom).
In order to characterize your antenna system and evaluate its EMI/EMC, it has to be measured in an anechoic or reverberation chamber using one of three wellknown test antennas: the logperiodic, biconical, or doubleridged horn antenna. You can examine these test antennas in your simulation and enhance their performance. These antennas are also a type of wideband antennas and usually cover 20 MHz ~ 200 MHz, 200 MHz ~ 2 GHz, and 2 GHz ~ 20 GHz, respectively. When you simulate them, you need to set the mesh either based on the highest frequency or parametrically updated for each analysis frequency during the frequency sweep.
Test and measurement antennas: a logperiod (left) and biconical antenna (right). These antennas are popularly used in an anechoic chamber.
You test your antenna design in an anechoic chamber, but it will eventually be deployed in real systems. You also have to evaluate the antenna’s performance when it is implemented on different types of platforms. The effect of one radiating device on another device’s system can result in problems, such as radio frequency interference. You can address these interference problems through simulation. For instance, you can use simulation to see how a car windshield’s antenna affects a cable harness and evaluate antenna crosstalk on an airplane’s fuselage.
Simulation showing a car windshield antenna’s effect on a cable harness (left) and antenna crosstalk on an airplane’s fuselage (right).
With the RF Module, you can also expedite antenna modeling with the body of revolution approach, which doesn’t demand a lot of computational resources. With this approach, you can quickly model antennas like the corrugated horn antenna and conical horn lens antenna.
Fast numerical modeling of a corrugated horn antenna (left) and a conical horn lens antenna (right).
In this blog post, we briefly reviewed efficient antenna modeling techniques and several different types of antennas that can be designed using the RF Module. Each antenna model begins with a simple geometry configuration and boundary conditions. You then gradually add more complex parts and elements to your simulation. This way, you can easily debug and tune your antenna model. Using this information and the featured examples, you can start designing antennas in COMSOL Multiphysics with optimized computational efficiency and speed.
Every year, the International Microwave Symposium (IMS) brings together researchers and engineers from around the world, giving them a preview of what’s up and coming in the RF and microwave industries. While attending last year’s event, my colleague Jiyoun Munn noted that there was a lot of buzz surrounding 5G and the Internet of Things (IoT). Throughout the science and engineering community, this sentiment has rang true, with various research in the works to advance the potential of these innovative concepts.
I recently had a chance to speak with Jiyoun about his experience attending IMS2016. During our discussion, he mentioned a technology that was referenced as a point of focus in the RF and microwave industries in the year ahead, one that could have large implications: autonomous vehicles.
A selfdriving car with U.S. Secretary of State John Kerry inside. Image in the public domain, via Wikimedia Commons.
Fully autonomous cars would essentially change all aspects of driving. People with disabilities that prevent them from driving could now travel on their own. Those who already drive, meanwhile, would have more time to dedicate to other tasks, while using less gas and staying safer.
Selfdriving cars would also have a significant impact on our roads and highways. Faster reactions, for instance, would reduce the need for greater distance between vehicles as well as overall traffic. This, in turn, could enable more cars to be on the road and allow for higher speed limits. Further, since these cars would automatically stick to the speed limit, police officers would no longer have to patrol highways, giving them more time to focus on other types of crime.
But what does it mean for a car to be fully autonomous? The National Highway Traffic Safety Administration has set out guidelines for classifying automation levels in vehicles. You can find a summary of these different levels highlighted below.
The Society for Automative Engineers (SAE) has put together a similar list, which you can view here. SAE’s standards include an additional level for cars that can drive only in specific situations, but are able to maintain control if the human driver doesn’t respond appropriately to a request to intervene.
While some semiautonomous cars are already on the road, there are still challenges that must be overcome before fully autonomous vehicles can become a reality. Let’s have a look…
Addressing the reliability and security of a fully autonomous vehicle presents various challenges. While laws don’t necessarily exclude these cars, they don’t directly address them either. So far, only 16 states have introduced legislature regarding autonomous vehicles. For autonomous cars to move from the testing phase to widespread use, more states have to pass laws that not only permit them on the road, but also account for who is liable in the case of an accident.
There’s also the question of privacy as the software needed to automate cars would have access to a lot of information about their drivers. Anything from their preferred coffee shops to their location at any given time could be stored. While this is convenient for the driver, the car companies would have access to such data as well. How can this information be used? It’s a question that the companies and lawmakers must decide on.
There are several challenges to consider as well when it comes to the reliability and security of the car’s software. Autonomous vehicles run on complicated software, which, like any other electronic device, can crash. Unlike other devices, this would put the driver in a dangerous and potentially lifethreatening situation. The software itself can also be hacked, which could lead to anything from cars being held for ransom to the hackers remotely driving vehicles.
For the actual design of autonomous cars, companies are using some combination of LIDAR, radar, and stereo cameras to “see”. LIDAR provides a 360° view using lasers and is the most precise of the three technologies, but the sensor can be easily tricked and its accuracy is compromised in weather such as heavy rain, snow, and fog. Radar and cameras are more limited in scope, but the first measures relative speed and range, while the second recognizes when objects are moving laterally in front of a vehicle.
An image taken by LIDAR, showing the road contour, elevation, and vegetation. Image by Oregon Department of Transportation — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
Along with the sensors, autonomous vehicles also need a GPS to navigate to the driver’s intended location. While we use this technology on a regular basis, a GPS depends on its map’s accuracy. If you’ve ever had one tell you to turn the wrong way on a oneway street or to use a park’s walkway as a road, you can imagine what might happen if cars relied solely on this system. Additionally, we need a GPS that provides realtime updates to account for closed roads and construction.
Most importantly, though, is that these cars actually reach Level 4 automation. A car this advanced could distinguish between potholes and plastic bags as well as know to avoid one and ignore the other. If a pedestrian suddenly moves into the street, the car must decide whether to brake or swerve out of the way. Making the jump to Level 4 means developing software that encompasses all possible driving situations and then, once this software is in place, considering how to reduce the expenses of such vehicles.
While a builtin chauffeur is still a long ways away, there have been many significant advancements in autonomous technology over the years. Cars are now equipped with the ability to automatically brake, maintain speeds, stay centered in a lane, and park themselves. With so many companies in the midst of designing autonomous cars, there is great promise that this technology will continue to advance until full autonomy is achieved.
With a better understanding of how the technology works, multiphysics simulation can serve as a powerful tool for advancing the design of autonomous vehicles. Sensors, for instance, are an important piece of the puzzle. Through modeling, we could test changes to the sensor’s design and placement in the vehicle, determining the configuration and location that delivers the optimal performance. We look forward to the continued improvements in autonomous vehicle design and simulation’s potential in helping to further its performance and capabilities.
The two simulation methods that we will discuss in today’s blog post are the asymptotic waveform evaluation (AWE) and frequencydomain modal methods. Both are designed to help you overcome the conventional issue of a longer simulation time when using a very fine frequency resolution or running a very wide band simulation. The AWE is quite efficient when it comes to describing smooth frequency responses with a single resonance or no resonance at all. The frequencydomain modal method, meanwhile, is useful for quickly analyzing multistage filters or filters of a high number of elements that have multiple resonances in a target passband.
For our purposes, it would be too technical to talk about the numerical characteristics of the asymptotic waveform evaluation (AWE) — a reducedorder modeling technique. Instead, we will go over how to use this method in the RF Module. The solver performs fastfrequency adaptive sweeping using an AWE that is configured in the Frequency Domain study settings, shown below.
Study Extensions section of Frequency Domain study settings.
When the Use asymptotic waveform evaluation check box is selected, it will trigger an AWE solver. By default, the solver uses Padé approximations.
The AWE method is very useful when simulating resonant circuits, especially bandpassfilter type devices with many frequency points. For instance, the Evanescent Mode Cylindrical Cavity Filter tutorial model, available in the Application Library, sweeps the simulation frequency between 3.45 GHz and 3.61 GHz with a 5 MHz frequency step.
The Evanescent Mode Cylindrical Cavity Filter tutorial model (left) and its discrete frequency sweep results (right). The Sparameter plot does not look smooth around the resonant frequency.
Say you run the simulation again with a much finer frequency resolution, such as a 100 kHz frequency step that is 50 times finer. You can expect that the simulation will take 50 times longer to finish. When using the AWE option in this particular example model, the simulation time is almost the same as the regular frequency sweep case, but we can obtain all of the computed solutions on the dependent variable with the 100 kHz frequency step.
The simulation time also varies with regards to the user input in the AWE expressions. Any model variable works as an AWE expression, so long as it generates a smooth resulting plot like a Gaussian pulse or a smooth curve as a function of frequency. The absolute value of S21 (abs(emw.S21)), for example, works as the input for the AWE expression in the case of a twoport bandpass filter. For oneport devices like antennas, S11 still works. If the frequency response of the AWE expression contains an infinite gradient — the case for the S11 value of an antenna, with excellent impedance matching at a single frequency point — the simulation will take longer to complete.
AWE expression using Sparameters (S21) for a twoport filter simulation.
Following the 100 kHz frequency step simulation, the solutions contain a ton of data. As a result, the model file size will increase tremendously when it is saved. When only Sparameters are of interest, a common theme in most passive RF and microwave device designs, it is not necessary to store all of the field solutions. By selecting the Store fields in output check box in the Values of Dependent Variables section, we can control the part of the model on which the computed solution is saved. We only add the selection containing these boundaries where the Sparameters are calculated. The lumped port size is typically very small compared to the entire modeling domain, and the saved file size with the AWE is more or less that of the regular discrete frequency sweep model when only the solutions on the port boundaries are stored.
Settings window for the lumped port boundary (left) and the explicit selection generated by the lumped port (right).
It is easy to add an explicit selection when setting up the lumped port. When you specify the lumped port boundary selection, click the Create Selection button. This will add an explicit selection with the boundary you just added for the lumped port. By repeating the same step for the other port, you will obtain all of the selections to use for storing only those results that you need to plot the Sparameters.
Values of Dependent Variables section of the Frequency Domain study settings.
In the Values of Dependent Variables section, change the selection in the Store fields in output combo box from All to For selections. You can then add the explicit selections created from the lumped ports.
Now you are ready to run the AWE frequency sweep. Don’t forget to use the finer frequency step in the study settings. You can do so in one of two ways: Directly type in the step you want, or click the Range button next to the input field to use the Range dialog box.
Updating the simulation frequency step via the Range dialog box.
Once the simulation is complete, you will notice that the simulation time for the AWE frequency sweep with a much finer step is almost the same as the discrete sweep. Let’s compare the computed Sparameters. Since the AWE performed a frequency sweep that was 50 times finer, its frequency response (Sparameters) plot consequently looks much nicer. Not only do you save precious time with this approach, but as the plot below illustrates, you also still obtain accurate and goodlooking results.
Sparameter plot of the AWE and discrete frequency sweep simulations.
Bandpassfrequency responses of a passive circuit result from a combination of multiple resonances. Eigenfrequency analysis is key to capturing the resonance frequencies of an arbitrary shape of a device. Once we obtain all of the necessary information from the eigenfrequency analysis, we can reuse it in the frequencydomain modal study. Doing so enables us to optimize the efficiency of the simulation when a finer frequency resolution is required to more accurately describe the frequency response, as illustrated in the AWE method.
To perform a frequencydomain modal analysis seamlessly, there are two important simulation steps to keep in mind. The first of these steps involves refining the eigenfrequency study results. The output of the eigenfrequency study is purely numerical and even includes nonphysical results. By using the manual eigenfrequency searching method, those unwanted, lowfrequency residues can be filtered out. The manual eigenfrequency searching process is constrained by a series of items: Eigenfrequency search method around shift, Desired number of eigenfrequencies, and Search for eigenfrequencies around. For the last item, the lowest passband frequency works as a ballpark value.
Manual search method in the Eigenfrequency settings.
The second step involves using the linper operator on the excitation voltage. The frequencydependent modal method requires using a linper operator as a load on the dependent variable of a model. This operator can be used by the excitation voltage input of a lumped port in the Electromagnetic Wave, Frequency Domain physics interface.
The linper operator is used in the voltage input of the lumped port settings.
To try this out, let’s take a look at the Coupled Line Filter tutorial model, available in our Application Library. You’ll want to add a new study with the eigenfrequency and frequencydomain modal study steps, configuring the settings for each study step as described above. Repeat this again with a frequency step that is 50 times finer. The Store fields in output check box in the Values of Dependent Variables section can also be applied to the frequencydomain modal study — if you are interested only in Sparameters, this is the way to go. By storing solutions only on the lumped port boundary, it is possible to further reduce simulation time.
Sparameter plot of the regular frequency sweep and frequencydomain modal simulations.
Sparameter plot of the regular frequency sweep and frequencydomain modal simulations with a frequency step that is 50 times finer.
Note that the eigenfrequency analysis contains a lumped port that impacts the simulation as an extra loading factor, so the phase of the computed Sparameters is different from that of the regular frequency sweep model. The results are compatible only with phaseindependent Sparameter values such as dBscaled, absolute value, reflectivity, or transmittivity.
The simulation methods presented in today’s blog post are powerful tools for enabling faster, more efficient modeling of passive RF and microwave devices. While these methods have been available since version 5.2, our latest release — COMSOL Multiphysics® version 5.2a — features polished Application Library examples that help further guide you in how to utilize these techniques. You can find such examples highlighted below:
Interested in learning about other improvements and upgrades in version 5.2a? Head over to our Release Highlights page for additional details.
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As we discussed in a previous blog post, there are many required advancements and design considerations for the 5G mobile network. One of the improvements that RF engineers need to work toward is increasing antenna gain to serve the much higher frequencies on which 5G will operate.
An isotropic lowgain antenna used in older networks versus a directive highgain antenna used for 5G.
Another requirement of the 5G mobile network is that we improve phase progression technology. This shapes the radiation pattern and steers the beam of an antenna array to control the input signal and address angular coverage issues.
A monopole phased antenna array can steer a beam toward the desired direction.
One device, a slotcoupled microstrip patch antenna array, can be incorporated into designs to address these coverage issues. However, there are many complex design parameters that must be considered in order to build a device that is optimized for 5G wireless communications.
Simulation can help by offering the capability to evaluate and implement physical effects that can not be easily tested in a design lab or through prototyping, such as extreme temperature variation, structural deformation, and chemical reactions. Unfortunately, not every engineer working on a design may be an expert in simulation, requiring the simulation expert on the team to be involved in every step of the design process whenever there are new changes to an antenna array design or simulation environment.
The Application Builder addresses these difficulties by further enhancing the capabilities of simulation. Now, an otherwise complex and tedious numerical model of an RF design can be turned into an interactive and userfriendly tool for experts and end users alike. Today, let’s explore the SlotCoupled Microstrip Patch Antenna Array Synthesizer simulation app and how it can help us optimize phased array antenna designs for 5G and IoT.
Active electronically scanned arrays (AESA), also known as phased antenna arrays, are conventionally used in the military for radar and satellite applications. These arrays are also used in a new application — commercial purposes — due to the growing needs of higher data rates in communication devices. The size of this simple component can easily exceed tens of a wavelength, causing simulation design to be very memory intensive. As a result, computation takes a very long time, even when only approximated values are needed to evaluate a proofofconcept model. Faster prototyping would help to analyze performance and determine design parameters quickly.
The SlotCoupled Microstrip Patch Antenna Array Synthesizer is based on a full finite element method (FEM) model of a single microstrip patch antenna built on lowtemperature cofired ceramic (LTCC) layers. The device initially operates at 30 GHz, and the radiation pattern and directivity analysis of the entire array structure are integrated using the very powerful postprocessing functionality of COMSOL Multiphysics. The Application Builder works as a shortcut to provide a variety of ways to design and build the userfriendly graphical user interface (GUI), transforming an ordinary mathematical model into an intuitive simulation tool.
The top view of a slotcoupled mictrostrip patch antenna.
The Application Builder offers two essential tools for creating our app: the Form Editor and Method Editor. The Form Editor enables us to design the GUI with simple functionality by adding form objects to a custom interface. The Method Editor helps to implement more advanced and customized functions over the form objects. After an accurate simulation of a single microstrip patch antenna, we find the twodimensional antenna array factor
This corresponds to user input, such as array size; arithmetic phase progression; and angular resolution, which is imposed on the single antenna radiation pattern data (emw.normEfar
).
The Method Editor moves beyond a simple simulation, where visualization is limited to the predefined postprocessing variables, and allows for further customization.
A preview of the main form to show form objects.
Using the Method Editor to create custom actions for form objects.
In this app, there are many design parameters that we can test for our microstrip patch antenna array design, including:
The array dimension, phase progression and spacing, and the distance between each element primarily characterize the shape and direction of the array antenna radiation pattern. The angular resolution also adds a delicate touch on the 3D and 2D radiation pattern visualization. Note that when the antenna directivity is higher, using a finer resolution will more accurately describe the sidelobes.
The GUI of the SlotCoupled Microstrip Patch Antenna Array Synthesizer app.
After the analysis, the app reports whether the single antenna design parameters are optimal by using the computed Sparameter (S11) value compared to the pass/fail target criterion that the app user specifies before running the simulation. The app depicts the electric field distribution on each dielectric and metallic layer and also visualizes the entire view of the array to give app users a better feel for the performance of the design. You can also choose to include a complete simulation result report and documentation that concisely explains how the app works.
There are infinite ways to convert your model into a customized tool by using the Application Builder, but what’s next? You can launch and use your simulation app with the core COMSOL Multiphysics® software. As long as you have an internet connection, you can run your app using a common web browser and even deploy it to colleagues and customers via COMSOL Server™ product.
In the Application Gallery, even more apps are available for you to download and explore, covering physics areas such as electrical, mechanical, fluid, chemical, and more. These demo apps can serve as a guide for you to build useful apps of your own.
The Frequency Selective Surface Simulator demo app (left) and the Plasmonic Wire Grating Simulator demo app (right).
Whether you are building an app to enhance RF designs for the 5G network, or working in another application area, get started building simulation apps and optimizing your design workflow and product performance today.
Both of these interfaces solve the frequencydomain form of Maxwell’s equations, but they do it in slightly different ways. The Electromagnetic Waves, Frequency Domain interface, which is available in both the RF and Wave Optics modules, solves directly for the complex electric field everywhere in the simulation. The Electromagnetic Waves, Beam Envelopes interface, which is available solely in the Wave Optics Module, will solve for the complex envelope of the electric field for a given wave vector. For the remainder of this post, we will refer to the Electromagnetic Waves, Frequency Domain interface as a FullWave simulation and the Electromagnetic Waves, Beam Envelopes interface as a BeamEnvelope simulation.
To see why the distinction between FullWave and BeamEnvelope is important, we will begin by discussing the trivial example of a plane wave propagating in free space, as shown in the image below. We will then apply the lessons learned to the dielectric slab.
A graphical representation of a plane wave propagating in free space, where the red, green, and blue lines represent the electric field, magnetic field, and Poynting vector, respectively.
To properly resolve the harmonic nature of the solution in a FullWave simulation, we need to mesh finer than the oscillations in the field. This is discussed further in these previous blog posts on tools for solving wave electromagnetics problems and modeling their materials. To simulate a plane wave propagating in free space, the number of mesh elements will then scale with the size of the free space domain in which we are interested. But what about the BeamEnvelopes simulation?
The BeamEnvelopes method is particularly wellsuited for models where we have good prior knowledge of the wave vector, . Practically speaking, this means that we are solving for the fields using the ansatz . Notice that the only unknown in the ansatz is the envelope function . This is the quantity that needs to be meshed to obtain a full solution, hence the mention of beam envelopes in the name of the interface. In the case of a plane wave in free space, the form of the ansatz matches exactly with the analytical solution. We know that the envelope function will be a constant, as shown by the green line in the figure below, so how many mesh elements do we need to resolve the solution? Just one.
The electric field and phase of a plane wave propagating in free space. In the field plot (left), the blue and green lines show the real part and absolute value of E(r), which are and , respectively. The phase plot (right) shows the argument of E(r). In both plots, the xaxis is normalized to a wavelength, so this represents one full oscillation of the wave.
In practice, BeamEnvelopes simulations are more flexible than the ansatz we just used. This is for two reasons. First, instead of specifying a wave vector, we can specify a userdefined phase function, . Second, there is also a bidirectional option that allows for a second propagating wave and a full ansatz of . This is the functionality that we will take advantage of in modeling the dielectric slab (also called a FabryPérot etalon).
The points discussed here will come up again in the dielectric slab example, and so we highlight them again for clarity. The size of mesh elements in a FullWave simulation is proportional to the wavelength because we are solving directly for the full field, while the mesh element size in a BeamEnvelopes simulation can be independent of the wavelength because we are solving for the envelope function of a given phase/wave vector. You can greatly reduce the number of mesh elements for large structures if a BeamEnvelopes simulation can be performed instead of a FullWave simulation, but this is only possible if you have prior knowledge of the wave vector (or phase function) everywhere in the simulation. Since the degrees of freedom, memory used, and simulation time all depend on the number of mesh elements, this can have a large influence on the computational requirements of your simulation.
Using the 2D geometry shown below, we can clearly see the different waves that need to be accounted for in a simulation of a dielectric slab illuminated by a plane wave. On the left of the slab, we have to account for the incoming wave traveling to the right, as well as the reflected wave traveling to the left. Because of internal reflections inside the slab itself, we have to account for both left and righttraveling waves in the slab, and finally, the transmitted waves on the right. We also choose a specific example so that we can use concrete numbers.
Let’s make the dielectric slab an undoped silicon (Si) wafer that is 525 µm thick. We will simulate the response to terahertz (THz) radiation (i.e., submillimeter waves), which encompasses wavelengths of approximately 1 mm to 100 µm and is increasingly used for classifying semiconductor properties. The refractive index of undoped Si in this range is a constant n = 3.42. We choose the domain length to be 15 mm in the direction of propagation.
The simulation geometry. Red arrows indicate incident and reflected waves. The left and right regions are air with n = 1 and the Si slab in the center has a refractive index n = 3.42. The x_{i}s on the bottom denote the spatial location of the planes. The slab is centered in the simulation domain, such that x_{1} = (15 mm – 525 µm)/2. Note that this image is not to scale.
For a 2D FullWave simulation, we set a maximum element size of to ensure the solution is well resolved. The simulation is invariant in the y direction and so we choose our simulation height to be . Because we have constrained the wave to travel along the xaxis, we choose a mapped mesh to generate rectangular elements. The mesh will then be one mesh element thick in the y direction, with a mesh element size in the x direction of , where n depends on whether it is air or Si. Again, note that this is a wavelengthdependent mesh.
Before setting up the mesh for a BeamEnvelopes simulation, we first need to specify our userdefined phase function. The Gaussian Beam Incident at the Brewster Angle example in the Application Gallery demonstrates how to define a userdefined phase function for each domain through the use of variables, and we will use the same technique here. Referring to x_{0}, x_{1}, and x_{2} in the geometry figure above, we define the phase function for a plane wave traveling left to right in the three domains as
where n = 3.42 and the first line corresponds to in the leftmost domain, the second line is in the Si slab, and the bottom line is in the rightmost domain. We then use this variable for the phase of the first wave, and its negative for the phase of the second wave. Because we have completely captured the full phase variation of the solution in the ansatz, this allows a mapped mesh of only three elements for the entire model — one for each domain. Let’s examine what the mesh looks like in the Si slab for these two interfaces at two different wavelengths, corresponding to 1 mm and 250 µm.
The mesh in the Si (dielectric) slab. From left to right, we have the FullWave mesh at 1 mm, the FullWave mesh at 250 µm, and the BeamEnvelopes mesh at any wavelength. Note that the FullWave mesh density clearly increases with decreasing wavelength, while the BeamEnvelopes mesh is a single rectangular element at any wavelength.
Yes, that is the correct mesh for the Si slab in the BeamEnvelopes simulation. Because the ansatz matches the solution exactly, we only need three total elements for the entire simulation: one for the Si slab and one each for the two air domains on either side of it. This is independent of wavelength. On the other hand, the mesh for the FullWave simulation is approximately four times more dense at = 250 µm than at = 1 mm. Let’s look at this in concrete numbers for the degrees of freedom (DOF) solved for in these simulations.
Wavelength Simulated 
FullWave Simulation DOF Used 
BeamEnvelopes Simulation DOF Used 

1 mm  4,134  74 
250 µm  16,444  74 
The number of degrees of freedom (DOF) used at two different wavelengths for the FullWave and BeamEnvelopes simulations.
Again, it is important to point out that this does not mean that one interface is better or worse than another. They are different techniques and choosing the appropriate option is an important simulation decision. However, it is fair to say that a FullWave simulation is more general, since we did not need to supply it with a wave vector or phase function. It can solve a wider class of problems than BeamEnvelopes simulations, but BeamEnvelopes simulations can greatly reduce the DOF when the wave vector is known. As we have seen in a previous blog post, memory usage in a simulation strongly depends on the number of DOF. Do not blindly use a BeamEnvelopes simulation everywhere though! Let’s take a look at another example where we intentionally make a bad choice for the wave vector and see what happens.
In the hypothetical free space example above, we chose a unidirectional wave vector. Here, we will do the same for the Si slab. It is important to emphasize that choosing a single wave vector where we know that the solution will be a superposition of left and righttraveling waves is an exceptionally bad choice, and we do this here solely for demonstration purposes. Instead of using the bidirectional formulation with a userdefined phase function, let’s naively choose a single “guess” wave vector of and see what the damage is. Using our ansatz, inside of the dielectric slab we have
where the lefthand side is the solution we are computing and the righthand side is exact. Now, we manipulate the equation slightly to examine the spatial variation in the solution.
We intentionally chose the case where , which means we can simplify to
Since and are constants determined by the Fresnel relations at the boundaries of the dielectric slab, this means that the only spatial variation in the computed solution will come from . The minimum mesh requirement in the slab is then determined by the “effective” wavelength of this oscillating term
which is half of the original wavelength. Not only have we made the BeamEnvelopes mesh wavelength dependent, but the required mesh in the dielectric slab for this choice of wave vector needs to be twice as dense as the mesh for a FullWave simulation. We have actually made the situation worse with the poor choice of a single wave vector for a simulation with multiple reflections. We could, of course, simply double the mesh density and obtain the correct solution, but that would defeat the purpose of choosing the BeamEnvelopes simulation in the first place. Make smart choices!
Another practical question is how do the results of a FullWave and BeamEnvelopes simulation compare? They are both solving Maxwell’s equations on the same geometry with the same material properties, and so the various results (transmission, reflection, field values) agree as you would expect. There are slight differences though.
If you want to evaluate the electric field of the rightpropagating wave in the dielectric slab, you can do that in the BeamEnvelopes simulation. This is, of course, because we solved for both right and leftpropagating waves and obtained the total field by summing these two contributions. This could be extracted from the FullWave simulation in this case as well, but it would require additional userdefined postprocessing and may not be possible in all cases. It may seem counterintuitive in that we actually have more information readily available from a BeamEnvelopes simulation, even though it is computationally less expensive. We must remember, however, that this is simply the result of solving the model using the ansatz we specified initially.
We have examined the simple case of a dielectric slab in free space using both the Electromagnetic Waves, Frequency Domain and Electromagnetic Waves, Beam Envelopes interfaces. In comparing FullWave and BeamEnvelopes simulations, we showed that a BeamEnvelopes simulation can handle much larger simulations, but only in cases where we have good knowledge of the wave vector (or phase function) everywhere in the simulation. This knowledge is not required for a FullWave simulation, but the simulation must then be meshed on the order of a wavelength, as opposed to meshing the change in the envelope function in a BeamEnvelopes simulation. It is also worth mentioning that most BeamEnvelopes meshes will need more than the three elements shown here. This was only possible here because we chose a textbook example with an analytical solution to use as a teaching model. For more realistic simulations, you can refer to the MachZehnder Modulator or SelfFocusing Gaussian Beam examples in the Application Gallery.
Note that the Electromagnetic Waves, Frequency Domain interface is available in both the RF and Wave Optics modules, although with slightly different features. The FullWave simulation discussed in this post could be performed in either module, although the BeamEnvelopes simulation requires the Wave Optics Module. For a full list of differences between the RF and Wave Optics modules, you can refer to this specification chart for COMSOL Multiphysics products.
5G is expected to be available for use by the year 2020, surpassing 4G LTE, 4G, 3G, and the networks before it. A new wireless network is supposed to be developed every ten years and with less than four years to go, we need to continue developing the technology needed to make 5G a reality for consumers. The ideal wireless network is constantly improving, and these expected advancements describe the 5G network, as well as the future networks that will follow.
You probably own a smartphone that you use multiple times throughout the day. According to recent data from the Pew Research Center, 64% of Americans own a smartphone of some kind, and about two thirds of these smartphone owners access the internet using their phones. In fact, the numbers even suggest that younger smartdevice owners use their phones for text messaging and internet access statistically more than they do for making traditional phone calls. In a survey published in April 2015, 97% of 18 to 29yearolds used their cellphones to go online, while 93% used their devices to make a call. These survey results mark a shift in the primary use of cellphones, creating a need for more data to keep the increasingly crowded mobile communications highway up and running.
Smartphone users browse the internet on their devices for everyday tasks like looking up bus routes and times. Image by Metropolitan Transportation Authority of the State of New York — Bus Time Manhattan Launch. Licensed under CC BY 2.0, via Wikimedia Commons.
Industry leaders in mobile communications have agreed on a list of specific requirements for the 5G network that cater to a new age in mobile communications. As mentioned above, the high use of mobile data has prompted a need for ultrahigh download rates, with almost no latency. Although you could argue that your current download rate is fast, for 5G, it must be close to instantaneous. Because of this increase in data needs, 5G should also have minimized signal traffic to handle the heavy load of mobile users, which is always increasing. The ideal wireless network should additionally provide reliable service everywhere. This means that you will be able to have fast and clear service in rural and desolate areas, not just urban and suburban locations, as is the case with 4G and 4G LTE.
5Gcompatible devices themselves should be relatively low cost and consume less energy so that they have a longer battery life than current devices. The way you charge your phone may also change. Some smartphones already have the ability to charge without a power cord — they are placed on a power base and charged through induction by transferring power wirelessly between the base and the phone. More advanced charging options are also being investigated, such as longerdistance wireless power transfer. In this case, a power base transmits small signals to charge a device from within a certain distance. Instead of plugging your phone into a wall, or even placing it on a base, you may soon be able to charge your phone while it sits in your pocket. Such advancements may be integrated into certain 5G technologies.
A wireless charging station transfers power to mobile devices. Image by Veredai from Powermat Technologies — Own work. Licensed under CC BYSA 3.0, via Wikimedia Commons.
5G must be able to facilitate communication between connected Internet of Things (IoT) devices. Sources predict that by 2020, the year in which we should be able to support 5G, there will be 25 billion connected devices in our world. 5G needs to be able to handle the enormous data traffic of this web of connected smart devices, as well as that from the large amount of cellphone users worldwide. Besides the high volume of data traffic, 5G further requires enough speed to quickly analyze the data collected from IoT devices for practical use by consumers.
As an engineer working toward the development of the 5G network, you understand that there are many design elements to consider. The RF capabilities of COMSOL Multiphysics can shed light on this burgeoning technology.
Currently, our wireless network cannot handle the amount of data we download — at least not at an optimal speed. To achieve the required data rate to handle the download speeds of 5G, our wireless network needs to operate in a wider frequency range. The frequency spectrum for today’s wireless network is around 1 GHz to 3 GHz. 5G not only has to operate at a frequency above 6 GHz, but it has to be able to handle a span of up to 100 GHz.
For 5G, a frequency of around 30 GHz may serve as the backbone for the mobile communication network, meaning it will mostly operate around this range. The need for frequencies of up to 100 GHz is more complementary, for reasons such as extra capacity for the system and wider bandwidths for denser areas. You will recognize the importance of this requirement if you have ever tried to access the internet on your device while at a music concert or sporting event, where thousands of other people are trying to do the same thing. A complementary frequency range will also help for surges of use in natural disasters, when many people are trying to contact loved ones, and provide the extra “push” of service to more remote areas that may have never had decent mobile service before.
A diplexer, one of the many components that will be used in 5G mobile systems, can help play a role in improving this issue. Diplexers split signals into two different frequency ranges that are designed for the wide range required for the mobile network. Within a diplexer, a lower frequency “listens”, or receives a signal, while the higher frequency “talks”, or transmits a signal. Simulation is a simple way to test different iterations of a diplexer design to determine the best settings. By studying the Sparameters and electric field of a waveguide diplexer, which we can compute through a simulationbased approach, we can see if the design will work well with the 5G mobile network.
A simulated waveguide diplexer.
In the WR28 waveguide diplexer model shown above, which is for Kaband applications, a lower and upper bandpass are set to 28 GHz (left) and 30.4 GHz (right), respectively. The simulation shows that the input power at each passband is separately distributed without being coupled to one another.
Another way to bring about the development of 5G technologies is to increase antenna gain in new mobile devices. This does not amplify the size of a cell signal, but rather increases the distance the signal can travel toward cell towers. If you think back to older, primitive phones where you would have to pull up the antenna to make a call, the circuit of the device would act as a quarterlength monopole antenna. These antennas had the same gain in all azimuthal directions, meaning the electromagnetic waves propagate isotropically on an Hplane so that the signal operates equally toward all directions and can more reliably reach cell towers, no matter the user’s location.
A 3D farfield radiation pattern of a planar invertedF antenna (PIFA) in a mobile device.
As cellphones continued to develop in both function and design, the antenna was miniaturized and embedded inside the phone’s body structure, thus distorting the ideal isotropic radiation pattern. Mobile communication progressed with the development of 3G, 4G, and 4G LTE, and cellphones started to include miniaturized multiband antennas instead of the quarterlength monopole version.
One wellknown problem with this era of communication was the spotty service and dropped calls that occurred when using your phone in certain areas. In some instances, it could even make a difference of standing in one area of a room versus another part of the same room. My colleague Jiyoun Munn, who develops the RF Module, explains why this would often happen: “When you are talking on a cellphone, you usually do not know exactly where the cell tower is located in relation to your phone. The occurrence of multipath fading from indoor propagation also contributes to this problem.”
The 5G network requires much higher frequencies. As Jiyoun mentions: “Since the attenuation in the air is more severe at higher frequencies while electromagnetic waves propagate, antennas will need to have an increased gain to reach a longer distance. Higher antenna gain means more directionality of its radiation pattern. As a consequence, the antenna’s visibility, or angular coverage, is very narrow.” Because of this, cellphones would see the base stations in a very limited range.
A quarterlength monopole antenna (above) propagates isotropically and operates on a lower frequency, while a phased array antenna (below) scans for cell tower signals with a farther gain at higher frequencies.
In order to overcome the shortcoming of highgain antennas regarding their angular coverage, it is necessary to use an active electronically scanned array (AESA), also known as a phased array, concept that shapes the radiation pattern and steers the beam from an antenna array by controlling the relative phases and magnitudes of the input signal. “The arithmetic phase progression on each antenna element in the antenna array changes the maximum radiation direction,” notes Jiyoun. “The direction of maximum radiation is normal to the equiphase plane, so the radiation pattern is tilted to the direction of the faster antenna element in terms of phase.” This is the basic idea of a phased array, which can steer the beam toward a desired direction.
The farfield radiation pattern of a monopole antenna array.
The optimal antenna for 5G technologies is a phased array antenna that can be built with microstrip patch antennas, which are made up of a cluster of regular antennas. By utilizing phase progression and the weight factor on each array element, the angular coverage and gain for the 5G network can be optimized. Using simulation, we can evaluate the design of a phased antenna array for 5G performance. Computer simulation makes it simple to compute the farfield radiation pattern and perform a fullscale farfield analysis for a variety of input parameters.
Simulation results for an 8×8 phased array antenna.
The SlotCoupled Microstrip Patch Antenna Array Synthesizer demo app has a simplified user interface that can be used to run quick design tests to develop prototypes for 5G antenna designs. The app can be launched and run using a web browser, even from a remote place. “With this app,” Jiyoun says, “design engineers can examine the asymptotic solution of the antenna, and they can also share the app with other colleagues on their team in order to work together to build the optimal device.” Because the app is intuitive and specialized for this specific use, your own antenna design can be tested and retested in just 90 seconds, rather than the two days it would take to run a full computer simulation. Building simulation apps is an effective and simple way to perform these analyses.
The SlotCoupled Microstrip Patch Antenna Array Synthesizer app.
With the arrival of 5G, we will find ourselves making room for an even more technologybased society. The Internet of Things, which also goes by monikers such as the Internet of Everything and the Industrial Internet, is the term used to describe this new age of smart devices and information sharing. The popularity of the Internet of Things may be because of the fact that nearly every industry can utilize some form of IoT technology.
Home automation and wearables, such as fitness tracking devices and smart watches, are popular uses of IoT that are able to track a person’s activity and internal statistics, connect this information to accessible apps on their smartphones, and analyze the data into knowledge that the individual consumer can apply to their home or lifestyle. As the 5G network continues to develop, more novel uses of IoT technology become available. In healthcare, IoT devices automatically disperse medicine and monitor patients based on their statistics and activity. In the media, smart devices track our entertainment and shopping preferences to automatically message us about related product information.
A fitness tracking bracelet, one example of how consumers use the Internet of Things.
The data collected by these smart devices — whether tracking temperature, footsteps, environmental conditions, or various other factors — needs to be analyzed for the entire system of devices to be useful. The IoT system must then take the analyzed information and “tell” the smart object or objects what to do with it (turn a device on or off, send a message, disperse a medication, etc.) to complete the cycle.
By studying the RF interference between these smart devices, we can create the best Internet of Things possible. There is almost no limit to the different applications of the Internet of Things, and by contributing to the development of the 5G network, we can optimize how each device communicates with each other for efficient operation. With a streamlined 5G network, the Internet of Things will be able to work to its fullest potential and become a global reality in just a matter of years.
The arrival of the 5G network is only a few years away. By working with RF applications such as antenna gain, frequency range, and beam progression in more detail, we can ensure that by the year 2020, the technology of the future will be readily accessible to us all. Using the power of computer simulation and simulation apps, we can ensure that we have a hand in creating exceptionally fast and reliable technology for a new era in global communication. Let’s get started developing 5G technology — and a more connected world — today.
Electrical cables, also called transmission lines, are used everywhere in the modern world to transmit both power and data. If you are reading this on a cell phone or tablet computer that is “wireless”, there are still transmission lines within it connecting the various electrical components together. When you return home this evening, you will likely plug your device into a power cable to charge it.
Various transmission lines range from the small, such as coplanar waveguides on a printed circuit board (PCB), to the very large, like high voltage power lines. They also need to function in a variety of situations and conditions, from transatlantic telegraph cables to wiring in spacecraft, as shown in the image below. Transmission lines must be specially designed to ensure that they function appropriately in their environments, and may also be subject to further design goals, including required mechanical strength and weight minimization.
Transmission wires in the payload bay of the OV095 at the Shuttle Avionics Integration Laboratory (SAIL).
When designing and using cables, engineers often refer to parameters per unit length for the series resistance (R), series inductance (L), shunt capacitance (C), and shunt conductance (G). These parameters can then be used to calculate cable performance, characteristic impedance, and propagation losses. It is important to keep in mind that these parameters come from the electromagnetic field solutions to Maxwell’s equations. We can use COMSOL Multiphysics to solve for the electromagnetic fields, as well as consider multiphysics effects to see how the cable parameters and performance change under different loads and environmental conditions. This could then be converted into an easytouse app, like this example that calculates the parameters for commonly used transmission lines.
Here, we examine a coaxial cable — a fundamental problem that is often covered in a standard curriculum for microwave engineering or transmission lines. The coaxial cable is so fundamental that Oliver Heaviside patented it in 1880, just a few years after Maxwell published his famous equations. For the students of scientific history, this is the same Oliver Heaviside who formulated Maxwell’s equations in the vector form that we are familiar with today; first used the term “impedance”; and helped develop transmission line theory.
Let us begin by considering a coaxial cable with dimensions as shown in the crosssectional sketch below. The dielectric core between the inner and outer conductors has a relative permittivity () of 2.25 – j*0.01, a relative permeability () of 1, and a conductivity of zero, while the inner and outer conductors have a conductivity () of 5.98e7 S/m.
The 2D cross section of the coaxial cable, where we have chosen a = 0.405 mm, b = 1.45 mm, and t = 0.1 mm. Note that this tutorial model is available for download in our Application Gallery.
A standard method for solving transmission lines is to assume that the electric fields will oscillate and attenuate in the direction of propagation, while the crosssectional profile of the fields will remain unchanged. If we then find a valid solution, uniqueness theorems ensure that the solution we have found is correct. Mathematically, this is equivalent to solving Maxwell’s equations using an ansatz of the form , where () is the complex propagation constant and and are the attenuation and propagation constants, respectively. In cylindrical coordinates for a coaxial cable, this results in the wellknown field solution of
which then yields the parameters per unit length of
where is the sheet resistance and is the skin depth.
While the equations for capacitance and shunt conductance are valid at any frequency, it is extremely important to point out that the equations for the resistance and inductance depend on the skin depth and are therefore only valid at frequencies where the skin depth is much smaller than the physical thickness of the conductor. This is also why the second term in the inductance equation, called the internal inductance, may be unfamiliar to some readers, as it can be neglected when the metal is treated as a perfect conductor. The term represents inductance due to the penetration of the magnetic field into a metal of finite conductivity and is negligible at sufficiently high frequencies. (The term can also be expressed as .)
For further comparison, we can compute the DC resistance directly from the conductivity and crosssectional area of the metal. The analytical equation for the DC inductance is a little more involved, and so we quote it here for reference.
Now that we have values for C and G at all frequencies, DC values for R and L, and asymptotic values for their highfrequency behavior, we have excellent benchmarks for our computational results.
When setting up any numerical simulation, it is important to consider whether or not symmetry can be used to reduce the model size and increase the computational speed. As we saw earlier, the exact solution will be of the form . Because the spatial variation of interest is primarily in the xyplane, we just want to simulate a 2D cross section of the cable. One issue, however, is that the 2D governing equations used in the AC/DC Module assume that the fields are invariant in the outofplane direction. This means that we will not be able to capture the variation of the ansatz in a single 2D AC/DC simulation. We can find the variation with two simulations, though! This is because the series resistance and inductance depend on the current and energy stored in the magnetic fields, while the shunt conductance and capacitance depend on the energy in the electric field. Let’s take a closer look.
Since the shunt conductance and capacitance can be calculated from the electric fields, we begin by using the Electric Currents interface.
Boundary conditions and material properties for the Electric Currents interface simulation.
Once the geometry and material properties are assigned, we assume that the conductors are equipotential (a safe assumption, since the conductivity difference between the conductor and the dielectric will generally be near 20 orders of magnitude) and set up the physics by applying V_{0} to the inner conductor and grounding the outer conductor to solve for the electric potential in the dielectric. The above analytical equation for capacitance comes from the following more general equations
where the first equation is from electromagnetic theory and the second from circuit theory.
The first and second equations are combined to obtain the third equation. By inserting the known fields from above, we obtain the previous analytical result for C in a coaxial cable. More generally, these equations provide us with a method for obtaining the capacitance from the fields for any cable. From the simulation, we can compute the integral of the electric energy density, which gives us a capacitance of 98.142 pF/m and matches with theory. Since G and C are related by the equation
we now have two of the four parameters.
At this point, it is also worth reiterating that we have assumed that the conductivity of the dielectric region is zero. This is typically done in the textbook derivation, and we have maintained that convention here because it does not significantly impact the physics — unlike our inclusion of the internal inductance term discussed earlier. Many dielectric core materials do have a nonzero conductivity and that can be accounted for in simulation by simply updating the material properties. To ensure that proper matching with theory is maintained, the appropriate derivations would need to be updated as well.
In a similar fashion, the series resistance and inductance can be calculated through simulation using the AC/DC Module’s Magnetic Fields interface. The simulation setup is straightforward, as demonstrated in the figure below.
The conductor domains are added to a SingleTurn Coil node with the Coil Group feature, and the reversed current direction option ensures that the direction of current through the inner conductor is the opposite of the outer conductor, as indicated by the dots and crosses. The singleturn coil will account for the frequency dependence of the current distribution in the conductors, as opposed to the arbitrary distribution shown in the figure.
We refer to the following equations, which are the magnetic analog of the previous equations, to calculate the inductance.
To calculate the resistance, we use a slightly different technique. First, we integrate the resistive loss to determine the power dissipation per unit length. We can then use the familiar to calculate the resistance. Since R and L vary with frequency, let’s take a look at the calculated values and the analytical solutions in the DC and highfrequency (HF) limit.
“Analytic (DC)” and “Analytic (HF)” refer to the analytical equations in the DC and highfrequency limits, respectively, which were discussed earlier. Note that these are both on loglog plots.
We can clearly see that the computed values transition smoothly from the DC solution at low frequencies to the highfrequency solution, which is valid when the skin depth is much smaller than the thickness of the conductor. We anticipate that the transition region will be approximately located where the skin depth and conductor thickness are within one order of magnitude. This range is 4.2e3 Hz to 4.2e7 Hz, which is exactly what we see in the results.
Now that we have completed the heavy lifting to calculate R, L, C, and G, there are two other significant parameters that can be determined. They are the characteristic impedance (Z_{c}) and complex propagation constant (), where is the attenuation constant and is the propagation constant.
In the figure below, we see these values calculated using the analytical formulas for both the DC and highfrequency regime as well as the values determined from our simulation. We have also included a fourth line: the impedance calculated using COMSOL Multiphysics and the RF Module, which we will discuss shortly. As can be seen, our computations agree with the analytical solutions in their respective limits, as well as yielding the correct values through the transition region.
A comparison of the characteristic impedance, determined using the analytical equations and COMSOL Multiphysics. The analytical equations plotted are from the DC and highfrequency (HF) equations discussed earlier, while the COMSOL Multiphysics results use the AC/DC and RF Modules. For clarity, the width of the “RF Module” line has been intentionally increased.
Electromagnetic energy travels as waves, which means that the frequency of operation and wavelength are inversely proportional. As we continue to solve at higher and higher frequencies, we need to be aware of the relative size of the wavelength and electrical size of the cable. As discussed in a previous blog post, we should switch from the AC/DC to RF Module at an electrical size of approximately λ/100. If we use the cable diameter as the electrical size and the speed of light inside the dielectric core of the cable, this yields a transition frequency of approximately 690 MHz.
At these higher frequencies, the cable is more appropriately treated as a waveguide and the cable excitation as a waveguide mode. Using waveguide terminology, the mode we have been examining is a special type of mode called TEM that can propagate at any frequency. When the cross section and wavelength are comparable, we also need to account for the possibility of higherorder modes. Unlike a TEM mode, most waveguide modes can only propagate above a characteristic cutoff frequency. Due to the cylindrical symmetry in our example model, there is an equation for the cutoff frequency of the first higherorder mode, which is a TE11 mode. This cutoff frequency is f_{c} = 35.3 GHz, but even with the relatively simple geometry, the cutoff frequency comes from a transcendental equation that we will not examine further in this post.
So what does this cutoff frequency mean for our results? Above that frequency, the energy carried in the TEM mode that we are interested in has the potential to couple to the TE11 mode. In a perfect geometry, like we have simulated here, there will be no coupling. In the real world, however, any imperfections in the cable could cause mode coupling above the cutoff frequency. This could result from a number of sources, from fabrication tolerances to gradients in the material properties. Such a situation is often avoided by designing cables to operate below the cutoff frequency of higherorder modes so that only one mode can propagate. If that is of interest, you can also use COMSOL Multiphysics to simulate the coupling between higherorder modes, as with this Directional Coupler tutorial model (although beyond the scope of today’s post).
Simulation of higherorder modes is ideally suited for a Mode Analysis study using the RF or Wave Optics modules. This is because the governing equation is , which is exactly the form that we are interested in. As a result, Mode Analysis will directly solve for the spatial field and complex propagation constant for a predefined number of modes. We can use the same geometry as before, except that we only need to simulate the dielectric core and can use an Impedance boundary condition for the metal conductor.
The results for the attenuation constant and effective mode index from a Mode Analysis. The analytic line in the left plot, “Attenuation Constant vs Frequency”, is computed using the same equations as the highfrequency (HF) lines used for comparison with the results of the AC/DC Module simulations. The analytic line in the right plot, “Effective Refractive Index vs Frequency”, is simply . For clarity, the size of the “COMSOL — TEM” lines has been intentionally increased in both plots.
We can clearly see that the Mode Analysis results of the TEM mode match the analytic theory, and that the computed higherorder mode has its onset at the previously determined cutoff frequency. It is also incredibly convenient that the complex propagation constant is a direct output of this simulation and does not require calculations of R, L, C, and G. This is because is explicitly included and solved for in the Mode Analysis governing equation. These other parameters can be calculated for the TEM mode, if desired, and more information can be found in this demonstration in the Application Gallery. It is also worth pointing out that this same Mode Analysis technique can be used for dielectric waveguides, like fiber optics.
At this point, we have thoroughly analyzed a coaxial cable. We have calculated the distributed parameters from the DC to highfrequency limit and examined the first higherorder mode. Importantly, the Mode Analysis results only depend on the geometry and material properties of the cable. The AC/DC results require the additional knowledge of how the cable is excited, but hopefully you know what you’re attaching your cable to! We used analytic theory solely to compare our simulation results against a wellknown benchmark model. This means that the analysis could be extended to other cables, as well as coupled to multiphysics simulations that include temperature change and structural deformation.
For those of you who are interested in the fine details, here are a few extra points in the form of hypothetical questions.
The following texts were referred to during the writing of this post and are excellent sources of additional information: