SVD methods applied to wire antennae

Pelagia Neocleous

Kings College London

IPAM, Lake Arrowhead Meeting

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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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Background

• 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.

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Description of the problem

•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

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.

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More on the model

• 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.

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Pocklington’s integral field equation

Pocklington’s equation for the thin wire is:

where G(z,z’) is the free space Green function:

and is the distance between the source and the observation points.

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Reduced and Exact kernel Formulation

Assume the current is on the wire axis while the boundary conditions are applied on the surface

Nearly singular when

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.

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'.zz

Ill-posedness and ill-conditioning

• 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).

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Regularization methods

• 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

• 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.

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Proposed solution: Truncated SVD

• 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.

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An example of the resultsKING’S

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Ill-conditioning dependence on the wire thickness

• 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.

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The real decomposition

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.

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The Pocklington eigen modes KING’SCollege

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Problems…

• 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.

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Polar decomposition and rotation of subspaces

• 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.

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Some concerns about the use Pocklington’s equation

• 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.

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Considerations for improvement of the model

• 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.

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Transmission Line Theory

in fdV z

ZI z V δ z zdz

dI zYV z

dz

Vin

z

z = zf

z = 0 l

l

Transmission line differential equations

where Z and V are the impedance and

admittance per unit length.

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Analytic Expressions

• Antenna current on the axis

• Far field radiation pattern

sinsin ,

sin 2

sinsin ,

sin 2

finf

finf

β l zYVβ l z l z z

β βlI z

β l zYVβ l z z z l

β βl

is the complex phase constant.where β β jβ

cos2 2 2

cos cos

sinsin

4 sin 2 cos

sin sin 2 f

jkrin jkl θ

θ f

jkl θ jkz θf

jωμ e YV θE θ β l z e

π r βl β k θ

β l z e βl e

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Analytic results

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|I| (

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Periodic Green’s FunctionKING’S

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)(),()( 2200 xxxx G

),,(),( 001 xxxex GG ), 0, 1( 1 e),,(),( 2 00 xxxex GG

). 1, 0( 2e

Consider the Green’s function solution for the Helmholtz operatorin 2D with doubly periodic boundary conditions:

Due to periodicity we only need to compute G on a fundamental cell , where For any

}.2/1,,),({ 21 yxyxxB11 BB

)()(),( yxxxx 00 GGG

11),( BB 0xx

with the new variable .2 1By

Conclusions

• 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.

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Future Work

• 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.

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Acknowledgements

• Professor E.R. Pike

• Dr. D. Chana

Kings College London

• Dr. G. D. DeVilliers

QinetiQ, Malvern

• IPAM, UCLA

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