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Coarse Outline• Day 1

– Review EM Theory– Review Antennas– Review Antenna Arrays– Review Scattering– Introduction to FEKO Lite

• Day 2– 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)

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)

D

DJHt

0 B

BEt

Faraday’s Law: Ampère’s Circuital Law:

Gauss’ Laws:

HB ED

Constitutive Equations:

Maxwell’s Equations

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

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

Postprocessing

Maxwell’sEquations

AnalyticalModel

BoundaryConditions

MaterialProperties

User Data

Computation

ComputerProgram

Results

MathematicalFormulation

AnalyticalPreprocessing

Discretization

User Interface

Problem-Dependent

Modern Electromagnetic Solution

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

Postprocessing

Maxwell’sEquations

NumericalModel

Huygens’Principle

VariationalPrinciple

Boundary ConditionsMaterial Properties

Computation

ComputerProgram

Results

NumericalFormulation

AndDiscretization

User Interface

Problem-Independent

Conventional Microwave Design

Initial Design

Circuit/AntennaSpecifications

DesignData

Modifications

Measurements

LaboratoryModel

Loop

Expensive,Time-ConsumingNot Automated

Conventional Microwave Design

Compare

FinalFabrication

Pass

Fail

RapidPrototyping

Computer-Aided Design (CAD)

VirtualPrototyping

Initial Design

Circuit/AntennaSpecifications

Design DataSynthesis Methods

Modifications

Measurements

Analysis

OptimizationLoop

Compare

PrototypeFabrication

Pass

Fail

SensitivityAnalysis

Models

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

Inexpensive,Fast,Automated

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 Computer-

Aided 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 MethodsMethods 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 inmind 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

Accuracy ChecklistTo 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)

Computational Hierarchy

computationalelectromagnetics

High frequencynumerical methods

IE DE

MoMFDTDTLM

field based current based

GO/GTD PO/PTD

TD FD TD FD

TWTD FDFD

Classification of Methods• Maxwell’s Equations-based methods

• Optics-based methods

19

Method Frequency Domain

Time Domain

Boundary Element

Method of Moments (MoM)

Finite Element FEMFinite Difference FDTD

Finite Volume FVTD

MethodPhysical Optics

GTD/UTD

Classification of Methods

• 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 beclassified according to the number of dimensions uponwhich the field and source functions depend.

Frequency Domain Methods(Time-Harmonic)

Time Domain Methods(Transient)

FourierTransform

1D Methods: Fields and voltage/current vary in one spacedimension (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 domain3D Methods: Fields and currents vary in three space

dimensions (General propagation, scatteringand radiation problems)

(…HFSS, FEKO, CST, XFDTD, GEMS, GEMACS…)

Classification of Methods

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 ClassificationsQuasi-TEM or full-wave?Quasi-TEM use notions of effective dielectric constant, singlemode impedance, axial current, dispersionless propagation.

Circuit or antenna?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

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

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

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 theFD (S-Parameters, loss tangent, dispersion)• Time domain information can be obtained byinverse Fourier Transform (Causality issues!)• Dispersive materials and boundaries are easilydescribed 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.