Electromagnetic field solvers are complex applications, and sometimes they require significant computational experience in order to gain highly accurate results. The use of these applications has started to become domain-specific, where the applications have specific solution algorithms and specific boundary definitions that are broadly applicable to particular systems. In electromagnetics, this is especially true when we look at simulation software for electronic devices, both for PBCs and component packaging.
If you plan to qualify your system using electromagnetic simulations, then you will need to apply electromagnetic boundary conditions in your simulation. The boundary conditions in a simulation are applied within simulation regions such that the electric and magnetic fields are bounded to certain values. This helps to ensure your simulation results reflect reality, and it ensures the numerical routines used in electromagnetic simulators can produce converging solutions.
Types of Electromagnetic Boundary Conditions
Electromagnetic boundary conditions define the values of the electric and magnetic fields at the interface between two dielectrics, a dielectric and a conductor, or a dielectric and vacuum. In general, because the electric and magnetic field form a self-reinforcing two-component wave, we have four electromagnetic boundary conditions that define the directions of the electric fields as they travel between media.
In a real simulation, we have a particular simulation space that must be considered in the simulation. Essentially, when you create an electromagnetic simulation model, you are solving Maxwell’s equations inside a finite box that mimics the extent of the simulation region. For example, if you are simulating the radiated emissions from a PCB, you are actually solving Maxwell’s equations inside a box that outlines your PCB.
Simulation region defined by a bounding box.
This is where boundary conditions come into play; the boundary conditions are applied within the simulation region and at the border of the simulation region such that the simulation accurately approximates the real world.
Perfect Electrical Conductor
Perfect electrical conductor (PEC) boundary conditions represent idealized materials that have an infinite electrical conductivity. This means that they can conduct an unlimited amount of electrical current without any resistance or energy loss. In reality, no material can achieve infinite conductivity, but this is a useful approximation for many electromagnetics problems.
In terms of mathematics, this boundary condition is used to approximate conductor boundaries in a system so that they produce very nicely terminated fields at boundaries. This condition is broadly applicable in closed structures. In this case, the normal electric field terminates to zero exactly at the interface of the system, as does the tangential component of the magnetic field. Meanwhile, the orthogonal components of the electric and magnetic field are related to charge and current sources, respectively:
These four conditions apply to perfect conductors that form the boundary along an arbitrary dielectric. In general, the dielectric forming an insulator or semiconductor in a system could be inhomogeneous, anisotropic, and/or nonlinear; the D and H terms account for this potentiality and will describe additional phenomena for inhomogeneous and anisotropic media.
When a wave travels between two different dielectrics, a portion of the wave reflects and a portion transmits, such as would be described by Fresnel’s equations from optics. This is due to a deeper boundary condition, where the electric and magnetic fields must be continuous across the boundary where two dielectrics meet.
This is visualized in the image below. Here, an incoming wave must have matched amplitude and matched derivatives across the boundary. In other words, the field amplitudes must match right at the boundary, and the slope of the wave on each side of the dielectric must match on each side of the boundary.
Wave traveling across the boundary between two dielectrics.
Note that the continuity across the boundary between the two dielectrics will create reflection as determined by the wave impedances in each medium. This type of boundary condition would be applied internally in your simulation volume, but it would not be applied at the exterior to bound the simulation volume. If PEC boundary conditions do not provide the approximate behavior of a bounded system, then there are alternative dielectric-based boundary conditions that can be applied.
Perfectly Matched Layer
Perfectly matched layer (PML) boundary conditions represent a type of dielectric matching where waves are allowed to propagate out of the system such that there is minimum reflection. As waves are allowed to propagate out of the simulation region, they then need to terminate to zero very quickly by enforcing absorption in the matched layer. These boundary conditions are sometimes referred to as a subset term used in computational electromagnetics, known as an absorbing boundary condition.
Conceptually, PMLs allow propagation out of the simulation volume regardless of the angle of incidence or frequency. The goal is to approximate the case where the system is allowing waves to propagate outside the system and into an infinite medium, where they will no longer interact with the simulation volume.
Open or Fixed Boundary Conditions
A higher-level set of boundary conditions, of which PMLs are one type, are open boundary conditions. There are various types of open boundary conditions:
- Periodic boundary conditions
- Radiating (or Sommerfeld) boundary conditions
- Absorbing boundary conditions
- Fixed impedance boundary conditions
- Symmetric and antisymmetric boundary conditions
Another set of boundary conditions defines fixed values of the electric and/or magnetic field at the boundary of a system. This could be used to, for example, represent the voltage (and thus the normal electric field) between two points in the system by prescribing some value at a boundary. This would be equivalent to specifying a value and direction for the terminating electric field in the above PEC conditions. In either case, they are used when the value of the field, its derivative, or both is to be fixed along some specific boundary.
Top: Dirichlet boundary conditions; middle: Neumann boundary conditions; bottom: Robin boundary conditions. When both are enforced together, these are known as mixed boundary conditions. When these are taken as a linear combination, they are known as Robin boundary conditions.
What About Initial Conditions?
Electromagnetic wave problems are always propagation problems, meaning there will be some behavior in the time domain that can be determined from numerical simulations. This would normally be a transient electromagnetic simulation that is calculated using FDTD. To determine the time domain behavior of the system, you need an initial condition and the initial rate of change of the electromagnetic field:
Note that A and B could be functions of space. With these two conditions, you can determine the temporal evolution of the system over time. These initial conditions are required if you want to know the time evolution of the electromagnetic field, and particularly the approach to steady state.
What about using a Fourier transform of the wave equation to determine the electromagnetic field in the frequency domain? In this case, you would determine the solution in the frequency domain using finite difference frequency domain (FDFD), but you still need the initial conditions to determine the particular solution in the frequency domain. After applying the inverse Fourier transform to the FDFD, you can apply the initial conditions as normal.
Here we must have two initial conditions because the wave equation defining the behavior of the electromagnetic field is of 2nd order (meaning it has a 2nd order derivative). These types of initial conditions are used in the exact same format in other types of problems in science and engineering. For example, these are used in multiphysics problems, mechanical problems, thermal problems, and CFD problems.
In some cases the initial conditions can be complex and they are essentially defining a boundary condition, as alluded to above in the above definition of initial conditions. A more complex case where the initial condition has a functional format would look like the following definition:
This type of inhomogeneous initial condition is related directly to the boundary conditions in that we can match the particular spatial solution to the initial condition at all points inside the system being simulated. The resulting particular solution to this inhomogeneous problem is defined as a linear combination of the set of eigenfunctions defining the homogeneous particular solution. By exploiting orthonormality of solutions to Maxwell’s equations, the expansion coefficients defining the particular solution can be determined with a single integral:
This integration is typically automatically defined in your simulation application and does not need to be performed manually. In some applications, these functional initial conditions can be imperfect, meaning you do not need to have the electromagnetic field defined at every single point in the solution space; these applications can use interpolation to “fill in gaps” in the functional definition of the initial condition. As a simulation software user, you do need to define things like the spatial discretization, solution cutoff (maximum n value), interpolation styles, and convergence conditions.
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