# The Basics of Electromagnetic Analysis Using Finite-Difference Time-Domain Simulations

### Key Takeaways

• A standard set of techniques for examining electromagnetic behavior in complex structures is finite-difference time-domain simulations.

• These simulations can be used to calculate the electromagnetic field distribution in a complex structure, such as a waveguide, unique antenna, or IC package.

• Electromagnetic analysis using finite-difference time-domain methods helps designers evaluate complex systems before prototyping and production.

Microwave circuits and components can be evaluated with electromagnetic analysis using finite-difference time-domain simulations

Electromagnetic analysis encompasses a broad range of techniques for evaluating and understanding the behavior of the electromagnetic field in complex structures. Examples include anything from large PCBs to small integrated circuit packages where signals can propagate and radiate. These systems have complex geometries that cannot be easily treated with closed-form equations, so a numerical technique is needed to calculate the behavior of the electromagnetic field in these structures.

Finite-difference time-domain (FDTD) is used to solve Maxwell’s equations in a system with arbitrary 3D geometry and evaluate their evolution in the time-domain. There is also a related technique, called finite-difference frequency-domain (FDFD), which solves Maxwell’s equations in the frequency domain. This numerical technique is powerful in that it can treat arbitrary geometry, nonlinear media, inhomogeneous media, and anisotropic media. Many designers may not be familiar with the finer points of electromagnetic analysis using finite-difference time-domain simulations, but newer CAD tools help simplify the creation and execution of these simulations.

## How Does Electromagnetic Analysis Using Finite-Difference Time-Domain Simulations Work?

In electromagnetic analysis, finite-difference time-domain simulations are used to solve Maxwell’s equations in an arbitrary geometry for a given set of initial conditions and boundary conditions. These simulations enforce a discretization scheme on the governing differential equations in the system (Maxwell’s equations), boundary conditions, and initial conditions. This converts the differential equations that need to be solved into an iterative arithmetic problem that is solved at each point in space and each time step.

All finite-difference time-domain simulations apply spatial discretization by defining a “mesh” for the electric and magnetic fields in space. This mesh is used to define the set of points in space where the solution will be calculated. Time is also discretized in this simulation, which is used as the iteration basis in standard FDTD solution algorithms. Due to the transverse nature of the electric and magnetic fields, the typical discretization scheme is known as Yee’s scheme, as defined below:

Yee’s scheme for a cubic mesh in FDTD simulations

The process for solving these problems is conceptually simple:

1. Solve the spatial portion of the problem.
2. Calculate the solution evolution at each point in space.
3. Iterate to the next time step.

Although these simulations follow a simple iterative workflow, they can be intractable to solve by hand, thus they are implemented on a computer. Typical simulation times can range from minutes to days, depending on the level of discretization required and the number of time steps. Electromagnetic analysis using finite-difference time-domain has a specific solution algorithm that is implemented in field solver applications.

### The FDTD Solution Algorithm for Maxwell’s Equations

The solution algorithm in FDTD with Maxwell’s equations is a bit different from the typical algorithm used for other partial differential equations. Because Maxwell’s equations are a coupled set of partial differential equations describing the electromagnetic field, they must be solved iteratively, starting from the initial condition evaluated at the boundary of the system. The flowchart below shows the order in which Maxwell’s equations are solved in an FDTD simulation. The loop in time represents an iteration to the next time step, after which the entire system is solved again in space.

FDTD solution algorithm as applied to Maxwell’s equations

Note that the divergence relation for the magnetic field is absent from this flowchart, as it is uncoupled from the remainder of Maxwell’s equations. The power of the above process is that it applies a simple iterative scheme to a complex set of differential equations that could, in principle, be solved by hand. However, due to the size of these problems, designers who want to use FDTD should consider some measures to balance the accuracy of the results with computation time.

### Reducing Convergence Time While Maintaining Accuracy

Obviously, everyone would like perfectly accurate solutions to be available instantly in a 3D field solver. Anyone that wants to use a field solver application to solve electromagnetics problems should consider some steps to reducing the total simulation time without sacrificing accuracy. The simulation time is a linear function of the number of iterations (in time) and nodes (in space), so finding creative ways to reduce both quantities will decrease the total simulation time.

• Transient analysis: Instead of running FDTD simulations for long time periods, a typical strategy is to examine transient behavior and the transition to steady state behavior. The steady state can be simulated with a finite element method (FEM) simulation, while only the transient response is simulated with FDTD.
• Adaptive meshing: This is used to apply a dense mesh to a region that requires a very highly accurate simulation result, while other regions are set to have a coarse mesh. The various mesh sizes can be based on spatial frequency in the structure (i.e., k = ⍵/c), feature sizes, temporal frequency requirements in each region, or some other factor.
• Truncation: This is used to cut off the region where the simulation is performed and define continuity to a solution outside the simulation region that is known to be valid. In simulations involving flux across a boundary, this can be done by setting a non-Hermitian boundary condition. In FDTD solver applications, this is enforced with a perfectly matched layer at the boundary of the simulation region.
• Take advantage of symmetry: If a problem has symmetry in space, then use this to your advantage. Some problems with symmetry can have a dimension eliminated, which reduces the size of the problem and the total simulation time.

Electromagnetic analysis using finite-difference time-domain is a complex topic and continues to be an active area of research. Electronics designers who need to evaluate the functionality of their systems should use Cadence’s PCB design and analysis software and the Clarity 3D EM Solver. This pair of applications is ideal for evaluating a new design before prototyping and production. Only Cadence’s software suite gives you access to a range of simulation features you can use in electromagnetic analysis, giving you everything you need to evaluate your system’s functionality.