AE04

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P. Knott, Antenna Engineering

Antenna Modelling and Design

Numerical EMModelling Methods

Full Wave Asymptotic

Local(Differential Eqn.)

FD, FIT, FEM

Global(Integral Eqn.)

MoM

Field Based

GO / GTD

Source Based

PO / PTD

Hybrid Methods

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Antenna Design Cicle

Design (CAD)Model Building, Meshing

AnalysisNumerical Calculation of Output Data (Impedance, Far Field, ...)

CheckDoes design fulfill

specs?

Selection ofAntenna Type,Initial Parameters

Modify Design Parameters

VerificationPrototype

Measurement

yesno

Specification(Far Field, Polarisation,

Gain, Size, ...)

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History of Computational Electromagnetics

Before 1960: Only analytical methods

� Simple problems / canonical geometries

Middle of 20th century: Asymptotic Methods

� Physical Optics (PO), Physical Theory of Diffraction (PTD)

� Geometrical Optics (GO), Geometrical Theory of Diffraction (GTD)

� Ray Tracing

Late 20th century: Full wave analysis

� Finite Element Method (FEM)

� Boundary Integral / Method of Moments (MoM)

� Periodic Structures, Frequency Selective Surfaces (FSS)

� Finite-Difference Time-Domain Method (FDTD)

Today: Design Suites (CAD, Analysis, Hybrid Solvers, Circuit Simulators, ...)

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Solutions to Partial Differential Equations

Finding a function u that satisfies (Laplace Equation)

� most important partial differential equation (PDE) in physics!

Solution not unique without a set of boundary conditions!

� on S (Dirichlet)

� on S (Neumann)

0),,(2 =∇ zyxu

02 =∇ u

0lim

02

==∇

∞→u

u

r

Inner Problem Outer Problem

0=u

0=∂∂n

u

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Influence of Frequency

� Example of slot diffraction

f0

d

Low frequency region

d << λ

(slot is not effective)

Resonance region

d ≈ λ

Wave diffraction

High frequency region

d >> λ

Ray optical path

subject of current researchsubject of current research

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Finite Element Method (FEM)

Subdivision of (inhomogeneous) problem into finite number of segments

(triangle, prism, tetrahedron)

Harmonic time dependency – one frequency, steady state

Decomposition of E, H for each subdomain with unknown amplitudes

Energy conservation

� Maxwell Equations

� Field distribution is stationary

Boundary conditions (natural and artificial) and solution of α

∫∫∫⋅−+=

V

22

222dV

j

EJEHF

ωεµ

rr

( )EfHrr

=

0=∂∂=

∂∂

αF

E

F

∑=

=N

iiiEE

1

rrα

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FEM Example – Coplanar stripline

=

⋅

−

−

NNNNNN

NN

J

J

J

YY

Y

YY

YYY

YY

MM

L

OM

M

L

2

1

2

1

1,

,1

3332

232221

1211

00

00

00

α

αα

Sparse matrix

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Finite Differences Time Domain (FDTD)

Solution of Maxwell‘s Equations in differential Form

� non-harmonic time dependence

� discretisation of time function (time steps)

� approximation of differential operators by difference quotients

� discretisation of (in)homogenous problem volume with rectangular grid

of homogenous cubes (Nyquist / fmax)

� introduction of artificial boundary conditions for “open problems”

12

12 )()(

xx

xfxf

x

f

−−=

∂∂

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Finite Differences Time Domain – Unit Cell

Kane S. Yee, Numerical Solution of InitialBoundary Value Problems Involving Maxwell’s Equations in Isotropic Media, IEEE Transactions on Antennas and Propagation, Vol. 14, No. 3, May 1966, pp. 302 - 307Unit (Yee) Cell

Unit cell specification(e.g. Eps / Z)

Boundary conditions for every cell wall

Time step processing

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Artificial Boundary Conditions - Perfectly Matched Layer

Absorb all energy at the boundary of the problem volume

� Minimise reflections normal to boundary

For wideband PML, multiplelayers have to be used

Jean-Pierre Berenger, A Perfectly Matched Layer for the Absorption of Electromagnetic Waves, Journal of Computational Physics, Vol. 114, pp. 185-200, 1994

)/(1

)/(1

cossin

sin1

sin1

0*

0

22

2

2

0

2

1

1

0

1

ωµσωεσ

θθ

θωε

σθωε

σ

x

xx

iyiixii

xx

j

jw

wwG

Gj

Gj

−−=

+=

−=

−

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2D FDTD Example – Horn Antenna and Obstacle

� Simple MATLAB program100 x 100 unit cells (dx = dy = 2.5 mm)

� f = 9.8 GHz / Time step 4.23 ps

� PML walls

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Full Wave Analysis Example – CST Microwave Studio

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Boundary Element Method / Method of Moments (MoM)

Solution of an integral equation with respect to the tangential E-field

Green‘s Function

� common solution of a differential equation for point source (current)

� includes information on the geometry without metallisation

Discretisation of metallic/conducting surfaces using triangles/rectangles (Nyquist, fmax)

Approximation of surface current J using piecewise linear / local basis functions

)()'()',()( itan

tottan rEadrJrrrE

S

rrrrrrrr+⋅⋅= ∫∫G

∑=

=N

iii JJ

1

rrα

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Method of Moments Procedure

Typical basis functions: Dirac, Rooftop, RWG, Higher order, ...

Boundary conditions on metallic surface

Multiplication with test function (weighting function)

� Galerkin‘s method: test function = basis function

Solve linear system of equations

=

⋅

NNNNN

N

V

V

V

ZZ

ZZ

ZZZ

MM

L

OM

M

L

2

1

2

1

1

2221

11211

α

αα

∫∫

∫∫ ∫∫

⋅=

⋅⋅=

m

m n

S

mi

mm

mnn

S S

mmn

adrErJV

adadrJrrrJZ

rrrrr

rrrrrrrr

)()(

)'()',()(

tan

GFull matrix

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Comparison of Full Wave Methods

Challenges

� Electrically large objects – Fast (and accurate) methods

� Complex geometries and materials

� Multi-physics design suites

efficient for purely metallic problems,

antennas, scattering

efficient for small volumes

low frequencies / largebandwith

applied to many physical fields, e.g. mechanical analysis, fluid / thermo-dynamics

surface meshvolume meshsurface/volume mesh

frequency domaintime domainfrequency domain

MoMFDTDFEM

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Asymptotic Methods (High Frequency Region)

Geometrical optics (GO) method

� Transport of energy (intensities)

� Ray-tracing

� Phase according to path length

Physical optics (PO) method

� Equivalent currents

� Illuminated / shadowed regions

� Far field integration (Green‘s Function)

×=×=

0

)(2 itot

PO

HnHnJ

rrrrr

∫∫ ⋅−=S

adJGjErrr

POS ωµ ∫∫ ⋅×∇=

S

adJGHrrr

POS

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Ray-Tracing with PO/PTD

� Function

Shooting and bouncing rays

Edge diffraction

PO/PTD incl. dielectric media

Curved surface treatment

Source

Simulation Object ObservationPoints

� Applications

Scattering from Large / ComplexObjects

Propagation / ElectromagneticCompatibility (EMC)

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Ray-Tracing with PO/PTDExample – Scattering by Aircraft (PEC Model M 1:32)

VV-Polarisation HH-Polarisation

Model: 500,000 triangles

Imaging with calculated monostatic RCS data

Da = 0,25° (721 angles)f = 25 .. 39 GHzDf = 50 MHz(281 frequencies) N = 200 million rays

CPU time: < 1 month(36 processors 1.4 GHz)

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Ray-Tracing with PO/PTDExample – Scattering by Car (PEC)

f = 6 … 10 GHz, ∆f = 10 MHz (401 frequencies)∆φ = 0.2° (=2*900 angles)

N = 10 million rays / aspect angle

CPU time: ≈ 1.5 h per angle

Model of original scale car:

Metallic object without wheels

108.000 triangular surface elements

Only surface contributions

Object length: > 100 λ

Simulation:

Polarisation: HH

PO contributions only

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Ray-Tracing Simulation of Propagation in Urban TerrainRelativeRCS in dB

Simulation number (500 corresponds to 1 km length)

Ran

ge

bin

f = 8 … 8.25 GHz∆f = 0.25 GHz(1001 frequencies)Horizontal polarization

Results show HRR profilesfor transmitter/receivermoving across the scene

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Hybrid MethodsBoundary Integral-Finite Elements-UTD-Hybrid Method

Modelling of Complex Objects

� Dielectric / Metallic components

� Numerically Rigorous Treatment

Hybrid Ansatz

� Finite Element Method (FEM)

� Fast BI Method: Multiple Level

Fast Multipole (MLFMM)

Environment

� Asymptotic Method (UTD)

� Electrically Large / Metallic Objects

Fast Near Field Calculation

dielectric

FEBI objectUTD object

��

rOEinc, Hinc

volume VA

r�rR

rD

closed surface A

metal

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Hybrid BI-FE-UTD MethodExample: Antenna Installed on Truck

P-MLFMM-UTD on 50,451 observation points13 min on Athlon 2800+ PC

Variation of instantaneous near field (E comp.)

- FEBI simulation: 913182 unknowns3.8 h, 2 GB RAM on Opteron 2.2 GHz CPU

- delta-gap voltage source excitation ofmonopole

- f=1.5 GHz

8.3 m

2.6 m

3.3 m

x

z

y�

�

λλλλ/4 monopole

antenna

|E | (V/m)y=0.0 m

x (m)

z (m)

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

Installed Antenna Performance

Monopole / patch antenne on truck

Influence of carrier platform

Influence of ground

Original Patch Antenna

AE04

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