If you’re not a pro physicist, or you simply aren’t a math enthusiast, looking at Maxwell’s equations can feel daunting. This set of four coupled partial differential equations with two quantities to determine and multiple sources appears very complex. To make things more complex, these equations can be packaged in multiple forms, and the inclusion of material effects in these equations add four additional variables (displacement field, polarization, auxiliary magnetic field, magnetization).
Simply put, these equations involve a lot of bookkeeping and solving them requires a graduate-level mathematics or physics class. For the engineer that doesn’t have time to solve everything by hand, there is a useful software application to help you out: electromagnetic field solvers. These applications can break a complex set of differential equations into a mathematically simple arithmetic problem, which can then be solved with brute force by a computer. This article will give an overview of this process and the mathematics behind it.
A Look Inside of Electromagnetic Field Solvers
All electromagnetic field solvers have a single goal: to solve Maxwell’s equations given a set of inputs from the system designer. The main inputs into an electromagnetic field solver include:
- A physical model representing the design, which is exported from CAD tools
- Boundary conditions prescribed at each surface in the model, as well as far away from the model
- Initial conditions (values of the electromagnetic field, voltage, and currents at t = 0)
- Any voltage/current/field sources in the system
- Material constants for each body or component in the system
Obviously, things are getting very complex very quickly. An electromagnetic field solver’s job is to map out the electromagnetic field strength, voltage, and current throughout the system while accounting for all of these inputs. The results are normally displayed in a color map, or they can be exported as numerical data and used in another analysis application. An example color map is shown below.
Field Solvers Rely on Discretization
An electromagnetic field solver is doing something very important behind the scenes: it is breaking up a complex problem into its equivalent numerical problem. Computers cannot directly solve Maxwells equations, but if a partial differential equation is converted to an arithmetic problem, then a computer can solve the problem very easily and without mathematical errors.
Discretization is used to do exactly this. Various discretization techniques are used to break up Maxwell’s equations into simpler arithmetic problems using finite difference methods. Essentially, the derivatives in Maxwell’s equations are approximated as numerical differences, just like you would do in an introductory calculus class. The result is millions or billions of arithmetic calculations that need to be performed to solve for the electric and magnetic field in the system. This is an intractable problem for a human, but a decently powerful computer can solve the problem.
The standard discretization methods in electromagnetic field solvers include:
- Finite-difference time-domain
- Finite-difference frequency-domain
- Finite element method
- Finite volume method
- Method of moments
- Boundary element method
Avoid Garbage In, Garbage Out (GIGO)
The idea in “garbage in, garbage out” (GIGO) refers to an important problem in simulation in general. The inputs in your simulation need to accurately reflect reality in order for the simulation results to provide any predictive power or useful insight into the system. If the inputs are wrong, then the outputs from the field solver will also be wrong. The result of the field solver calculation will still be mathematically correct, it just might not have any reflection to reality in your specific system.
How can simulation application users avoid GIGO? This requires simulating a reference case in your system to verify that the simulation inputs are correct. This can involve recreating an experiment in the simulator, or it involves simulating a use case to which you already know the answer. The result should match intuition and experimental results. This will allow you to conclude that the model you have built from your electronics system in the simulation application is correct and the results will have some predictive power.
More Than Electromagnetics
The great thing about electromagnetic field solvers is that the underlying mathematics is very similar to computational solvers used in other areas of science and engineering. The discretization schemes may be different in these other disciplines, but the central idea is the same: break the equations up into a large set of arithmetic calculations, and allow the computer to execute these calculations at high speed.
- Thermal problems, such as the heat equation or thermal transients
- Computational fluid dynamics (laminar and turbulent)
- Mechanical problems involving deformation, vibration, or shock
- Multiphysics problems, where coupling may exist between physical phenomena
The last of these simulation types, multiphysics simulators, are the best tools for considering real interactions between different phenomena, such as the temperature dependence of material constants. These solvers are best equipped to simulate real systems that are deployed in challenging environments, and systems designers depend on these applications as an important qualification step.
Design teams working on some of the most advanced electronics systems trust the complete set of system analysis tools from Cadence to build their systems and evaluate functionality. Only Cadence offers a comprehensive set of circuit, IC, and PCB design tools for any application and any level of complexity. Cadence also provides users with a powerful set of electromagnetic field solvers for use in PCB design, IC design, and RF systems design.