Computational Electromagnetics

Slides From ATI Professional Development Short Course Computational Electromagnetics

Instructor: Dr. Keefe Coburn

ATI Course Schedule: ATI's ComputationalElectromagnetics

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Coarse Outline •

Day 1 – – – – –

•

Day 2 – – – – – –

•

Review EM Theory Review Antennas Review Antenna Arrays Review Scattering Introduction to FEKO Lite CEM Introduction FEM Introduction MoM Introduction FDTD Introduction FVTD, TLM and FIT Examples

Day 3 – FEM Tutorial – MoM and FDTD Tutorial – Summary and Advanced Topics • High-frequency Methods • Hybrid Methods

– Discussion and Examples

Electromagnetics (EM) Maxwell’s Equations Faraday’s Law:

E  

 B t

Gauss’ Laws:

B  0

D  

Ampère’s Circuital Law:

  H  J

 D t

Constitutive Equations:

B  H

DE

Actual solution for realistic problems is complex and requires simplifying assumptions and/or numerical approximations Solutions to Maxwell’s equations using numerical approximations is known as the study of Computational Electromagnetics (CEM)

Why Computer-Aided Methods Are Used?

Importance • Key to analysis, design & optimization of RF to optical systems. • Basis for field-theory based and process-oriented CAD (virtual prototyping). • Key to economical success of a product through shortening of development time. • Only means for dealing with complex (non-canonical) electromagnetic structures. • Theoretical models must be validated by experiments. • Theoretical and experimental work are of equal importance.

Classic Electromagnetic Solution Mathematical Formulation

Maxwell’s Equations

Boundary Conditions

Material Properties

Analytical Preprocessing

Analytical Model

Problem-Dependent

Discretization

Computer Program User Data Computation User Interface

Postprocessing

Results

Examples: • Closed-form expressions for microstrip/transmission lines; • Spectral domain program for coplanar waveguides; • Eigenvalue solvers for guided waves and cavities;

Modern Electromagnetic Solution Numerical Formulation And Discretization

Maxwell’s Equations

Huygens’ Principle

Variational Principle

Numerical Model

Problem-Independent

Computer Program Boundary Conditions Material Properties Computation User Interface Postprocessing

Results

Examples: • Finite Element or MoM Frequency Domain Solver • FVTD or FDTD Time Domain Numerical Simulation

Conventional Microwave Design Circuit/Antenna Specifications Design Data Initial Design

Rapid Prototyping

Laboratory Model

Modifications Loop

Measurements

Fail

Expensive, Time-Consuming Not Automated

Compare Pass Final Fabrication

Computer-Aided Design (CAD) Circuit/Antenna Specifications Design Data Synthesis Methods Initial Design

Models Virtual Prototyping

Analysis

Modifications Optimization Loop

Measurements • Physical Modeling Error • Discretization Error • Numerical Modeling Error • Measurement Error

Sensitivity Analysis

Inexpensive, Fast, Automated

Fail Compare Pass Prototype Fabrication

Electromagnetic Simulators • An Electromagnetic Simulator is a modeling tool that: – solves electromagnetic field problems by numerical analysis; – extracts engineering parameters from the field solution and visualize fields and parameters; – allows design by means of analysis combined with optimization (PSO, GA, parameterized models, etc.). • The field solver engine employs one or several numerical methods obtained through the practice of CEM: – is the theory and practice of solving electromagnetic field problems on digital computers; – provides the only viable approach to solving “real world” field problems; – enables Computer-Aided Engineering (CAE) and ComputerAided Design (CAD) of EM components and systems.

Solving EM Field Problems • Find electromagnetic field and/or source functions such that they – – – –

obey Maxwell’s equations, satisfy all boundary conditions, satisfy all interface and material conditions, satisfy all excitation conditions.

(In both time and space, or at one frequency in space) • Field solutions are then unique when tangential fields on conductors and initial conditions are known • But numerical solution depends on • Physical Modeling Error • Discretization Error • Numerical Modeling Error • Measurement Error

Field-Solving Methods Methods for solving Maxwell’s Equations: • Analytical Methods – Exact explicit solutions (only a few ideal cases)

• Semi-Analytical Methods – Explicit solutions requiring final numerical evaluation – Numerical solutions with analytical “preprocessing”

• Approximate analytical models – Approximate analytical solutions for simplified structures (provides physical insight) – Only practical way to handle very large electrical structures

• Numerical Methods – Differential or integral equations are transformed into matrix equations by numerical approximations (sampling) and solved iteratively or by matrix inversion

Overview of CEM • Central to all CEM techniques is the idea of discretizing some unknown EM property, for example: – In MoM the surface current is typically used – In FE, the Electric Field – In FDTD, the Electric and Magnetic Field

• Meshing is used to subdivide a large geometry into a number of nonoverlapping subregions or elements, for example: – In two dimensional regions triangles maybe used – In three dimensional geometries a tetrahedral shape may be used

• Within each element, a simple functional dependence (basis functions) is assumed for the spatial variation of the unknown • CEM is a modeling process and therefore a study in acceptable approximation and numerical solution In other words, CEM replaces a real field problem with an approximate one which causes physical (geometric) and numerical limitations that one must keep in mind

Overview of CEM •

•

•

•

During the creation of the approximate “model”, assumptions and simplifications are generally introduced so limitations on the solution accuracy, for example: – Assuming an infinite ground plane or substrate in an antenna structure – Simplifying a thin wire by a current filament (dipole impedance) When analyzing solutions generated from a CEM techniques keep in mind limitations of the solution introduced by manufacturing tolerances: – Small changes in dimensions may affect the performance – Frequency dependent (or unknown) material properties Limitations are also introduced by the finite discretization – The mesh must be fine enough so that the basis functions can adequately represent the electromagnetic fields – Fine mesh required for critical variations such as source region Numerical approximations and finite precision will limit the analysis – Limited computational resources (i.e., mesh size) – Double precision accuracy will not help if the problem is ill conditioned – Finer mesh is often the only choice (multi-scale codes)

Fundamentals of Simulations

Fundamentals of Simulations

Fundamentals of Simulations

To use verified CEM tools successfully requires: • Knowledge and understanding of electromagnetics and RF engineering • Validation of simulation results (analytical models, code-to-code, experiments, prototyping)

Accuracy Checklist

Computational Hierarchy computational electromagnetics numerical methods IE

High frequency DE field based

TD

TWTD

FD

TD

MoM

FDTD TLM

FD

FDFD

GO/GTD

current based

PO/PTD

Classification of Methods • Maxwell’s Equations-based methods Method

Frequency Domain

Boundary Element

Method of Moments (MoM)

Finite Element

FEM

Time Domain

Finite Difference

FDTD

Finite Volume

FVTD

• Optics-based methods Method Physical Optics GTD/UTD 19

Classification of Methods Frequency Domain Methods (Time-Harmonic)

Time Domain Methods (Transient)

Fourier Transform • This distinction is based more on human experience than on physical or mathematical considerations. • The time dimension can be treated as a fourth dimension in Minkowski space in the form jct, where c is the speed of light. • The user requires knowledge of different methods to be able to choose the most suitable design tool and setup the calculation correctly. • In the most general sense, solution methods can thus be classified according to the number of dimensions upon which the field and source functions depend.

Classification of Methods 1D Methods:

Fields and voltage/current vary in one space dimension (Transmission Line Problems) (…Touchstone, Supercompact, SPICE…) 2D Methods: Fields and currents vary in two space dimensions (Cross-section problems, TEn0 waveguide problems) (…FEM-2D, MEFiSTo-2D…) 2 1/2 D Methods: Fields vary in three space dimensions, currents vary in two space dimensions (Planar multilayer circuits) (…Sonnet, Momentum, Ensemble...) frequency domain 3D Methods: Fields and currents vary in three space dimensions (General propagation, scattering and radiation problems) (…HFSS, FEKO, CST, XFDTD, GEMS, GEMACS…)

What Have All Methods In Common? 1. In all methods, the unknown solution is expressed as a sum of known functions (expansion functions or basis functions). 2. The weight (coefficient) of each expansion function is determined for best fit.

What distinguishes them? • the electromagnetic quantity approximated, • the expansion functions used, • the strategy employed for determining the coefficients of the expansion functions, and • the numerical solution method.

Other Classifications Quasi-TEM or full-wave? Quasi-TEM use notions of effective dielectric constant, single mode impedance, axial current, dispersionless propagation.

Circuit or antenna? Static and quasi-static techniques are applicable to analysis of circuits, except for spurious effects: radiation & surface waves.

Open or closed problem space? Circuits are at some point usually enclosed within a metal box, similarly, antennas are designed to operate in free space. One would expect that techniques for boxed in structures would be used to analyse circuits and those techniques for open structures for antennas. This is not what happens in practice! Common assumptions are: infinite dielectrics and ground planes, zero thickness of strips, etc.

Frequency and Time Domain Concepts • Complex Frequency • Phase angle • Complex Dielectric Constant • Complex Reflection/ Transmission Coeff. • Complex Impedance • Q-Factor • Complex multiplication

• Time dependence • Delay • Real permittivity and conductivity • Reflection/Transmission time response • Impulse response • Decay time • Time domain convolution

Why Model In The Frequency Domain? • Most microwave engineers are more familiar with FD concepts than with TD concepts • Frequency domain simulations are steady-state • Complex notation is elegant and efficient • Specifications are traditionally formulated in the FD (S-Parameters, loss tangent, dispersion) • Time domain information can be obtained by inverse Fourier Transform (Causality issues!) • Dispersive materials and boundaries are easily described by frequency-dependent parameters

Finite Element Method (FEM)

Finite Element Method (FEM)

Method of Moments (MoM)

Method of Moments (MoM)

Why Model In The Time Domain? • Time domain simulations are “life-like” and allow visualization of signal propagation • Virtual experiments are set up as in the lab (Source, reference planes, output probes) • Cause and effect can be distinguished • One simulation can cover a wide bandwidth • Transient phenomena can be simulated • Nonlinear behavior is modeled naturally • Dispersive materials and boundaries are modeled in a more physical manner • Frequency domain information can be obtained via Fourier transform

Finite Difference Time Domain (FDTD)

Finite Difference Time Domain (FDTD)

Comparison

Comparison

Summary and Conclusions • Numerical Methods allow us to solve real life EM problems (within certain limits). They form the engine(s) of electromagnetic simulators. • Electromagnetic simulators are not merely Maxwell equation solvers, but powerful simulation and design tools with visualization capabilities. • Understanding EM phenomena and knowledge of radio engineering are necessary for successful use of codes. • Understanding the underlying numerical methods is essential in assessing the accuracy, performance and limitations of a particular simulation tool. • Electromagnetic simulators are the heart of modern CAD tools for analog microwave, digital high-speed and mixed signal design, EMC and signal integrity engineering and other applications of electromagnetic fields and waves.

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