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ING’S
College
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LONDON
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SVD methods applied to wire
antennae
Pelagia Neocleous
Kings College London
IPAM, Lake Arrowhead Meeting
ING’S
Overview
• Antenna design as an inverse problem
• The wire antenna (Background)
• Ill-posedness and ill-conditioning of the Pocklington IEFE
• Regularization methods commonly used
• The singular value decomposition approach
• Considerations for improvement of the Pocklington model
– Transmission line theory and periodic Green’s functions
• Future directions
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ING’S
Background
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•
Antenna design is the inverse problem of finding the structures that give rise to
specific far-field radiation patterns. The radiation pattern of a wire antenna can
easily be calculated from the current distribution across its length.
•
Determining the current distribution on a wire given an incident harmonic
electromagnetic field is a hard problem with a long history of 90 years.
•
Analytic solutions are known for the Hertzian dipole and the infinitely long
and thin wire, but there is no mathematical theory to provide a solution
between the two extremes.
•
Pocklington in 1907 constructed a family of asymptotic solutions to the
infinite thin wire, which approach perfect sinusoids.
•
Harrington in 1967 proposed a way to solve the integral equation numerically
using matrix methods.
ING’S
Description of the problem
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•Problem: Determine the charge distribution J(z’) on a metal scatterer, given the
incident field Ei(z) on its surface.
•The linear operator is derived from solving Maxwell’s equations subject to the
boundary condition:
where Es(z) is the scattered field.
•The resulting Fredholm equation of the first kind is:
where K(z,z’) is a linear kernel.
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The thin wire
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Assumptions:
• The tangential component of the
electric field at the surface of the
wire is zero.
• The wire has infinite conductivity.
• The current vanishes at the open
ends of the conductor.
• The radius
and
• The excitation field source is a
generator.
.
ING’S
More on the model
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• Based on these assumptions a thin linear antenna can be treated as a
series of hertzian dipoles of charge density J(z’) , the electromagnetic
fields of which, superimpose at any given point in space.
• The radiation field is the result of an one-dimensional integration over
all the elementary dipoles across the antenna.
•
The source is modelled as a constant voltage applied on a gap of
length  at the centre of the wire.
Pocklington’s integral field
equation
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Pocklington’s equation for the thin wire is:
where G(z,z’) is the free space Green function:
and
points.
is the distance between the source and the observation
ING’S
Reduced and Exact kernel Formulation
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Assume the current is on the wire axis while the boundary conditions are
applied on the surface
Nearly singular when
z  z'.
The current is modelled as the sum of rings of azimuthally symmetric
current density constructing the surface of the wire
Has a removable singularity at the origin.
ING’S
Ill-posedness and ill-conditioning
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•
Pocklington’s equation with the reduced kernel is shown to be ill-posed
(Rynne 2000). The integral is a compact operator, hence its inverse is
unbounded.
•
The electric field data do not depend continuously on the solution for the
current distribution.
•
This results to difficulties such as the appearance of rapid oscillations near the
driving point when the number or basis functions becomes larger than L/2a.
•
The accuracy of the solution is limited to a discretization of a mesh size h not
smaller than the wire thickness. If h < a an oscillating error is introduced near
the endpoints (Fikioris 2002).
ING’S
Regularization methods
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• Restrict the solutions to specific Sobolev-type function spaces, defined
so that the solution satisfies the smoothness conditions at the endpoints
(Rynne 2003)
• Numerical methods expanding the solution in appropriate function
subspaces
– Method of Moments
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The method of moments
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• Matrix methods for solving linear equations. First generalised and
applied to electromagnetism by R.F. Harrington (1967)
• As in the moment definition (Moment = Force x Arm), it takes the
moments by multiplying with weighting functions and integrating.
• Transform the integral equation in a matrix form
• Expand g and f in appropriate basis (bn) and weighting functions (wn)
such that:
• The basis and weighting functions should be linearly independent and
chosen so that they can approximate the solution domain f sufficiently
well.
ING’S
Proposed solution: Truncated
SVD
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• Expand solutions in the most information dense orthogonal subspaces
• Overcome ill-conditioning by truncation
• There are no limits on the number of points representing the integral
operator.
• Provide information about the noise subspace
• The singular vector subspaces only need to be calculated once and
used as basis for the expansion of the solution in all applications.
ING’S
An example of the results
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ING’S
Ill-conditioning dependence on
the wire thickness
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• The singular values
drop rapidly as the
radius to length ratio
increases.
• The condition number
grows exponentially
with the matrix size N
and is a rapidly
increasing function of
of the radius to length
ratio.
ING’S
The real decomposition
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The problem is simplified by looking at the product of the
Pocklington operator K multiplied with its adjoint:
K is Hermitian the problem can be solved with just one real
SVD.
The Pocklington eigen modes
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ING’S
Problems…
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• The decomposition is non unique on the complex plane. The analogue
of the sign parity in the real case, is translated into a phase parity on
the complex 2D plane.
• The eigen vectors of the real decomposition and the solution they yield
is real. How does that relate to the complex results?
• In order to relate the two solutions we need to understand the phase
rotations which show how the two subspaces interact.
ING’S
Polar decomposition and rotation of
subspaces
•
•
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The mapping of one subspace into another can be studied using polar
decomposition.
The polar decomposition is given by:
where
is Hermitian and
is positive semidefinite.
It is analogous to the complex number factorisation
and reveals
information on the effect of the transformation to the magnitude and phase of a
complex vector.
The calculation of the polar factor for both cases, allows mapping from one
subspace to the other.
ING’S
Some concerns about the use Pocklington’s
equation
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• The implication that the currents are point sources along the wire
instead of accelerated charges is a poor model for the use of Maxwell’s
equations.
• Pocklington’s equation is a result of an infinite sinusoidal expansion of
the solution to a one dimensional infinitely long perfect conductor.
• The reduced kernel implies a dimensional collapse of the cylinder and
is a poor approximation to real wires of finite thickness.
ING’S
Considerations for improvement
of the model
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• Use transmission line theory
– Compare the antenna to a lumped wave guide and apply Kirchoff’s
electric circuit equations for given input impedances.
• Use periodic Green’s function
– Surface charge is evaluated on periodic boundary conditions on the
rings of the Green’s function.
ING’S
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Transmission Line Theory
z
Transmission line differential equations
dV  z 
 ZI  z   Vin δ  z  zf 
dz
dI  z 
 YV  z 
dz
l
Vin
l
where Z and V are the impedance and
admittance per unit length.
z = zf
z=0
ING’S
Analytic Expressions
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• Antenna current on the axis
 YVin sin  β  l  zf  
sin  β  l  z   , l  z  zf
 β
sin  2 βl 
Iz  
 YVin sin  β  l  zf   sin  β  l  z   , z  z  l
f
 β
sin  2 βl 
where β  β   jβ  is the complex phase constant.
• Far field radiation pattern
jωμ e  jkr
YVin sin θ
Eθ  θ  
sin  β  l  zf   e jkl cos θ

2
2
2
4π r sin  2 βl   β  k cos θ 
 sin  β  l  zf   e  jkl cos θ  sin  2βl  e jkzf cos θ 
ING’S
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Analytic results
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200
Analytical
MoM
8
Analytical
MoM
150
100
ÐI (deg)
|I| (mA)
6
4
50
0
-50
-100
2
-150
0
-0.75
-0.5
-0.25
0
z/ l
0.25
0.5
-200
0.75 -0.75
90
-0.5
800
120
60
600
400
150
-0.25
Analytical
MoM
30
200
180
0
210
330
240
300
270
0
z/ l
0.25
0.5
0.75
ING’S
Periodic Green’s Function
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Consider the Green’s function solution for the Helmholtz operator
in 2D with doubly periodic boundary conditions:
( 2   2 )G (x, x 0 )   (x  x 0 )
G(x  e1 , x0 )  G(x, x0 ),
G(x  e 2 , x0 )  G(x, x0 ),
e1  (1,0),
e2  (0,1).
Due to periodicity we only need to compute G on a fundamental cell
B1  {x  ( x, y )   2 x , y ,  1 / 2}.
B1  B1 , where
For any
(x, x0 )  B1  B1
G(x, x0 )  G(x  x0 )  G(y)
with the new variable y  2B1.
ING’S
Conclusions
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•
Blind application of numerical methods can often give good results, but a
thorough understanding of the difficulties associated with the application is
needed to determine whether to trust or distrust one’s results.
•
Both ill-posedness and matrix ill-conditioning need to be carefully considered
when solving Pocklington’s equation.
•
The singular value decomposition approach yields good results for realistic
finitely thin wires.
•
Pocklington’s equation has severe limitations as a model for the finite wire
antenna.
•
The area of antenna design can be benefited from the application of inverse
problems methods.
ING’S
Future Work
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• Extend the method to more complicated wire structures such as H
aerials and Yagi antennas.
• Apply the SVD method to linear arrays of dipoles and solve the array
design problem by expanding in the appropriate eigen modes.
• Use periodic Green functions to extend the results to linear arrays of
dipoles.
• Ultimate goal of the project is to provide a rigorous method based on
electromagnetic theory, for gaining understanding of the mutual
coupling between neighbouring elements in radar array design.
ING’S
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Acknowledgements
• Professor E.R. Pike
• Dr. D. Chana
Kings College London
• Dr. G. D. DeVilliers
QinetiQ, Malvern
• IPAM, UCLA