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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Advanced composites are used extensively throughout the Boeing 787 Dreamliner, as shown in the diagram below. Also known as carbon fiber reinforced plastic (CFRP), the composites are formed from a lightweight polymer binder with dispersed carbon fiber filler to produce materials with high strength-to-weight ratios. Many wing components, for example, are made of CFRP, ensuring that they can support the load imposed during flight while minimizing their overall contribution to the weight of an aircraft.

*Advanced composites are used throughout the body of the Boeing 787. Copyright © Boeing.*

Despite their remarkable strength and light weight, CFRPs are generally not conductive like their aluminum counterparts, thus making them susceptible to lightning strike damage. Therefore, electrically conductive expanded metal foil (EMF) is added to the composite structure layup, shown in the figure below, to dissipate the high current and heat generated by a lightning strike.

*The composite structure layup shown at left consists of an expanded metal foil layer shown at right. This figure is a screenshot from the COMSOL Multiphysics® software model featured in this blog post. Copyright © Boeing.*

The figure also shows the additional coatings on top of the EMF, which are in place to protect it from moisture and environmental species that cause corrosion. Corrosive damage to the EMF could result in lower conductivity, thereby reducing its ability to protect aircraft structures from lightning strike damage. Temperature variations due to the ground-to-air flight cycle can, however, lead to the formation of cracks in the surface protection scheme, reducing its effectiveness.

During takeoff and landing, aircraft structures are subjected to cooling and heating, respectively. Thermal stress manifests as the expansion and compression — or ultimately the displacement — of adjacent layers throughout the depth of the composite structure. Although a single round-trip is not likely to pose a significant risk, over time, each layer of the composite structure contributes to fatigue damage buildup. Repetitive thermal stress results in cumulative strain and higher displacements, which are, in turn, associated with an increased risk of crack formation. The stresses in a material depend on its mechanical properties quantified by measurable attributes such as yield strength, Young’s modulus, and Poisson’s ratio.

By taking the thermal and mechanical properties of materials into account, it is possible to use simulation to design and optimize a surface protection scheme for aircraft composites that minimizes stress, displacement, and the risk of crack formation.

Evaluating the thermal performance of each layer in the surface protection scheme is essential in order to reduce the risks and maintenance costs associated with damage to the protective coating and EMF. Therefore, researchers at Boeing Research & Technology (BR&T), pictured below, are using multiphysics simulation and physical measurements to investigate the effect of the EMF design parameters on stress and displacement throughout the composite structure layup.

*The research team at Boeing Research & Technology from left to right: Patrice Ackerman, Jeffrey Morgan, Robert Greegor, and Quynhgiao Le. Copyright © Boeing.*

In their work, the researchers at BR&T have developed a coefficient of thermal expansion (CTE) model in COMSOL Multiphysics® simulation software. The figure shown above that presents the composite structure layup and EMF is a screenshot acquired from the model geometry used for their simulations in COMSOL Multiphysics.

The CTE model was used to evaluate heating of the aircraft composite structure as experienced upon descent, where the final and initial temperatures used in the simulations represent the ground and altitude temperatures, respectively. The *Thermal Stress* interface, which couples heat transfer and solid mechanics, was used in the model to simulate thermal expansion and solve for the displacement throughout the structure.

The material properties of each layer in the surface protection scheme as well as of the composites are custom-defined in the CTE model. The relative values of the coefficient of thermal expansion, heat capacity, density, thermal conductivity, Young’s modulus, and Poisson’s ratio are presented in the chart below.

*This graph presents the ratio of each material parameter relative to the paint layer. Copyright © Boeing.*

From the graph, trends can be identified that provide early insight into the behavior of the materials, which aids in making design decisions. For example, the paint layer is characterized by higher values of CTE, heat capacity, and Poisson’s ratio, thus indicating that it will undergo compressive stress and tensile strain upon heating and cooling.

Multiphysics simulation takes this predictive design capability one big step forward by quantifying the resulting displacement due to thermal stress throughout the entire composite structure layup simultaneously, taking into account the properties of all materials. The following figure shows an example of BR&T’s simulation results and presents the stress distribution and displacement throughout the composite structure.

*Left: Top-down and cross-sectional views of the von Mises stress and displacement in a one-inch square sample of a composite structure layup. Right: Transparency was used to show regions of higher stress, in red. Lower stress is shown in blue. Copyright © Boeing.*

In the plots at the left above, the displacement pattern caused by the EMF is evident through the paint layer at the top of the composite structure while a magnified cross-sectional view shows the variations in displacement above the mesh and voids of the EMF. The cross section also makes it easy to see the stress distribution through the depth of the composite structure, where there is a trend toward lower stress in the topmost layers. Transparency was used in the plot shown at the right to depict the regions of high stress in the composites and EMF, which is noticeably higher at the intersection of the mesh wires. Stress was plotted through the depth of the composite structure layup along the vertical red line shown in the center of the plot. The figure below shows the relative stress in each layer of the composite structure layup for different metallic compositions of the EMF.

*Relative stress in arbitrary units was plotted through the depth of the composite structure layups containing either aluminum (left) or copper EMF (right). Copyright © Boeing.*

The samples vary by the presence of a fiberglass corrosion isolation layer when aluminum is used as the material for the EMF. The fiberglass acts as a buffer resulting in lower stress in the aluminum EMF, when compared with the copper.

From lightning strike protection to the structural integrity of the composite protection scheme, it all relies on the design of the expanded metal foil layer. The design of the EMF layer can vary by its metallic composition, height, width of the mesh wire, and the mesh aspect ratio. For any EMF design parameter, there is a trade-off between current-carrying capacity, displacement, and weight. By using the CTE model, the researchers at BR&T found that increasing the mesh width and decreasing the aspect ratio are better strategies for increasing the current-carrying capacity of the EMF that minimize its impact on displacement in the composite structure.

The metal chosen for the EMF can also have a significant effect on stress and displacement in the composite structure, which was investigated using simulation and physical testing. Two composite structures, one with aluminum and the other with copper EMF, underwent thermal cycling with prolonged exposure to moisture in an environmental test chamber. In the results, shown below, the protective layers remained intact for the composite structure with copper EMF. However, for the layup with aluminum, cracking occurred in the primer, at the edges, on surfaces, and was particularly substantial in the mesh overlap regions.

*Photo micrographs of the composite structure layup after exposure to moisture and thermal cycling. A crack in the vicinity of the aluminum EMF is contained within the red ellipse. Copyright © Boeing.*

Simulations confirm the experiment results. Shown below, displacements are noticeably higher throughout the composite structure layup when aluminum is used for the EMF layer, where higher displacements are associated with an increased risk for developing cracks. The higher displacement is easiest to observe in the bottom plots, which show displacement ratios for each EMF height.

*Effect of varying the EMF height on displacement in each layer of the surface protection scheme. Copyright © Boeing.*

The larger displacements caused by the aluminum EMF can be attributed in part to its higher CTE when compared with copper, which exemplifies how important the properties of materials are to the thermal stability of the aircraft composite structures.

In the early design stages and along with experimental testing, multiphysics simulation offers a reliable means to evaluate the relative impact of the EMF design parameters on stress and displacement throughout the composite structures. An optimized EMF design is essential to minimizing the risk of crack formation in the composite surface protection scheme, which reduces maintenance costs and allows the EMF to perform its important protective function of mitigating lightning strike damage.

Refer to page 4 of *COMSOL News* 2014 to read the original article, “Boeing Simulates Thermal Expansion in Composites with Expanded Metal Foil for Lightning Strike Protection of Aircraft Structures”.

This article was based on the following publicly available resources from Boeing:

- The Boeing Company. “787 Advanced Composite Design.” 2008-2013.
- J.D. Morgan, R.B. Greegor, P.K. Ackerman, Q.N. Le, “Thermal Simulation and Testing of Expanded Metal Foils Used for Lightning Protection of Composite Aircraft Structures,” SAE Int. J. Aerosp. 6(2):371-377, 2013, doi:10.4271/2013-01-2132.
- R.B. Greegor, J.D. Morgan, Q.N. Le, P.K. Ackerman, “Finite Element Modeling and Testing of Expanded Metal Foils Used for Lightning Protection of Composite Aircraft Structures,” Proceedings of 2013 ICOLSE Conference; Seattle, WA, September 18-20, 2013.

To learn more about adding material property data to your COMSOL Multiphysics® simulations, read the following blog post series on *Obtaining Material Data for Structural Mechanics Simulations from Measurements* by my colleague Henrik Sönnerlind:

General information about aircraft design and structures can be found in chapter 1 of this handbook on aircraft maintenance from the Federal Aviation Administration.

*BOEING, Dreamliner, and 787 Dreamliner are registered trademarks of The Boeing Company Corporation in the U.S. and other countries.*

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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Imagine that you’re driving in the desert. As night falls, the temperature falls with it. You slow down and shift into park, stopping the car at your destination.

When you shifted gears, the car sensed it and reacted properly. For this to happen, parts like automatic gear shifters need to function correctly. A device called a *position sensor* is used in automatic gear shifter modules and in other parts of cars, such as seat position adjusters. These position sensors need to work well in the varied environments in which cars operate.

*Some uses of position sensors in cars. Image by A. K. Palit, and taken from his COMSOL Conference 2014 Cambridge presentation.*

A researcher from Lemfoerder Electronic GmbH (ZF-Friedrichshafen AG. Group) in Germany talked about his simulation of an inductive position sensor at the COMSOL Conference 2014 in Cambridge. He chose to simulate a non-contact inductive position sensor because of its temperature stability and low cost. The typically used Hall sensors are inferior in these aspects since temperature changes alter their performance and they are not a cheap option.

In order to analyze the workings of an inductive position sensor, the researcher used COMSOL Multiphysics finite element analysis (FEA) software to create a frequency-domain model of a planar inductive coil.

*An image depicting an inductive position sensor, with the red shape representing the activator. Image by A. K. Palit, and taken from his COMSOL Conference 2014 Cambridge presentation.*

The spiral inductive coils in the model are planar and they form a row on top of the printed circuit board (PCB). The planar coils can have different geometrical shapes: square, rectangular, and circular.

His model also includes an activator (shaped here as a rhombus), which is created from a thin copper plate. The activator rests 0.2 millimeters to 0.3 millimeters above the planar coils. The activator can move, or slide, horizontally over the planar coils. It’s important to note that although the activator can move horizontally, the vertical distance is fixed.

When the copper activator slides close to a planar coil, eddy currents reduce the inductance of that planar coil. In the diagram above, for instance, coils 2, 3, and 4 have a change in inductance due to their placement in relation to the activator. The change in inductance is converted into a corresponding voltage signal that allows you to approximate the activator’s location.

The model that has been described so far focuses on a very limited area of the PCB. This allows for only a small size and number of turns in the planar coils, which can make the sensor incapable of reliable position sensing due to the resulting reduced inductance. To avoid this issue, a model of double-layer planar coil was also created and compared to the single-layer version.

Overall, the position sensor can be visualized as the leaky or loosely coupled primary and secondary windings of a transformer that uses air as the core material.

For more detailed information on the modeling of this device, please check out the paper and presentation.

This simulation helped to visualize the real-world functionality of a position sensor by determining the planar coil’s change of inductance in relation to the copper activator’s horizontal movement over the coil. In the position sensor, a change in the horizontal Xoff position correlates to a shift in the physical position of the copper activator. Traditionally, Xoff is a signal to pause or stop a transmission, and in this simulation, it signals the reduction of the planar coil’s inductance. In these graphs, Xoff = 0 indicates the center of the planar coil. At this point, the inductance should be at its lowest value because this Xoff position stops the most inductance. The graphs below visualize how a sliding copper activator with a constant vertical distance produced inductance changes at a frequency of 10 MHz.

*Left: The copper activator’s horizontal Xoff position (mm) vs. coil inductance (nH) of a single-layer planar coil. Right: The copper activator’s horizontal Xoff position (mm) vs. coil inductance (nH) of a double-layer planar coil. Images by A. K. Palit, and taken from his COMSOL Conference 2014 Cambridge presentation.*

Although both curves look similar, the results have some significant differences. In the graph of a single-layer planar coil, the sliding activator caused an inductance change of around 49% for the coil. However, the double-layer planar coil’s inductance had a higher change of around 53%. The double-layer planar coil also had a much higher inductance value. These are expected results, as a greater coil area is assumed to result in more inductance. The higher amount of inductance (and higher amount of inductance change) in the double-layer planar coil is more likely to be sufficient for reliable position sensing, because the position sensor can better estimate the location of the copper activator.

*Left: Magnetic flux density norm of a single-layer planar coil. Right: Magnetic flux density norm of a double-layer planar coil. Images by A. K. Palit, and taken from his COMSOL Conference 2014 Cambridge presentation.*

The magnetic flux density norm (mT) was also calculated at a frequency of 10 MHz for both the single-layer and double-layer models. By calculating this, we can see the varying magnetic flux in the model’s air core. Although these two images cannot be directly compared, as the double-layer image is zoomed in, the flux density arrows do appear to be more prevalent in the double-layer coil. This indicates that the magnetic field of the double-layer planar coil is greater than the alternative. This is further evidence that a double-layer planar coil will be superior for accurate position sensing.

Through the modeling of an inductive position sensor and all of the information gained from it, the researcher was able to move forward and create an optimized design that functions in the varied conditions cars experience. This design was optimized by learning what impacts the functionality of an inductive position sensor. He looked into how factors like distance between different elements (namely, the vertical and horizontal location of the activator), geometry, and size affect the efficiency and sensitivity of the inductive sensor. One thing the researcher discovered was that coil inductance could be reduced when the activator is at the center of the planar coil with a vertical distance between the activator and planar coils of 0.2 millimeters or lower. The results of this simulation were directly used to develop an automatic gear shifter module in a German car.

- Read the paper and presentation: “Frequency Response Modeling of Inductive Position Sensor with Finite Element Tools“

When first introduced in COMSOL Multiphysics version 5.0, the Application Builder marked a shift in the simulation industry, widening its scope with easy-to-use, customized apps based on models. The recently released COMSOL Multiphysics version 5.1 introduced dramatic improvements to this revolutionary tool, enhancing the simulation experience for you — the creator of the app — as well as those using it.

Applications are designed to seamlessly interact with their underlying model to enable dynamic variation of the model parameters as well as the highly customized visualization of model results. This opens up endless possibilities for easily implementing advanced functionality. Such an approach is particularly useful for semiconductor simulations, where an intricate analysis of the solution results is often desired.

We will begin by introducing the InGaN/AlGaN Double Heterostructure LED tutorial model and then demonstrate how to turn this model into an app suitable for designing LEDs to emit within a specified wavelength range. The advantage of building an app is that it takes the advanced physics functionality behind the model and makes it available in a user-friendly interface. A single technical expert can create a complicated finite element or finite volume simulation tool and deploy it to individuals within the design staff who may not otherwise have access to or knowledge of advanced simulation and numerical analysis techniques.

*Left: The user interface (UI) of the underlying model shows all of the physics features and the geometry of the device. Right: The application’s UI. While the app has access to all of the model’s physics features, only the settings that need to be varied to enhance the design of the output wavelength are made available. This app offers a much simpler UI without compromising performance.*

As mentioned above, applications utilize all of the same physics from the underlying model, while benefiting from a more intuitive user interface and results display. Although the performance and accuracy of the simulation are not affected, the compromise for this increased usability is that apps are less versatile than the models they are based on. Apps should be carefully created for a specific purpose and present users with only the functionality that is needed to satisfy their design requirements.

The Wavelength Tunable LED demo app can be used to assess the emission properties of a AlGaN/InGaN light-emitting diode in order to assist in the design of LEDs that emit within a user-specified wavelength range. The process of converting the InGaN/AlGaN Double Heterostructure LED tutorial model into an app as well as more general tips and tricks for making your own semiconductor applications are discussed in detail here.

From the previous screenshot, it is clear that the required functionality for simulating a wavelength tunable LED has been made available through utilizing several different “panels”. For example, the *Specify Material Composition*, *Operation Voltage*, and *Desired Spectral Range* panels enable input design decisions about the LED device, with results displayed in tabular and graphical forms in the *Results* and *Results Viewer* panels, respectively.

Additionally, the material properties of the LED can be varied as well as the voltage applied to the device, and the app calculates the emission intensity, electroluminescence spectrum, and device efficiency. The emission spectrum is analyzed to determine if the emission falls within the desired spectral range. If the app is used to design an LED that emits in the visible spectrum, the electroluminescence is converted into an RGB color ratio and displayed to provide immediate feedback on the emission color of the device.

When designing an application, you have to consider which physics features need to be accessible and how to link the important physics principles to your app’s UI. For example, in the Wavelength Tunable LED app, modification of the emission wavelength is necessary. As the emission wavelength from semiconductors is highly dependent on the bandgap energy, this is achieved by allowing the material composition of the LED to be varied.

The active region of the LED from which light is emitted is made of InGaN, which is a semiconductor alloy comprised of indium nitride and gallium nitride. Indium nitride has a small bandgap of around 0.75 eV and emits infrared light; gallium nitride has a much larger bandgap (approximately 3.4 eV) and emits in the far ultraviolet range. By varying the blend of the InGaN, it is possible to create a material that emits between these two extremes, thus any color of visible light is possible.

*The slider settings in the Application Builder. It is straightforward to link a graphical input, such as the slider bar, to a model parameter. In this case, the slider bar is used to set the fraction of indium in the InGaN alloy.*

The fraction of indium in the active region material can be varied using an interactive slider bar within the *Specify Material Composition* panel of the Application window. The slider bar is linked to a parameter, `In_x`

, in the underlying model that is used to set the fraction of indium in the InGaN material. When the slider is moved, the value of `In_x`

is automatically varied between the minimum value and the maximum value (shown under the settings for the slider). The value of `In_x`

is then used “behind the scenes” to calculate the bandgap energy for the InGaN material used in the light-emitting region of the device.

The app provides dynamic feedback by updating the indium composition and bandgap energy displays in real time as the slider is changed. The screenshot below shows how two data displays, one for each value, are used to achieve this. Each data display is linked to a parameter that is calculated using the value in `In_x`

. When the slider is moved, the data display automatically updates to indicate the new corresponding values.

*Settings for the bandgap energy data display. It is straightforward to link a data display with any model parameter. Here, the data display is linked to the parameter that represents the bandgap energy of the InGaN material.*

Using a similar approach, the input box for the operational voltage is linked to a model parameter. The voltage that is applied across the LED can be set. In general, it is good practice to create models where the user-controlled variables are accessible via parameters, as this is the easiest way for others involved in the design process to interact with them through apps. For semiconductor simulations, quantities required by the physics features are often functions of several user inputs, and it is important to carefully consider which inputs should be included in the UI.

In addition to offering a simplified user interface, applications also allow custom methods to be created. Custom methods are extremely powerful and can be used to automate any of the functionality in COMSOL Multiphysics. They can also be used to manipulate solution data to perform an analysis in order to extract important figures of merit. This is particularly significant for semiconductor devices, where the device operation can often be evaluated by extracting device-specific values, such as turn-on voltage for a MOSFET transistor or the current gain for a bipolar transistor.

In this app, a custom method is used to analyze the emission spectrum from the LED. The wavelength of peak emission is extracted from the spectrum and compared to the selection in the *Desired Spectral Range* panel. This panel contains a button that enables the selection of either infrared, visible, or ultraviolet as the spectral desired emission range and a dynamic feedback that displays the corresponding wavelength range. The screenshot below shows the settings window for the radio button. The radio button is configured by linking it to a choice list, which, in this case, contains the three spectral options for selection as well as a source variable in the declarations. The source variable is used to set the initial value of the radio button. When the radio button is modified, the corresponding value is written into the source variable.

*The configuration of the radio button from the* Desired Spectral Range *panel. The blue boxes highlight the selected choice list, which contains the three possible options. The red boxes indicate the output_card variable, which is used as a source for the button. This variable holds a value that corresponds to the radio button’s selected option.*

The source variable — in this case, `output_card`

— can then be used within custom methods. The following screenshot shows an example section of the custom method *f_compute_and_plot*. This method is executed by clicking the *Compute* button in the ribbon and solves the underlying model by using the current input settings and performing an analysis of the results. The section depicted in the screenshot compares the wavelength of emission to the radio button selection stored in `output_card`

. If the peak emission wavelength falls within the selected spectral range, a flag variable, `check[n]`

, is set to `1`

. The flag variable is later used to control the message displayed in the *Summary* section that indicates whether the desired emission wavelength has been achieved.

*A section of the f_compute_and_plot custom method that evaluates if the peak emission wavelength falls within the selected desired spectral range.*

We have just demonstrated how a user-specified input and a value extracted from a solution can be combined with a custom method to implement highly flexible functionality within the application. The radio button within the *Operation Voltage* panel is used similarly in other logical conditions that control which analysis to perform on the solution. When a single voltage is selected, only the emission spectrum is plotted as a graph. However, when a voltage range is selected, the other performance metrics from the *Results* panel are also plotted as a function of voltage.

In addition to performing logic operations, you can also use custom methods to execute Java® code to perform a more complicated analysis of the results. The screenshot below shows a private method — nested within the `f_compute_and_plot`

method — that converts the emission spectrum into an RGB value. Each color component is obtained by multiplying the spectrum by the corresponding color sensitivity curve and integrating it with respect to the wavelength. You can explore this in further detail within the application by clicking the *About* button.

*The RGB_converter() method. This is a private method within the f_compute_and_plot method that is used to convert the electroluminescence spectrum into an approximate RGB value to display the emission color if it falls within the visible range. It is an example of how custom functionality can be implemented using methods.*

With this example, we can see how advanced operations can be performed on solution data to create custom displays or feedback within applications. The screenshot below shows the app after solving for two different values of the indium composition. InGaN with 12.7% indium emits blue light; InGaN with 23.7% indium emits green light. This offers immediate feedback from the app when designing an LED for visible emission.

*A screenshot of the app after solving with two different slider bar positions. The left image represents the default case of 12.7% indium composition. The right image illustrates an increased indium composition of 23.7%.*

To ensure that your application can be used effectively, it is often a good idea to document your app. This can easily be done by utilizing an embedded file that is linked to a button, such that it can be opened with a single click (shown in the screenshot below). Here, you can include a summary of the physics behind your semiconductor simulation along with some instructions on your app’s purpose and how it should be used. COMSOL includes a Report Generator feature, which can be used to create your documentation. You might also consider implementing tooltips on buttons and user input fields to provide more immediate help to those using your app.

*The settings window for the About button in the application ribbon. The button is set to open the embedded PDF file that contains the app’s documentation. The text in the tooltip input field is displayed when the cursor hovers over the About button.*

Before you begin designing your own app, it is helpful to ask yourself the following questions:

*What is the purpose of the application?*- Apps work best when they are designed to address specific simulation goals.

*What physics features does the application need to have access to?*- Try to parameterize any variables that the app needs to modify.

*How should the user interface be structured?*- Try to present only necessary functionality. Consider using panels in the layout for user inputs and selections.

*How should the user interact with variables?*- Slider bars work well when a quantity varies between two known limits.
- Radio buttons are helpful in making selections among several predefined options.
- Input boxes can be used to allow custom values to be entered.

*What results need to be displayed to provide the required feedback?*- Graphs, tables, or display boxes are suitable for displaying simulation results.

*Can custom methods be used to enhance the functionality of the application?*- Use custom methods to perform a bespoke analysis on the solution data, if required.

*What documentation is required in order for the application to be used effectively?*- Documents can be embedded inside application files. Include PDF instructions on how to use your app along with a brief explanation of the physics involved.

Considering these questions will help you maximize the usability of your app and create powerful tools for performing semiconductor simulations. COMSOL Multiphysics version 5.1 makes it easy to quickly package advanced semiconductor simulations into easy-to-use analysis tools that can be used by those who are not experts in simulation to optimize the design of the device.

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

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.

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Piezoelectric valves are common in medical and laboratory applications because they offer many advantages, such as energy efficiency, durability, and fast response times. To open and close the valve featured in this tutorial, there is a hyperelastic material with a piezoelectric actuator sitting on top of it. When a voltage is applied to the stacked piezoelectric actuator, it deforms in a way that either pushes the hyperelastic material against the opening of the valve to seal it or moves it away from the valve to open it.

*Valve, piezoelectric actuator, and seal.*

Stacked piezoelectric actuators consist of two actuators stacked on top of each other. Each of the two actuators is made up of alternating layers of piezoelectric material, PZT, and very thin metal conducting layers between them. Every second metal layer is grounded, while every other layer receives an applied voltage. Similarly, the stacked PZT layers have alternating polarization directions.

*Close-ups of the actuator and seal with alternating layers of PZT and metal highlighted. The top images show the PZT layers of alternating polarization directions. The bottom images show the metal substrate with an applied voltage to every other layer and the others set to a ground.*

The bimorph actuator under consideration can be thought of as two stacked actuators placed one on top of the other. For a positive applied voltage, the upper and lower actuators are designed to expand laterally and contract laterally, respectively. This results in a bending of the structure (in this case, a disc), such that the center of the disc arches downwards. This forces the hyperelastic seal into contact with the valve seat — closing the valve. In the surface plot below, the stress is indicated by the color scale.

*The von Mises stresses in a piezoelectric valve with a bimorph disc actuator.*

The Piezoelectric Valve tutorial model, a new addition to the Application Gallery with COMSOL Multiphysics 5.1, demonstrates how to model a stacked piezoelectric bimorph disc actuator in a pneumatic valve. The MEMS Module and Nonlinear Structural Materials Module are used for this simulation.

The valve model consists of a multilayer stacked piezoelectric actuator, which in itself is a complex structure of stacked layers and electrodes. The model also includes a stainless steel substrate and a seal of hyperelastic material over the through hole of the valve.

For the simulation, we apply 50 volts to the layers. The contact pressure is determined here at the two contact pressure points of the seal. We can see that deformation of the disc is greatest at the center, which compresses the hyperelastic seal against the valve’s opening and closes the valve.

*Left: The strain at the two contact surfaces of the valve’s seal. Here, we can see that the deformation of the disc is greatest at the center, which closes the valve. Right: The contact pressure at the two surface points of the valve’s seal.*

Modeling a piezoelectric valve allows us to analyze the operation of the stacked piezoelectric actuator and evaluate the stress and strain in the seal and the surrounding materials. The analysis could be extended to estimate the performance of the seal with different pressure differentials applied across the valve in the closed state.

- Tutorial Download: Piezoelectric Valve