Electrical cables, also called transmission lines, are used everywhere in the modern world to transmit both power and data. If you are reading this on a cell phone or tablet computer that is “wireless”, there are still transmission lines within it connecting the various electrical components together. When you return home this evening, you will likely plug your device into a power cable to charge it.

Various transmission lines range from the small, such as coplanar waveguides on a printed circuit board (PCB), to the very large, like high voltage power lines. They also need to function in a variety of situations and conditions, from transatlantic telegraph cables to wiring in spacecraft, as shown in the image below. Transmission lines must be specially designed to ensure that they function appropriately in their environments, and may also be subject to further design goals, including required mechanical strength and weight minimization.

*Transmission wires in the payload bay of the OV-095 at the Shuttle Avionics Integration Laboratory (SAIL).*

When designing and using cables, engineers often refer to parameters per unit length for the series resistance (R), series inductance (L), shunt capacitance (C), and shunt conductance (G). These parameters can then be used to calculate cable performance, characteristic impedance, and propagation losses. It is important to keep in mind that these parameters come from the electromagnetic field solutions to Maxwell’s equations. We can use COMSOL Multiphysics to solve for the electromagnetic fields, as well as consider multiphysics effects to see how the cable parameters and performance change under different loads and environmental conditions. This could then be converted into an easy-to-use app, like this example that calculates the parameters for commonly used transmission lines.

Here, we examine a coaxial cable — a fundamental problem that is often covered in a standard curriculum for microwave engineering or transmission lines. The coaxial cable is so fundamental that Oliver Heaviside patented it in 1880, just a few years after Maxwell published his famous equations. For the students of scientific history, this is the same Oliver Heaviside who formulated Maxwell’s equations in the vector form that we are familiar with today; first used the term “impedance”; and helped develop transmission line theory.

Let us begin by considering a coaxial cable with dimensions as shown in the cross-sectional sketch below. The dielectric core between the inner and outer conductors has a relative permittivity (\epsilon_r = \epsilon' -j\epsilon'') of 2.25 – j*0.01, a relative permeability (\mu_r) of 1, and a conductivity of zero, while the inner and outer conductors have a conductivity (\sigma) of 5.98e7 S/m.

*The 2D cross section of the coaxial cable, where we have chosen a = 0.405 mm, b = 1.45 mm, and t = 0.1 mm. *

A standard method for solving transmission lines is to assume that the electric fields will oscillate and attenuate in the direction of propagation, while the cross-sectional profile of the fields will remain unchanged. If we then find a valid solution, uniqueness theorems ensure that the solution we have found is correct. Mathematically, this is equivalent to solving Maxwell’s equations using an *ansatz* of the form \mathbf{E}\left(x,y,z\right) = \mathbf{\tilde{E}}\left(x,y\right)e^{-\gamma z}, where (\gamma = \alpha + j\beta) is the complex propagation constant and \alpha and \beta are the attenuation and propagation constants, respectively. In cylindrical coordinates for a coaxial cable, this results in the well-known field solution of

\begin{align}

\mathbf{E}&= \frac{V_0\hat{r}}{rln(b/a)}e^{-\gamma z}\\

\mathbf{H}&= \frac{I_0\hat{\phi}}{2\pi r}e^{-\gamma z}

\end{align}

\mathbf{E}&= \frac{V_0\hat{r}}{rln(b/a)}e^{-\gamma z}\\

\mathbf{H}&= \frac{I_0\hat{\phi}}{2\pi r}e^{-\gamma z}

\end{align}

which then yields the parameters per unit length of

\begin{align}

L& = \frac{\mu_0\mu_r}{2\pi}ln\frac{b}{a} + \frac{\mu_0\mu_r\delta}{4\pi}(\frac{1}{a}+\frac{1}{b})\\

C& = \frac{2\pi\epsilon_0\epsilon'}{ln(b/a)}\\

R& = \frac{R_s}{2\pi}(\frac{1}{a}+\frac{1}{b})\\

G& = \frac{2\pi\omega\epsilon_0\epsilon''}{ln(b/a)}

\end{align}

L& = \frac{\mu_0\mu_r}{2\pi}ln\frac{b}{a} + \frac{\mu_0\mu_r\delta}{4\pi}(\frac{1}{a}+\frac{1}{b})\\

C& = \frac{2\pi\epsilon_0\epsilon'}{ln(b/a)}\\

R& = \frac{R_s}{2\pi}(\frac{1}{a}+\frac{1}{b})\\

G& = \frac{2\pi\omega\epsilon_0\epsilon''}{ln(b/a)}

\end{align}

where R_s = 1/\sigma\delta is the sheet resistance and \delta = \sqrt{2/\mu_0\mu_r\omega\sigma} is the skin depth.

While the equations for capacitance and shunt conductance are valid at any frequency, it is extremely important to point out that the equations for the resistance and inductance depend on the skin depth and are therefore only valid at frequencies where the skin depth is much smaller than the physical thickness of the conductor. This is also why the second term in the inductance equation, called the *internal inductance*, may be unfamiliar to some readers, as it can be neglected when the metal is treated as a perfect conductor. The term represents inductance due to the penetration of the magnetic field into a metal of finite conductivity and is negligible at sufficiently high frequencies. (The term can also be expressed as L_{Internal} = R/\omega.)

For further comparison, we can compute the DC resistance directly from the conductivity and cross-sectional area of the metal. The analytical equation for the DC inductance is a little more involved, and so we quote it here for reference.

L_{DC} = \frac{\mu}{2\pi}\left\{ln\left(\frac{b+t}{a}\right) + \frac{2\left(\frac{b}{a}\right)^2}{1- \left(\frac{b}{a}\right)^2} ln\left(\frac{b+t}{b}\right) – \frac{3}{4} + \frac{\frac{\left(b+t\right)^4}{4} – \left(b+t\right)^2a^2+a^4\left(\frac{3}{4} + ln\frac{\left(b+t\right)}{a}\right) }{\left(\left(b+t\right)^2-a^2\right)^2}\right\}

Now that we have values for C and G at all frequencies, DC values for R and L, and asymptotic values for their high-frequency behavior, we have excellent benchmarks for our computational results.

When setting up any numerical simulation, it is important to consider whether or not symmetry can be used to reduce the model size and increase the computational speed. As we saw earlier, the exact solution will be of the form \mathbf{E}\left(x,y,z\right) = \mathbf{\tilde{E}}\left(x,y\right)e^{-\gamma z}. Because the spatial variation of interest is primarily in the *xy*-plane, we just want to simulate a 2D cross section of the cable. One issue, however, is that the 2D governing equations used in the AC/DC Module assume that the fields are invariant in the out-of-plane direction. This means that we will not be able to capture the variation of the ansatz in a single 2D AC/DC simulation. We can find the variation with two simulations, though! This is because the series resistance and inductance depend on the current and energy stored in the magnetic fields, while the shunt conductance and capacitance depend on the energy in the electric field. Let’s take a closer look.

Since the shunt conductance and capacitance can be calculated from the electric fields, we begin by using the *Electric Currents* interface.

*Boundary conditions and material properties for the *Electric Currents* interface simulation.*

Once the geometry and material properties are assigned, we assume that the conductors are equipotential (a safe assumption, since the conductivity difference between the conductor and the dielectric will generally be near 20 orders of magnitude) and set up the physics by applying V_{0} to the inner conductor and grounding the outer conductor to solve for the electric potential in the dielectric. The above analytical equation for capacitance comes from the following more general equations

\begin{align}

W_e& = \frac{1}{4}\int_{S}{}\mathbf{E}\cdot \mathbf{D^\ast}d\mathbf{S}\\

W_e& = \frac{C|V_0|^2}{4}\\

C& = \frac{1}{|V_0|^2}\int_{S}{}\mathbf{E}\cdot \mathbf{D^\ast}d\mathbf{S}

\end{align}

W_e& = \frac{1}{4}\int_{S}{}\mathbf{E}\cdot \mathbf{D^\ast}d\mathbf{S}\\

W_e& = \frac{C|V_0|^2}{4}\\

C& = \frac{1}{|V_0|^2}\int_{S}{}\mathbf{E}\cdot \mathbf{D^\ast}d\mathbf{S}

\end{align}

where the first equation is from electromagnetic theory and the second from circuit theory.

The first and second equations are combined to obtain the third equation. By inserting the known fields from above, we obtain the previous analytical result for C in a coaxial cable. More generally, these equations provide us with a method for obtaining the capacitance from the fields for any cable. From the simulation, we can compute the integral of the electric energy density, which gives us a capacitance of 98.142 pF/m and matches with theory. Since G and C are related by the equation

G=\frac{\omega\epsilon'' C}{\epsilon'}

we now have two of the four parameters.

At this point, it is also worth reiterating that we have assumed that the conductivity of the dielectric region is zero. This is typically done in the textbook derivation, and we have maintained that convention here because it does not significantly impact the physics — unlike our inclusion of the internal inductance term discussed earlier. Many dielectric core materials do have a nonzero conductivity and that can be accounted for in simulation by simply updating the material properties. To ensure that proper matching with theory is maintained, the appropriate derivations would need to be updated as well.

In a similar fashion, the series resistance and inductance can be calculated through simulation using the AC/DC Module’s *Magnetic Fields* interface. The simulation setup is straightforward, as demonstrated in the figure below.

*The conductor domains are added to a *Single-Turn Coil* node with the *Coil Group* feature, and the reversed current direction option ensures that the direction of current through the inner conductor is the opposite of the outer conductor, as indicated by the dots and crosses. The single-turn coil will account for the frequency dependence of the current distribution in the conductors, as opposed to the arbitrary distribution shown in the figure.*

We refer to the following equations, which are the magnetic analog of the previous equations, to calculate the inductance.

\begin{align}

W_m& = \frac{1}{4}\int_{S}{}\mathbf{B}\cdot \mathbf{H^\ast}d\mathbf{S}\\

W_m& = \frac{L|I_0|^2}{4}\\

L& = \frac{1}{|I_0|^2}\int_{S}{}\mathbf{B}\cdot \mathbf{H^\ast}d\mathbf{S}

\end{align}

W_m& = \frac{1}{4}\int_{S}{}\mathbf{B}\cdot \mathbf{H^\ast}d\mathbf{S}\\

W_m& = \frac{L|I_0|^2}{4}\\

L& = \frac{1}{|I_0|^2}\int_{S}{}\mathbf{B}\cdot \mathbf{H^\ast}d\mathbf{S}

\end{align}

To calculate the resistance, we use a slightly different technique. First, we integrate the resistive loss to determine the power dissipation per unit length. We can then use the familiar P = I_0^2R/2 to calculate the resistance. Since R and L vary with frequency, let’s take a look at the calculated values and the analytical solutions in the DC and high-frequency (HF) limit.

*“Analytic (DC)” and “Analytic (HF)” refer to the analytical equations in the DC and high-frequency limits, respectively, which were discussed earlier. Note that these are both on log-log plots.*

We can clearly see that the computed values transition smoothly from the DC solution at low frequencies to the high-frequency solution, which is valid when the skin depth is much smaller than the thickness of the conductor. We anticipate that the transition region will be approximately located where the skin depth and conductor thickness are within one order of magnitude. This range is 4.2e3 Hz to 4.2e7 Hz, which is exactly what we see in the results.

Now that we have completed the heavy lifting to calculate R, L, C, and G, there are two other significant parameters that can be determined. They are the characteristic impedance (Z_{c}) and complex propagation constant (\gamma = \alpha + j\beta), where \alpha is the attenuation constant and \beta is the propagation constant.

\begin{align}

Z_c& = \sqrt{\frac{(R+j\omega L)}{(G+j\omega C)}}\\

\gamma& = \sqrt{(R+j\omega L)(G+j\omega C)}

\end{align}

Z_c& = \sqrt{\frac{(R+j\omega L)}{(G+j\omega C)}}\\

\gamma& = \sqrt{(R+j\omega L)(G+j\omega C)}

\end{align}

In the figure below, we see these values calculated using the analytical formulas for both the DC and high-frequency regime as well as the values determined from our simulation. We have also included a fourth line: the impedance calculated using COMSOL Multiphysics and the RF Module, which we will discuss shortly. As can be seen, our computations agree with the analytical solutions in their respective limits, as well as yielding the correct values through the transition region.

*A comparison of the characteristic impedance, determined using the analytical equations and COMSOL Multiphysics. The analytical equations plotted are from the DC and high-frequency (HF) equations discussed earlier, while the COMSOL Multiphysics results use the AC/DC and RF Modules. For clarity, the width of the “RF Module” line has been intentionally increased.*

Electromagnetic energy travels as waves, which means that the frequency of operation and wavelength are inversely proportional. As we continue to solve at higher and higher frequencies, we need to be aware of the relative size of the wavelength and electrical size of the cable. As discussed in a previous blog post, we should switch from the AC/DC to RF Module at an electrical size of approximately λ/100. If we use the cable diameter as the electrical size and the speed of light inside the dielectric core of the cable, this yields a transition frequency of approximately 690 MHz.

At these higher frequencies, the cable is more appropriately treated as a waveguide and the cable excitation as a waveguide mode. Using waveguide terminology, the mode we have been examining is a special type of mode called *TEM* that can propagate at any frequency. When the cross section and wavelength are comparable, we also need to account for the possibility of higher-order modes. Unlike a TEM mode, most waveguide modes can only propagate above a characteristic cut-off frequency. Due to the cylindrical symmetry in our example model, there is an equation for the cut-off frequency of the first higher-order mode, which is a TE11 mode. This cut-off frequency is f_{c} = 35.3 GHz, but even with the relatively simple geometry, the cut-off frequency comes from a transcendental equation that we will not examine further in this post.

So what does this cut-off frequency mean for our results? Above that frequency, the energy carried in the TEM mode that we are interested in has the potential to couple to the TE11 mode. In a perfect geometry, like we have simulated here, there will be no coupling. In the real world, however, any imperfections in the cable could cause mode coupling above the cut-off frequency. This could result from a number of sources, from fabrication tolerances to gradients in the material properties. Such a situation is often avoided by designing cables to operate below the cut-off frequency of higher-order modes so that only one mode can propagate. If that is of interest, you can also use COMSOL Multiphysics to simulate the coupling between higher-order modes, as with this Directional Coupler tutorial model (although beyond the scope of today’s post).

Simulation of higher-order modes is ideally suited for a Mode Analysis study using the RF or Wave Optics modules. This is because the governing equation is \mathbf{E}\left(x,y,z\right) = \mathbf{\tilde{E}}\left(x,y\right)e^{-\gamma z}, which is exactly the form that we are interested in. As a result, Mode Analysis will directly solve for the spatial field and complex propagation constant for a predefined number of modes. We can use the same geometry as before, except that we only need to simulate the dielectric core and can use an Impedance boundary condition for the metal conductor.

*The results for the attenuation constant and effective mode index from a Mode Analysis. The analytic line in the left plot, “Attenuation Constant vs Frequency”, is computed using the same equations as the high-frequency (HF) lines used for comparison with the results of the AC/DC Module simulations. The analytic line in the right plot, “Effective Refractive Index vs Frequency”, is simply n = \sqrt{\epsilon_r\mu_r}. For clarity, the size of the “COMSOL — TEM” lines has been intentionally increased in both plots.*

We can clearly see that the Mode Analysis results of the TEM mode match the analytic theory, and that the computed higher-order mode has its onset at the previously determined cut-off frequency. It is also incredibly convenient that the complex propagation constant is a direct output of this simulation and does not require calculations of R, L, C, and G. This is because \gamma is explicitly included and solved for in the Mode Analysis governing equation. These other parameters can be calculated for the TEM mode, if desired, and more information can be found in this demonstration in the Application Gallery. It is also worth pointing out that this same Mode Analysis technique can be used for dielectric waveguides, like fiber optics.

At this point, we have thoroughly analyzed a coaxial cable. We have calculated the distributed parameters from the DC to high-frequency limit and examined the first higher-order mode. Importantly, the Mode Analysis results only depend on the geometry and material properties of the cable. The AC/DC results require the additional knowledge of how the cable is excited, but hopefully you know what you’re attaching your cable to! We used analytic theory solely to compare our simulation results against a well-known benchmark model. This means that the analysis could be extended to other cables, as well as coupled to multiphysics simulations that include temperature change and structural deformation.

For those of you who are interested in the fine details, here are a few extra points in the form of hypothetical questions.

- “Why didn’t you mention and/or plot all of the characteristic impedance and distributed parameters for the TE11 mode?”
- This is because only TEM modes have a uniquely defined voltage, current, and characteristic impedance. It is still possible to assign some of these values for higher-order modes, and this is discussed further in texts on transmission line theory and microwave engineering.

- “When I solve for modes using a Mode Analysis study, they are labeled by the value of their effective index. Where did TEM and TE11 come from?”
- These names come from the analytic theory and were used for convenience when discussing the results. This name assignment may not be possible for an arbitrary geometry, but what’s in a name? Would not a mode by any other name still carry electromagnetic energy (excluding nontunneling evanescent waves, of course)?

- “Why is there an extra factor of ½ in several of your calculations?”
- This comes up when solving electromagnetics in the frequency domain, notably when multiplying two complex quantities. When taking the time average, there is an extra factor of ½ as opposed to the equation in the time domain (or at DC). For more information, you can refer to a text on classical electromagnetics.

The following texts were referred to during the writing of this post and are excellent sources of additional information:

*Microwave Engineering*, by David M. Pozar*Foundations for Microwave Engineering*, by Robert E. Collin*Inductance Calculations*, by Frederick W. Grover*Classical Electrodynamics*, by John D. Jackson

The detection and removal of landmines and IEDs is important for both humanitarian and military purposes. While the term for the process of detecting these mines — *minesweeping* — is the same in both cases, the removal process is referred to as *demining* in times of relative peace and *mine clearance* during times of war. The latter case refers to when mines are removed from active combat zones for tactical reasons as well as for the safety of soldiers.

When a war ends, landmines may still be in the ground and detonate under civilians, leading to casualties. The majority of the mines are located in developing countries that are trying to recover from recent wars. Aside from being politically unstable, these countries are unable to farm viable land that is strewn with IEDs, keeping their economies in poor positions. Unfortunately, finding and removing the dangerous devices can be rather difficult.

*A U.S. Army detection vehicle digs up an IED during a training exercise.*

In efforts to locate and remove landmines, a mechanical approach is one option. With this method, an area with known landmines is bombed or plowed using sturdy, mine-resilient tanks to detonate them safely. For a more natural approach, dogs, rats, and even honeybees are trained to detect landmines with their sense of smell, and they are usually too light to trigger detonation. Biological detection methods offer another option, utilizing plants and bacteria that change color or become fluorescent in the presence of certain explosive materials. Once the mines are detected, they are safely removed from the area.

*A trained rat searches for landmines in a field.*

One method can provide more knowledge about an area that contains IEDs: *electromagnetic detection*. An important element within electromagnetic detection is a process called *ground-penetrating radar* (GPR), which uses electromagnetic waves to create an image of a subsurface, revealing the buried objects.

GPR involves sending electromagnetic waves into a subsurface (the ground) through an antenna. The transmitter of the antenna sends the waves, and the receiver collects the energy reflected off of the different objects in the subsurface, recording the patterns as real-time data.

*Data from a traditional GPR scan of a historic cemetery.*

With recent developments in landmine cloaking technology, identifying buried objects through traditional GPR has become more challenging. Dr. Reginald Eze and George Sivulka from the City University of New York — LaGuardia Community College and Regis High School sought to improve electromagnetic IED detection by testing the method under different variables and environmental situations. By creating an intelligent subsurface sensing template with the help of COMSOL Multiphysics, the research team was able to determine better ways to safely locate and remove landmines and IEDs.

Let’s dive a bit deeper into their simulation research, which was presented at the COMSOL Conference 2015 Boston.

When setting up their model of the mine-strewn area, the researchers needed to ensure that they were accurately portraying a real-world landmine scenario. They started with a basic 2D geometry and defined the target objects and boundaries. The different layers of the model featured:

- A homogenous soil surface with varying levels of moisture
- Air
- The landmine

The physical parameters in the model included relative permittivity; relative permeability; and the conductivity of the air, dry soil, wet soil, and TNT (the explosive material used in the landmine).

Using the *Electromagnetic Waves, Frequency Domain* interface in the RF Module, the team built a model consisting of air, soil, and the landmine. Additionally, a perfectly matched layer (PML) was used to truncate the modeling domain and act as a transparent boundary to outgoing radiation, thus allowing for a small computational domain. A transverse electric (TE) plane wave was applied to the computational domain in the downward direction. The scattering results were analyzed via LiveLink™ *for* MATLAB®.

*The scattering effect of a wave on a landmine in wet soil (left) compared to dry soil (right).*

The research team studied the radar cross section (RCS), which quantifies the scattering of the waves off of various objects. Their studies were based on five key factors:

- Projected cross section
- Reflectivity
- Directivity
- Contrast between the landmine and the background materials
- Shapes of the landmine and the ground surface

With each adjustment to an environmental parameter, a parametric sweep was performed every 0.5 GHz from 0.5 GHz to 3.0 GHz. The parametric sweeps enabled an educated selection of the optimal frequency for IED detection in every possible environmental scenario.

*A parametric sweep used to identify the optimal frequency for a landmine detection system.*

The simulation results pointed out the differences in scattering patterns depending on the parameters. For example, as the depth of the target increased, the scattering effects became more negligible. The relation between how deep the mine was buried and the scattering showed a clear connection to the soil’s interference with the wave.

The results also showed that dry soil has more interference with the RF signal than wet soil. Both the size and depth of the mine were related to the amount of scattering. For instance, the more shallow the mine was buried, the more easily it was detected. The parameter sweep of the frequencies indicated that the optimal frequency to detect anomalies in the subsurface scan was 2 GHz.

*The scattering amplitude for a landmine buried in an air/wet soil/dry soil layer combination (left) compared to air/dry soil/wet soil (right).*

Studying the parameters and their effects on the scattering patterns of the waves offers insight into the objects that are being detected, including their chemical composition. Such knowledge makes it easier to identify an object, whether a TNT-based landmine, another type of IED, a rock, or a tree root.

Through simulation analyses, the researchers gained a more comprehensive understanding of the microphysical parameters and their impact on the scattering of waves off of different objects. This gave them a better idea of the remote sensing behavior, offering potential for increased accuracy in landmine detection and removal. Such advancements could lead to safer environments, particularly within developing areas of the world.

- Read the full paper: “Remote Sensing of Electromagnetically Penetrable Objects: Landmine and IED Detection“
- View the research poster, which received the Popular Choice Poster award at the COMSOL Conference 2015 Boston

*MATLAB is a registered trademark of The MathWorks, Inc.*

Skin cancer affects numerous people around the world and is recognized as the most common form of cancer in the United States. Despite its prominence, this disease is highly treatable when skin tumors are detected early and removed. These tumors can be identified during monthly self-examinations and with the help of medical professionals. However, noninvasive skin tumor detection tools, such as *dielectric probes*, are emerging as an alternative.

To identify tumors, dielectric probes can utilize a millimeter wave with frequencies of either 35 GHz or 95 GHz. This millimeter wave has a sensitive reflective response to water content, which it uses as a means of detecting skin tumors. Such tumors possess a different *scattering parameter* or *S-parameter* than that of healthy skin, and the probes locate tumors by identifying these abnormal S-parameters.

Through simulation, we can evaluate the functionality of a conical dielectric probe and ensure its safety as an alternative for detecting skin tumors.

Our 2D axisymmetric tutorial model consists of a metallic circular waveguide, a tapered PTFE dielectric rod, a skin phantom, an air domain, and perfectly matched layers (PMLs).

In this example, we model our waveguide as a perfect electric conductor (PEC) and assume that its conductivity is high enough to negate any loss. The waveguide terminates at a circular port on one end and is connected to the dielectric rod on the other end. The dielectric rod is designed for impedance matching between the waveguide and the air domain. It is symmetrically tapered and supported on the rim of the waveguide by a ring structure. The tip of the rod touches the skin phantom, and the whole device uses a low-power 35 GHz Ka-band millimeter wave when operating.

*Left: Dielectric probe model. Right: The probe interacting with a skin tumor.*

To analyze the validity of the probe design, we first observe the electromagnetic properties of the circular waveguide and dielectric probe without the skin phantom. From the simulation results, we can conclude that the probe is functional.

*The dielectric rod’s wave propagation without the skin phantom.*

Next, we increase the complexity of our model with two additions: a healthy skin phantom and a skin phantom containing a tumor. This enables us to calculate and compare the S-parameters for each of these cases. Our findings show that the S_{11} value of the healthy phantom is -9.84 dB, while the phantom containing a tumor features an S-parameter value of -8.87 dB. These values indicate that more reflection occurs when the probe touches the skin phantom with a tumor. We can expect such a result, as tumors have a higher moisture content than healthy skin.

While we found the S-parameter approach to be functional, we also want to ensure that it is safe. To do so, we study the temperature distribution over the skin phantom surface in order to find the fraction of necrotic (damaged due to heat) tissue.

*Left: Temperature variation on a skin phantom with a tumor. Right: Plot of the necrotic tissue.*

Our analysis of a skin phantom with a tumor shows that, after ten minutes of low-powered millimeter wave exposure, the temperature change is within 0.06°C. Even at the relatively hotter spot, the temperature remains very close to the initial temperature of 34°C. With this information, we can assume that there are no harmful temperature differences. Furthermore, our results show that the fraction of necrotic tissue is negligibly small, indicating that the temperature rise induced by the probe has a negligible effect on the tissue.

- Download the tutorial: Modeling a Conical Dielectric Probe for Skin Cancer Diagnosis
- Read this blog post to learn about the use of simulation in one form of cancer treatment: Hyperthermic Oncology: Hyperthermia for Cancer Treatment

Electronic devices are a key component in our day-to-day lives. Imagine if it were possible to charge these devices without the need for wires or cords. The development of wireless power transfer (WPT) technology is making this possible, offering a simplified approach to charging electronics, including the ability to charge multiple devices at once. As the technology continues to grow, charging electronic devices wirelessly is becoming a reality in more and more applications, from phones to electric cars.

*Wireless charging spots at a coffee shop. Image by Veredai from Powermat Technologies — Own work, via Wikimedia Commons.*

As previously noted, WPT offers ubiquitous power by transmitting electrical power without the use of solid wires or conductors. Generally, this is achieved by using an electromagnetic field to transfer energy between two separate objects. Here, a power transmitting unit (PTU), which is connected to a power source, generates a magnetic field and a power receiving unit (PRU) captures this energy and converts it into usable power.

*A simple illustration of wireless power transfer. The PTU is shown on the left and the PRU is shown on the right.*

For some WPT systems, an important concern is that the orientation between the PTU and PRU can greatly affect the energy coupling. As such, a device may have to be carefully aligned on a PTU in order for it to charge. But at what point does a lack of alignment between a PTU and a PRU impede the energy coupling?

Here, we’ll use simulation to investigate how a change in orientation affects wireless power transfer antennas.

The wireless power transfer tutorial presented today analyzes the energy coupling between two circular loop antennas. In this example, the antennas are made of a polytetrafluoroethylene (PTFE) board and have a thin copper layer on top, which is modeled as a perfect electric conductor (PEC). Each of the devices feature a lumped inductor and a lumped port that can excite or terminate the antenna.

The antennas have a UHF RFID operating frequency of 915 MHz and a shape that inherently provides the ability to perform inductive coupling.

*The model geometry. Note that the air domain and perfectly matched layers (PMLs) are not included here.*

In our simulation, the receiving antenna is rotating and the transmitting antenna maintains a fixed location. This set up is similar to having a fixed position for a charger, but placing a mobile phone on it at different angles.

The changing orientation allows us to identify the impact a change in position has on the energy coupling. To visualize this effect, we model the E-field norm distribution and the power flow from the transmitting antenna at different rotation angles of the receiving antenna.

*The E-Field norm and power flow (arrow plot) of our wireless power transfer antennas.*

The results show that when the antennas are facing each other, which occurs when the receiving antenna’s angle of rotation is 0 degrees, the fields are strongly coupled — an indication of successful wireless power transfer. However, by the time the receiving antenna reaches 90 degrees of rotation, we don’t observe any distortion when the power flow penetrates the receiving antenna. At this rotation, there is almost no coupling or hot coupling area. As such, we can conclude that the power transfer between these two antennas is greatly reduced at this angle.

In the future, we can increase the functionality of WPT antennas by creating systems that function at a wide rage of orientations, enabling the charging of electronics without concern for their particular placement.

- Download the tutorial model: Simulating Wireless Power Transfer in Circular Loop Antennas
- Read a user story about wireless power transfer on page 8 of the 2015
*Multiphysics Simulation*IEEE insert

By now, you are likely familiar with the material known as graphene. Much of the excitement surrounding graphene is due to its exotic material properties. These properties manifest themselves because graphene is a 2D sheet of carbon atoms that is one atomic layer thick. Graphene is discussed as a 2D material, but is it *really* 2D or is it just incredibly thin like a very fine piece of paper? It is one atom thick, so it must have thickness, right?

*A schematic of graphene.*

This is a complex question that is better directed towards researchers within the field. It does, however, lead us to another important question within the simulation environment — should we simulate graphene as a 2D sheet or a thin 3D volume?

To answer this, there are various important contributions that must first be discussed.

From a simulation stand-point, we want our model to accurately represent reality. This is accomplished through verification and validation procedures that often involve comparisons with analytical solutions. In open areas of research such as the investigation of novel materials like graphene, the verification and validation process depends on several interlocking pieces. This is due to the fact that there may not be any benchmarks or analytical results for comparison, and the theoretical predictions may be hypotheses that are awaiting experimental verification.

For graphene, the process begins with a theory — like the random phase approximation (RPA) — that describes the material properties. Graphene of a sufficiently high quality must then be reliably fabricated, and done so in large enough sample sizes for experimental measurements to be conducted. Lastly, the experiments themselves must be performed, with the results analyzed and compared to the theoretical predictions. The process is then repeated as required.

Numerical simulation is an integral part of every stage within the research process. Here, we will focus solely on its use in the comparison of theoretical predictions and experimental results. Theoretical predictions do not always come in simple and straightforward equations. In such cases, the theory can be solved numerically with COMSOL Multiphysics, offering a closer comparison with experimental results.

When performing simulations in active research areas, it is important to keep the previously mentioned research cycle in mind. A simulation can be set up correctly, but if it uses incorrect theoretical predictions for the material properties, the simulation results will not show reliable agreement with the experimental results. Similarly, accurate theoretical predictions must be properly implemented in simulations in order to yield meaningful results — a particularly important concern when modeling graphene, the world’s first 2D material.

So what does it mean for an object to be 2D and how do we correctly implement it in simulation? This brings us back to our original question of whether it is better to model graphene as a 2D layer or a thin 3D material. Perhaps you can see the answer more clearly now. The simulation technique itself needs to be verified during the research process!

Let’s now turn to the experts.

Led by Associate Professor Alexander V. Kildishev, researchers at Purdue University’s Birck Nanotechnology Center are at the forefront of graphene research. Among their many works are graphene devices that are designed in COMSOL Multiphysics and then fabricated and tested experimentally. Professor Kildishev recently joined us for a webinar, “Simulating Graphene-Based Photonic and Optoelectronic Devices”, where he discussed important elements behind the modeling of graphene.

*When designing graphene and graphene-based devices, simulation helps to enhance design and optimization, achieving the highest possible performance.*

During the webinar, Kildishev showed simulation results in which graphene was treated as both a thin 3D volume and a 2D sheet. When conducting this research with his colleagues, he found that the best agreement between simulation results and experimental results is achieved through modeling graphene as a 2D layer. Using COMSOL Multiphysics, Kildishev also showcased simulations of graphene in the frequency and time domains.

To learn more about the simulation of graphene, you can watch the webinar here. We also encourage you to visit the Model Exchange section of our website, where you can download the models featured in the webinar and perform your own simulations of 2D graphene.

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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 PTFE 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}=<\mathbf{E}_x,\mathbf{E}_y, \mathbf{E}_z>. All other quantities (such as magnetic fields, currents, and power flow) can be derived from the electric field. It is also possible to reformulate the above equation as an eigenvalue problem, where a model is solved for the resonant frequencies of the system, rather than the response of the system at a particular frequency.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\sum \theta_i \sigma_i

\sum \theta_i \sigma_i

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

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

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

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

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

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

\prod\sigma_i^{\theta_i }

\prod\sigma_i^{\theta_i }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where the coefficients B_k and C_k determine the relative permittivity.

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

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

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

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

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

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

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

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

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

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

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