Whenever a lossy material, such as a conductor, is exposed to a timevarying electromagnetic field, there will be currents induced in that material. These currents then produce a magnetic field that alters the current distribution within the material, with the net result being that the induced currents will preferentially flow at the surfaces of any lossy material. This is known as the skin effect.
We can model this effect using the AC/DC Module and any of the physics interfaces that solve for the magnetic fields and currents, such as the:
All of these physics interfaces support modeling in the frequency domain under the assumption that the magnetic fields, and all other fields, are varying sinusoidally in time. The Magnetic Fields interface — as well as the Rotating Machinery Magnetic and the Magnetic Field Formulation interfaces — support full timedomain analysis, where the fields can vary nonsinusoidally over time.
To start off, we focus on the frequency domain, a very reasonable thing to do, since most electric devices operate at a known frequency or range of frequencies. Based upon the operating frequency, we can then compute the skin depth, δ, in the material:
where is the operating frequency, is the permeability, is the permittivity of free space, and are the relative permeability and permittivity of the material, and is the material electrical conductivity.
For conductors, this equation can be simplified to:
Strictly speaking, the skin depth is defined in terms of the exponential decrease in the induced currents within a planar, semiinfinite conductor, but it is still a useful measure regardless of the shape of the geometry. First, we always estimate the skin depth of all of the materials in the simulation, since this value governs how we will model them. To understand this key point, let’s look at a simple case of a closed loop of wire (with a wire radius of 1 cm and loop radius of 10 cm) exposed to a uniform background magnetic field at different frequencies, as shown schematically below.
A loop of copper wire exposed to a sinusoidally timevarying magnetic field.
The above illustrated problem can be modeled using a 2D axisymmetric model, as seen below. The infinite element domain is used to truncate the modeling space for the reasons outlined in this previous blog post.
A schematic of the coil model.
Let’s look at the results at several different frequencies. The image below shows the magnitude of the current flow in the coil. At higher frequencies, we can observe how the current primarily flows near the surface of the coil. In fact, at the highest frequency, the current in the center of the coil is, for all practical purposes, zero. That is, the skin effect has shielded the inside of the conductor.
The current flow within the coil cross section at different frequencies.
To appropriately model these various cases, we also need to be aware of the finite element mesh used. At higher frequencies, as the current is being driven closer and closer to the boundaries, we need to have a finer mesh to resolve the spatial variation of the fields. However, the fields are only varying strongly in the direction normal to the boundary and varying quite gradually along the perimeter of the coil.
In these cases, it is appropriate to use the boundary layer meshing functionality, which will insert thin elements normal to the boundary, as shown in the image below. Depending upon how accurately you want to resolve the current distribution in the coil, you can adjust the thickness of the boundary layer elements to be between one half to one skin depth and use as few as two or as many as eight or more boundary layer elements. On the other hand, at frequencies that are low enough, the boundary layer meshing is not needed at all.
The mesh within the wire at varying frequencies, corresponding to the previous plots of current distribution.
Now, we can also see from the image above that at higher frequencies, the current distributions extend very negligibly into the coil interior. Therefore, for all practical purposes, we can say that at frequencies that are high enough, the currents flow on the surface. In these cases, we can use the Impedance boundary condition and not model any of the coil interior at all, as shown in the image below.
Schematic and mesh of a model using the Impedance boundary condition.
This approach saves us a significant amount of computational effort, since we now only need to mesh the air domain and apply the Impedance boundary condition. Obviously, we do lose some information here: the current distribution within the conductor. But if we do not care about that, then this approach can be a superior alternative to meshing the conductor interior. The first plot below shows the losses in the coil versus frequency, computed via the Impedance boundary condition and via a domain model of the coil with a boundary layer mesh.
Plot of the computed losses via a domain model and the Impedance boundary condition.
The next figure plots the ratio of the losses computed by using the Impedance boundary condition to the losses computed by explicitly modeling the domains, and this ratio is plotted against the ratio of the wire radius to the skin depth. As the characteristic dimension of the parts (in this case, the radius) approaches ten times the skin depth, the losses computed via these two approaches are similar.
Plot of the ratio of computed losses versus the ratio of object size to skin depth.
We can conclude from the plot above that the Impedance boundary condition gives us an accurate prediction of total losses as long as the skin depth is relatively small compared to the dimensions of the conductors that we want to model. This is an important result, as it can significantly simplify a lot of our frequencydomain modeling within the AC/DC Module.
Let’s finish up this topic by addressing timedomain simulations. The Impedance boundary condition is not available in this case, since this boundary condition is formulated based upon the frequencydomain form of Maxwell’s equations. For timedomain simulations, we must model and mesh the interior of all conductors. Boundary layer meshing is still appropriate, but you will want to adjust the thicknesses of the elements based on both the average and maximum expected frequency content of your timedomain excitations. This can sometimes lead to a much more computationally expensive model, of course, so try to model in the frequency domain as much as possible.
And what if you have materials that are nonlinear with respect to field strength or materials that you might think you have to model in the time domain? What do you do then? If you’re imagining a ferromagnetic material with a nonlinear magnetic permeability, you can still model the magnetic material in the frequency domain using the effective HB curve functionality.
Having a good understanding of how to model conductive, lossy materials interacting with timevarying magnetic fields is important for effective use of the AC/DC Module. You can either explicitly model the conductive domains or model the conductors via the Impedance boundary condition at higher frequencies. If you do model the domains explicitly, you will need to use a boundary layer mesh to give a good resolution of the currents at higher frequencies, which does increase the computational requirements. If you use the Impedance boundary condition, you introduce an approximation, but you will not need to model the interior of the conductor domains at all, saving you significant computational resources.
To learn more about the specialized features and functionality for electromagnetics modeling available in the AC/DC Module, click the button below.
To work through a set of supplemental exercises on this topic, see the following tutorials:
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MT is a passive geophysical method that generates electrical resistivity profiles or reciprocal electrical conductivity profiles for specific areas of Earth’s subsurface and crust. This technique uses the natural electromagnetic source generated by the ionosphere.
When performing MT surveys, researchers often align the electromagnetic sensors by placing them in a line that runs perpendicular to an expected fault line or other anticipated strike of the formation. By simultaneously measuring the electromagnetic field as a function of time across different points in an area of interest, it’s possible to statistically evaluate the local electromagnetic impedance as a function of frequency. Using this impedance, researchers can predict the electrical resistivity as a function of depth.
Mount Erebus, an active volcano studied with MT. Image by jeaneeem. Licensed under CC BY 2.0, via Flickr Creative Commons.
Since the rocks within Earth’s crust have different resistivities, the calculated profiles can help determine what types of rocks are located in a studied area — important information for scientists studying geological structures and processes. Besides rock composition, MT can also be used to investigate factors like porosity, permeability, and temperature.
Due to these abilities, MT has a quite a few different applications, including:
To improve MT for these applications, the technique needs to be thoroughly analyzed and optimized. This is where simulation comes into play…
The model featured in this blog post represents a 70by70km piece of Earth’s crust. It contains three “fault lines” along with three layers with different conductivities:
There are two rectangular, highconductivity inserts in the top layer of the model, where measurements are taken for realworld MT surveys. Using this model, you can estimate the conductivities of these inserts via the MT method.
Geometry for the MT model.
To represent the magnetic field source, you can use a plane wave, which generates horizontal currents throughout the model geometry. Here, the MT analysis is based on the decomposition of the incoming plane waves into two waves that have perpendicular polarizations.
Note: While we don’t go into detail about setting up and solving the model in this blog post, you can find all of that information in the Magnetotellurics tutorial model. The accompanying documentation contains stepbystep instructions and you can even download the MPHfile for this example as long as you have a COMSOL Access account and a valid software license.
The geometry and parameters for this model are taken from the study by Zhdanov et al. (ref. 1 in the model documentation) and the COMMEMI3D2 model, which is a benchmark for modeling MT.
The first results show the components of the apparent resistivity, which can be extracted from the top layer and turned into surface plots for the analyzed frequencies. For uniform halfspace or 1D models, the apparent resistivity at a specified position equals the resistivity below that point. Thus, at points where halfspace approximations are valid, the apparent resistivity values should equal the material resistivity. This is because the skin depth is small at high frequencies. Below, you can see that locations with uniform resistivities that are far from fault lines have an apparent resistivity nearly equal to the resistivity of the material underneath them.
Top views of the logarithms of the apparent resistivity in the 3D domain.
Moving along, the next plot shows data from a line that crosses the fault lines in the model for two frequencies: 0.1 and 0.01 Hz. The results generated here are comparable with the reference literature. However, at lower frequencies, an increased discrepancy between apparent and material resistivity occurs.
Apparent resistivity across strikes for frequencies of 0.1 Hz and 0.01 Hz.
Lastly is the tipper plot, where nonzero values indicate that there is a large variation in resistivity between the two sides of a fault line. By moving in the direction of the field vector at a certain phase value, engineers can use the tipper sign to determine if they are transitioning from a region with higher resistivity to a region with lower resistivity (where the tipper is positive) or vice versa (where the tipper is negative).
Tipper plot for a frequency of 0.01 Hz.
For simplicity’s sake, the model highlighted here only solves for two frequencies. However, MT data analysis is often gathered at frequencies ranging from 0.1 mHz to 10 Hz. It’s possible to add more frequencies to this study by adjusting the model size and mesh.
For more information about this example (or to try it yourself), click the button below. This will take you to the Application Gallery, where you can download the model documentation and MPHfile for the Magnetotellurics model.
EBGs are periodic structures that pass or stop electromagnetic waves at certain frequency bands, a useful ability for a wide range of applications. Some examples include increasing the isolation between antenna arrays in highfrequency designs, improving the signaltonoise ratio in highprecision GPS, and acting as a band reject for ultrawideband devices. Similar to frequency selective surfaces, these band gaps also help to reduce issues from EMI as well as improve EMC for highspeed communication systems. In addition, the shielding property of these structures could potentially be used to reduce the specific absorption rate from mobile phones in the human brain.
The specific absorption rate (surface) and temperature variation (contour).
Frequency selective surface using complementary split ring resonators.
The typical process for designing EBGs for these and other applications includes the following steps:
However, there’s a problem with this approach: The properties of an EBG are slightly different in a finite space as compared to an infinite space. Since applying the wrong frequency, polarization, or coupling plane configuration can increase the unexpected coupling between antennas, it’s important to analyze EBG structures in real space…
To examine the decoupling effect of an EBG structure, you can use the RF Module and COMSOL Multiphysics. To start, it’s important to simulate antennas without a band gap, as this provides a baseline for the performance of an antenna with a band gap. EBGs are usually placed between antenna arrays, but for simplicity’s sake, only two antennas are included here. The antennas are made of metal strips that are fed by a coaxial cable and sit above a dielectric substrate and a ground plane. While these antenna elements aren’t exactly typical, they highlight the effectiveness of the isolation from the EBG structure.
Geometry of the EBG model.
After running the model with just antennas, it’s time to add the EBG. The structure has a band gap centered at 1.85 GHz and consists of what look like small metallic mushrooms. One row of these “mushrooms” is placed between the antennas. Note that this model is run over a range of frequencies around the band gap to visualize the change in S_{21}, which describes the amount of coupling between two ports connected to each end of the coaxial cables. Observing this Sparameter helps to determine the receiving antenna isolation from the excited source antenna.
First up are the results for the frequency response of the coupling between the two antennas. As can be seen below, the isolation measured from S_{21} is much better when adding the EBG. While the band gap (2.2 GHz) is not quite at the frequency determined by the analysis (1.85 GHz), this is likely because only one row of EBG structures is used.
The frequency response of the coupling between antennas with and without the EBG.
With the EBG, the coupling is actually stronger in some areas of the frequency band and the decoupling bandwidth isn’t as wide. The decoupling center frequency and bandwidth depend on the coupling plane configuration with respect to the polarization as well as the number of EBG elements — so, as we can see from this example, including an EBG doesn’t always increase isolation.
The electric field around the antennas with (left) and without (right) the EBG structure.
With RF simulation, engineers can gain valuable insight into the effectiveness of an EBG structure in suppressing a surface wave and decoupling antennas. Using this information, they can then optimize the structure’s design for a specific application.
Interested in trying this model for yourself? Click the button below to head to the Application Gallery, where you can find the tutorial documentation for this example. There, you can also download the MPHfile if you have a valid software license and COMSOL Access account.
If you lived in Glasgow in the 1830s, one of the best ways to get around Scotland was a horsedrawn flyboat on the Glasgow, Paisley and Ardrossan Canal. Flyboats were lightweight, long, and narrow. They were pulled through the shallow water of the canal by horses.
A modern view of Glasgow, Scotland, (left) and a canal in central Scotland, home of the rotating boat lift known as the Falkirk Wheel (right).
One day, William Houston, an owner of one of the flyboat companies, was traveling down a canal when the horse pulling his flyboat was startled and bolted. Houston noticed something strange: The water exhibited no resistance. The ship glided fast (similar to what we now call “hydroplaning”), and the turbulence from the boat’s movement didn’t damage the shores of the canal. (Ref. 1)
Enter John Scott Russell, a naval architect from Glasgow. When he heard about the phenomenon Houston observed, he thought that witnessing it for himself could give him insight into boat design. While observing the canal, one of the horses suddenly stopped. A wave formed under the middle of the boat, moved to the prow, and then took off past the boat entirely. Scott Russell followed the wave, first on foot and then on his horse. He was astounded to see that the wave kept going at the same size and pace. He later called it “the wave of translation” and described the event as “the happiest day of my life.” (Ref. 2)
John Scott Russell in 1847. Image in the public domain in the United States, via Wikimedia Commons.
Scott Russell devoted two years to replicating the wave of translation so he could study it further. He even built a 30foot basin to test his different theories. Eventually, he observed some unique properties of these waves, which he now called solitary waves. According to Scott Russell, a solitary wave:
Animation showing the behavior of a solitary wave.
He also categorized these waves into four types:
Scott Russell presented his findings at a British Science Association meeting in Edinburgh, describing the waves and the mechanics behind them. His work contained a few misinterpretations of the foundations of mechanical laws — understandably so, as his background was in architecture, not physics. Scientists at the time balked at his theories and distrusted his lack of engineering and physics expertise.
And so began a lifetime — actually, many lifetimes — of investigating the behavior of solitary waves.
At first, Scott Russell didn’t earn many fans in the scientific community regarding his theory of solitary waves. George Biddell Airy, who studied the behavior of waves in relation to the tides, did not hold Scott Russell in high regard and believed that solitary waves contradicted PierreSimon Laplace’s theory of fluid mechanics. Laplace’s equations, as we know, are integral to the study of fluid dynamics today. (Ref. 1)
George Gabriel Stokes, of the famed NavierStokes equations, also did not support the possibility of solitary waves and initially condemned their relation to tides and sound. Over time, as he researched finite oscillatory waves, he changed his stance and admitted that a theoretical solitary wave was plausible. (Ref. 1)
Scott Russell never gave up the subject of solitary waves, but he continued what his main job description entailed: building ships. He used his research into waves to design a special ship prow, which was based on the shape of a versed sine wave, that could better handle water resistance.
In the 1850s, Isambard Kingdom Brunel asked for Scott Russell’s help regarding a steamship called the Great Eastern. Brunel had designed the ship, which was slated to be the largest ship of its time (he affectionately called it his “Great Babe”). The ship could purportedly travel from England to Australia without needing to refuel.
The Great Eastern before its first launch. Image in the public domain in the United States, via Wikimedia Commons.
Scott Russell was successful in taking Brunel’s designs and building the Great Eastern, which eventually made many transatlantic voyages. However, his success was tarnished when Brunel passed away shortly before launch and a major accident occurred during the ship’s maiden voyage. Around this time, Scott Russell was also having trouble with his finances and was dealing with the seizure of his assets.
Toward the end of his life, Scott Russell’s solitary wave was still rebuked by scientists, and he was bankrupt. Fortunately, things were about to change.
Joseph Valentin Boussinesq, a protégé of Adhémar Jean Claude Barré de SaintVenant known for his contributions to the field of fluid dynamics, did not write off Scott Russell like others at the time. Instead, he subjected every aspect of waves and tides to mathematical analysis. In 1872, Boussinesq attempted to explain shallow water waves, which led to an equation that proved solitary waves are theoretically possible. He even mentioned Scott Russell in his paper on the subject in 1877. Lord Rayleigh independently developed similar theories on waves and also supported Scott Russell. (Ref. 1)
Finally, two scientists had spoken out in support of his work! Then, in 1882, Scott Russell passed away at the age of 74 on the Isle of Wight, England. But the story doesn’t end there.
In 1895, Diederik Korteweg and Gustav de Vries expanded on Boussinesq’s work and developed an equation that proves solitary waves are theoretically possible. The KdV equation doesn’t introduce dissipation, which means it can be used to describe waves that travel long distances while retaining their shape and speed. The equation is also simpler than Boussinesq’s version and gives a better solution. (Ref. 2) Because of these advantages, the KdV equation is still used to understand wave behavior — in its many forms — today.
Research into solitary waves picked up in 1965, when researchers Martin Krustal and Norman Zabusky studied the KdV equation in more detail. They found that solitary waves can occur not just theoretically but also naturally, coining the term solitons to describe them. In addition, solitons were not just thought of in the context of water waves anymore — with research into applications for optics, acoustics, and other areas.
Krustal, along with researchers Gardner, Greene, and Miura, developed the inverse scattering transform in 1967. This method can be used to find the exact solution of the KdV equation and also demonstrates the elastic collisions between waves, originally observed by Krustal and Zabusky. (Ref. 2)
A hydrodynamic soliton. Image by Christophe.Finot et Kamal HAMMANI. Licensed under CC BYSA 2.5, via Wikimedia Commons.
Moving forward, scientists Zakharov and Shabat used a formulation developed by Peter Lax to solve the nonlinear Schrödinger equation, which describes the evolution of the slowly varying envelope of a general wave packet. Ablowitz, Kaup, Newell, and Segur later came up with a more systematic approach to solving the nonlinear Schrödinger equation, which is now known as the AKNS method. (Ref. 2)
All of this mathematical activity around solitons caught the attention of the scientific community in a way that Scott Russell couldn’t. Over the next 30 years, solitons in a wide range of different fields were researched, including geomechanics, oceanography, astrophysics, quantum mechanics, and more.
Optical fibers are an important and practical application area for solitons. The linear dispersion properties of a fiber level out a soliton, while its nonlinear properties help the soliton achieve a focusing effect. The result is a very stable pulse that can travel for what seems like forever.
The behavior was first observed by a group at Bell Labs led by Linn Mollenauer in the 1980s that aimed to apply solitons to longdistance telecommunication systems. In the 1990s, a team of MIT researchers added optical filters to a transmission system in an attempt to maximize an optical soliton’s propagation distance. Using this method, Mollenauer’s group sent a 10Gbit/s signal more than 20,000 km — impressive for the time. (Ref. 2) During the 2000s, optical soliton research ventured into the field of vector solitons, which have two distinct polarization components.
Current research from Lahav et al. aims to create solitons that are stable in all three dimensions, known as “light bullets”. This requires the simultaneous cancellation of diffraction and dispersion, which has been achieved with a highly structured material, but not with an unstructured one that can be used in practical applications. The Lahav group has investigated the fundamental properties of 3D solitons to develop more technological applications of solitons for fiber optics. They have also developed a method that involves shining a repetitive string of light pulses into a special material called “strontium barium niobate” to create a selfguided beam and cancel out the dispersion. This method creates a string of 3D solitons that could potentially be used in advanced nonlinear optics and optical information processing. (Ref. 3)
The solution to the KdV equation tells us that the speed of a soliton determines its amplitude and width. By investigating this effect, we can better predict the behavior and limitations of solitons for optical applications. Simulation can be used to visualize soliton behavior beyond numerical equations, without setting up resourceintensive or costly optical experiments. Besides demonstrating how speed influences the amplitude and width of a wave, simulation also shows how solitons collide and reappear while maintaining their shape (like the solitary waves Scott Russell observed “overtaking” each other in the ocean).
Simulation results that show solitons colliding and reappearing. Image from The KdV Equation and Solitons tutorial model.
Predefined physics settings are an efficient and easy option for straightforward modeling tasks; however, optical solitons are anything but. Equationbased modeling enables you to expand what is normally possible with simulation for problems that require flexibility and creativity. Using equationbased modeling, you can seamlessly implement the KdV equation into the COMSOL Multiphysics® software by adding partial differential equations (PDEs) and ordinary differential equations (ODEs). You can even create a physics interface from your custom settings so that you don’t have to start from scratch the next time you need to set up a model.
Adding a PDE to a soliton model in the COMSOL Multiphysics graphical user interface (GUI), an example of equationbased modeling.
Equationbased modeling functionality makes it possible to simulate an initial pulse in an optical fiber as well as the resulting waves or solitons.
In 1885, Scott Russell’s book The Wave of Translation in the Oceans of Water, Air and Ether was published posthumously. It included his speculations on the physics of matter and how we can find the depths of the atmosphere and universe by computing the velocity of sound and light, respectively. (Ref. 4) Even at the end of his life, Scott Russell continued to theorize on how we can apply mathematics to the observable world as well as the significance of solitons in modern physics. If only he could have seen the development of the KdV equation or recent advancements in optics.
One of Scott Russell’s supporters, Osborne Reynolds, made a fitting observation in his own research of solitary waves: In deep water, groups of waves travel faster than the individual waves from which they were made. (Ref. 1) Perhaps we can think of John Scott Russell as the individual wave, inspiring others to keep moving toward a common goal.
Learn more about equationbased modeling, and the other features and functionality in COMSOL Multiphysics, by clicking the button below.
An example of a moderately complex optical system is a Petzval lens with a fieldflattening lens. A multielement lens system with a focal length of 100 mm and a focal ratio of approximately f/2.4 is shown in the figure below.
A Petzval lens that includes a fieldflattening lens.
The optical prescription of this camera lens (as described in Ref. 1) is given in the following table. The geometry consists of two doublet lenses together with a fieldflattening lens. Altogether, the rays refract across eight different surfaces. Two additional surfaces for the aperture stop and image plane are also shown.
Surface Index

Element Number

Name

Radius of Curvature (mm)

Thickness (mm)

Clear Radius (mm)

Material


0

N/A

Object

∞

∞

N/A

N/A

1

1

Lens 1

99.56266

13.00000

28.478

SBK7

2

2

Lens 2

86.84002

4.00000

26.276

SBASF12

3

N/A

N/A

1187.63858

40.00000

22.020

N/A

4

3

Stop

∞

40.00000

16.631

N/A

5

4

Lens 3

57.47191

12.00000

20.543

SSK2

6

5

Lens 4

54.61865

3.00000

20.074

SSF5

7

N/A

N/A

614.68633

46.82210

16.492

N/A

8

6

Lens 5

38.17110

2.00000

17.297

SSF5

9

N/A

N/A

∞

1.87179

18.940

N/A

10

N/A

Image

∞

N/A

17.904

N/A

Now, before we can perform a ray optics simulation with COMSOL Multiphysics and the addon Ray Optics Module, it is necessary to create the model geometry. In the following section, we discuss some options for doing this.
There are a variety of ways to create a model geometry in COMSOL Multiphysics. One possibility is to use combinations of builtin geometry primitives such as spheres, cylinders, polygons, and line segments. You can combine these simple geometric entities into more specialized shapes using Boolean, partition, transformation, and conversion operations.
Various geometry primitives (left) and Boolean and partition operations (right).
It will quickly become apparent that the creation of even the simplest optical geometry (such as a single lens) requires a significant number of operations. The full model geometry might require the same sequence of operations to be repeated several times with different numeric values. For example, different lenses might require the same steps but with different surface radii of curvature. To avoid setting up the same geometry sequence several times in a row, you can define some Geometry Parts. These parts are parameterized subsequences that can be called as if they are geometry primitives. By this means, each repetition of a lens can be accomplished using a Part Instance node, where the input parameters are the relevant details from the optical prescription.
An example of a userdefined geometry part can be seen below. In this case, the part defines a single lens with arbitrary front and rear radii of curvature as well as various definitions for the full, surface, and clear aperture diameters.
An example of a userdefined geometry part.
The Ray Optics Module includes a set of predefined parts for some of the most common shapes used in optical simulation, including a lens similar to the one shown above. These are available in the Part Libraries. The Part Library available in the Ray Optics Module is discussed in the following section.
A variety of 2D and 3D parts have been created for use in the Ray Optics Module. One of the more advanced examples is the Spherical General Lens 3D (shown below in the left image). This part can be used to create a wide variety of lens forms, including any combination of concave and convex lens surfaces. Indeed, it is flexible enough that it can be used for all of the lenses in both a Petzval lens as well as a double Gauss lens. The Circular Planar Annulus part is also shown below. This part is a convenient way to create aperture stops and to define the image plane.
Examples of parts in the Part Library in the Ray Optics Module: Spherical General Lens 3D (left) and Circular Planar Annulus (right).
Parts found in the Ray Optics Module’s Part Library often contain one or more variants. These variants are intended to accommodate the use of the same basic part in a variety of ways. For example, the Spherical General Lens 3D can be specified either in terms of the clear aperture diameter or in terms of the clear aperture fraction. Other parts have variations that permit specifying either the central thickness or the edge thickness. It might also be possible to give the effective focal length (and refractive index) in place of the radius of curvature.
We now have almost everything in place to create the Petzval lens model geometry. We proceed by inserting instances of the Spherical General Lens 3D and subsequently entering the geometry parameters (that is, the radii of curvatures, central thickness, and diameters) directly from the optical prescription table. However, this process can be simplified by reformatting the lens prescription so that it can be loaded in its entirety (for example, from a text file) into the Global Definitions node of the current model. That way, if you wish to adjust the optical prescription later, you can make all of the changes in one place. The optical prescription parameters for the Petzval lens can be seen below.
The optical prescription parameters for the Petzval lens model.
Having already loaded the two geometry parts from the Part Library (that is, the Spherical General Lens 3D and the Circular Planar Annulus parts), we insert the first element of the Petzval lens (“Lens 1″). The optical prescription parameters we defined above can be used as input parameters for the Part Instance node. Each lens parameter has been given a unique name (such as “Tc_1″, “R1_1″, “R2_1″, etc.) so that these values can be entered directly into the Expression fields. This process can also be simplified by loading a text file that contains definitions of the relevant expressions for each lens element.
The first element (Lens 1) of the Petzval lens model geometry.
Note that in addition to allowing definitions of the surface clear diameters, the Spherical General Lens 3D part has parameters that can be used to specify the physical diameters of each surface as well as the overall lens diameter. This capability makes it possible to construct a geometry that closely matches the geometry that will be manufactured. For example, this makes it easier to set up a highfidelity model that includes the structural and thermal effects on the lenses and the surrounding barrel.
Next, as shown below, we insert a second instance of the Spherical General Lens 3D, which is used to define “Lens 2″. This element must, of course, be placed in the correct absolute location within the model geometry. A standard optical prescription usually specifies the distance from the entrance surface of the current lens to the exit surface of the preceding lens. Therefore, the definition of the Spherical General Lens 3D part includes work planes that locate the entrance and exit vertices of each lens surface. Each subsequent lens part instance can then be placed with respect to the work plane located on the exit vertex of the preceding lens. As demonstrated below, “T_1″ is the distance between the exit surface of Lens 1 and the entrance surface of Lens 2. (In this case, the separation happens to be 0. Therefore, a doublet lens is created, but the same principle applies for any given separation.)
The second element (Lens 2) of the Petzval lens model geometry.
Along with the lens elements, additional surfaces can be added to the Petzval lens model geometry that define the aperture stop and the image plane. For convenience, the complete model geometry can be saved as a geometry sequence so that it can be used in other COMSOL Multiphysics simulations. The full model (neglecting stray light apertures) is shown in the figure below. Here, we see that a unique feature of the Ray Optics Module is its ability to perform a ray trace on a fully meshed geometry. The same mesh can be used to solve for other quantities like temperature and structural displacement in the lens geometry, streamlining the setup of bidirectionally coupled multiphysics models.
The completed model geometry for the Petzval lens, showing the surface mesh elements.
Other features can be built into the definition of the parts. For example, work planes have also been defined in the Spherical General Lens 3D part and indicate the location of the front and rear edges of the lens. These work planes can be used to precisely locate apertures for blocking stray light or to locate mechanical mounting features. In addition, selections have been defined, separating the clear apertures, obstructions, and edges of each lens part. These selections can be used to quickly define the various physics features required for a geometrical optics simulation, as seen in the figures below.
The selections used in the Petzval lens to define the clear apertures (left) and the obstructions (right).
For this ray tracing example, we generated results via a gridbased ray trace at a single wavelength (550 nm) and field angle (onaxis). The results shown below make use of custom color expressions, added during postprocessing for better visualization. The color expression of the left plot (the ray trace) is based on the ray locations on the image plane. The color expression of the right plot (the spot diagram) is based on the ray locations at the entrance pupil. Each of these color expressions makes it possible to easily visualize the relative contribution of the rays at the entrance pupil to the final image quality.
Ray trace (left) and spot diagram (right) of the Petzval lens.
The example featured above demonstrates how to create an optical geometry prior to performing a ray tracing simulation. To try this model yourself, click the button below. This takes you to the Application Gallery, where you can download the MPHfile if you have a COMSOL Access account and valid software license.
Helical antennas are named for their spiral geometry, consisting of one or more conducting wires wound in a helix. Due to their shape, helical antennas are able to emit circularly polarized fields. Their design is simple but effective and can be applied in a variety of ways, including in highly compact antennas for use in smart implants and other RFID devices.
Larger helical antennas are used in radio, GPS, and ballistic missile system applications, as well as extraterrestrial communications with satellites and space probes orbiting Earth and the moon.
A helical antenna used for SatCom. Image in the public domain in the United States, via Wikimedia Commons.
One current project is boldly taking a helical antenna where no helical antenna has gone before: Mars. NASA’s “Interior Exploration using Seismic Investigations, Geodesy and Heat Transport (InSight)” mission aims to study Mars’ interior structure, including its crust, mantle, and core. The Mars lander will gather surface data, such as temperature and heat flow, to provide scientists with more insight into how rocky planets form. Among other instruments that measure and transmit information, the lander is equipped with a helical UHF antenna, which will be used for communication with orbital relay spacecraft.
An artistic depiction of the InSight Mars lander. Image by NASA/JPLCaltech. Image in the public domain in the United States, via Wikimedia Commons.
Optimizing helical antenna designs for these applications requires an understanding of their operating modes. This can be accomplished using RF simulation.
Helical antennas can be set up with two arms (conducting wires) to account for their two major operating modes:
Similar to a monopole antenna, the normalmode antenna is linearly polarized, but because it’s in a helical shape, it’s shorter and more compact. The antenna is considered a normalmode helix when the circumference of the helix is significantly less than the wavelength and its pitch is significantly less than a quarter wavelength. In the normal, or perpendicular mode of radiation, the antenna’s farfield radiation pattern is similar to the torusshaped pattern of the classic dipole antenna.
In the axial, or endfire, mode of radiation, the antenna radiates circularly polarized waves. One of the benefits of circularly polarized waves is that they are less vulnerable to multipath fading and have less polarization dependency than linearly polarized waves. The antenna is considered an axialmode helix when the helix circumference is near the wavelength of operation. The helical antenna is at a much higher frequency band than at the normal mode, working similarly to an endfire array along the helical axis, generating a directive radiation pattern.
An advantage of helical antenna designs with two arms is the ability to match impedance. With a singlehelix antenna, the impedance is much lower when it is resonant at the normal mode. If you add a second antenna that is shorted to the ground, it acts as a folded antenna that has an input resistance four times bigger than a dipole antenna of the same size, which means that you can adjust the low impedance close to the reference impedance of the coaxial cable, 50 Ω.
A model of a twoarm helical antenna and its axial mode radiation pattern. Only a half of the radiation pattern is visualized.
This example of a twoarm helical antenna is modeled using the RF Module, which is an addon to the COMSOL Multiphysics® software.
As shown below, the model geometry includes a twoarm helix radiator, a circular ground plate (shown in blue), a tuning stub, a coaxial cable, and a perfectly matched layer (PML) enclosing the air region. In addition, two helix structures are wound along the zaxis and meet at the top end.
For this example, all metal parts are modeled as perfect electric conductors (PECs) and the space between the inner and outer conductor of the coaxial cable is filled with polytetrafluoroethylene (PTFE). To excite the antenna, a coaxialtype lumped port is used. In addition, all domains are meshed (except the PML) by a tetrahedral mesh with around five elements per wavelength, and the PML is swept along the absorbing direction automatically through the physicscontrolled mesh.
In addition to the second antenna, which helps adjust for impedance, you can also add an impedancematching stub for the axial mode located at the center of the ground plate. Note that the ground plate, the PML sphere shell, and the maximum mesh size are adjusted automatically as a function of wavelength for each operational mode.
The Sparameters and farfield patterns are calculated at both of the operating modes: 0.385 GHz at the normal mode and 4.77 GHz at the axial mode. The results shown below plot the logscaled electric field magnitudes for each mode. You can see a difference in field intensity around the antenna between the normal mode (left) and the axial mode (right).
Logscaled electric field magnitude around the antenna at 0.385 GHz (normal mode, left) and 4.77 GHz (axial mode, right).
Next, let’s take a look at a polar plot of the 2D radiation patterns in the yzplane. This plot shows both modes of operation. As expected, you can see a classic Eplane pattern of a dipole antenna at the normal mode (blue) and a directional radiation pattern at the axial mode (green).
Polar plot of the farfield pattern in the yzplane, with the normal mode shown in blue and the axial mode shown in green.
You also get a visualization of the radiation patterns in a 3D farfield radiation plot for each mode. The Sparameters for both modes are less than 10 dB. When you look at the 3D farfield patterns, these results once again verify the dipole antenna shape for the normal mode and the endfire array shape for the axial mode.
The 3D farfield pattern at the normal mode (left) resembles that of a dipole antenna. The 3D farfield pattern at the axial mode (right) resembles that of an endfire array antenna along the zaxis, backed by a ground plane.
Below, the axial ratio plot shows how much the antenna is circularly polarized. When it is characterized as the perfect circular polarization, the axial ratio is 1 or 0 dB. When it is below 3 dB (inside the red circle), it is typically regarded as a circular polarization. In the figure, the axial ratio is lower than 3 dB at the antenna boresight that is the major direction of propagation of the axial mode, parallel to the helical twist axis.
Axial ratio in dB scale (blue) plotted with 3 dB line (red).
By modeling a twoarm helical antenna, you can effectively analyze the normal and axial operating modes, which can help you improve antenna designs for terrestrial and extraterrestrial applications.
Want to try modeling a helical antenna? Get started by clicking the button below. This button takes you to the Application Gallery, where you can use your COMSOL Access account to download the MPHfile (with a valid software license).
To learn more about antenna modeling, check out these blog posts:
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One of the basic tasks of any navigation system is to keep track of an object’s position and orientation, as well as their rates of change. Extreme accuracy may be required, particularly in space travel. For example, a communications satellite can be sensitive to angular velocities as small as one thousandth of a degree per hour.
While this accuracy requirement may seem daunting, this fundamental task of attitude control can be posed as a simple question: How do I determine how fast I’m spinning, and about what axis?
In principle, this task is the same for any observer in any rotating frame of reference — even a guest in the revolving restaurant shown below.
A photograph of the revolving restaurant at Ambassador Hotel, the oldest revolving restaurant in India. Image by AryaSnow — Own work. Licensed under CC BYSA 4.0, via Wikimedia Commons.
Suppose you’re the guest in such a rotating restaurant, trying to determine its angular velocity, Ω (unit: rad/s).
The simplest approach is to look outside. Pick a stationary object, like a building or a tree, and see if its location is changing over time in your field of view.
The above image shows the position of a tree in the observer’s field of view (i.e., through the window) at an initial time t_{1} and a later time t_{2}. Let θ (unit: rad) be the angle between these two lines of sight. If the tree is very far away compared to the size of the restaurant, you could estimate the angular velocity as
Because space travel is so much more demanding than the restaurant example described above, there are a few caveats to consider. In space, the idea of a “stationary object” is a bit tricky. For example, when using a Sun sensor for attitude control of a satellite in a geostationary orbit, you must also account for the relative motion as the earth orbits around the sun. A star sensor, on the other hand, can be extremely accurate because stars other than the sun can be considered fixed in space for many purposes, and because a star more closely approximates a point of light rather than a continuous source over some finite angle.
Because of the accuracy requirements of spacecraft attitude detection and control, the finite size of the object being observed must also be considered. For a line of sight to the sun, for example, you need to know what part of the sun you’re looking at. For arbitrary rotation in 3D, at least two objects are needed, because you don’t always know how the axis of rotation is oriented.
Next, suppose we’re back at the restaurant, except that all of the windows are covered. Because your view of the outdoors is blocked, you can’t rely on any stationary object to get information about your rotating frame of reference.
There are several experiments you could perform within your rotating frame of reference to determine its angular velocity. For example, you could put a ball on the floor and see if it rolls, presumably due to a centrifugal force. (This requires that you know where the axis of rotation is — not always a given in space travel!) Another approach would be to use a mechanical gyroscope.
A third approach, explained in the following section, is to exploit the unique properties of light; namely, its uniform speed in a vacuum in all frames of reference. When light propagates in a rotating frame of reference, it reveals a phenomenon known as the Sagnac effect. A ring laser gyroscope takes advantage of this effect. Such gyroscopes have become a popular alternative to traditional mechanical gyroscopes, which use rotating masses, because the ring laser gyro has no moving parts, thus reducing the cost of maintenance.
The easiest way to visualize the Sagnac effect is to consider two counterpropagating light rays — that is, two rays going in opposite directions — that are constrained to move in a ring. The ring is rotating counterclockwise with a constant angular velocity Ω. (The SI unit is radians per second, but for inertial navigation systems, we might work in degrees per hour instead.)
The two rays are initially released at a point P_{0} along the ring. The rays go around the ring at the speed of light in opposite directions, while the release point rotates with the frame of reference. By the time the clockwise ray returns to the release position, it has moved to a new location shown by P_{1}, and the distance it has traveled is somewhat less than one full circle. By the time the counterclockwise ray returns to the release position, it has moved to a different location, P_{2}, and the distance it has traveled is greater than one full circle.
Of course, the movement shown here is greatly exaggerated. In reality, the displacement of P_{1} and P_{2} from P_{0} (and from each other) might be 10 billion times smaller! Even then, the tiny difference in the distance traveled (and similarly, the transit time) between the two rays is detectable because it’s accompanied by a phase shift, which produces an interference pattern between the rays. If we let ΔL represent the difference in the distance traveled by the two rays, then
(1)
where A is the area of the ring and c_{0} = 299,792,458 m/s is the speed of light in a vacuum.
As it turns out, Eq. (1) isn’t just true for circular paths, but for other shapes as well. The optical path difference only depends on the area enclosed by the loop and not by its shape. A more general derivation of Eq. (1) can be accomplished using the principles of general relativity. At its core, the Sagnac effect is a relativistic phenomenon, for which a classical derivation gives the same results to first order. For a more rigorous application of the theory, see Refs. 1–2.
In this section, we examine a model of a basic Sagnac interferometer. This shares the same fundamental operating principle as the ring laser gyro, but is simpler to set up because we don’t need to consider the presence of a lasing medium along the beam path. (Besides the intensity gain, such a lasing medium can introduce many other complications, such as dispersive effects, that we can ignore for illustrative purposes.) However, a Sagnac interferometer with a given geometry will introduce the same optical path difference and phase delay as a ring laser gyro with the same arrangement of mirrors, so we can still learn quite a lot from it.
The basic Sagnac interferometer geometry consists of a beam splitter, two mirrors, and an obstruction to absorb the outgoing rays. It is illustrated below.
A few geometry parameters for this model are tabulated below.
Name  Expression  Value  Description 

λ_{0}  N/A  632.8 nm  Vacuum wavelength 
R  N/A  10 cm  Ring radius 
b  17.3 cm  Triangle side length  
P  52.0 cm  Triangle perimeter  
A  130 cm^{2}  Triangle area 
The geometry is sometimes designed in a square rather than a triangle, with mirrors at three vertices and a beam splitter at the other. Rays are traced through the system with directions indicated by arrows. Because the whole apparatus is rotating counterclockwise, the rays going counterclockwise propagate a slightly longer distance than the rays going clockwise, before reaching the obstruction.
To better visualize this phenomenon, see the two animations below. (Note again that the rotation here is exaggerated by a factor of about ten billion!)
In the left animation, the observer stands in an inertial (nonaccelerating) frame of reference. Thus the rays go along straight paths but they hit the mirrors at different times. In the right animation, the observer is “riding” the spacecraft and is thus in a noninertial frame. (Strictly speaking, even in this rotating frame, the counterpropagating rays go at the same speed; the speed of light is the same in any frame of reference!)
For the geometry parameters given above, applying Eq. (1) gives the optical path difference between the counterpropagating rays as about 8 × 10^{16}, or 0.8 femtometers. This is about the radius of a proton; clearly a difficult quantity to measure! Rather than report path length directly, Sagnac interferometers and ring laser gyros usually report the frequency difference or beat frequency Δν, given by
(2)
where ν (in Hz) is the frequency of the light, and L is the optical path length for light going around the perimeter of the triangle.
Note that L is not necessarily the perimeter of the triangle itself, since there might be a comoving medium such as a lasing medium with n ≠ 1 along the beam path. In this example, we assume the space between the mirrors and beam splitter is a vacuum. The beat frequency is on the order of 1 Hz, which is certainly much easier to measure than a distance equal to the proton radius.
This model uses the Geometrical Optics interface to trace rays through the Sagnac interferometer geometry. The two mirrors are given the dedicated Mirror boundary condition, which causes specular reflection. The beam splitter uses the Material Discontinuity boundary condition with a userdefined reflectance of 0.5, so that both of the counterpropagating beams have the same intensity.
To rotate the apparatus, use the Rotating Domain feature, as shown below.
The resulting plot shows the rays propagating in both directions through the system of mirrors, but because the mirrors move so slowly relative to the speed of light, the two paths are indistinguishable in this image. If we zoomed in by a factor of 10 billion or so, we’d be able to discern two triangles spaced a tiny distance apart.
In the following plot, the beat frequency is given as a function of the angular velocity of the interferometer. As expected from Eqs. (1)–(2), this relationship is linear. Some numerical noise is visible in the bottomleft corner of the plot. This is due to numerical precision and is explained in greater detail in the model documentation.
The Sagnac interferometer described above, along with related devices like ring laser gyros and fiber optic gyros, are examples of inertial navigation systems, which predict an object’s position and orientation by starting from a known position and then integrating the translational and angular velocity over time. In practice, inertial navigation systems are usually combined with absolute measurements of position and orientation relative to some other object in space. This absolute measurement might be done with an Earth sensor, Sun sensor, or star sensor; with RF beacons at known locations on the earth’s surface; with measurements of the earth’s magnetic field; or with any combination of these.
The uncertainty of an inertial navigation system grows over time due to small errors in the measurement of the translational and angular velocity. Periodically taking an absolute measurement using one of the sensors described above resets this uncertainty to a more reasonable value. A prediction of the uncertainty over time might look like the following graph.
We’ve successfully demonstrated the Sagnac effect in a simple interferometer using ray optics simulation. The resulting beat frequency agrees with the more rigorous theory, which is based on general relativity, as long as the velocity of all of the moving parts is much smaller than the speed of light. The magnitude of the optical path difference due in a Sagnac interferometer or ring laser gyro depends only on the area enclosed by the counterpropagating beams, not on the geometry of the loop.
Explore the Sagnac interferometer model by clicking the button below. Once in the Application Gallery, you can log into your COMSOL Access account and download the MPHfile (with a valid software license) as well as a tutorial for this model.
RFID technology is common in many industries. In healthcare, though, there is a major design challenge: size. RFID tags on the smaller end are about the size of a grain of rice, but for cellularlevel uses (such as research and diagnostics), the designs need to be even smaller.
A group of Stanford researchers developed an RFID tag that is small enough to be implanted in a cell, such as a skin or cancer cell. The size of the tag is about onefifth the thickness of a human hair. It works in conjunction with a specialized RFID reader that interprets the data and monitors the cell’s activity in real time. These tiny RFID tags also have the potential to be linked to sensors for advanced biomedical treatment, such as antibody detection and the destruction of cancer cells.
A surgeon implants an RFID microchip into the hand of a doctor. Soon, these tags could be implanted into single cells. Image by Paul Hughes — Own work. Licensed under CC BYSA 4.0, via Wikimedia Commons.
No matter how good a doctor’s bedside manner is, patients don’t particularly enjoy being poked and prodded in order to get their vital signs taken. At Cornell University, researchers design ultrahigh frequency (UHF) RFID tags that can be used to monitor vital signs, such as heart rate, breathing, and blood pressure, without even touching a patient. The tags, which can be put into hospital wristbands or even sewn into clothes, communicate with an RFID reader that monitors multiple people in range simultaneously. The system relies on backend software to manage, interpret, and monitor the data. This way, doctors get a clearer picture of each patient’s vital system performance, medical professionals save time and energy taking vitals, and patients are happier.
“Smart fabrics” are one potential application area of RFID systems. Image by Joshua Dickens. Licensed under CC BYSA 2.0, via Wikimedia Commons.
Sleep disorders and sleep apnea are areas of biomedicine that often go untreated. Although they can lead to myriad health and safety issues, overnight sleep tests are costly and disruptive to a patient’s schedule, and athome tests can be difficult to use. (I’ve personally undergone an athome sleep test and it was extremely uncomfortable and awkward to strap the system around my torso, tape the breathing tubes to my face, and keep the monitor on my finger from falling off.)
To offer help in this area, researchers from RADIO6ENSE, the University of Palermo, and the University of Roma developed a passive RFID system for tracking sleep patterns remotely and in real time. The userfriendly passive RFID system involves RFID tags sewn into pajamas that don’t require any batteries and operate on a low power level, making them safe to use as wearables while they accurately collect sleep pattern data.
Electromagnetic interference (EMI) and electromagnetic compatibility (EMC) are common phenomena in electromagnetics applications and can be analyzed with, for instance, EMC/EMI testing.
An anechoic chamber is one test facility used to test antennas for EMI/EMC.
EMI is of particular concern when it comes to RFID tags for biomedical applications, as an undesired mutual inductance can occur between devices and negatively affect performance, operation, and reliability. For instance, 2011 research from the National Center for Biotechnology Information says that RFID systems could be affected by contact with water, metal, or other devices (which is completely plausible in a medical scenario) — or RFID tags could negatively affect other medical devices. Additionally, a 2017 FDA report on RFID warns that EMI is a potential hazard when RFID systems interact with other medical devices.
When it comes to a patient’s wellbeing and safety, “potential hazard” is not a phrase healthcare professionals want to hear. That’s where simulation comes in…
Designers of RFID tags for biomedical applications need to account for the performance of tag and reader designs as well as how RFID affects other medical devices and systems. By characterizing a single device, such as an RFID tag, these designers have a good starting point for an EMI analysis. Electromagnetics simulation can be used to compute the mutual inductance of the RFID system design.
Passive UHF RFID tags, like those mentioned at the beginning of this blog post, are preferred over their lowfrequency and highfrequency counterparts because UHF tags can be detected at near and far distances from a reader, and can thus be used for a wide variety of applications. These tags also transfer data at fast rates and are more costeffective to produce.
To evaluate the detection and read range of a UHF RFID tag, you can use the RF Module, an addon to the COMSOL Multiphysics® software. RF simulation enables us to determine the default electric field norm, or Efield, for a tag design. This value can be used to predict the ideal placement of tags on a patient as well as the ideal placement of an RFID reader for tracking multiple patients at once.
Analyzing the Efield (left) and farfield radiation pattern (right) of a UHF RFID tag can improve the detection and range of the device.
Simulation analysis can also be used to find the farfield radiation pattern of a tag. For instance, in the model above, we can see that the radiation pattern is nearly the same in every direction on the plane of the tag. These results show that the RFID tag design is optimized for performance and has a long read range.
Consider a model of a basic RFID system, which consists of a readertransponder with two main parts:
Geometries of a reader (left) and RFID tag (right).
The system works as such: The reader generates an electromagnetic (EM) field that energizes the chip inside the RFID tag. The EM field is altered by the tag’s circuit and the altered signal is recovered by the RFID reader’s antenna.
Using the AC/DC Module, an addon product to COMSOL Multiphysics, and the Magnetic Fields interface, designers can simulate the inductive coupling between the reader and the tag. By finding the total magnetic flux intercepted by one antenna for current flowing in the system’s other antenna, you can compute the mutual inductance.
The simulation results below show the magnetic flux lines and the magnetic flux intensity between the RFID tag and reader. These results can be used to determine the mutual inductance of the system.
The magnetic flux density of an RFID system.
By finding the mutual inductance of an RFID system, you are able to predict its EMI with other medical devices. What’s more, you can design RFID tags that can be safely used to enhance healthcare in a variety of ways.
Find out what’s possible with the builtin functionality in the RF Module by clicking the button below.
One of the first physical laws that we learn as engineers is Ohm’s law: The current through a device equals the applied voltage difference divided by the device electrical resistance, or I = V/R_{e}, where the electrical resistance, R_{e}, is a function of the device geometry and material’s electrical conductivity.
Shortly after learning that law, we probably also learned about the dissipated power within the device, which equals the current times the voltage difference, or Q = IV, which we could also write as Q = I^{2}R_{e} or Q = V^{2}/R_{e}. Maybe a little bit later in our studies, we also learned about thermal conductivity and equivalent device thermal resistance, R_{t}, which let us compute, in a lumped sense, the rise in temperature of a device via ΔT = QR_{t} relative to ambient conditions. We can then determine the absolute device temperature using .
We start our discussion from this point and consider a completely lumped model of a device. (Yes, we’re starting so simple that we don’t even need to use the COMSOL Multiphysics® software for this part!) Let’s consider a lumped device with electrical resistance of R_{e} = 1 Ω and thermal resistance of R_{t} = 1 K/W. We can drive this device with either a constant voltage and compute the temperature as or drive it via constant current where the peak temperature is .
We choose an ambient temperature of 300 K, or 27°C, which is about room temperature. Let’s now plot out the device lumped temperature as a function of increasing voltage (from 0 to 10 V) and current (from 0 to 10 A), as shown in the image below. Unsurprisingly, we see a quadratic increase in temperature.
Device temperature as a function of applied voltage (left) and applied current (right), assuming constant properties.
We might think that we can use the curve to predict a wider range of operating conditions. Suppose we want to drive the device up to its failure temperature, where the material melts or vaporizes. Let’s say that this material will vaporize when its temperature gets up to 700 K (427°C). Based on this curve, some simple math would indicate that the maximum voltage is 20 V and the maximum current is 20 A, but this is quite wrong!
At this point, you’re probably ready to point out the simple mistake that we’ve made: Electrical resistance is not constant with temperature. For most metals, the electrical conductivity goes down with an increasing temperature and since resistivity is the inverse of conductivity, the device resistivity goes up. So, let’s introduce a temperature dependence for the resistivity:
This is known as a linearized resistivity model, where ρ_{0} is the reference resistivity at , the reference temperature, and α_{e} is the temperature coefficient of electrical resistivity.
Let’s choose ρ_{0} = 1 Ω, = 300 K, and α_{e} = 1/200 K. Now, the resistance is 1 Ω at a device temperature of 300 K and 2 Ω at a temperature rise of 200 K above the set temperature. The equations for lumped temperature as a function of voltage and current now become:
and
These equations are a bit more complicated (the first is a quadratic equation in terms of T) but still possible to solve by hand. The plots of temperature as a function of increasing voltage and current are displayed below.
Device temperature as a function of applied voltage (left) and applied current (right) with the electrical resistivity as a function of temperature.
For the voltagedriven problem, as the temperature rises, the resistance rises. Since the resistance occurs in the denominator of the temperature expression, higher resistance lowers the temperature and we see that the temperature will be lower than that for the constant resistivity case. If we drive the device with constant current, the temperaturedependent resistance appears in the numerator. As we increase the current, the resistive heating will be higher than that for the linear material case.
We might be tempted at this point to compute the maximum voltage or current that this device can sustain, but you are probably already realizing the second mistake we’ve made: We also need to incorporate the temperature dependence of the thermal resistance. For metals, it’s reasonable to assume that the electrical and thermal conductivity will show the same trends. Thus, let’s use a nonlinear expression that is similar to what we used before:
Now, our voltagedriven and currentdriven equations for temperature become:
and
Although only slightly different from before, these nonlinear equations are now quite a bit more difficult to solve. Simulation software is starting to look more attractive! Once we do solve these equations (let’s set r_{0} = 1 K/W, α_{t} = 1/400 K, and = 300 K), we can plot the device temperature, as shown below.
Device temperature as a function of applied voltage (left) and applied current (right) with the electrical and thermal resistivity as a function of temperature.
Observe that for the currentdriven case, the temperature rises asymptotically. Since both the electrical and the thermal resistance increase with an increasing temperature, the device temperature rises very sharply as the current is increased. As the temperature rises to infinity, the problem becomes unsolvable. This is actually entirely expected; in fact, this is how your basic car fuse works. Now, if we were solving this problem in COMSOL Multiphysics, we could also solve this as a transient model (incorporating the thermal mass due to the device density and specific heat) and predict the time that it takes for the device temperature to rise to its failure point.
Things are luckily a bit simpler for the voltagedriven case. Here, we also see a predictable behavior: The rising thermal resistivity drives the temperature higher than when we only considered a temperaturedependent electrical conductivity. Now, the interesting point here is the temperature is still lower than for the constant resistivity case. This also sometimes confuses people, but just keep in mind that one of these nonlinearities is driving the temperature down while the other is driving the temperature up. In general, for a more complicated model (such as one you would build in COMSOL Multiphysics), you don’t know which nonlinearity will predominate.
What other mistake might we have made here? We have used a positive temperature coefficient of thermal resistivity. This is certainly true for most metals, but it turns out that the opposite is true for some insulators, glass being a common example. Usually, the total device thermal resistance is mostly a function of the insulators rather than the electrically conductive domains. In addition, the device’s thermal resistance should incorporate the effects of the cooling to the surrounding ambient environment. So, the effects of free convection (which increases with the temperature difference) and radiation (which has a fourthorder dependence on temperature difference) could also be lumped into this single thermal resistance. For now, though, let’s keep the problem (relatively) simple and just switch the sign of the temperature coefficient of thermal resistivity, α_{t} = 1/400 K, and directly compare the voltage and currentdriven cases for driving voltage up to 100 V and current up to 100 A.
Device temperature as a function of applied voltage (pink) and applied current (blue) with a negative temperature coefficient of thermal resistivity.
We now see some results that are quite different. Observe that for both the voltage and currentdriven cases, the temperature increases approximately quadratically at low loads, but at higher loads, the temperature increase starts to flatten out due to the decreasing thermal resistivity. The slope, although always positive, decreases in magnitude. The currentdriven case starts to asymptotically approach T = 700 K, but the voltagedriven case stays significantly below the failure temperature.
This is an important result and highlights another common mistake. The nonlinear material models we used here for electrical and thermal resistivity are approximations that start to become invalid if we get too close to 700 K. If we anticipate operating in this regime, we should go back to the literature and find a more sophisticated material model. Although our existing nonlinear material models did solve, we always need to check that they are still valid at the computed operating temperature. Of course, if we are not close to these operating conditions, we can use the linearized resistivity model (one of the builtin material models within COMSOL Multiphysics). Then, our model will be valid.
We can hopefully now see from all of this data that the temperature has a very complicated relationship with respect to the driving voltage or current. When nonlinear materials are considered, the temperature might be higher or lower than when using constant properties, and the slope of the temperature response can switch from quite steep to quite shallow just based on the operating conditions.
Have these results thoroughly confused you yet? What if we went back and changed one of the coefficients in the resistance expressions? Certain materials have negative temperature coefficients of electrical and thermal resistivity. What if we used an even more complicated nonlinearity? Would you feel confident in saying anything about the expected temperature variations in even this simple lumped device case, or would you rather check it against a rigorous calculation?
What about the case of a realworld device? One that has a combination of many different materials, different electrical and thermal conductivities as a function of temperature, and complex shapes? Would you model this under steadystate conditions only or in the time domain, to find out how long it takes for the temperature to rise? Maybe — in fact, most likely — there will also be nonlinear boundary conditions such as radiation and free convection that we don’t want to approximate via a single lumped thermal resistance. What can you expect then? Almost anything! And how do you analyze it? Well, with COMSOL Multiphysics, of course!
Evaluate how COMSOL Multiphysics can help you meet your multiphysics modeling and analysis goals. Contact us via the button below.
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SMA connectors are used for a variety of applications, including cellphone antennas and printed circuit board (PCB) testing. To ensure that SMA connectors perform well when used for these and other applications, it’s important for engineers to evaluate their designs.
One option for studying these connectors is with fullwave electromagnetics simulation. This process is made easier when using the RF Module, an addon to COMSOL Multiphysics, as it includes three parameterized types of SMA connector geometries (4hole, 2hole, and vertical mount) in the Part Libraries to enhance the modeling experience.
Left: Photograph of an SMA connector. Image by Meggar. Licensed under CC BYSA 3.0, via Wikimedia Commons. Right: Model of a fourhole and verticalmount SMA connector. The connectors are linked via a 50Ω meander microstrip line.
The example highlighted here examines an SMA connector located on a grounded coplanar waveguide (GCPW). Coplanar waveguides are often found in microwave circuits and can be fabricated on PCBs, which are used in almost all modern electronic devices. Here, the SMA connector is soldered onto one end of a GCPW line and a lumped port terminates the opposite end of the line with 50 Ω.
SMA connector geometry.
This model has two main goals. The first is to study how a signal from a coaxial cable travels through an SMA connector and excites a GCPW. The signal is carried through the coaxial cable by a TEM mode at 1 GHz. The electric field of this TEM mode is pointed radially between the inner and outer conductors within the cable. After moving through the cable, the signal travels through the connector and eventually excites the symmetric mode of the GCPW. Depending on the phase, the symmetric electric field in the GCPW either points from the center conductor to the outer conductors or vice versa.
The second goal is to model the 50Ω passive termination of the GCPW using a lumped port and air bridge. It’s possible to study this passive termination using three different approaches:
Three methods for adding a port onto a GCPW.
Through the electromagnetics analysis, you can find the norm of the electric field on the top surface of the GCPW circuit, as shown in the images below. Please note that the results here have been adjusted to emphasize the field confined within the ground plane of the GCPW.
Electric field norm plots for the SMA connector. All three configurations match approximately well to the 50Ω reference impedance, and no standing wave due to reflection is observed.
Since the electric field norm plots do not provide all of the necessary information for this study, it’s also helpful to evaluate the Sparameters for the design. In this case, the results indicate that the passive termination methods have Sparameters (S_{11}) of around 24 dB. With this knowledge, you can evaluate the performance of SMA connectors and improve their designs.
You can try the SMA Connector on a Grounded Coplanar Waveguide tutorial by clicking the button below. Doing so will take you to the Application Gallery, where you can download the MPHfile as long as you have both a COMSOL Access account and a valid software license.
10 years after making a breakthrough in electric motors, Michael Faraday created the first electric generator in 1831. The setup for this device (later called a homopolar generator) was simple, consisting of a copper disc that rotated in between the poles of a permanent magnet. While Faraday’s generator successfully demonstrated electromagnetic induction, it was too inefficient to be practical, as the current tended to circulate backward and create counterflows.
Sketch of an early homopolar generator, also known as a Faraday disc. Image in public domain in the United States, via Wikimedia Commons.
Over the years, other scientists tried their hand at improving the performance of homopolar generators. A famous example is Nikola Tesla, who developed a design in which a metallic belt separated parallel discs with parallel shafts. This arrangement helped reduce frictional losses, an important factor in enhancing their efficiency.
In the 1950s, it was discovered that homopolar generators are particularly good for pulsed power applications, as they can store energy over long periods of time and release it in bursts. This finding led to renewed interest in the generator, with scientists creating large versions of the device. One such example was built by Sir Mark Oliphant for the Australian National University. This supersized generator could supply up to 2 MA of current and was used for over 20 years.
Parts of the homopolar generator created by Sir Oliphant, which has been disassembled and put on display. Image by Martyman at English language Wikipedia. Licensed under CC BYSA 3.0 Unported, via Wikimedia Commons.
While homopolar generators have come a long way — and been called many names — since Faraday’s time, researchers and engineers are still working to improve their performance. To investigate potential designs, one approach is electromagnetics simulation…
This example models a simple 3D homopolar generator, which consists of a rotating disc with a 10cm radius placed in a uniform magnetic field of 1 T. A conductor connects the disc’s rim to the center, passing through the magnetic field and generating Lorentz currents in the disc.
The geometry of the homopolar generator model.
Note that the angular velocity of the disc is 1200 rpm and that ~45.16 kA of current flows through the conductor.
To model the rotation of the disc, you can use a Lorentz term for two reasons:
Here, the current distribution stays stationary and does not follow the disc’s rotation.
After using a stationary solver, it’s possible to examine the flow of the current moving through both the disc and conductor. By looking at the current density norm and current direction, you can find ways of improving the current flow in a homopolar generator design.
Current density norm (left) and direction of the current (right) in the conductor and disc.
In addition, it’s possible to examine the effect of the magnetic field on a design, such as how it affects rotation. As an example, you can see the total and induced magnetic flux densities below.
The homopolar generator causes perturbations in the Bfield as shown by the arrow plot in the volume surrounding the generator. The wheel velocity is shown with arrows and colors on the surface.
Since resistive losses play a key role in the efficiency of these generators, it’s important to minimize them as much as possible. As can be seen in the image below, simulation provides a way to determine the losses in the conductive parts of a generator design.
The resistive losses in the disc and conductor.
Using electromagnetics modeling, engineers can improve homopolar generator designs, enhancing their performance by reducing frictional losses or altering the magnetic field.
To get started with modeling homopolar generators, click the button below. Doing so will take you to the Application Gallery, where you can sign into your COMSOL Access account and then download the MPHfile and tutorial documentation for this example.