Phononic crystals are rather unique materials that can be engineered with a particular band gap. As the demand for these materials continues to grow, so does the interest in simulating them, specifically to optimize their band gaps. COMSOL Multiphysics, as we’ll show you here, can be used to perform such studies.

A *phononic crystal* is an artificially manufactured structure, or material, with periodic constitutive or geometric properties that are designed to influence the characteristics of mechanical wave propagation. When engineering these crystals, it is possible to isolate vibration within a certain frequency range. Vibration within this selected frequency range, referred to as the *band gap*, is attenuated by a mechanism of wave interferences within the periodic system. Such behavior is similar to that of a more widely known nanostructure that is used in semiconductor applications: a *photonic crystal*.

Optimizing the band gap of a phononic crystal can be challenging. We at Veryst Engineering have found COMSOL Multiphysics to be a valuable tool in helping to address such difficulties.

When it comes to creating a band gap in a periodic structure, one way to do so is to use a unit cell composed of a stiff inner core and a softer outer matrix material. This configuration is shown in the figure below.

*A schematic of a unit cell. The cell is composed of a stiff core material and a softer outer matrix material.*

Evaluating the frequency response of a phononic crystal simply requires an analysis of the periodic unit cell, with Bloch periodic boundary conditions spanning a range of wave vectors. It is sufficient to span a relatively small range of wave vectors covering the edges of the so-called *irreducible Brillouin zone* (IBZ). For rectangular 2D structures, the IBZ (shown below) spans from Γ to X to M and then back to Γ.

*The irreducible Brillouin zone for 2D square periodic structures.*

The Bloch boundary conditions (known as the Floquet boundary conditions in 1D), which constrain the boundary displacements of the periodic structure, are as follows:

u_{destination} = exp[-i\pmb{k}_{F} \cdot (r_{destination} - r_{source})] u_{source}

where **k**_{F} is the wave vector.

The source and destination are applied once to the left and right edges of the unit cell and once to the top and bottom edges. This type of boundary condition is available in COMSOL Multiphysics. Due to the nature of the boundary conditions, a complex eigensolver is needed. The system of equations, however, is Hermitian. As such, the resulting eigenvalues are real, assuming that no damping is incorporated into the model. The COMSOL software makes this step rather easy, as it automatically handles the calculation.

We set up our eigensolver analysis as a parametric sweep involving one parameter, *k*, which varies from 0 to 3. Here, 0 to 1 defines a wave number spanning the Γ-X edge, 1 to 2 defines a wave number spanning the X-M edge, and 2 to 3 defines a wave number spanning the diagonal M-Γ edge of the IBZ. For each parameter, we solve for the lowest natural frequencies. We then plot the wave propagation frequencies at each value of *k*. A band gap appears in the plot as a region in which no wave propagation branches exist. Aside from very complex unit cell models, completing the analysis takes just a few minutes. We can therefore conclude that this approach is an efficient technique for optimization if you are targeting a certain band gap location or if you want to maximize band gap width.

To illustrate such an application, we model the periodic structure shown above, with a unit cell size of 1 cm × 1 cm and a core material size of 4 mm × 4 mm. The matrix material features a modulus of 2 GPa and a density of 1000 kg/m^{3}. The core material, meanwhile, has a modulus of 200 GPa and a density of 8000 kg/m^{3}. The figure below shows no wave propagation frequencies in the range of 60 to 72 kHz.

*The frequency band diagram for selected unit cell parameters.*

To demonstrate the use of the band gap concept for vibration isolation, we simulate a structure consisting of 11 x 11 cells from the periodic structure analyzed above. These cells are subjected to an excitation frequency of 67.5 kHz (in the band gap).

*The structure used to illustrate vibration isolation for an applied frequency in the band gap.*

The animation below highlights the response of the cells. From the results, we can gather how effective the periodic structure is at isolating the rest of the structure from the applied vibrations. The vibration isolation is still practically efficient, even if fewer periodic cells are used.

*An animation of the vibration response at 67.5 kHz.*

Note that at frequencies outside of the band gap, the periodic structure does not isolate the vibrations. These responses are depicted in the figures below.

*The vibration response at frequencies outside of the band gap. Left: 27 kHz. Right: 88 kHz.*

To learn more about the 2D band gap model presented here, head over to the COMSOL Exchange, where it is available for download.

- P. Deymier (Editor),
*Acoustic Metamaterials and Phononic Crystals*, Springer, 2013. - M. Hussein, M. Leamy, and M. Ruzzene,
*Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook*, Appl. Mech. Rev 66(4), 2014.

Nagi Elabbasi, PhD, is a managing engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is the modeling and simulation of multiphysics systems. He has extensive experience in the finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, failure analysis, and material testing and modeling, and is a COMSOL Certified Consultant.

]]>

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.*

Take a CD in your hand. As the sun reflects off it, point the CD at a white wall. As you look at the wall, you will notice that a color reflection appears. What you are seeing is a result of small pits on one side of the CD that are arranged in a spiral. This is just one everyday example of diffraction grating.

*Image by Luis Fernández García – Own work, via Wikimedia Commons.*

Often utilized in monochromators and spectrometers, *diffraction gratings* are optical components with a periodic structure that reflects and transmits different wavelengths of light in different directions. The spacing and structure of the grating determines the directions and the relative magnitudes of the reflection and transmission. This reflection and transmission is also a function of the wavelength as well as the angle of incidence of the incoming light. As such, it is important that the grating is configured to ensure proper diffraction efficiency and thus enhance the overall performance of the optical instrument.

Testing different grating configurations can be costly and time consuming when done experimentally. Instead, simulation is a more cost-effective and efficient approach to achieving the optimal design. This virtual testing environment provides greater flexibility in analyzing different design scenarios, while eliminating costs associated with having to build prototypes to analyze each new modification.

With the Application Builder in COMSOL Multiphysics, you can now further simplify your simulation process by creating an easy-to-use app. Customized to fit your own design needs, simulation apps can be distributed throughout your organization, enabling others to run their own simulation tests. Our Plasmonic Wire Grating Analyzer demo app offers a helpful foundation for building an app of your own.

Let’s begin by discussing the model underlying the app. In the model, an electromagnetic wave is incident on a wire grating on a dielectric substrate. The example is designed for one unit cell of the grating, with Floquet boundary conditions used to describe the periodicity.

The Plasmonic Wire Grating Analyzer demo app takes the physics and functionality behind this model and makes it available in a simplified format. With this app, users can easily compute diffraction efficiencies for the transmitted and reflected waves as well as the first and second diffraction orders as functions of the angle of incidence. Additionally, this simulation app enables visualization of the electric field norm plot for various grating periods for a specific angle of incidence.

*A diffraction efficiency plot shown in the app.*

The figure above provides an overview of the app’s user interface. The left side of the interface features user-defined parameters, which are broken down into four different sections. The radius of a wire and the periodicity can be defined in the *Geometry Parameters* section, with the relative permittivity of the wire grating and the refractive index of the substrate arranged in the *Material Properties* section. The wavelength and the orientation of polarization are indicated in the *Wave Properties* section, and the current status of the app is referenced in the *Information* section.

Looking to the right side of the interface, there is a command toolbar comprised of six buttons — *Analyze, Reset Parameters, Simulation Report, Electric Field Norm Plot, Diffraction Plot,* and *Open PDF Document*. In their respective order, these buttons enable app users to run the simulation, revert input parameters back to their default values, make a simulation report, plot the electric field norm and the diffraction efficiency, and open the documentation. All results can be visualized within the graphics window in the center of the app’s interface.

When designing your own app, you can customize the look and feel of the user interface to fit your simulation needs. By including only those parameters and features that are relevant to your analysis, you can help hide the complexity of your model and create a user-friendly experience for app users.

Simulation apps offer a revolutionary approach to design that prompts greater involvement in the simulation process and thus delivers faster results. In the case of a wire grating, building an app simplifies the analysis of diffraction efficiencies, helping to identify a grating configuration that offers the optimal efficiency for its dedicated use. We encourage you to use our Plasmonic Wire Grating demo app as a resource in developing your own app.

- Download the Plasmonic Wire Grating Analyzer demo app

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.

]]>

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.

]]>

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!

]]>

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.

]]>

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!

]]>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

]]>

When modeling electromagnetic structures (e.g., antennas, waveguides, cavities, filters, and transmission lines), we can often limit our analysis to one small part of the entire system. Consider, for example, a coaxial splitter as shown here, which splits the signal from one coaxial cable (coax) equally into two. We know that the electromagnetic fields in the incoming and outgoing cables will have a certain form and that the energy is propagating in the direction normal to the cross section of the coax.

There are many other such cases where we know the form (but not the magnitude or phase) of the electromagnetic fields at some boundaries of our modeling domain. These situations call for the use of the Lumped Port and the Port boundary conditions. Let us look at what these boundary conditions mean and when they should be used.

We can begin our discussion of the Lumped Port boundary condition by looking at the fields in a coaxial cable. A coaxial cable is a waveguide composed of an inner and outer conductor with a dielectric in between. Over its range of operating frequencies, a coax operates in Tranverse Electro-Magnetic (TEM) mode, meaning that the electric and the magnetic field vectors have no component in the direction of wave propagation along the cable. That is, the electric and magnetic fields both lie entirely in the cross-sectional plane. Within COMSOL Multiphysics, we can compute these fields and the impedance of a coax, as illustrated here.

However, there also exists an analytic solution for this problem. This solution shows that the electric field drops off proportional to 1/r between the inner and outer conductor. So, since we know the shape of the electric field at the cross section of a coax, we can apply this as a boundary condition using the *Lumped Port, Coaxial* boundary condition. The excitation options for this condition are that the excitation can be specified in terms of a cable impedance along with an applied voltage and phase, in terms of the applied current, or as a connection to an externally defined circuit. Regardless of these three options, the electric field will always vary as 1/r times a complex-valued number that represents the sum of the (user-specified) incoming and the (unknown) outgoing wave.

*The electric field in a coaxial cable.*

For a coaxial cable, we always need to apply the boundary condition at an annular face, but we can also use the Lumped Port boundary condition in other cases. There are also a Uniform and a User-Defined option for the Lumped Port condition. The Uniform option can be used if you have a geometry as shown below: a surface bridging the gap between two electrically conductive faces. The electric field is assumed to be uniform in magnitude between the bounding faces, and the software automatically computes the height and width of the Lumped Port face, which should always be much smaller than the wavelength in the surrounding material. Uniform Lumped Ports are commonly used to excite striplines and coplanar waveguides, as discussed in detail here.

*A typical Uniform Lumped Port geometry.*

The User-Defined option allows you to manually enter the height and width of the feed, as well as the direction of the electric field vector. This option is appropriate if you need to manually enter these settings, like in the geometry shown below and as demonstrated in this example of a dipole antenna.

*An example of a User-Defined Lumped Port geometry.*

Another use of the Lumped Port condition is to model a small electrical element such as a resistor, capacitor, or inductor bonded onto a microwave circuit. The Lumped Port can be used to specify an effective impedance between two conductive boundaries within the modeling domain. There is an additional Lumped Element boundary condition that is identical in formulation to the Lumped Port, but has a customized user interface and different postprocessing options. The example of a Wilkinson power divider demonstrates this functionality.

Once the solution of a model using Lumped Ports is computed, COMSOL Multiphysics will also automatically postprocess the S-parameters, as well as the impedance at each Lumped Port in the model. The impedance can be computed for TEM mode waveguides only. It is also possible to compute an approximate impedance for a structure that is very nearly TEM, as shown here. But once there is a significant electric or magnetic field in the direction of propagation, then we can no longer use the Lumped Port condition. Instead, we must use the Port boundary condition.

To begin discussing the Port boundary condition, let’s examine the fields within a rectangular waveguide. Again, there are analytic solutions for propagating fields in waveguide. These solutions are classified as either Transverse Electric (TE) or Transverse Magnetic (TM), meaning there is no electric or magnetic field in the direction of propagation, respectively.

Let’s examine a waveguide with TE modes only, which can be modeled in the 2D plane. The geometry we will consider is of two straight sections of different cross-sectional area. At the operating frequency, the wider section supports both TE10 and TE20 modes, while the narrower supports only the TE10 mode. The waveguide is excited with a TE10 mode on the wider section. As the wave propagates down the waveguide and hits the junction, part of the wave will be reflected back towards the source as a TE10 mode, part will continue along into the narrower section again as a TE10 mode, and part will be converted to a TE20 mode, and then propagate back towards the source boundary. We want to properly model this and compute the split into these various modes.

The Port boundary conditions are formulated slightly differently from the Lumped Port boundary conditions in that you can add multiple types of ports to the same boundary. That is, the Port boundary conditions each *contribute to* (as opposed to the Lumped Ports, which *override*) other boundary conditions. The Port boundary conditions also specify the magnitude of the incoming wave in terms of the power in each mode.

*Sketch of the waveguide system being considered.*

The image below shows the solution to the above model with three Port boundary conditions, along with the analytic solution for the TE10 and TE20 modes for the electric field shape. Computing the correct solution to this problem does require adding all three of these ports. After computing the solution, the software also makes the S-parameters available for postprocessing, which indicates the relative split and phase shift of the incoming to outgoing signals.

*Solution showing the different port modes and the computed electric field.*

The Port boundary condition also supports Circular and Coaxial waveguide shapes, since these cases have analytic solutions. However, most waveguide cross sections do not. In such cases, the Numeric Port boundary condition must be used. This condition can be applied to an arbitrary waveguide cross section. When solving a model with a Numeric Port, it is also necessary to first solve for the fields at the boundary. For examples of this modeling technique, please see this example first, which compares against a semi-analytic case, followed by this example, which can only be solved by numerically computing the field shape at the ports.

*Rectangular, Coaxial, and Circular Ports are predefined.*

*Numeric Ports can be used to define arbitrary waveguide cross sections.*

The last case, when using the Port boundary condition, is appropriate for the modeling of plane waves incident upon quasi-infinite periodic structures such as diffraction gratings. In this case, we know that any incoming and outgoing waves must be plane waves. The outgoing plane waves will be going in many different directions (different diffraction orders) and we can determine ahead of time the directions, albeit not the relative magnitudes. In such instances, you can use the Periodic Port boundary condition, which allows you to specify the incoming plane wave polarization and direction. The software will then automatically compute all the directions of the various diffracted orders and how much power goes into each diffracted order.

For an extensive discussion of the Periodic Port boundary condition, please read this previous blog post on periodic structures. For a quick introduction to the use of these boundary conditions, please see this model of plasmonic wire grating.

We have introduced the Lumped Port and the Port boundary conditions for modeling boundaries at which an electromagnetic wave can pass without reflection and where we know something about the shape of the fields. Alternative options for the modeling of boundaries that are non-reflecting in cases where we do not know the shape of the fields can be found here.

The Lumped Port boundary condition is available solely in the RF Module, while the Port boundary condition is available in the *Electromagnetic Waves* interface in the RF Module and the Wave Optics Module as well as the Beam Envelopes formulation in the Wave Optics Module. This previous blog post provides an extensive description of the differences between these modules.

But what about those boundaries that are not transparent, such as the conductive walls of the waveguide we have looked at today? These boundaries will reflect almost all of the wave and require a different set of boundary conditions, which we will look at next.

]]>