ADVANCED ELECTROMAGNETIC THEORY

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]).

mailto:[email protected]


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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


3

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


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.


6

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.


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


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.


7

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

( )( ) √

√


8

( )( ) √

√

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

( )


9

( )

( )

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

∑

( )( )


11

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


13

∑ ( )

( )

∑ ( )

( )

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.


14

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 ( ) ( )


15

( )

∑

* (

( )) (

( ))+

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.


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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.


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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


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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

( ⁄ )

ADVANCED ELECTROMAGNETIC THEORY

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