advanced electromagnetic theory
TRANSCRIPT
Advanced Electromagnetic Theory Dr. Serkan Aksoy
Advanced
Electromagnetic
Theory
Lecture Notes
Dr. Serkan Aksoy
These lecture notes are heavily based on the book of Advanced Engineering Electromagnetics (C. A. Balanis), 2012. For
future version or any proposals, please contact with Dr. Serkan Aksoy ([email protected]).
Advanced Electromagnetic Theory Dr. Serkan Aksoy
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Content
1. THEOREM & PRINCIPLES ------------------------------------------4
1.1. Duality Theorem ------------------------------------------------------------------------------------------ 4 1.2. Uniqueness Theorem ------------------------------------------------------------------------------------ 4 1.3. Image Theory ---------------------------------------------------------------------------------------------- 4 1.4. Reciprocity Theorem ------------------------------------------------------------------------------------ 4 1.5. Reaction Theorem ---------------------------------------------------------------------------------------- 5 1.6. Volume Equivalence Theorem ------------------------------------------------------------------------ 5 1.7. Surface Equivalence Theorem ------------------------------------------------------------------------ 6 1.8. Induction Equivalent ------------------------------------------------------------------------------------ 6 1.9. Physical Optics Equivalent ---------------------------------------------------------------------------- 6 1.10. Equivalency Evaluation --------------------------------------------------------------------------------- 6
2. SCATTERING ------------------------------------------------------------7
2.1. LINE SOURCE - CYLINDRICAL WAVE ---------------------------------------------------------- 7 2.1.1. Electrical Line Source ( Mode) ---------------------------------------------------------------------- 7 2.1.2. Magnetic Line Source -------------------------------------------------------------------------------------- 8 2.1.3. Electrical Line Source above Infinite PEC ------------------------------------------------------------- 8
2.2. PLANE WAVE SCATTERING (PWS) -------------------------------------------------------------- 8 2.2.1. PWS by Planar Structures --------------------------------------------------------------------------------- 8 2.2.2. PWS from a Strip ------------------------------------------------------------------------------------- 8 2.2.3. PWS from a Flat Plate -------------------------------------------------------------------------------- 9
2.3. CYLINDRIC WAVE TRANSFORM ----------------------------------------------------------------- 9 2.3.1. Plane Waves by Cylindrical Wave ---------------------------------------------------------------------- 9 2.3.2. Addition Theorem of Bessel Function ------------------------------------------------------------------ 9 2.3.3. Addition Theorem of Hankel Function ---------------------------------------------------------------- 9
2.4. CIRCULAR CYLINDER SCATTERING ---------------------------------------------------------- 10 2.4.1. Normal Incidence PWS: Polarization ---------------------------------------------------------- 10 2.4.2. Normal Incidence PWS: Polarization ---------------------------------------------------------- 10 2.4.3. Oblique Incidence PWS: Polarization --------------------------------------------------------- 10 2.4.4. Oblique Incidence PWS: Polarization ---------------------------------------------------------- 10 2.4.5. Electric Line Scattering: Polarization ---------------------------------------------------------- 10 2.4.6. Magnetic Line Scattering: Polarization -------------------------------------------------------- 11
2.5. CONDUCTING WEDGE SCATTERING --------------------------------------------------------- 11 2.5.1. Electric Line Scattering: Polarization ---------------------------------------------------------- 11 2.5.2. Magnetic Line Scattering: Polarization -------------------------------------------------------- 11 2.5.3. Electric & Magnetic Line Scattering ------------------------------------------------------------------ 11
2.6. SPHERICAL WAVE ORTOGONOLITY ---------------------------------------------------------- 11 2.6.1. Vertical Dipole Spherical Wave Radiation ---------------------------------------------------------- 11 2.6.2. Orthogonality Relations --------------------------------------------------------------------------------- 12 2.6.3. Wave Transformations & Theorems ------------------------------------------------------------------ 12
2.7. CONDUCTING SPHERE SCATTERING -------------------------------------------------------- 12
3. DIFFRACTION --------------------------------------------------------- 13
3.1. GEOMETRICAL OPTICS ----------------------------------------------------------------------------- 13 3.1.1. Amplitude Relation --------------------------------------------------------------------------------------- 13 3.1.2. Phase & Polarization Relation -------------------------------------------------------------------------- 14 3.1.3. Reflection from Surfaces --------------------------------------------------------------------------------- 14
3.2. GEOMETRIC THEORY of DIFFRACTION ------------------------------------------------------ 14 3.2.1. Amplitude, Phase & Polarization --------------------------------------------------------------------- 14 3.2.2. Straight Edge Diffraction & Normal ------------------------------------------------------------------ 14 3.2.3. Straight Edge Diffraction & Oblique ----------------------------------------------------------------- 15 3.2.4. Curves Edge Diffraction & Oblique ------------------------------------------------------------------ 15 3.2.5. Slope Diffraction ------------------------------------------------------------------------------------------ 15
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3.2.6. Multiple Diffractions ------------------------------------------------------------------------------------- 15 3.2.7. Equivalent Diffraction Current ------------------------------------------------------------------------ 15
4. INTEGRAL EQUATIONS ------------------------------------------ 16
4.1. POINT MATCHING METHOD -------------------------------------------------------------------- 16 4.1.1. Basis Functions -------------------------------------------------------------------------------------------- 16 4.1.2. Subdomain Functions ------------------------------------------------------------------------------------ 16
4.2. METHOD of MOMENTS ----------------------------------------------------------------------------- 17 4.3. EFIE and MFIE ------------------------------------------------------------------------------------------- 17
4.3.1. Electric Field Integral Equation ------------------------------------------------------------------------ 17 4.3.2. Magnetic Field Integral Equation --------------------------------------------------------------------- 17
4.4. FAST MULTIPOLE METHOD ---------------------------------------------------------------------- 17 4.5. FINITE DIAMETER WIRES -------------------------------------------------------------------------- 17
4.5.1. Pocklington’s Integral Equation ----------------------------------------------------------------------- 18 4.5.2. Hallen’s Integral Equation ------------------------------------------------------------------------------ 18 4.5.3. Source Modeling ------------------------------------------------------------------------------------------ 18
Advanced Electromagnetic Theory Dr. Serkan Aksoy
1. THEOREM & PRINCIPLES
1.1. Duality Theorem
A sample circuit with its reciprocal circuit is given for better
understanding the duality theorem.
and are written from the circuit as
where , , show the reciprocal values that
solution of both circuits are equal to each other. Similar to
this, and fields are also dual of each other. It means that
solution of fields can be found by using solution of fields.
In other words, with proper arrangements, a solution for
( ) , ( ) can be used for a solution for
( ) , ( ) . These arrangements are given
below.
( ) , ( ) ( ) , ( )
( ) ( ) ( )
( )
⁄
( ) ( )
( ) ( )
⁄
According to the duality theorem, the problems can be solved
by interchanging the quantities. For example, a duality of a
problem of is equal to . The duality theorem gives chance to calculate fields
excited by equivalent magnetic sources using fields excited by
electric field sources (such as that fields of magnetic dipole
can be calculated by using electric dipole fields).
1.2. Uniqueness Theorem
It is necessary to know that the uniqueness of the solution of
electromagnetic problems are provided under which condition
(mean they have no other solutions). In a lossy medium with
and parameters, let and fields are excited by
sources of and . In both cases, Maxwell’s equations
,
,
where subtracting the second equation from the first
( ) ( )
( ) ( ) ( )
where and , then, the second equation
( ) ( )( )
where defining and , it is clear
that the condition for a unique solution is .
Using energy conservation, it is possible to prove for lossy medium. For lossless medium, the same can be
proved as a limit case. In lossy medium, the following special
conditions are appeared in prove of uniqueness:
- If is defined on a surface (such as zero), fields
are unique valued. The reason for this is the tangential
components, but the normal components are ineffective.
- If is defined on a surface (such as zero), fields
are unique valued. The reason for this is the tangential
components, but the normal components are ineffective.
- If is defined on some certain section of surface, and
is defined on the other part of the same surface,
fields are unique valued.
1.3. Image Theory
It can be applied to PEC, flat and infinite extended surfaces.
Since the boundary conditions on the tangential electric field
components are satisfied over a closed surface ( to ), the
solution is unique. The uniqueness and boundary conditions
give a chance to define equivalent source. Images acts as a
source of reflected rays. Below the conductor, the equivalent
system does not give the correct fields but are not concerned.
The direction of current can be evaluated from the polarization
of the reflected fields or boundary condition on the surface.
1.4. Reciprocity Theorem
Reciprocity theorem means that ideal voltage sources (zero
internal impedance) can be changed with ideal current source
s (infinite internal impedance) without changing value read on
ampermeter in a linear electric circuit. For electromagnetic
theory, the reciprocity theorem means that location of
transmitter and receiver can be changed. Thus, difficulties of
cylindrical or spherical wave propagation from a source can
be mitigated by transforming the problem plane wave
propagation. To explain the reciprocity theorem, let us
consider that independent , and , sources having
same frequencies excite , and , fields in a same
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linear and isotropic medium. In this case, Maxwell’s equations
,
,
where multiplying the first equation with and the last
equation with
is found. Subtracting the first equation from the second
where using a relation of ( ) ( ) ( ), it becomes
( )
In a similar manner, multiplying the second equation with
and the third equation with
where subtracting the second equation from the first
where again using a relation of ( ) ( ) ( ), it becomes
( )
where subtracting this equation from ( ) equation
( )
is found. This equation is known as a differential Lorentz
reciprocity principle. Applying volumetric integral and using
divergence thermo to this equation
∬( )
∭( )
is found. This equation is known as a integral Lorentz
reciprocity principle. Specially, if there are no sources in a
region as , then, the following
equations are satisfied.
( )=0
∬( )
Reciprocity theorem must be satisfied for two different wave
guide modes and so on.
1.5. Reaction Theorem
The reciprocity relation can not be evaluated in the sense of
power because of no conjugate, but it may be preted a reaction
(coupling) between sources and fields as
∭( )
∭( )
The first equation relates the coupling of the fields , to
the sources , , . Due to
reciprocity theorem
.
The reaction theorem is used to calculate the mutual
impedance and admittance between aperture antennas and can
also be expressed by voltages and currents.
1.6. Volume Equivalence Theorem
The sources of and can create the fields and in
free space or and in a dielectric medium. In that case,
Maxwell's equations
After subtraction second one from the first one, one can obtain
( ) ( )
( ) ( )
If and are defined as scattered
(disturbance) fields, then rearranging
, ( )- ( )
, ( )- ( )
using the definition ( ) and
( ) , the equations
where and are only exist in the
material and give chance to formulate Integral Equations for
scattering by dielectric objects. Moreover the surface
equivalence theorem is more useful for scattering by PEC
surfaces and for analysis of antenna aperture radiations.
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1.7. Surface Equivalence Theorem
Inside Outside On surface of
I , by , , by , -
II , by , ( )
( )
III , by ,
IV , PEC , by ,
V , PMC , by ,
This theorem is also called Huygen's Principle and the step III
is known as Love's Equivalence Principle. One of the key
point in the theorem is that the medium is the same ( and
) and homogeneous inside and outside the up to step IV.
The different applications of the theorem are also given at
below for magnetic source radiation near a PEC material and
for a waveguide aperture mounted on an infinite flat electric
ground plane.
Magnetic source radiation near a PEC material.
A waveguide aperture on an infinite flat electric ground plane.
1.8. Induction Equivalent
The Induction (or Induction Equivalent) Theore is related to
Surface Equivalence Theorem and used for scattering problem
from aperture.
Inside Outside On surface of
I , by , , by , -
II
-
III ( )
( )
IV
V , PEC
where are transmitted fields in is different medium
having and after step I. Whereas in outside and .
and are due to the obstacle . In the step V, means the shorted by conductor. As an example, the induction
equivalent for scattering by conducting surface of infinite
extend is given at below:
The induction equivalent for scattering by conducting surface
of infinite extend.
1.9. Physical Optics Equivalent
If one can consider the Induction Theorem for a PEC
Inside Outside On surface of
I
-
II
( ) ( )
Considering boundary condition,
( ) ( )
where, the currents can be formed for equivalence problem as
( ) ( ( ) )
( ) ( ( ) )
This equivalency is known as Physical Equivalent and is
based on the EFIE and MFIE for unknown current densities. If
the conducting obstacle is an infinite, flat and PEC conductor,
the Physical Equivalent problem can be stated as
| ( )| |
with following figure.
Physical equivalent problem for PEC materials.
1.10. Equivalency Evaluation
At a first glance, the solution of a scattering problem
by the Induction Equivalent or Physical Optics Equivalent will
be the same in any sense. Whenever the Induction Equivalent
gives a known current placed on the surface of
the obstacle can not be used for scattering calculation because
the medium within and outside the obstacle is not the same.
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The new boundary value problem (as difficult as the original
one) must be solved to know the currents on the surface of the
obstacles. Nevertheless the Physical Equivalent currents
can be used for calculation of the scattered field
because the same medium is present within and outside of the
surface. The fundamental difficulty in this way is not to know
the current density on the surface of the obstacles and
generally as difficult as to find the original solution of the
problem with requiring the knowledge of the total fields in
the original problem.
If one assumes that the PEC obstacle is enough large
electrically (locally flat), the image theory can be used for
Induction Equivalent and the known current can
be used to calculate scattered fields. In many cases, a closed
form solution can not be obtained easily due to the inability
for integration along closed surface of the obstacle. In this
case, the integration can be performed only part of the surface
(visible for transmitter) that the current density is intense for
major contribution of the scattered field. With the electrically
large assumption, the Physical Equivalent currents can be used
with the form of known as Physical Optics
Approximation. For backscattering calculations, the Induction
Equivalent and the Physical Equivalent give the same results
and these approaches must be extended for conductors (not
PEC) and dielectrics.
2. SCATTERING
Modal Solution: Needs orthogonal systems ( ), poor
convergent series.
IE, MoM: Arbitrary shapes, not to many wavelengths.
GO (GTD), PO (PTD): Many wavelengths.
2.1. LINE SOURCE - CYLINDRICAL WAVE
2.1.1. Electrical Line Source ( Mode)
: No variation (infinite length of wire)
The field solution is the form of
[ ( )( )
( )( )]
, ( ) ( )-
,
-
Outward direction,
Lowest order mode,
No variation,
In that case, the solution
( ) ( )( )
Then
( )( )
,
( )( )
The calculation of in the field equation from Amper law
∫ ⏟
∫
( )( )
where
( )( ) , then the fields
( )( ) √
√
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( )( ) √
√
where ⁄ is impedance and
( )( ) √
,
( )( ) √
.
2.1.2. Magnetic Line Source
Physically not realizable, but used for modeling of radiating
apertures. Using Duality Principle
( )( ) √
√
( )( )
√
√
2.1.3. Electrical Line Source above Infinite PEC
Original problem Near-field equivalent Far-field equivalent
{
[
( )( ) ( )( )]
}
In the asymptotic case:
{ (
√
√
)√
}
At large distance: for amplitude and
, for phase, the far-field approximation
( )
{ √
( )
√
}
2.2. PLANE WAVE SCATTERING (PWS)
2.2.1. PWS by Planar Structures
Radar Cross Section (RCS): Area intercepting the amount of
power that, when scattered isotropically, produced at the
receiver a density that is equal to the density scattered by the
actual target. ( or ) is known as scattering width
(RCS per unit length), ( or ) is RCS. RCS pattern
is a function of space coordinates.
*
+
[
| |
| | ]
[
| |
| | ]
*
+
[ | |
| | ]
[
| |
| | ]
Relation between and ( is the target length)
√ |
and definition of RCS can be approximated when the
target is placed in the far-field of the source (Specular
reflections satisfies Snell’s law).
- Monostatic (Backscattered): Tx and Rx are in same place.
- Bistatic: Tx and Rx are in the different place.
2.2.2. PWS from a Strip
( )
( )
( )
( )
( )
( )
for finite strip, for infinite strip, and for PEC material. Thus reflected fields are
( )
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( )
( )
Using Physical Optics
|
( )|
|
( )
To calculate far-zone scattered field
,
∫ ( )
Then
∫ [ ∫
√| | | |
√| | | |
]
It is known that
∫ √
√
( )( )
Then
∫
( )( | |)
For far-zone observation | | ( )
( )( | |) √
( )
√
√
Then and by using it, ,
and can be calculated.
2.2.3. PWS from a Flat Plate
( ) ( )
( )
Induced current on the surface
|
Using the far-field transformation ( ), the field
components ,
, ,
, ,
are calculated.
2.3. CYLINDRIC WAVE TRANSFORM
2.3.1. Plane Waves by Cylindrical Wave
∑ ( )
where the infinite sum shows the cylindrical wave function
and it can be proved that .
2.3.2. Addition Theorem of Bessel Function
( )
[
( )( ) ( )( )]
( | |)
{
∑ ( ) ( )
( )
∑ ( ) ( ) ( )
}
( )
[
( )( ) ( )( )]
( | |)
{
∑ ( ) ( )
( )
∑ ( ) ( ) ( )
}
2.3.3. Addition Theorem of Hankel Function
It is based on the equivalence of ( )
and ( )
in the far field.
( )( | |)
{
∑ ( )
( )( ) ( )
∑ ( ) ( )( ) ( )
}
( )( | |)
{
∑ ( )
( )( ) ( )
∑ ( ) ( )( ) ( )
}
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2.4. CIRCULAR CYLINDER SCATTERING
2.4.1. Normal Incidence PWS: Polarization
∑ ( )
In outward direction
∑ ( )( )
Using the boundary condition
|
( )
( )( )
The induced current with small radius approximation ( )
meaning the first term is dominant.
|
(
)
Far zone scattered field:
√
√ ∑
( )
( )( )
2.4.2. Normal Incidence PWS: Polarization
The solution is similar to the section 11.5.1 for .
2.4.3. Oblique Incidence PWS: Polarization
( )
Using the transformation
∑ ( )
The outward direction
∑ ( )( )
Applying the boundary condition
|
|
|
,
( ) ( )( )⁄
Far zone scattered field:
( )( ) √
2.4.4. Oblique Incidence PWS: Polarization
The solution is similar to the section 11.5.3 for .
2.4.5. Electric Line Scattering: Polarization
( )( | |)
Using the addition theorem
{
∑ ( )
( )( ) ( )
∑ ( ) ( )( ) ( )
}
where ( ) is chosen for because the field should be
finite everywhere including and ( )( ) is chosen for
because the travelling nature of the wave. Then
∑
( )( )
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Applying the boundary condition as
|
|
|
may be found. Then the field components ,
,
and
, ,
are found. The current
density |
can be found.
2.4.6. Magnetic Line Scattering: Polarization
The solution is similar to the electric line scattering.
2.5. CONDUCTING WEDGE SCATTERING
2.5.1. Electric Line Scattering: Polarization
{
∑ ( )
( )( ) ( )
∑ ( ) ( )( ) ( )
}
{
∑ ( )
( ) , ( )- , ( )-
∑ ( )
( )( ) , ( )- , ( )-
}
when , two must be identical, then and
( ) ( )( ), ( ) ( ). When and
, will vanish. It means that
( ) ⁄ . Then can be calculated
by Maxwell’s equation. Induced current density
|
∑
, ( )- , ( )-
Since the Fourier series for a current located at and
, then
( ) ∑ , ( )- , ( )-
where ( )⁄ and depends on source type.
Far zone field ( ) or PWS ( ) can
be found by using the asymptotic form of Hankel function.
2.5.2. Magnetic Line Scattering: Polarization
{
∑ ( )
( )( )
, ( )-
, ( )-
∑ ( )
( )( )
, ( )-
, ( )-
}
Using Maxwell’s equations is calculated and boundary
condition is applied at and with .
Then with allowable ( ) ⁄ , since
source is magnetic field ( ( )⁄ ) where
or . Far zone field ( ) or PWS ( ) can be found by using the asymptotic
Hankel function.
2.5.3. Electric & Magnetic Line Scattering
Using a new coordinate system, it is possible to write
( , soft) and ( , hard) as
( )
( )
where ( ) is Green function.
2.6. SPHERICAL WAVE ORTOGONOLITY
2.6.1. Vertical Dipole Spherical Wave Radiation
Using spherical Hankel function ( )( ) ⁄
( )( )
For a linear magnetic current element using duality
( )( )
If the source is removed from origin as
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( )( | |)
( )( | |)
where ( )( | |) | | | |⁄ .
2.6.2. Orthogonality Relations
In spherical coordinates, Legendre ( ) functions and
Associated Legendre (Zonal Harmonics) functions ( )
form a complete orthogonal set for . Therefore,
their series form Legendre polynoms
( ) ( )
( ) ( )
are used to represent arbitrary functions. The polynoms
(Tesseral Harmonics) form a complete set on the sphere
surface. These form also Fourier-Legendre series
( ) ∑ ( )
where ( ⁄ ) ∫ ( )
( ) and the
following condition holds
∫ ( )
( ) {
( )⁄ }
2.6.3. Wave Transformations & Theorems
∑ ( ) ( )
Using the orthogonality relation ( ).
Addition theorem of spherical wave functions
( )( )
{
∑
( ) ( ) ( )( )
( , ( )-)
∑( ) ( )
( )( )
( , ( )-)
}
( )( )
{
∑
( ) ( ) ( )( ) ( )
( , ( )-)
∑( ) (
) ( )( ) ( )
( , ( )-)
}
2.7. CONDUCTING SPHERE SCATTERING
Using the transformation for
where is also can be written as
Using the spherical transformation
∑ ( ) ( )
, ( )-
∑ ( ) ( ) ( )
∑ ( ) ( ) ( )
The solution can be constructed as
:
:
Using the , it can be proven that
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∑ ( )
( )
∑ ( )
( )
where ( ) ( )⁄ and ( ) ( ) ⁄ .
Due to the field component are uniform plane waves, and
can be constructed for scattered field as
∑
( )( ) ( )
∑
( )( ) ( )
To determine and , the boundary conditions are applied
( )
( )
Then, it can be proved that
( )
( ) ( )
( )
( )( )
Then ,
, and can be calculated. Using the spherical
Hankel function and Hankel function relations
( )( ) √
( ) ( )
In the far field region ( )
[ | |
| | ]
,| |
| | -
The RCS of a sphere is given below.
3. DIFFRACTION GTD: Keller, extended by Pathak (diffraction)
PTD: Extended by Ufimtsev (nonuniform fringe edge current)
Diffraction is a local phenomenon depends on
- the geometry of the object (edge, vertex, curve)
- the amplitude, phase and polarization of the field
3.1. GEOMETRICAL OPTICS
Phase of rays is assumed that equals to product of optical ray
length (with ) from reference. Amplitude of rays is assumed
that vary in a narrow tube with the principle of energy
conservation. The phases at the caustics should be rearranged.
Specular Reflection is only allowed (Snell law). The Fermat
principle equations is used for GO as
∫ ( )
where is variational differential, ( ) ( ) ⁄ is
refraction index. If ( ) is constant, paths are straight lines.
3.1.1. Amplitude Relation
Due to the energy conservation .
( )
( )
( )
( )
where because ( ⁄ )| | , then
| |
| |
√
The areas can be written by the radii of curvature as
Spherical | |
| | √
Cylindiral | |
| | √
√
√
Plane | |
| | √
Phase and polarization information are absent yet and the
caustic problem is present.
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3.1.2. Phase & Polarization Relation
Luneberg Kline series is given as
( ) ( ) ∑ ( )
( )
Substituting this equation to wave equation, Eikonal,
Transport and Conditional Equation can be found as
Eikonal equation | |
Transport equation
,
-
Conditional equation
where ⁄ and is wave front surfaces (may be
plane, cylindrical, spherical), then using the transport equation
and reference field values
( ) ( ) ( )⏟
√
⏟
⏟
The solution from Luneburg-Kline series predicts phase and
polarization information. More accurate results may be
obtained by higher order terms. But diffraction mechanism
can not be treated. In the caustics, the field is singular and
another approach is used.
3.1.3. Reflection from Surfaces
Snell law is applied. Near the reflection point
- Reflecting surface is approximated to plane,
- Incident wave front is assumed to be planar.
Then, the reflected field is given by
( ) ( )⏟
⏟
√
( ) (
)⏟
⏟
where and
can be approximated by and
with focal
point and . If incident ray is spherical, spherical wave is
treated. The principal radii of curvature (Principal Plane) are
defined. It is assumed that the reflecting surface is well-
behaved (smooth and continuous).
3.2. GEOMETRIC THEORY of DIFFRACTION
Edge diffraction can be evaluated by a diffraction coefficient
needs canonical problem. Applying Fermat principle, Law of
Diffraction is obtained. GO fails
- Incident rays are tangent to curved surface
- Present an edge, vertex, corner
- Present caustics
New semi-heuristic approach must be proposed.
3.2.1. Amplitude, Phase & Polarization
√ ( )
where is eikonal surface, ( ) is diffracted field factor.
Substituting to wave equations
( ) ( )√
( ) ( )
where ( ) [ ( ) √ ⁄ ] ( ) is diffracted field at
reference. The diffracted field has to satisfy the following
relation
( ) √ ( )
Thus, the diffracted field
( ) ( )⏟
⏟
( )⏟
⏟
where the spreading factor for a curved surface ( )
√ ( )⁄ . And is a function of wavefront curvature
angles of incident and diffraction radius of edge curvature.
If the edge is straight
( ) Incident Wave
√
For plane and conical
wave incidences
√
√
For cylindrical wave
incidences
√
( )
√
For spherical wave
incidences
The important key is to find diffraction coefficient.
3.2.2. Straight Edge Diffraction & Normal
The radiation mechanism nearby the edges needs to separate
space surrounding wedge into three different regions with
reflection shadow boundary and incident shadow boundary.
To remove discontinuity on boundaries (modify the fields for
physically realizable field), diffraction has to be included.
Diffraction coefficient can be extracted by steps as
- Find Green’s function in series form of far field region
- Convert series form Green’s function to integral form
- Evaluation of the integral by Steepest Descent method
In the case of electrical (or magnetic) line source
where the asymptotic solution of ( ) ( )
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( )
∑
* (
( )) (
( ))+
where depends on Dirichlet or Nuemann boundary
condition. The series converges rapidly when small , but
slowly converges when large . To overcome this problem,
the series are transformed to integral form
( ) ∫ ( ) * (
) (
)+
( ) ∫ ( ) (
)
⏞ ( )
∫ ( ) (
)
⏟ ( )
( ) ∫ ⏟
∫ ∫ ⏟
( ) ∫ ⏟
∫ ∫ ⏟
After contour integration, Steepest Descent and Modified
Steepest Descent methods, the solution can be arranged as the
( ) ( ) ( ). After evaluating the
solution, diffraction coefficient can be extracted for incident
and reflected waves in the sense of polarizations. Thus the
incident and reflection shadow boundaries can be clarified.
3.2.3. Straight Edge Diffraction & Oblique
In reality, not only for principal pattern, but also for all pattern
(directions), the diffraction coefficient has to be calculated.
Although in the case of normal incidence, the diffraction
coefficient becomes scalar, it becomes Dyadic form for
oblique incidence.
3.2.4. Curves Edge Diffraction & Oblique
Because the diffraction is a local phenomenon, curved edges
(Convex, Concave) can be approximated as a wedge and
application of the wedge diffraction theory meaning the scalar
diffraction coefficients.
3.2.5. Slope Diffraction
In classical evaluation, the diffraction coefficient is zero when
the field is zero at the point of diffraction. But in fact if the
normal derivative (slope) of the incident field is also causes a
higher order diffraction known as Slope Diffraction creating
the currents on the wedge surface. The slope diffraction
coefficients can be calculated separately for different
polarizations.
3.2.6. Multiple Diffractions
If the structures have multiple edges, coupling between edges
should be considered. Especially this is important for close
edges. The multiple interactions are known as Multiple
Diffraction (higher order diffraction). To account especially
third and even higher order diffraction, a procedure adopted
for accounting all diffraction is used and known as Self
Consisting Method. It is based on the multiple reflection and
transmission coefficients and series representations of
Multiple Diffractions.
3.2.7. Equivalent Diffraction Current
In contrast to straight edges diffraction, curved edges
diffraction creates caustics. To correct field near to caustics,
Equivalent Current (an equivalent two dimensional electric
(soft polarization) or magnetic (hard polarization) line current
technique can be used. In that case, by equated the currents
and diffracted fields, the currents
√
( )
√
( )
If the wedge has finite length , the currents have also finite.
The field created by the currents can be calculated using
standard way. Moreover in the case oblique angle, the
Equivalent Currents should be modified. In the case of curved
edge diffraction, Equivalent Currents are modeled by quarter-
wavelength monopole mounted on a circular ground plane in
which rim of it is modeled as a ring radiator.
Advanced Electromagnetic Theory Dr. Serkan Aksoy
16
4. INTEGRAL EQUATIONS The solution of realistic finite sized objects can be found by
Integral Equation, IE (numerically Method of Moments). The
total current density can be Physical Optics (with fringe wave)
currents. Using the current with IE, the fields can be
calculated. But because the current is not known, current is
approximated to series solution with a basis function and an
unknown coefficient for a boundary value of the field. The set
of linear equation is obtained as a matrix equation.
4.1. POINT MATCHING METHOD
Electrical line source above a segmented strip can produce the
scattered electric field because of line source can be written as
( )
( )( )
where because of segmentation, the current is ( ) (
) , then
∑ (
)
( )( )
If every segment is so small (
) , then
∫ ( )
⁄
⁄
( )( | |)
Using the boundary condition for electric field |
( )( ) ∫ ( )
⁄
⁄
( )( | |)
This equation is known as Electric Field Integral Equations
(EFIE). If one expands the current density ( ) as
( ) ∑ ( )
then the EFIE
( )( ) ∑
∫ ( )
⁄
⁄
( )( | |)
which takes the general form as ∑ ( ) where
is the known excitation function, is the linear integral
operator. If one considers the problem in observation
points, EFIE is
[ ( )( )]⏟
∑ ⏟
[ ∫ ( )
⁄
⁄
( )( |
|) ]
⏟
then , - , -, - in which , - is the unknown. The
system linear equation each with unknowns by applying
the boundary condition at discrete points technique is
known as Point Matching (or Collocation) method.
4.1.1. Basis Functions
Basis functions should be chosen to have ability for accurately
representation of anticipated unknown function while
minimizing the computational efforts. Basis functions can be
classified as a Subdomain (non-zero over a part of the
domain) and an Entire Domain (exist entire domain) basis
functions. Do not chose basis functions with smoother
properties than the unknown function.
4.1.2. Subdomain Functions
These are most common and can be used without prior
knowledge of the unknown function. The subdomain approach
based on the subdivision of the structure into no
overlapping segments. Some subdomain basis functions
- Pulse function,
- Triangle function,
- Piecewise sinusoid,
- Truncated cosine.
As an example, triangle function is overlap adjacent function
which is smoother than pulses but the cost of increased
computational complexity.
If ( ) is subdomain type (exist only over one segment)
means that ( ) only for , otherwise
is zero, then the integral is as
∫ ( )( |
|)
The closed form evaluation of this integral is not possible
because of the self term on surface (for
application of boundary condition on the surface) is zero
causing the singularity of Hankel function. To overcome this,
the observation point is chosen away from the surface. But
will still sufficiently small that the computation of
Hankel function may not be very accurate. The approximation
of Hankel function for small argument is used and
approximate closed form of the integral can be evaluated for
diagonal and nondiagonal terms. Specifically the average
value of arguments is considered for also curved space.
4.1.2.1. Entire Domain Functions
A common one is sinusoidal function (similar to Fourier
series) useful for modeling sinusoidal distribution as the wire
current. The main advantages of it are assumed a priori to
follow a known pattern. Such entire domain functions may
render the unknowns with a fewer terms. Because using a
finite number of functions, the modeling of arbitrarily or
complicated functions have difficulty by entire functions
which can be generated from polynomials.
Advanced Electromagnetic Theory Dr. Serkan Aksoy
17
4.2. METHOD of MOMENTS
Boundary conditions are satisfied only at discrete points in
Point Matching Method and between these points, boundary
conditions are not satisfied and deviation as a . To minimize residual that its overall average over
entire structures approaches zero, Method of Weighted
Residuals (MoM) is utilized and forces boundary conditions to
be satisfied in an average sense. To do this, Weighting
(Testing) functions * + form inner product as
⟨ ⟩ ∑
⟨ ( )⟩
This can be formulated as the matrix equation ( )
, - , -, - , - , - , -
where must be linearly independent (matrix equation be
also linearly independent) and should be chosen to minimize
computational load. If weighting and basis function are the
same, technique is known as Galerkin's Method (Others are
Point Collocation, Collocation, Least Squares). Specifically
choosing the set of Dirac weighting function (Point
Collocation) will reduce the computational requirements, but
is forced the boundary condition at discrete points, hence
name is Point Matching. The positioning of points (equally
spaced one yield good results) depends on basis function
choosing in some configuration (such as match point does not
coincide with peak of triangle basis functions due to it's not
differentiability) may cause errors. Point Matching is popular
testing technique due to its acceptable accuracy. For a strip
problem, a convenient inner product would be
⟨ ⟩ ∫ ( ) ( )
⁄
⁄
Applying the inner product to previously given EFIE
, - ,
-, -
If the weighting (testing) functions are Dirac functions as
( ) ( ), then and
. The
MoM is introduced to minimize average deviation from actual
values, of the boundary conditions over the entire structure.
4.3. EFIE and MFIE
4.3.1. Electric Field Integral Equation
EFIE enforces the boundary condition on total tangential
electric field as |
|
|
( )
where
∬ ( )
If the observation is restricted on the surface
|
[ ∬ ( )
∬ ( )
]
Because unknown is expressed by incident electric field, it
is referred as EFIE which is actually integro-differential
equation. It can be used for closed or open surfaces. After
calculating , the scattered field can be calculated. For open
surfaces, a boundary condition should be supplemented to
yield a unique solution for normal component of to vanish
on . The above given EFIE is for a general surface of 3D
problems but can be reduced to 2D case.
4.3.2. Magnetic Field Integral Equation
MFIE enforces boundary condition on tangential magnetic
field and similar to EFIE but based on incident magnetic field
| ( )|
( )
where
∬ ( )
If the observation is restricted on the surface
|
{ ∬ ( )
, ( )- } |
Because unknown is expressed by incident magnetic field,
it is referred as MFIE which is valid for only closed surfaces.
After calculating , the scattered field can be calculated. The
above given MFIE is for a general surface of 3D problems but
can be reduced to 2D case.
4.4. FAST MULTIPOLE METHOD
The MoM treats each of basis function resulting in an
( ) scaling of memory requirements for storing the
impedance matrix and in an ( ) CPU time to solve linear
set of equations (number of process) if the solution of the
matrix is performed by Gaussian elimination method. By
iterative method, number of process is ( ) where is
number of iteration. Fast Multipole Method (FMM) is a kind
of iterative method and uses memory and number of matrix-
vector multiplication as ( ⁄ ). EFIE solution by FMM is
not preferable because preconditioners have to be more
elements and use more memory. MFIE solution by FMM is
also not preferable because internal resonance problems.
Therefore ( ) ( ) solution by FMM is used and shown less memory and less
progress time. Moreover Multi Level FMM results in
( ) scaling in memory and , ( )- in CPU time.
4.5. FINITE DIAMETER WIRES
Three dimensional IE can be arranged to find the current
distribution on a conducting wire. The obtained forms are
known as Pocklington’s Integrodifferential Equation and
Hallen’s Integral Equation. For very thin wires, the current
Advanced Electromagnetic Theory Dr. Serkan Aksoy
18
distribution can be assumed as sinusoidal but for finite
diameter wires ( ), the sinusoidal distribution
assumption is not accurate.
4.5.1. Pocklington’s Integral Equation
The boundary condition on the wire surface
|
|
|
|
|
At any observation point, the scattered field can be calculated
by the vector potential. But at the wire surface, only | is
enough to calculate as
(
)
where
∫
∫ ∫
⁄
⁄
If wire is very thin, is not a function of azimuthal angle as
( ) ( )
where ( ) is equivalent line source current at . Than
∫
∫ ( )
⁄
⁄
Because of symmetry of the scatterer, fields are not function
of chosen as and observation at the surface ( )
∫ ( )
∫
⏟
( )
⁄
⁄
Then,
(
) ∫ ( ) ( )
⁄
⁄
By arranging with boundary condition
∫ ( ) *(
) ( )+
⁄
⁄
This is Pocklington’s Integral Equation. It can be used to
determine filamentary line source current of the wire. If the
wire is very thin ( ), then
( ) ( )
more convenient form of the Pocklington’s Integral Equation
can be obtained. Point Matching method can be used to solve
Pocklington’s Integral Equation. Matching points are to be at
the interior of the wire, but by reciprocity, matching points
can also be chosen on the wires.
4.5.2. Hallen’s Integral Equation
When and , neglecting end effects of the wire, the
only current flows along wire means that and
satisfies the equation
(
)
with boundary condition |
and its solution ( ) √ , ( ) ( | |)- with the definition of vector potential for a line
∫ ( )
⁄
⁄
√
, ( ) ( | |)-
This is Hallen's Integral Equation for a PEC wire. and
can be found from boundary conditions.
4.5.3. Source Modeling
To fed finite diameter wire, Delta-Gap and Equivalent
Magnetic Ring Current (Magnetic Frill) can be used as
Delta-Gap: It is simplest and widely used, but least accurate
for impedances. Excitation voltage at feed terminals is
constant and zero elsewhere. Therefore electric field is also
constant over the gap, and zero elsewhere. Then Equivalent
Magnetic Current Density ( ⁄ ⁄ )
Magnetic Frill: It is based on near as well as far zone fields
of coaxial apertures. Circumferentially directed magnetic
current density is replaced over an annular aperture. Then
Magnetic Source basing on transmission lines is
( ⁄ )