K. ING’S. College. LONDON. 8. Founded I 2. 9. SVD methods applied to wire antennae. Pelagia Neocleous Kings College London IPAM, Lake Arrowhead Meeting. K. ING’S. College. LONDON. 8. Founded I 2. 9. Overview. Antenna design as an inverse problem The wire antenna (Background)
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K
ING’S
College
LONDON
8
Founded I2
9
SVD methods applied to wire antennae
Pelagia Neocleous
Kings College London
IPAM, Lake Arrowhead Meeting
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
Assumptions:
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
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 theobservation points.
K
ING’S
College
LONDON
8
Founded I2
9
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.
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
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.
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
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.
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
z
Transmission line differential equations
l
Vin
z = zf
l
z = 0
where Z and V are the impedance and
admittance per unit length.
K
ING’S
College
LONDON
8
Founded I2
9
where
is the complex phase constant.
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
Consider the Green’s function solution for the Helmholtz operator
in 2D with doubly periodic boundary conditions:
Due to periodicity we only need to compute G on a fundamental cell
, where
For any
with the new variable
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
K
ING’S
College
LONDON
8
Founded I2
9
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