When characterizing electronic devices radiating electromagnetic waves, it is important to make sure that the radiated waves are not returning to the device under test (DUT). Infinite space without surrounding objects is ideal. Such an environment hides effects from reflection (i.e., multipath fading) that will cause a phase distortion when the reflected waves are added to the original wave. The closest equivalent to this setting on Earth is an open field, though there is still a significant effect from the ground.

*An antenna in the middle of an open field. Image by Dr Patty McAlpin, via Wikimedia Commons.*

If we know the exact spatial configuration between a transmitter and a receiver and are sure that the ground is the only object distorting the waves, we can remove the unwanted signal path using a time-gating feature with a network analyzer. It is, however, not ideal to have to haul heavy equipment over to the open field every time you need to take measurements. Instead, it would be more convenient if you had access to a lab providing effectively infinite space — that is, an *anechoic chamber*. The anechoic chamber wall absorbs incident waves and does not interfere with the DUT.

*Antenna measurement in an anechoic chamber. Image by Max Alexander / PromoMadrid, via Wikimedia Commons.*

In an earlier blog post, we demonstrated how to design microwave absorbers using COMSOL Multiphysics and the RF Module. The pyramidal shape of periodic lossy structures gradually attenuates incident waves and generates almost no reflection, making the chamber an interference-free environment.

So, can we use these absorbers to simulate an antenna in the anechoic chamber? Of course!

*A conventional microwave absorber used in an anechoic chamber.*

The geometry of the original pyramidal object is extended to adjust the operating frequency of the absorber for its use with a biconical antenna tuned for the UHF band. The size of the pyramidal object is proportional to the wavelength of interest for the measurement.

The steps for building a model of an anechoic chamber are much like those for building a real-life chamber. We begin by creating an empty room that is 3.9 meters by 3.9 meters by 3.2 meters. The outer wall is covered by a perfect electric conductor mimicking a conductive coating that is thick enough to block all incoming signals from outside the chamber. Absorbers are added on six sides of the walls.

At the center of the chamber, we place our tutorial model of a biconical antenna. Our findings show that the antenna’s performance is very similar to the results found in the example from our Application Gallery. The figure below offers a beautiful visualization of the contours of the magnitude of the electric field.

*Simulation of a biconical antenna in an anechoic chamber.*

Due to the chamber’s complicated geometry and size, this simulation requires more than 16 GB of memory. As we will demonstrate next, there is a way to simplify this process.

My colleague Walter Frei previously highlighted different approaches for modeling a domain with open boundaries — in particular, perfectly matched layers and scattering boundary conditions. Using perfectly matched layers (PMLs), we can create the perfect anechoic chamber within the simulation environment.

*The frames of a biconical antenna are modeled as boundaries. The surrounding air domain and perfectly matched layers are required for the simulation. Only half of the PMLs are shown in this figure.*

For this example, the operating frequency is in the conventional VHF range, which extends from 60 MHz to 240 MHz. To simplify modeling steps and reduce the required computational resources, we assume that the antenna frame structure is geometrically flat and very thin. Because the thickness is greater than the skin depth in the given frequency range, it is reasonable to model the structure as a perfect electric conductor.

A lumped port with a 50 Ω reference impedance is assigned to the gap located at the center of the two structures composed of hexagonal frames. The antenna is enclosed by a spherical air domain. The outermost layers of the air domain are configured as PMLs that absorb all outgoing radiation from the antenna and work as an anechoic chamber during the simulation.

*Electric field distribution on the* yz*-plane in dB at 70 MHz. The electric field is resonant over the entire antenna structure.*

*Voltage standing wave ratio (VSWR) plot with a log scale on the* y*-axis. It presents a VSWR of approximately 3:1 on average.*

The figure above illustrates the electric field distribution in dB, as well as an arrow plot depicting the directional properties of the field at 70 MHz. When the frequency is in the lower range, the electric field is confined to the entire structure. As the frequency increases, the reacting area gradually decreases. Thus, the part of the antenna structure that is responsive to electromagnetic waves becomes shorter around the center of the lumped port. The computed VSWR is approximately 3:1 on average. This is close to the performance of commercial off-the-shelf products of biconical antennas for EMI/EMC measurements.

*3D far-field pattern at 70 MHz. The pattern resembles that of a typical half-wave dipole antenna.*

The 3D far-field radiation pattern shows the same omnidirectional characteristics on the H-plane. The suggested modeling configuration requires less than 2 GB of memory to compute the far-field radiation pattern and the VSWR of a biconical antenna made of lightweight hexagonal frames. Thus, it is much easier and faster to set up this model than the full anechoic chamber simulation.

- Check out these related blog posts:
- Download these tutorial models from our Application Gallery:

Whenever we want to solve a modeling problem involving Maxwell’s equations under the assumption that:

- All material properties are constant with respect to field strength
- That the fields will change sinusoidally in time at a known frequency or range of frequencies

and

we can treat the problem as *Frequency Domain*. When the electromagnetic field solutions are wave-like, such as for resonant structures, radiating structures, or any problem where the effective wavelength is comparable to the sizes of the objects we are working with, then the problem can be treated as a *wave electromagnetic* problem.

COMSOL Multiphysics has a dedicated physics interface for this type of modeling — the *Electromagnetic Waves, Frequency Domain* interface. Available in the RF and Wave Optics modules, it uses the finite element method to solve the frequency domain form of Maxwell’s equations. Here’s a guide for when to use this interface:

The wave electromagnetic modeling approach is valid in the regime where the object sizes range from approximately \lambda/100 to 10 \lambda, regardless of the absolute frequency. Below this size, the Low Frequency regime is appropriate. In the Low Frequency regime, the object will not be acting as an antenna or resonant structure. If you want to build models in this regime, there are several different modules and interfaces that you could use. For details, please see this blog post.

The upper limit of \sim 10 \lambda comes from the memory requirements for solving large 3D models. Once your modeling domain size is greater than \sim 10\lambda in each direction, corresponding to a domain size of (10\lambda)^3 or 1000 cubic wavelengths, you will start to need significant computational resources to solve your models. For more details about this, please see this previous blog post. On the other hand, 2D models have far more modest memory requirements and can solve much larger problems.

For problems where the objects being modeled are much larger than the wavelength, there are two options:

- The beam envelopes formulation is appropriate if the device being simulated has relatively gradual variations in the structure — and magnitude of the electromagnetic fields — in the direction of beam propagation compared to the transverse directions. For details about this, please see this post.
- The Ray Optics Module formulation treats light as rays rather than waves. In terms of the above plot, there is a wide region of overlap between these two regimes. For an introduction to the ray optics approach, please see our introduction to the Ray Optics Module.

If you are interested in X-ray frequencies and above, then the electromagnetic wave will interact with and scatter from the atomic lattice of materials. This type of scattering is not appropriate to model with the wave electromagnetics approach, since it is assumed that within each modeling domain the material can be treated as a continuum.

So now that we understand what is meant by wave electromagnetics problems, let’s further classify the most common application areas of the *Electromagnetic Waves, Frequency Domain* interface and look at some examples of its usage. We will only look at a few representative examples here that are good starting points for learning the software. These applications are selected from the RF Module Application Library and online Application Gallery and the Wave Optics Module Application Library, as well as online.

An antenna is any device that radiates electromagnetic radiation for the purposes of signal (and sometimes power) transmission. There is an almost infinite number of ways to construct an antenna, but one of the simplest is a dipole antenna. On the other hand, a patch antenna is more compact and used in many applications. Quantities of interest include the S-parameters, antenna impedance, losses, and far-field patterns, as well as the interactions of the radiated fields with any surrounding structures, as seen in our Car Windshield Antenna Effect on a Cable Harness tutorial model.

Whereas an antenna radiates into free space, waveguides and transmission lines guide the electromagnetic wave along a predefined path. It is possible to compute the impedance of transmission lines and the propagation constants and S-parameters of both microwave and optical waveguides.

Rather than transmitting energy, a resonant cavity is a structure designed to store electromagnetic energy of a particular frequency within a small space. Such structures can be either closed cavities, such as a metallic enclosure, or an open structure like an RF coil or Fabry-Perot cavity. Quantities of interest include the resonant frequency and the Q-factor.

Conceptually speaking, the combination of a waveguide with a resonant structure results in a filter or coupler. Filters are meant to either prevent or allow certain frequencies propagating through a structure and couplers are meant to allow certain frequencies to pass from one waveguide to another. A microwave filter can be as simple as a series of connected rectangular cavities, as seen in our Waveguide Iris Bandpass Filter tutorial model.

A scattering problem can be thought of as the opposite of an antenna problem. Rather than finding the radiated field from an object, an object is modeled in a background field coming from a source outside of the modeling domain. The far-field scattering of the electromagnetic wave by the object is computed, as demonstrated in the benchmark example of a perfectly conducting sphere in a plane wave.

Some electromagnetics problems can be greatly simplified in complexity if it can be assumed that the structure is quasi-infinite. For example, it is possible to compute the band structure of a photonic crystal by considering a single unit cell. Structures that are periodic in one or two directions such as gratings and frequency selective surfaces can also be analyzed for their reflection and transmission.

Whenever there is a significant amount of power transmitted via radiation, any object that interacts with the electromagnetic waves can heat up. The microwave oven in your kitchen is a perfect example of where you would need to model the coupling between electromagnetic fields and heat transfer. Another good introductory example is RF heating, where the transient temperature rises and temperature-dependent material properties are considered.

Applying a large DC magnetic bias to a ferrimagnetic material results in a relative permeability that is anisotropic for small (with respect to the DC bias) AC fields. Such materials can be used in microwave circulators. The nonreciprocal behavior of the material provides isolation.

You should now have a general overview of the capabilities and applications of the RF and Wave Optics modules for frequency domain wave electromagnetics problems. The examples listed above, as well as the other examples in the Application Gallery, are a great starting point for learning to use the software, since they come with documentation and step-by-step modeling instructions.

Please also keep in mind that the RF and Wave Optics modules also include other functionality and formulations not described here, including transient electromagnetic wave interfaces for modeling of material nonlinearities, such as second harmonic generation and modeling of signal propagation time. The RF Module additionally includes a circuit modeling tool for connecting a finite element model of a system to a circuit model, as well as an interface for modeling the transmission line equations.

As you delve deeper into COMSOL Multiphysics and wave electromagnetics modeling, please also read our other blog posts on meshing and solving options; various material models that you are able to use; as well as the boundary conditions available for modeling metallic objects, waveguide ports, and open boundaries. These posts will provide you with the foundation you need to model wave electromagnetics problems with confidence.

If you have any questions about the capabilities of using COMSOL Multiphysics for wave electromagnetics and how it can be used for your modeling needs, please contact us.

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A number of tech-savvy consumers have already configured their homes to some degree of automation, such as by using mobile apps to start their dishwashers or adjust their thermostats. A relatively new phenomenon, the *Internet of Things* (IoT), also called the “Industrial Internet” or even the “Internet of Everything”, takes these simple automated tasks to the next level by enabling objects to communicate data with connected devices.

Eventually, IoT as we know it will progress from simple home and building automation to more advanced application areas. In the ideal Internet of Things, mobile devices will be able to collect and interpret data such as your location and known preferences, and communicate with “smart objects” — without requiring your input at all.

The wide range of possible future applications for the Internet of Things include:

- Media
- While driving past a billboard on the highway or watching a commercial on television, data collected by your devices shows that you like the type of product being advertised and you are automatically messaged with more information.

- Transportation
- Not only can your car parallel park, but it can navigate and drive on its own. Similar improvements to trains and aircraft are just a few of the ways that IoT can help us get around as technology progresses.

- Healthcare
- Medical devices automatically administer medications and monitor patients’ conditions as well as their overall well-being. Pacemakers, hearing aids, and heart monitors streamline medical care by communicating with doctors and patients.

- Environment and energy conservation
- Sensors optimize energy consumption by powering lights and electronics based on a person’s activity. On a larger scale, an improved IoT helps monitor water and air quality, among other environmental concerns.

- Infrastructure
- IoT sensors monitor the structural stability of bridges, railways, and waste management systems for safety and security.

With the wide use of smartphones in our society, the Internet of Things has an easy way to collect and use our personal data to communicate with objects and smart devices. But to get a jump-start on making IoT possible, we need to create the next generation of mobile technology to optimize smartphones.

*5G* is the chosen moniker for the next generation of wireless communication, of which a new version is released about every ten years. 5G is currently just a concept, but it is expected to be implemented by 2020 — giving the mobile industry a lot of work to do in a short amount of time.

Most wireless communication professionals agree that when 5G replaces 4G LTE, it should address three key needs:

- A decreased latency of under one second.
- Increased data rates of at least one gigabit per second for tens of thousands of users simultaneously.
- Increased energy efficiency.

Whatever improvements are made to wireless communication by the release of 5G, the main goal is for mobile technology, data collection, and wireless communication to be more seamlessly integrated through speed and efficiency. Without these features, the Internet of Things won’t work correctly and will be redundant.

Among the many developments researchers from across the globe are already working on for 5G, the optimization of mobile device antennas is an important topic to study. Though 5G applications have not been standardized yet and many researchers are developing a range of devices to expand the world of IoT, we can start by looking at a basic introductory model showing how to design a small antenna in a mobile device in our Modeling of a Mobile Device Antenna tutorial.

A mobile device antenna must be small and lightweight enough to fit in the limited amount of space allotted for it in a smartphone’s design. Planar inverted-F antennas (PIFA) are a good choice for mobile communication because they are small, powerful, and efficient. These antennas can cover multiple frequency bands for cellular devices, WiFi, and Bluetooth® technology — which makes them a great choice for IoT compatible objects and devices.

The mobile device antenna simulated in this tutorial includes a 4G device made up of a PIFA on a PTFE block with an FR4 circuit board, ABS housing, and glass with a composite silicon substrate. The antenna itself is made up of the PFTE block with a thin copper layer for high conductivity, a lumped port between a perfect electric conductor (PEC) ground plane and feeding strip, as well as another strip shorted to the ground plane and adjacent to the feeding strip for impedance matching purposes. It also includes an impedance matching gap that matches the antenna to the reference impedance of 50 Ω.

*Model geometry of a planar inverted-F antenna in a mobile device.*

For the simulation, this antenna can be modeled using PEC boundaries because of the low downlink frequency range. The losses on the metal are inconsequential due to the high conductivity of the copper layer. The PIFA is modeled in a spherical domain that is enclosed by perfectly matched layers (PML) to absorb its outgoing radiation. The lumped port, with a reference impedance of 50 Ω, is used to excite the PIFA and evaluate its input impedance.

Through simulation, we are able to calculate the field distribution plot for the PIFA. Results show that the field is strong at one end of the metallic surface at the top of the model, far from the feeding strip. These measurements actually resemble those of a quarter wave monopole antenna, a design from which the PIFA is derived.

*Results plot of the electric field distribution at the top of the PIFA.*

The simulation also calculates the polar-formatted far-field radiation pattern. The azimuthal radiation pattern is no longer omnidirectional since the antenna is now miniaturized and located on only one corner of the ground plane.

From the S-parameters, we can see that the voltage standing wave ratio (VSWR) is less than 2:1. This means that the antenna input impedance is well matched to the reference impedance, which is a typical measurement in network analyzers and other common measurement systems.

*The S-parameters of the given AWS downlink frequency range are calculated.*

Going beyond the results of 2D far-field calculations, you are also able to review the simulation in a 3D radiation pattern to show maximum radiation and null.

*The far-field radiation pattern of the PIFA plotted in 3D.*

To address 5G applications, there are many developments to be considered above the introductory model. In order to handle higher data rates, the operating frequency has to be increased to a millimeter range from which we can achieve a wider bandwidth. This will result in higher path loss between transmitters and receivers, so antennas need to provide higher gain to reach a longer distance.

However, this will significantly reduce the covering range in terms of angle because the radiation pattern will be very sharp. Consequently, phased array antennas are required to get over the limit of angular dependency of high-gain antennas using the ability to steer a radiation beam toward wanted directions.

By optimizing the design and performance of mobile device antennas, including those just mentioned, the ideal Internet of Things will be here before we know it, and we’ll be ready to embrace the new technology.

- Download the tutorial: Modeling of a Mobile Device Antenna
*Wired*article: “In the Programmable World, All Our Objects Will Act As One“*Tech Republic*article: “The race to 5G: Inside the fight for the future of mobile as we know it“

While many different types of laser light sources exist, they are all quite similar in terms of their outputs. Laser light is very nearly single frequency (single wavelength) and coherent. Typically, the output of a laser is also focused into a narrow collimated beam. This collimated, coherent, and single frequency light source can be used as a very precise heat source in a wide range of applications, including cancer treatment, welding, annealing, material research, and semiconductor processing.

When laser light hits a solid material, part of the energy is absorbed, leading to localized heating. Liquids and gases (and plasmas), of course, can also be heated by lasers, but the heating of fluids almost always leads to significant convective effects. Within this blog post, we will neglect convection and concern ourselves only with the heating of solid materials.

Solid materials can be either partially transparent or completely opaque to light at the laser wavelength. Depending upon the degree of transparency, different approaches for modeling the laser heat source are appropriate. Additionally, we must concern ourselves with the relative scale as compared to the wavelength of light. If the laser is very tightly focused, then a different approach is needed compared to a relatively wide beam. If the material interacting with the beam has geometric features that are comparable to the wavelength, we must additionally consider exactly how the beam will interact with these small structures.

Before starting to model any laser-material interactions, you should first determine the optical properties of the material that you are modeling, both at the laser wavelength and in the infrared regime. You should also know the relative sizes of the objects you want to heat, as well as the laser wavelength and beam characteristics. This information will be useful in guiding you toward the appropriate approach for your modeling needs.

In cases where the material is opaque, or very nearly so, at the laser wavelength, it is appropriate to treat the laser as a surface heat source. This is most easily done with the *Deposited Beam Power* feature (shown below), which is available with the Heat Transfer Module as of COMSOL Multiphysics version 5.1. It is, however, also quite easy to manually set up such a surface heat load using only the COMSOL Multiphysics core package, as shown in the example here.

A surface heat source assumes that the energy in the beam is absorbed over a negligibly small distance into the material relative to the size of the object that is heated. The finite element mesh only needs to be fine enough to resolve the temperature fields as well as the laser spot size. The laser itself is not explicitly modeled, and it is assumed that the fraction of laser light that is reflected off the material is never reflected back. When using a surface heat load, you must manually account for the absorptivity of the material at the laser wavelength and scale the deposited beam power appropriately.

*The Deposited Beam Power feature in the Heat Transfer Module is used to model two crossed laser beams. The resultant surface heat source is shown.*

In cases where the material is partially transparent, the laser power will be deposited within the domain, rather than at the surface, and any of the different approaches may be appropriate based on the relative geometric sizes and the wavelength.

If the heated objects are much larger than the wavelength, but the laser light itself is converging and diverging through a series of optical elements and is possibly reflected by mirrors, then the functionality in the Ray Optics Module is the best option. In this approach, light is treated as a ray that is traced through homogeneous, inhomogeneous, and lossy materials.

As the light passes through lossy materials (e.g., optical glasses) and strikes surfaces, some power deposition will heat up the material. The absorption within domains is modeled via a complex-valued refractive index. At surfaces, you can use a reflection or an absorption coefficient. Any of these properties can be temperature dependent. For those interested in using this approach, this tutorial model from our Application Gallery provides a great starting point.

*A laser beam focused through two lenses. The lenses heat up due to the high-intensity laser light, shifting the focal point.*

If the heated objects and the spot size of the laser are much larger than the wavelength, then it is appropriate to use the Beer-Lambert law to model the absorption of the light within the material. This approach assumes that the laser light beam is perfectly parallel and unidirectional.

When using the Beer-Lambert law approach, the absorption coefficient of the material and reflection at the material surface must be known. Both of these material properties can be functions of temperature. The appropriate way to set up such a model is described in our earlier blog entry “Modeling Laser-Material Interactions with the Beer-Lambert Law“.

You can use the Beer-Lambert law approach if you know the incident laser intensity and if there are no reflections of the light within the material or at the boundaries.

*Laser heating of a semitransparent solid modeled with the Beer-Lambert law.*

If the heated domain is large, but the laser beam is tightly focused within it, neither the ray optics nor the Beer-Lambert law modeling approach can accurately solve for the fields and losses near the focus. These techniques do not directly solve Maxwell’s equations, but instead treat light as rays. The beam envelope method, available within the Wave Optics Module, is the most appropriate choice in this case.

The beam envelope method solves the full Maxwell’s equations when the field envelope is slowly varying. The approach is appropriate if the wave vector is approximately known throughout the modeling domain and whenever you know approximately the direction in which light is traveling. This is the case when modeling a focused laser light as well as waveguide structures like a Mach-Zehnder modulator or a ring resonator. Since the beam direction is known, the finite element mesh can be very coarse in the propagation direction, thereby reducing computational costs.

*A laser beam focused in a cylindrical material domain. The intensity at the incident side and within the material are plotted, along with the mesh.*

The beam envelope method can be combined with the *Heat Transfer in Solids* interface via the *Electromagnetic Heat Source* multiphysics couplings. These couplings are automatically set up when you add the *Laser Heating* interface under *Add Physics*.

*The* Laser Heating *interface adds the* Beam Envelopes *and the* Heat Transfer in Solids *interfaces and the multiphysics couplings between them.*

Finally, if the heated structure has dimensions comparable to the wavelength, it is necessary to solve the full Maxwell’s equations without assuming any propagation direction of the laser light within the modeling space. Here, we need to use the *Electromagnetic Waves, Frequency Domain* interface, which is available in both the Wave Optics Module and the RF Module. Additionally, the RF Module offers a *Microwave Heating* interface (similar to the *Laser Heating* interface described above) and couples the *Electromagnetic Waves, Frequency Domain* interface to the *Heat Transfer in Solids* interface. Despite the nomenclature, the RF Module and the *Microwave Heating* interface are appropriate over a wide frequency band.

The full-wave approach requires a finite element mesh that is fine enough to resolve the wavelength of the laser light. Since the beam may scatter in all directions, the mesh must be reasonably uniform in size. A good example of using the *Electromagnetic Waves, Frequency Domain* interface: Modeling the losses in a gold nanosphere illuminated by a plane wave, as illustrated below.

*Laser light heating a gold nanosphere. The losses in the sphere and the surrounding electric field magnitude are plotted, along with the mesh.*

You can use any of the previous five approaches to model the power deposition from a laser source in a solid material. Modeling the temperature rise and heat flux within and around the material additionally requires the *Heat Transfer in Solids* interface. Available in the core COMSOL Multiphysics package, this interface is suitable for modeling heat transfer in solids and features fixed temperature, insulating, and heat flux boundary conditions. The interface also includes various boundary conditions for modeling convective heat transfer to the surrounding atmosphere or fluid, as well as modeling radiative cooling to ambient at a known temperature.

In some cases, you may expect that there is also a fluid that provides significant heating or cooling to the problem and cannot be approximated with a boundary condition. For this, you will want to explicitly model the fluid flow using the Heat Transfer Module or the CFD Module, which can solve for both the temperature and flow fields. Both modules can solve for laminar and turbulent fluid flow. The CFD Module, however, has certain additional turbulent flow modeling capabilities, which are described in detail in this previous blog post.

For instances where you are expecting significant radiation between the heated object and any surrounding objects at varying temperatures, the Heat Transfer Module has the additional ability to compute gray body radiative view factors and radiative heat transfer. This is demonstrated in our Rapid Thermal Annealing tutorial model. When you expect the temperature variations to be significant, you may also need to consider the wavelength-dependent surface emissivity.

If the materials under consideration are transparent to laser light, it is likely that they are also partially transparent to thermal (infrared-band) radiation. This infrared light will be neither coherent nor collimated, so we cannot use any of the above approaches to describe the reradiation within semitransparent media. Instead, we can use the radiation in participating media approach. This technique is suitable for modeling heat transfer within a material, where there is significant heat flux inside the material due to radiation. An example of this approach from our Application Gallery can be found here.

In this blog post, we have looked at the various modeling techniques available in the COMSOL Multiphysics environment for modeling the laser heating of a solid material. Surface heating and volumetric heating approaches are presented, along with a brief overview of the heat transfer modeling capabilities. Thus far, we have only considered the heating of a solid material that does not change phase. The heating of liquids and gases — and the modeling of phase change — will be covered in a future blog post. Stay tuned!

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COMSOL Multiphysics uses the finite element method to solve for the electromagnetic fields within the modeling domains. Under the assumption that the fields vary sinusoidally in time at a known angular frequency \omega = 2 \pi f and that all material properties are linear with respect to field strength, the governing Maxwell’s equations in three dimensions reduce to:

\nabla \times \left( \mu_r^{-1} \nabla \times \mathbf{E} \right)-\frac{\omega^2}{c_0^2} \left( \epsilon_r -\frac{i \sigma}{\omega \epsilon_0} \right) \mathbf{E}= 0

where the material properties are \mu_r, the relative permeability; \epsilon_r, the relative permittivity; and \sigma , the electrical conductivity.

With the speed of light in vacuum, c_0, the above equation is solved for the electric field, \mathbf{E}=\mathbf{E}(x,y,z), throughout the modeling domain, where \mathbf{E} is a vector with components \mathbf{E}=. All other quantities (such as magnetic fields, currents, and power flow) can be derived from the electric field. It is also possible to reformulate the above equation as an eigenvalue problem, where a model is solved for the resonant frequencies of the system, rather than the response of the system at a particular frequency.

The above equation is solved via the finite element method. For a conceptual introduction to this method, please see our blog series on the weak form, and for a more in-depth reference, which will explain the nuances related to electromagnetic wave problems, please see *The Finite Element Method in Electromagnetics* by Jian-Ming Jin. From the point of view of this blog post, however, we can break down the finite element method into these four steps:

**Model Set-Up:**Defining the equations to solve, creating the model geometry, defining the material properties, setting up metallic and radiating boundaries, and connecting the model to other devices.**Meshing:**Discretizing the model space using finite elements.**Solving:**Solving a set of linear equations that describe the electric fields.**Postprocessing:**Extracting useful information from the computed electric fields.

Let’s now look at each one of these steps in more detail and describe the options available at each step.

The governing equation shown above is the frequency domain form of Maxwell’s equations for wave-type problems in its most general form. However, this equation can be reformulated for several special cases.

Let us first consider the case of a modeling domain in which there is a known background electric field and we wish to place some object into this background field. The background field can be a linearly polarized plane wave, a Gaussian beam, or any general user-defined beam that satisfies Maxwell’s equations in free space. Placing an object into this field will perturb the field and lead to scattering of the background field. In such a situation, you can use the *Scattered Field* formulation, which solves the above equation, but makes the following substitution for the electric field:

\mathbf{E} = \mathbf{E}_{relative} + \mathbf{E}_{background}

where the background electric field is known and the relative field is the field that, once added to the background field, gives the total field that satisfies the governing Maxwell’s equations. Rather than solving for the total field, it is the relative field that is being solved. Note that the relative field is *not* the scattered field.

For an example of the usage of this *Scattered Field* formulation, which considers the radar scattering off of a perfectly electrically conductive sphere in a background plane wave and compares it to the analytic solution, please see our Computing the Radar Cross Section of a Perfectly Conducting Sphere tutorial model.

Next, let’s consider modeling in a 2D plane, where we solve for \mathbf{E}=\mathbf{E}(x,y) and can additionally simplify the modeling by considering an electric field that is polarized either In-Plane or Out-of-Plane. The In-Plane case will assume that E_z=0, while the Out-of-Plane case assumes that E_x=E_y=0. These simplifications reduce the size of the problem being solved, compared to solving for all three components of the electric field vector.

For modeling in the 2D axisymmetric plane, we solve for \mathbf{E}=\mathbf{E}(r,z), where the vector \mathbf{E} has the components < E_r, E_\phi, E_z> and we can again simplify our modeling by considering the In-Plane and Out-of-Plane cases, which assume E_\phi=0 and E_r=E_z=0, respectively.

When using either the *2D* or the *2D axisymmetric In-Plane* formulations, it is also possible to specify an *Out-of-Plane Wave Number*. This is appropriate to use when there is a known out-of-plane propagation constant, or known number of azimuthal modes. For 2D problems, the electric field can be rewritten as:

\mathbf{E}(x,y,z)= \mathbf{\tilde E}(x,y)exp(-i k_z z)

and for 2D axisymmetric problems, the electric field can be rewritten as:

\mathbf{E}(r,\phi,z)= \mathbf{\tilde E}(r,z)exp(-i m \phi)

where k_z or m, the out-of-plane wave number, must be specified.

This modeling approach can greatly simplify the computational complexity for some types of models. For example, a structurally axisymmetric horn antenna will have a solution that varies in 3D but is composed of a sum of known azimuthal modes. It is possible to recover the 3D solution from a set of 2D axisymmetric analyses by solving for these out-of-plane modes at a much lower computational cost, as demonstrated in our Corrugated Circular Horn Antenna tutorial model.

Whenever solving a wave electromagnetics problem, you must keep in mind the mesh resolution. Any wave-type problem must have a mesh that is fine enough to resolve the wavelengths in all media being modeled. This idea is fundamentally similar to the concept of the *Nyquist frequency* in signal processing: The sampling size (the finite element mesh size) must be at least less than one-half of the wavelength being resolved.

By default, COMSOL Multiphysics uses second-order elements to discretize the governing equations. A minimum of two elements per wavelength are necessary to solve the problem, but such a coarse mesh would give quite poor accuracy. At least five second-order elements per wavelength are typically used to resolve a wave propagating through a dielectric medium. First-order and third-order discretization is also available, but these are generally of more academic interest, since the second-order elements tend to be the best compromise between accuracy and memory requirements.

The meshing of domains to fulfill the minimum criterion of five elements per wavelength in each medium is now automated within the software, as shown in this video, which shows not only the meshing of different dielectric domains, but also the automated meshing of Perfectly Matched Layer domains. The new automated meshing capability will also set up an appropriate periodic mesh for problems with periodic boundary conditions, as demonstrated in this Frequency Selective Surface, Periodic Complementary Split Ring Resonator tutorial model.

With respect to the type of elements used, tetrahedral (in 3D) or triangular (in 2D) elements are preferred over hexahedral and prismatic (in 3D) or rectangular (in 2D) elements due to their lower dispersion error. This is a consequence of the fact that the maximum distance within an element is approximately the same in all directions for a tetrahedral element, but for a hexahedral element, the ratio of the shortest to the longest line that fits within a perfect cubic element is \sqrt3. This leads to greater error when resolving the phase of a wave traveling diagonally through a hexahedral element.

It is only necessary to use hexahedral, prismatic, or rectangular elements when you are meshing a perfectly matched layer or have some foreknowledge that the solution is strongly anisotropic in one or two directions. When resolving a wave that is decaying due to absorption in a material, such as a wave impinging upon a lossy medium, it is additionally necessary to manually resolve the skin depth with the finite element mesh, typically using a boundary layer mesh, as described here.

Manual meshing is still recommended, and usually needed, for cases when the material properties will vary during the simulation. For example, during an electromagnetic heating simulation, the material properties can be made functions of temperature. This possible variation in material properties should be considered before the solution, during the meshing step, as it is often more computationally expensive to remesh during the solution than to start with a mesh that is fine enough to resolve the eventual variations in the fields. This can require a manual and iterative approach to meshing and solving.

When solving over a wide frequency band, you can consider one of three options:

- Solve over the entire frequency range using a mesh that will resolve the shortest wavelength (highest frequency) case. This avoids any computational cost associated with remeshing, but you will use an overly fine mesh for the lower frequencies.
- Remesh at each frequency, using the parametric solver. This is an attractive option if your increments in frequency space are quite widely spaced, and if the meshing cost is relatively low.
- Use different meshes in different frequency bands. This will reduce the meshing cost, and keep the solution cost relatively low. It is essentially a combination of the above two approaches, but requires the most user effort.

It is difficult to determine ahead of time which of the above three options will be the most efficient for a particular model.

Regardless of the initial mesh that you use, you will also always want to perform a mesh refinement study. That is, re-run the simulation with progressively finer meshes and observe how the solution changes. As you make the mesh finer, the solution will become more accurate, but at a greater computational cost. It is also possible to use adaptive mesh refinement if your mesh is composed entirely of tetrahedral or triangular elements.

Once you have properly defined the problem and meshed your domains, COMSOL Multiphysics will take this information and form a system of linear equations, which are solved using either a direct or iterative solver. These solvers differ only in their memory requirements and solution time, but there are several options that can make your modeling more efficient, since 3D electromagnetics models will often require a lot of RAM to solve.

The direct solvers will require more memory than the iterative solvers. They are used for problems with periodic boundary conditions, eigenvalue problems, and for all 2D models. Problems with periodic boundary conditions do require the use of a direct solver, and the software will automatically do so in such cases.

Eigenvalue problems will solve faster when using a direct solver as compared to using an iterative solver, but will use more memory. For this reason, it can often be attractive to reformulate an eigenvalue problem as a frequency domain problem excited over a range of frequencies near the approximate resonances. By solving in the frequency domain, it is possible to use the more memory-efficient iterative solvers. However, for systems with high Q-factors it becomes necessary to solve at many points in frequency space. For an example of reformulating an eigenvalue problem as a frequency domain problem, please see these examples of computing the Q-factor of an RF coil and the Q-factor of a Fabry-Perot cavity.

The iterative solvers used for frequency-domain simulations come with three different options defined by the Analysis Methodology settings of *Robust* (the default), *Intermediate*, or *Fast*, and can be changed within the physics interface settings. These different settings alter the type of iterative solver being used and the convergence tolerance. Most models will solve with any of these settings, and it can be worth comparing them to observe the differences in solution time and accuracy and choose the option most appropriate for your needs. Models that contain materials that have very large contrasts in the dielectric constants (~100:1) will need the *Robust* setting and may even require the use of the direct solver, if the iterative solver convergence is very slow.

Once you’ve solved your model, you will want to extract data from the computed electromagnetic fields. COMSOL Multiphysics will automatically produce a slice plot of the magnitude of the electric field, but there are many other postprocessing visualizations you can set up. Please see the Postprocessing & Visualization Handbook and our blog series on Postprocessing for guidance and to learn how to create images such as those shown below.

*Attractive visualizations can be created by plotting combinations of the solution fields, meshes, and geometry.*

Of course, good-looking images are not enough — we also want to extract numerical information from our models. COMSOL Multiphysics will automatically make available the S-parameters whenever using Ports or Lumped Ports, as well as the Lumped Port current, voltage, power, and impedance. For a model with multiple Ports or Lumped Ports, it is also possible to automatically set up a *Port Sweep*, as demonstrated in this tutorial model of a Ferrite Circulator, and write out a Touchstone file of the results. For eigenvalue problems, the resonant frequencies and Q-factors are automatically computed.

For models of antennas or for scattered field models, it is additionally possible to compute and plot the far-field radiated pattern, the gain, and the axial ratio.

*Far-field radiation pattern of a Vivaldi antenna.*

You can also integrate any derived quantity over domains, boundaries, and edges to compute, for example, the heat dissipated inside of lossy materials or the total electromagnetic energy within a cavity. Of course, there is a great deal more that you can do, and here we have just looked at the most commonly used postprocessing features.

We’ve looked at the various different formulations of the governing frequency domain form of Maxwell’s equations as applied to solving wave electromagnetics problems and when they should be used. The meshing requirements and capabilities have been discussed as well as the options for solving your models. You should also have a broad overview of the postprocessing functionality and where to go for more information about visualizing your data in COMSOL Multiphysics.

This information, along with the previous blog posts on defining the material properties, setting up metallic and radiating boundaries, and connecting the model to other devices should now give you a reasonably complete picture of what can be done with frequency domain electromagnetic wave modeling in the RF and Wave Optics modules. The software documentation, of course, goes into greater depth about all of the features and capabilities within the software.

If you are interested in using the RF or Wave Optics modules for your modeling needs, please contact us.

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Here, we will speak about the frequency-domain form of Maxwell’s equations in the *Electromagnetic Waves, Frequency Domain* interface available in the RF Module and the Wave Optics Module. The information presented here also applies to the *Electromagnetic Waves, Beam Envelopes* formulation in the Wave Optics Module.

Under the assumption that material response is linear with field strength, we formulate Maxwell’s equations in the frequency domain, so the governing equations can be written as:

\nabla \times \left( \mu_r^{-1} \nabla \times \mathbf{E} \right)-\frac{\omega^2}{c_0^2} \left( \epsilon_r -\frac{j \sigma}{\omega \epsilon_0} \right) \mathbf{E}= 0

This equation solves for the electric field, \mathbf{E}, at the operating (angular) frequency \omega = 2 \pi f (c_0 is the speed of light in vacuum). The other inputs are the material properties \mu_r, the relative permeability; \epsilon_r, the relative permittivity; and \sigma , the electrical conductivity. All of these material inputs can be positive or negative, real or complex-valued numbers, and they can be scalar or tensor quantities. These material properties can vary as a function of frequency as well, though it is not always necessary to consider this variation if we are only looking at a relatively narrow frequency range.

Let us now explore each of these material properties in detail.

The *electrical conductivity* quantifies how well a material conducts current — it is the inverse of the electrical resistivity. The material conductivity is measured under steady-state (DC) conditions, and we can see from the above equation that as the frequency increases, the effective resistivity of the material increases. We typically assume that the conductivity is constant with frequency, and later on we will examine different models for handling materials with frequency-dependent conductivity.

Any material with non-zero conductivity will conduct current in an applied electric field and dissipate energy as a resistive loss, also called *Joule heating*. This will often lead to a measurable rise in temperature, which will alter the conductivity. You can enter any function or tabular data for variation of conductivity with temperature, and there is also a built-in model for linearized resistivity.

*Linearized Resistivity* is a commonly used model for the variation of conductivity with temperature, given by:

\sigma = \frac{1}{\rho_0 (1 + \alpha ( T-T_{ref} )) }

where \rho_0 is the reference resistivity, T_{ref} is the reference temperature, and \alpha is the resistivity temperature coefficient. The spatially-varying temperature field, T, can either be specified or computed.

Conductivity is entered as a real-valued number, but it can be anisotropic, meaning that the material’s conductivity varies in different coordinate directions. This is an appropriate approach if you have, for example, a laminated material in which you do not want to explicitly model the individual layers. You can enter a homogenized conductivity for the composite material, which would be either experimentally determined or computed from a separate analysis.

Within the RF Module, there are two other options for computing a homogenized conductivity: Archie’s Law for computing effective conductivity of non-conductive porous media filled with conductive liquid and a Porous Media model for mixtures of materials.

*Archie’s Law* is a model typically used for the modeling of soils saturated with seawater or crude oil, fluids with relatively higher conductivity compared to the soil.

*Porous Media* refers to a model that has three different options for computing an effective conductivity for a mixture of up to five materials. First, the *Volume Average, Conductivity* formulation is:

\sigma_{eff}=\sideset{}{^n_{i=1}}

\sum \theta_i \sigma_i

\sum \theta_i \sigma_i

where \theta is the volume fraction of each material. This model is appropriate if the material conductivities are similar. If the conductivities are quite different, the *Volume Average, Resistivity* formulation is more appropriate:

\frac{1}{\sigma_{eff}} = \sideset{}{^n_{i=1}}

\sum\frac{\theta_i}{ \sigma_i}

\sum\frac{\theta_i}{ \sigma_i}

Lastly, the *Power Law* formulation will give a conductivity lying between the other two formulations:

\sigma_{eff} = \sideset{}{^n_{i=1}}

\prod\sigma_i^{\theta_i }

\prod\sigma_i^{\theta_i }

These models are all only appropriate to use if the length scale over which the material properties’ change is much smaller than the wavelength.

The *relative permittivity* quantifies how well a material is polarized in response to an applied electric field. It is typical to call any material with \epsilon_r>1 a *dielectric material*, though even vacuum (\epsilon_r=1) can be called a dielectric. It is also common to use the term *dielectric constant* to refer to a material’s relative permittivity.

A material’s relative permittivity is often given as a complex-valued number, where the negative imaginary component represents the loss in the material as the electric field changes direction over time. Any material experiencing a time-varying electric field will dissipate some of the electrical energy as heat. Known as *dielectric loss*, this results from the change in shape of the electron clouds around the atoms as the electric fields change. Dielectric loss is conceptually distinct from the resistive loss discussed earlier; however, from a mathematical point of view, they are actually handled identically — as a complex-valued term in the governing equation. Keep in mind that COMSOL Multiphysics follows the convention that a negative imaginary component (a positive-valued electrical conductivity) will lead to loss, while a positive complex component (a negative-valued electrical conductivity) will lead to gain within the material.

There are seven different material models for the relative permittivity. Let’s take a look at each of these models.

*Relative Permittivity* is the default option for the RF Module. A real- or complex-valued scalar or tensor value can be entered. The same Porous Media models described above for the electrical conductivity can be used for the relative permittivity.

*Refractive Index* is the default option for the Wave Optics Module. You separately enter the real and imaginary part of the refractive index, called n and k, and the relative permittivity is \epsilon_r=(n-jk)^2. This material model assumes zero conductivity and unit relative permeability.

*Loss Tangent* involves entering a real-valued relative permittivity, \epsilon_r', and a scalar loss tangent, \delta. The relative permittivity is computed via \epsilon_r=\epsilon_r'(1-j \tan \delta), and the material conductivity is zero.

*Dielectric Loss* is the option for entering the real and imaginary components of the relative permittivity \epsilon_r=\epsilon_r'-j \epsilon_r''. Be careful to note the sign: Entering a positive-valued real number for the imaginary component \epsilon_r'' when using this interface will lead to loss, since the multiplication by -j is done within the software. For an example of the appropriate usage of this material model, please see the Optical Scattering off of a Gold Nanosphere tutorial.

The *Drude-Lorentz Dispersion* model is a material model that was developed based upon the Drude free electron model and the Lorentz oscillator model. The Drude model (when \omega_0=0) is used for metals and doped semiconductors, while the Lorentz model describes resonant phenomena such as phonon modes and interband transitions. With the sum term, the combination of these two models can accurately describe a wide array of solid materials. It predicts the frequency-dependent variation of complex relative permittivity as:

\epsilon_r=\epsilon_{\infty}+\sideset{}{^M_{k=1}}

\sum\frac{f_k\omega_p^2}{\omega_{0k}^2-\omega^2+i\Gamma_k \omega}

\sum\frac{f_k\omega_p^2}{\omega_{0k}^2-\omega^2+i\Gamma_k \omega}

where \epsilon_{\infty} is the high-frequency contribution to the relative permittivity, \omega_p is the plasma frequency, f_k is the oscillator strength, \omega_{0k} is the resonance frequency, and \Gamma_k is the damping coefficient. Since this model computes a complex-valued permittivity, the conductivity inside of COMSOL Multiphysics is set to zero. This approach is one way of modeling frequency-dependent conductivity.

The *Debye Dispersion* model is a material model that was developed by Peter Debye and is based on polarization relaxation times. The model is primarily used for polar liquids. It predicts the frequency-dependent variation of complex relative permittivity as:

\epsilon_r=\epsilon_{\infty}+\sideset{}{^M_{k=1}}

\sum\frac{\Delta \epsilon_k}{1+i\omega \tau_k}

\sum\frac{\Delta \epsilon_k}{1+i\omega \tau_k}

where \epsilon_{\infty} is the high-frequency contribution to the relative permittivity, \Delta \epsilon_k is the contribution to the relative permittivity, and \tau_k is the relaxation time. Since this model computes a complex-valued permittivity, the conductivity is assumed to be zero. This is an alternate way to model frequency-dependent conductivity.

The *Sellmeier Dispersion* model is available in the Wave Optics Module and is typically used for optical materials. It assumes zero conductivity and unit relative permeability and defines the relative permittivity in terms of the operating wavelength, \lambda, rather than frequency:

\epsilon_r=1+\sideset{}{^M_{k=1}}

\sum\frac{B_k \lambda^2}{\lambda^2-C_k}

\sum\frac{B_k \lambda^2}{\lambda^2-C_k}

where the coefficients B_k and C_k determine the relative permittivity.

The choice between these seven models will be dictated by the way the material properties are available to you in the technical literature. Keep in mind that, mathematically speaking, they enter the governing equation identically.

The *relative permeability* quantifies how a material responds to a magnetic field. Any material with \mu_r>1 is typically referred to as a magnetic material. The most common magnetic material on Earth is iron, but pure iron is rarely used for RF or optical applications. It is more typical to work with materials that are ferrimagnetic. Such materials exhibit strong magnetic properties with an anisotropy that can be controlled by an applied DC magnetic field. Opposed to iron, ferrimagnetic materials have a very low conductivity, so that high-frequency electromagnetic fields are able to penetrate into and interact with the bulk material. This tutorial demonstrates how to model ferrimagnetic materials.

There are two options available for specifying relative permeability: The *Relative Permeability* model, which is the default for the RF Module, and the *Magnetic Losses* model. The Relative Permeability model allows you to enter a real- or complex-valued scalar or tensor value. The same Porous Media models described above for the electrical conductivity can be used for the relative permeability. The Magnetic Losses model is analogous to the Dielectric Loss model described above in that you enter the real and imaginary components of the relative permeability as real-valued numbers. An imaginary-valued permeability will lead to a magnetic loss in the material.

In any electromagnetic modeling, one of the most important things to keep in mind is the concept of *skin depth*, the distance into a material over which the fields fall off to 1/e of their value at the surface. Skin depth is defined as:

\delta=\left[ \operatorname{Re} \left( \sqrt{j \omega \mu_0 \mu_r (\sigma + j \omega \epsilon_0 \epsilon_r)} \right) \right] ^{-1}

where we have seen that relative permittivity and permeability can be complex-valued.

You should always check the skin depth and compare it to the characteristic size of the domains in your model. If the skin depth is much smaller than the object, you should instead model the domain as a boundary condition as described here: “Modeling Metallic Objects in Wave Electromagnetics Problems“. If the skin depth is comparable to or larger than the object size, then the electromagnetic fields will penetrate into the object and interact significantly within the domain.

*A plane wave incident upon objects of different conductivities and hence different skin depths. When the skin depth is smaller than the wavelength, a boundary layer mesh is used (right). The electric field is plotted.*

If the skin depth is smaller than the object, it is advised to use boundary layer meshing to resolve the strong variations in the fields in the direction normal to the boundary, with a minimum of one element per skin depth and a minimum of three boundary layer elements. If the skin depth is larger than the effective wavelength in the medium, it is sufficient to resolve the wavelength in the medium itself with five elements per wavelength, as shown in the left figure above.

In this blog post, we have looked at the various options available for defining the material properties within your electromagnetic wave models in COMSOL Multiphysics. We have seen that the material models for defining the relative permittivity are appropriate even for metals over a certain frequency range. On the other hand, we can also define metal domains via boundary conditions, as previously highlighted on the blog. Along with earlier blog posts on modeling open boundary conditions and modeling ports, we have now covered almost all of the fundamentals of modeling electromagnetic waves. There are, however, a few more points that remain. Stay tuned!

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When approaching the question of what a metal is, we can do so from the point of view of the governing Maxwell’s equations that are solved for electromagnetic wave problems. Consider the frequency-domain form of Maxwell’s equations:

\nabla \times \left( \mu_r^{-1} \nabla \times \mathbf{E} \right) – {-\frac{\omega^2}{c_0^2}} \left( \epsilon_r -\frac{i \sigma}{\omega \epsilon_0} \right) \mathbf{E}= 0

The above equation is solved in the *Electromagnetic Waves, Frequency Domain* interface available in the RF Module and the Wave Optics Module. This equation solves for the electric field, \mathbf{E}, at the operating (angular) frequency \omega = 2 \pi f. The other inputs are the material properties: \mu_r is the relative permeability, \epsilon_r is the relative permittivity, and \sigma is the electrical conductivity.

For the purposes of this discussion, we will say that a metal is any material that is both lossy and has a relatively small skin depth. A *lossy material* is any material that has a complex-valued permittivity or permeability or a non-zero conductivity. That is, a lossy material introduces an imaginary-valued term into the governing equation. This will lead to electric currents within the material, and the *skin depth* is a measure of the distance into the material over which this current flows.

At any non-zero operating frequency, inductive effects will drive any current flowing in a lossy material towards the boundary. The skin depth is the distance into the material within which approximately 63% of the current flows. It is given by:

\delta=\left[ \operatorname{Re} \left( \sqrt{i \omega \mu_0 \mu_r (\sigma + i \omega \epsilon_0 \epsilon_r)} \right) \right] ^{-1}

where both \mu_r and \epsilon_r can be complex-valued.

At very high frequencies, approaching the optical regime, we are near the material plasma resonance and do in fact represent metals via a complex-valued permittivity. But when modeling metals below these frequencies, we can say that the permittivity is unity, the permeability is real-valued, and the electrical conductivity is very high. So the above equation reduces to:

\delta=\sqrt{\frac{2}{\omega \mu_0 \mu_r \sigma }}

Before you even begin your modeling in COMSOL Multiphysics, you should compute or have some rough estimate of the skin depth of all of the materials you are modeling. The skin depth, along with your knowledge of the dimensions of the part, will determine if it is possible to use the Impedance boundary condition or the Transition boundary condition.

Now that we have the skin depth, we will want to compare this to the *characteristic size*, L_c, of the object we are simulating. There are different ways of defining L_c. Depending on the situation, the characteristic size can be defined as the ratio of volume to surface area or as the thickness of the thinnest part of the object being simulated.

Let’s consider an object in which L_c \gg \delta. That is, the object is much larger than the skin depth. Although there are currents flowing inside of the object, the skin effect drives these currents to the surface. So, from a modeling point of view, we can treat the currents as flowing *on* the surface. In this situation, it is appropriate to use the Impedance boundary condition, which treats any material “behind” the boundary as being infinitely large. From the point of view of the electromagnetic wave, this is true, since L_c \gg \delta means that the wave does not penetrate through the object.

*The Impedance boundary condition is appropriate if the skin depth is much smaller than the object.*

With the Impedance boundary condition (IBC), we are able to avoid modeling Maxwell’s equations in the interior of any of the model’s metal domains by assuming that the currents flow entirely on the surface. Thus, we can avoid meshing the interior of these domains and save significant computational effort. Additionally, the IBC computes losses due to the finite conductivity. For an example of the appropriate usage of the IBC and a comparison with analytic results, please see the Computing Q-Factors and Resonant Frequencies of Cavity Resonators tutorial.

The IBC becomes increasingly accurate as L_c / \delta \rightarrow \infty; however, it is accurate even when L_c / \delta \gt > 10 for smooth objects like spheres. Sharp-edged objects such as wedges will have some inaccuracy at the corners, but this is a local effect and also an inherent issue whenever a sharp corner is introduced into the model, as discussed in this previous blog post.

Now, what if we are dealing with an object that has one dimension that is much smaller than the others, perhaps a thin film of material like aluminum foil? In that case, the skin depth in one direction may actually be comparable to the thickness, so the electromagnetic fields will partially penetrate through the material. Here, the IBC is not appropriate. We will instead want to use the Transition boundary condition.

The Transition boundary condition (TBC) is appropriate for a layer of conductive material with a thickness relatively smaller than the characteristic size, and curvature, of the objects being modeled. The TBC can be used even if the thickness is many times greater than the skin depth.

The TBC takes the material properties as well as the thickness of the film as inputs, computing an impedance through the thickness of the film as well as a tangential impedance. These are used to relate the current flowing on the surface of either side of the film. That is, the TBC will lead to a drop in the transmitted electric field.

From a computational point of view, the number of degrees of freedom on the boundary is doubled to compute the electric field on either surface of the TBC, as shown below. Additionally, the total losses through the thickness of the film are computed. For an example of using this boundary condition, see the Beam Splitter tutorial, which models a thin layer of silver via a complex-valued permittivity.

*The Transition boundary condition computes a surface current on either side of the boundary.*

So far, with both the TBC and the IBC, we have assumed that the surfaces are perfect. A planar boundary is assumed to be geometrically perfect. Curved boundaries will be resolved to within the accuracy of the finite element mesh used, the geometric discretization error, as discussed here.

*Rough surfaces impede current flow compared to smooth surfaces.*

All real surfaces, however, have some roughness, which may be significant. Imperfections in the surface prevent the current from flowing purely tangentially and effectively reduce the conductivity of the surface (illustrated in the figure above). With COMSOL Multiphysics version 5.1, this effect can be accounted for with the *Surface Roughness* feature that can be added to the IBC and TBC conditions.

For the IBC, the input is the Root Mean Square (RMS) roughness of the surface height. For the TBC, the input is instead given in terms of the RMS of the thickness variation of the film. The magnitude of this roughness should be greater than the skin depth, but much smaller than the characteristic size of the part. The effective conductivity of the surface is decreased as the roughness increases, as described in “Accurate Models for Microstrip Computer-Aided Design” by E. Hammerstad and O. Jensen. There is a second roughness model available, known as the *Snowball model*, which uses the relationships described in *The Foundation of Signal Integrity* by P. G. Huray.

It is also worth looking at the idealized situation — the Perfect Electric Conductor (PEC) boundary condition. For many applications in the radio and microwave regime, the losses at metallic boundaries are quite small relative to the other losses within the system. In microwave circuits, for example, the losses in the dielectric substrate typically far exceed the losses at any metallization.

The PEC boundary condition is a surface without loss; it will reflect 100% of any incident wave. This boundary condition is good enough for many modeling purposes and can be used early in your model-building process. It is also sometimes interesting to see how well your device would perform without any material losses.

Additionally, the PEC boundary condition can be used as a symmetry condition to simplify your modeling. Depending on your foreknowledge of the fields, you can use the PEC boundary condition, as well as its complement — the Perfect Magnetic Conductor (PMC) boundary condition — to enforce symmetry of the electric fields. The Computing the Radar Cross Section of a Perfectly Conducting Sphere tutorial illustrates the use of the PEC and PMC boundary conditions as symmetry conditions.

Lastly, COMSOL Multiphysics also includes Surface Current, Magnetic Field, and Electric Field boundary conditions. These conditions are provided primarily for mathematical completeness, since the currents and fields at a surface are almost never known ahead of time.

In this blog post, we have highlighted how the Impedance, Transition, and Perfect Electric Conductor boundary conditions can be used for modeling metallic surfaces, helping to identify situations in which each should be used. But, what if you cannot use any of these boundary conditions? What if the characteristic size of the parts you are simulating are similar to the skin depth? In that case, you cannot use a boundary condition. You will have to model the metal domain explicitly, just as you would for any other material. This will be the next topic we focus on in this series, so stay tuned.

]]>What if you could enable non-experts to run your multiphysics simulations on their own? You would save time, for sure, and they would get easy access to your expertise. Turning your simulations into apps with customized and easy-to-use interfaces is now a reality. Here, I will explain why you should start creating apps and how to go about it. We’ll use the new Corrugated Circular Horn Antenna Simulator demo app to guide us.

Your designs and processes are affected by many physics, either wanted or unwanted, so you cannot design technologies as if they existed in a world of perfect conditions. Simulation is a powerful tool for studying multiple physics in your designs to learn how they impact the overall performance of the end-product. As a simulation engineer, you can then optimize the designs before moving into physical prototyping or production.

I suspect, however, that more often than not, your simulation process is not as simple as set up, solve, visualize, and move on to the next project. If you created the simulations for someone else — an antenna designer or someone else in the manufacturing process, maybe — chances are they will be back for more. Their subsequent requests are probably not major changes, either. Perhaps they are simple but repeated updates to the same parameters. “Please re-run the simulation but for a smaller waveguide radius,” he says. Or: “What if the flare angle was a couple of degrees wider?”

How many times are you willing to make the same menial simulation updates for your fellow antenna designer? Wouldn’t you rather enable him to make those changes himself? It’s time to join the simulation revolution.

As technologies continue to advance, the study of multiphysics elements has grown in importance, with tools becoming more complex in order to handle intricate analyses. With the COMSOL Multiphysics® software Application Builder, simulation capabilities are no longer limited to trained engineers. By creating apps with the Application Builder, your simulations can be used by non-experts and you can avoid re-running your simulations for various design iterations.

I will show you how to create a simulation app, but first, let’s take a closer look at the building blocks of a COMSOL® software simulation app.

Applications are easy-to-use simulations that are generated from other simulations or models. When creating apps, you can allow access only to the parameters that are important to the design or manufacture of the system or process, and add appropriate input and output fields to change and manipulate these parameters.

Within the resulting app user interface (UI), there are several components that we refer to as *form objects*. We can customize the UI by including and editing these form objects through the Form Editor, while optimizing their usability by using the Method Editor. The Method Editor offers a programming environment that supports Java® API, helping to further simulation capabilities and provide a more integrated approach to the workflow. If we use a cooking analogy, the form objects added through the Form Editor would be the main ingredients in your recipe and the Method Editor would serve as the spices, increasing the flavor of the dish and garnishing it.

Once you have created a customized (and intuitive) interface, you can distribute your apps to others involved in the design and manufacturing process. This enables both experts and non-experts to run simulations, modify parameters, and optimize designs to meet specific goals and specifications — all the while you are working on your next simulation project.

With the Application Builder, you can turn any simulation into a simulation app, making the possibilities truly endless. With the recent release of COMSOL Multiphysics version 5.1, we introduced 20 demo apps to help get you started on building and distributing your own apps. One of these demo apps, the Corrugated Circular Horn Antenna Simulator, provides a helpful overview of how to create an app.

Tip: You can also see more demo apps in action in this video introduction to the Application Builder.

The Corrugated Circular Horn Antenna Simulator demo app is based on a tutorial from our Application Gallery, which we have previously blogged about. In this tutorial, simulation techniques are used to solve for the electromagnetic fields (expedited by exploiting the symmetry of the structure), visualize the 2D axisymmetric geometry in a 3D geometry form, and evaluate and display the results in a 3D domain.

For a non-expert, this simple example can look quite complex. You wouldn’t want to put this raw model in the hands of your antenna designer; the geometry can be broken and simulation settings such as boundary conditions and mesh can easily malfunction if unreasonable geometric design parameters are selected. Only *you* may know what the safe input parameter range is for performing the simulation seamlessly and generating valid results.

In order to avoid said problems, you can create an app that provides the following:

- Intuitive UI
- Easily accessible results visualization
- Quick overview of the simulation’s validity compared to the user-specified target criterion
- App documentation
- Simulation report that can be accessed with just one click of a button

We will show you what these components are and how to include them in your app via our Corrugated Horn Antenna demo.

As mentioned earlier, form objects enable you to customize the user interface of an app and enhance its usability. Here is a screenshot of the objects you can choose to include in the UI of your app:

*Form objects allow you to create apps with easy-to-use UIs.*

You may be curious as to how these form objects were implemented in the app and how they work. Let’s take a quick look at the UI of the Corrugated Circular Horn Antenna Simulator demo app.

*An app showing the far-field radiation of a corrugated circular horn antenna. The geometry parameters and operating frequencies can be changed to optimize the antenna’s performance.*

The user interface for the application is composed of sections for the design parameters, analysis of the results, and information about the app. Furthermore, it includes a command section featuring 10 buttons as well as a graphics window. The command section at the top of the UI is comprised of many buttons implemented in a subform. The button object provides one-click action control and it initially looks like this:

To finalize the command section design as shown in the UI, the following steps are included:

- Implementing the button on a form
- Specifying the button’s name and size (large) and the image imposed on it
- Adding the background image of the form
- Adding the corresponding method for action

*The command section was built with form and button objects.*

The design parameters section includes the following objects: Image, Text Labels, Input Fields, Units, Data Displays, and Spacer. Using this component, the app collects the input parameters for the geometry and the accessible input conditions for the simulation. The All Input Field object is implemented with the Events feature to update the current status of the app when any change is made to the input parameters after a geometry view update or simulation.

*The design parameter form is added as an input console.*

In the result analysis and information sections, a notable form object is the Information Card Stack, which is added through a subform. The Information Card Stack provides conditional texts with icons based on the selected source variable state. Either one of the “cards” in the “stack” will be visible according to the state of the simulation. The suggested default state includes three cases:

- There is no solution
- The input data has changed from the previous simulation
- The last computation time

*The default state of the Information Card Stack.*

In this app, there are three information card stacks in the UI. According to the selected source variable state of the app, each stack notifies users of the following, respectively:

- Solution availability, any change in the user input parameters, and the last computation time
- Geometry view status
- Result pass/fail analysis compared to the simulation target criterion specified by the user

The graphics window is comprised of form objects called *Graphics*. The Graphics object enables the visualization of the antenna’s geometry, a variety of 2D and 3D results, and the layout information. In order to visualize several different views on the same location as tabs, each Graphics object is added via a Card Stack form object. According to the selected source variable of the card stack, the current Graphics object will be chosen.

Some of the form objects are not shown in the app’s initial launch, but they can be viewed at a later point when further using the app. Upon clicking the Compute button, a pop-up dialog window and a progress bar reports the solution status. When clicking the Simulation Report button, you will see a pop-up dialog window decorated with a radio button asking which format you’d like the report to be generated in.

*Progress bar and radio button implemented in a form.*

There are other useful form objects not used in this app, including the Check Box and Slider form objects. The Check Box object can be used to toggle a certain visualization condition to be on or off; for example, you may want to remove the field plot on the antenna aperture. The Slider object can change the color range of the visualization or it can be linked to one of the input parameters to easily update the value without having to type in numbers.

When customizing the UI, there are numerous ways to enhance it, with no limit to your imagination. The design is entirely up to you!

*Slider and Check Box form objects provide more functionality in the UI.*

You are now familiar with the innovative widgets we call form objects, resources that enable the customization of your UI. There are, however, other tools integrated in the Application Builder that also enable you to customize your UI: the Form Editor and the Method Editor. As I mentioned above, these editors allow you to develop specific simulation tools that can be shared with people throughout your company and across the design and development process.

Let’s take a look at the workflow behind creating the Corrugated Circular Horn Antenna Simulator demo app, which is based on this pre-existing tutorial model. To employ another analogy, the model acts as a seed that will grow into a dynamic app with the help of the Form Editor and the Method Editor. The original model’s geometry has been completely parameterized (i.e., parameters based on the coordinates represent the geometric measurements rather than integers) such that changes to the geometric inputs in the app will result in a consistent geometry even after these changes take place. Furthermore, the result plots have also been added to and reconfigured from the original model to include more information in the app.

First, to begin the process of turning a model into an app, click on the *Application Builder* button, which can be accessed through the *Home* tab of the ribbon.

*The Application Builder button on the Home tab will lead you to the Application Builder.*

The first screen within the Application Builder appears like the figure below:

*The Application Builder window when you start to build an app.*

The New Form button in the ribbon is a very useful tool for building a basic layout of the app and creating any app with only a few clicks of your mouse.

*The New Form button simplifies the process of making an app.*

When you click on the New Form button, a window opens with a wizard that takes you through the steps of creating an app.

You can add:

- User input fields with text labels inherited from the parameter descriptions
- A graphics window for geometry view and results analysis
- Buttons for actions such as running the simulation, plotting the results, and executing commands customized by the Method Editor

*The Form wizard will let you build an app very easily.*

After adding several form objects, our finalized main form looks like this:

*The finalized main form of the app consists of forms, card stacks, text labels, input fields, units, images, etc.*

Is the app ready? Not quite yet. We still have to discuss the methods implemented in the app. Since there are several methods, we will focus on only one at this time. The Button form object is a widget that can add commands. These commands are either predefined and suggested by the Application Builder or can be fully customized using the Method Editor.

Here, the Compute button has a customized method to run the simulation.

*The Button form object with an embedded method and an icon.*

Note the small icon at the bottom-right corner in the above image. This indicates that the form object has an embedded method. The method on this button will be executed when it is clicked and can be added from the *Choose Commands to Run* section of the settings window.

*With the Choose Commands to Run section, predefined commands or user-defined methods can be added.*

The method was created before it was embedded in the button. You can create a new method by clicking the New Method button in the ribbon.

*The New Method button will add an empty method.*

If you are familiar with Java® API, you should feel comfortable using this tool. For those less familiar with the program, that’s no problem. There is a helpful tool in the ribbon, called *Record Code*, which will record the actions you take in the Model Builder and then present these actions as the code they represent in the Method Editor.

Furthermore, there are many demo apps that show how to write and apply methods. Another valuable resource for learning more about these editors and the design and development of apps is the Introduction to Application Builder documentation (PDF).

The method embedded on the Compute button that is labeled *fRunAll* will perform the following actions:

- Open the progress window
- Update the geometry view
- Run the simulation
- Update all results plots
- Update text labels
- Calculate the cross-polarization ratio
- Update the state of all Information Card Stacks
- Close the progress window

We complete the app with more combinations of forms and methods until it works as an antenna simulation tool package that guides users to follow the appropriate input parameter range. The app provides the cut-off frequency and wavelength and allows users to reset the input parameters to the default values, visualize the 2D axisymmetric geometry as a 3D model, and plot the 2D antenna gain pattern and the 3D far-field radiation pattern. It also displays the electric field polarization to allow for analyzing the cross-polarization ratio on the input waveguide feed and the output antenna aperture. Furthermore, the app shows the antenna layout so that users can identify input parameters. When the app has finished running, it generates a simulation report. If app users want to learn more, they can open the embedded application documentation. It is even possible to email this report to your colleagues from the app by adding an extra method.

The original app designer and non-antenna simulation expert alike can easily tune the antenna geometry parameters to meet the specification without getting too involved in the simulation settings and underlying physics.

*The app is designed to prevent users from entering input values that are out of range.*

*Various features provided by one-click actions in the UI.*

*The customized report will help to summarize the simulation results.*

*Easy access to the application documentation.*

Let’s conclude this blog post by summarizing how to create an app.

The basic workflow involved in creating an app is as follows:

- Create a model
- Add forms
- Add form objects
- Add methods, as necessary

Though it may appear complicated the first time you open the Corrugated Circular Horn Antenna Simulator demo app, it is in reality just a combination of forms and methods. By designing a more intuitive UI, you can build better apps with greater potential in advancing the power and scope of your simulations. If you’ve already designed an interesting model, why not start building your own app today?

- Download the demo app: Corrugated Circular Horn Antenna Simulator
- Watch a video: Introducing the Application Builder in COMSOL Multiphysics
- Learn more about the Application Builder and the other demo apps that were released with COMSOL Multiphysics version 5.1

*Oracle and Java are registered trademarks of Oracle and/or its affiliates.*

Antennas play an important role in aviation. Pilots rely on them to ensure that their aircraft maintain safe alignment and position. This is especially important during times of reduced visibility, such as at night and in poor weather conditions. Therefore, to ensure safe flights, these antennas have to be highly reliable.

*A close-up of an airplane with antennas attached to the fuselage.*

Placing multiple antennas on a single airplane fuselage can be problematic, as antenna crosstalk, or *cosite interference*, may occur. As the name implies, this effect happens when electromagnetic signals from one antenna cross signals with another antenna and cause interference. When interference happens between a transmitting antenna that is transmitting *to* antennas on the ground and receiving devices that are receiving signals *from* antennas on the ground, it can affect the operation of the entire aircraft.

By simulating antenna crosstalk on an airplane’s fuselage, you can determine where to place the transmitting and receiving antennas to ensure safe and efficient communication. For this purpose, let’s look to a new tutorial that was released with COMSOL Multiphysics version 5.1.

The airplane model is made up of a metallic body and two antennas, one on top of the fuselage and one on the bottom. The metal surfaces are modeled with a perfect electric conductor (PEC) condition. The airplane is enclosed by a spherical domain of air that is encased by perfectly matched layers (PMLs), which mimics infinite free space. The PMLs ensure that there is no reflection from the boundary of the air domain.

*A schematic of an airplane with transmitting and receiving antennas highlighted. The size of the antennas are exaggerated.*

The antennas are made up of metal strips in a dielectric block. They’re miniaturized with a meander line design, which decreases their input impedance to be lower than the reference impedance of 50 Ohm. A folded monopole antenna design technique on a large ground plane is used to match the initial low impedance.

When simulating the antennas, we also place a lumped port in the gap between the meander line and the fuselage. S-parameters are calculated from these lumped ports to show the antennas’ matching properties as well as their level of interference with each other when placed in different configurations.

The simulation shows shadowed and lit areas of the fuselage. Shadow areas are regions where the structure of the aircraft, such as the wings, fins, and gear doors, hinder the transmitting power of the transmitting antenna. On the other hand, lit areas are not obstructed by any gear. This means that shadow areas are better suited locations to place the receiving antenna, because there will be less interference from the transmitting antenna, while lit regions are less ideal. The figures below show the electric field strength of the transmitting antenna and its effect on the receiving antenna in three different configurations.

*The transmitting antenna interferes with the receiving antenna at each of the three tested locations on the bottom of the airplane.*

The extent of the lit and shadow areas help in determining the best place for the antennas on an airplane’s fuselage.

*The lit and shadow areas on an airplane’s fuselage help determine the level of crosstalk between two antennas.*

In an alternate set of figures, below, the norm of the electric field is plotted on the dB scale as the receiving antenna is tested in three different locations going from the rear to the front of the fuselage. From these images, we can deduce that the shadowing is best when the antenna is placed at the rear of the plane. This is confirmed when comparing the computed S-parameters at these three different configurations.

*The shadow area of the receiving antenna when it is tested in three different locations on the bottom of the fuselage.*

By modeling antenna crosstalk with COMSOL Multiphysics version 5.1 and the RF Module, you can optimize the antenna placement on the airplane’s fuselage for a strong overall design that ensures safe travel.

- Try it yourself: Download the Antenna Crosstalk on an Airplane’s Fuselage tutorial model

Radio-frequency identification (RFID) tags allow objects to be identified without the use of a wired connection. They can be used in applications ranging from ID badges to animal maintenance (or a more creative and fictional use: as part of a robotic butterfly). Scientists also use RFID tags to track whale sharks, the largest fish in the sea. Through this process, they hope to learn more about these gentle creatures. A more common use of RFID tags is on farm animals. Farmers will tag their animals (such as cows) in order to verify that they own them, protect against theft, and identify animals in the case of a disease outbreak.

*A cow with an RFID tag in her ear. (By Man vyi. Licensed under public domain, via Wikimedia Commons).*

There are three kinds of RFID tags:

- Low-frequency (LF) range
- High-frequency (HF) range
- Ultra-high-frequency (UHF) range

A UHF RFID tag has many advantages over other RFIDs, making it suitable for monitoring animals. Unlike the HF alternative, the UHF RFID tag can be read at near and far distances and can thus work for a wider range of uses. These tags can generally also transfer data faster than their LF and HF counterparts and offer competitive prices.

With COMSOL Multiphysics version 5.1, we are offering a new tutorial model to help you get started with the simulation of passive UHF RFID tags. Let’s have a look at what is included in the tutorial.

As is a common practice in RFID tag design, the antenna in our model is folded in a meandered path in order to make the overall tag smaller. The antenna is made up of copper traces patterned on an FR4 board. A lumped port represents the microchip that will be used to excite the tag and analyze the input impedance of the tag’s antenna. Our lumped port has a reference impedance of 50 Ω. Next to the meander line is another copper strip, which we use to control the impedance.

All of these components are covered by a yellow low-dielectric PTFE casing to look similar to the tag we saw in the cow’s ear, above.

*The geometry of a UHF RFID tag with one half of the circuit board exposed. The lumped port represents the chip.*

Since we set our RFID tag’s operating frequency to 915 MHz and the copper traces are thicker than the skin depth, the metallic aspects of our design can be modeled as a perfect electrical conductor (PEC). You cannot see this in the figure above, but surrounding our whole model is a spherical air domain, which is itself encompassed within perfectly matched layers (PMLs). All outgoing radiation is absorbed by the PMLs.

We approach this simulation in a non-conventional way, by choosing a power wave reflection coefficient term to deal with the complex impedance. This allows us to evaluate the matching properties in our RFID tag.

Note that we will skip ahead to the results in the next section. To learn how to build this tutorial model step-by-step, please visit the Application Gallery.

After setting up and solving our model, we can analyze the results to evaluate how well the tag design performs. To understand how the electric field is distributed throughout the antenna, we can study the electric field norm plot. When the antenna is resonant, strong fields will be observed at each end and some of the outer edges of the meander line. The tag’s range is determined by the impedance match between the antenna and the chip.

First, we analyze the default electric field (E-field) norm on an *xy*-plane to determine where in the tag the field is confined. As we can see in the plot below, the electric field is symmetrical along the meander line and in the space between the meander line and the impedance matching strip. This is ideal because a symmetric field may generate a symmetric radiation pattern.

With a symmetric radiation pattern (in this model, it’s omni-directional on one plane), we can easily expect what the ideal angular configuration of the tag is. To put it in terms of our example use case, this information will guide us on where to attach the tag on the cow and where to locate the RFID reader in order to ensure that we monitor *all* of the cows.

*The E-field norm plot of our UHF RFID model for the full-sized circuit board.*

Next, we can take a look at the far-field radiation pattern. The pattern will be familiar to simulation engineers who work on other antenna designs too; the radiation pattern looks a lot like that of a half-wave dipole antenna.

*The far-field radiation pattern of our UHF RFID tag model.*

Through our simulation, we discover that the power wave reflection coefficient of the tag is below -15 dB with respect to the chip impedance 18+j124 Ω. This means that when the delivered power to the RFID chip re-radiates through the tag, less than -15 dB of power will be reflected back to the chip. Compare this with a commercial off-the-shelf antenna of mediocre performance, which would provide -10 dB. Given that, our design is not excellent, but it is acceptable. Since the radiation pattern is omni-directional around the meander line, its response is almost the same at any direction on one plane. This suggests that our tag is robust and reliable.

If you are a design engineer at an RFID tag company, you can turn to COMSOL Multiphysics simulation software to evaluate how your UHF RFID tag will perform. Get started by downloading the tutorial model below.

- Tutorial model: Numerical Modeling of a UHF RFID Tag
- Tutorial model: An RFID System
- Blog post: RFID Tag Read Range and Antenna Optimization
- Blog post: A Creative Take on RFID Tags

The Vivaldi antenna was invented by Peter Gibson. Gibson had a passion for music and he supposedly chose to name the antenna after Antonio Vivaldi, a Baroque composer. Although sources vary on this, the choice was probably due to the antenna resembling an instrument from Vivaldi’s era. The instrument in question may be a violin, cello, or a cross section of a trumpet.

The Vivaldi antenna is made to be malleable. Being created on a thin and flexible substrate gives it the ability to mold itself over a variety of surfaces. This means that it can have many different applications in a range of environments.

There’s a patent that utilizes a double Vivaldi element feed section for use on planes. This works because the antenna is able to form a streamlined shape. Another reason the Vivaldi antenna is a good fit for the airplane is that it can handle velocities up to Mach 2. This antenna is going places…

*Aircraft. ( By Björn — Own work. Licensed under Creative Commons Attribution Share Alike 2.0, via Wikimedia Commons)*

You can also find uses for the Vivaldi antenna inside of a hospital. By using this antenna in conjunction with microwave imaging, doctors may be able to better detect breast and brain cancer.

The breast cancer detection function of the Vivaldi antenna makes use of an interferometric called I-MUSIC. (This is a different kind of music than what the composer Vivaldi made…) I-MUSIC stands for interferometric Multiple Signal Classification and when combined with a Vivaldi antenna, it becomes a powerful method for detecting tumors.

The Vivaldi antenna (both as itself and when modified) has an incredibly wide range of functions. Security workers can use it to detect concealed weapons and military personnel can use it as a high-range radar when it’s placed in an array configuration.

This versatile antenna can do a lot, but first we have to analyze the design to see how well it works.

Using the Vivaldi, Tapered Slot Antenna (TSA) model, we can evaluate the antenna based on its far-field pattern and impedance.

Our model simulates a realistic Vivaldi antenna design by using a thin dielectric substrate for the antenna. On top of this substrate, a tapered slot is patterned with a perfect electric conductor (PEC) ground plane. We build the curves of the tapered slot by using the exponential function *e*^{0.044x}. The tapered slot itself does look something like a trumpet with a wide end curving into a narrow line. However, unlike a trumpet, the thin end connects to a circular slot while the wide end opens out to air, as seen below.

*The geometry of a Vivaldi antenna.*

Flipping the substrate over, there is a shorted 50 Ω microstrip feed line, which is modeled as PEC surfaces. It is a crucial element to the design since the lumped port located on the line is used to excite the antenna.

A perfectly matched layer (PML) is surrounding the whole modeling domain. This functions as an anechoic chamber and absorbs all of the radiated energy.

After setting up and solving our antenna model using COMSOL Multiphysics and the RF Module, we get the results back in various forms. The SWR plot reveals that our model has good wide-band impedance matching. It’s even better than 2:1 in the majority of our simulated range.

*The results of our simulation seen in the form of an SWR plot.*

We also create a model to display the far-field pattern. We know that a Vivaldi antenna forms a directive radiation pattern toward the wide end of its tapered slot. Our model shows that the far-field pattern does display a directional radiation pattern.

*A 3D far-field pattern, displayed at 5.5 GHz.*