1 Electromagnetic Principles

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The electromagnetic analysis requires an accurate and complete theory. In this chapter some concepts from electromagnetic field theory used in numerical

problems are shown. Integral and differential equations provides a base for many computational techniques. The aim of this chapter is to give an overview of the electromagnetism theory. For a more exhaustive treatment, the reader

1.1 Maxwell Equations

Any electromagnetic problem can

of the electric and magnetic field vectors (unknowns), these differential equations provide all the information on the electromagnetic field.

We consider an electromagnetic field as function of time

region space contained a body characterized by relative permittivity (Figure 1.1).

Figure 1.1 Inhomogeneous body illuminated by an incident electromagnetic field

In an linear and isotropic medium, the electric field

1 Electromagnetic Principles

The electromagnetic analysis requires an accurate and complete theory. In this chapter some concepts from electromagnetic field theory used in numerical formulation for scattering . Integral and differential equations provides a base for many computational techniques. The aim of this chapter is to give an overview of the electromagnetism theory. For a more exhaustive treatment, the reader is referred to [1

Any electromagnetic problem can be described from Maxwell equations. From the knowledge of the electric and magnetic field vectors (unknowns), these differential equations provide all the information on the electromagnetic field.

We consider an electromagnetic field as function of time j t

e ω that illuminate a source region space contained a body characterized by relative permittivity

ε

r and permeability

Inhomogeneous body illuminated by an incident electromagnetic field

In an linear and isotropic medium, the electric field E and the magnetic field

1

Electromagnetic Principles

The electromagnetic analysis requires an accurate and complete theory. In this chapter some formulation for scattering . Integral and differential equations provides a base for many computational techniques. The aim of this chapter is to give an overview of the

is referred to [1-3]

from Maxwell equations. From the knowledge of the electric and magnetic field vectors (unknowns), these differential equations provide all

that illuminate a source-free and permeability

µ

r

Inhomogeneous body illuminated by an incident electromagnetic field

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_________________________________________________________________________

________________________________________________________________________________ The solution of above equations leads at the knowledge of the field that insists on the object and of those scattered from the object in the possible directions. In th

conditions have a fundamental role to determinate a unique solution of the electromagnetic problem. The regions containing dielectric or magnetic material can be bounded by material with very high conductivity, which is often appro

material is called perfect electrical conductor (PEC). On this boundary the electric field and the magnetic field have to be satisfied the ensuing conditions:

where

n

ɵ

is the normal vector at the surface which points to the problem d in the Figure 1.2; while Js is the superficial current and

Figure

The continuity condition of electric and magnetic field on the interface between two homogenous regions, with

ε

r and

_________________________________________________________________________ ________________________________________________________________________________ 0 0 0 0 , , ( ) 0, ( ) 0. r r r r E j H H j E E H ωµ µ ωε ε ε ε µ µ ∇ × = − ∇ × = ∇ ⋅ = ∇ ⋅ =

The solution of above equations leads at the knowledge of the field that insists on the object and of those scattered from the object in the possible directions. In this context, the boundary conditions have a fundamental role to determinate a unique solution of the electromagnetic problem. The regions containing dielectric or magnetic material can be bounded by material with very high conductivity, which is often approximated with infinite conductivity. Such material is called perfect electrical conductor (PEC). On this boundary the electric field and the magnetic field have to be satisfied the ensuing conditions:

ɵ ɵ ɵ ɵ 0 0, , , 0; s s r n E n H J n E n H ρ ε ε × = × = ⋅ = ⋅ =

is the normal vector at the surface which points to the problem domain, how shown is the superficial current and

ρ

s the superficial density charge.

Figure 1.2 Electric current on PEC

The continuity condition of electric and magnetic field on the interface between two and

µ

r specified, are provided from the next equations:

________________________________________________________________________________

________________________________________________________________________________ (1.1)

The solution of above equations leads at the knowledge of the field that insists on the object is context, the boundary conditions have a fundamental role to determinate a unique solution of the electromagnetic problem. The regions containing dielectric or magnetic material can be bounded by material ximated with infinite conductivity. Such material is called perfect electrical conductor (PEC). On this boundary the electric field and the

(1.2)

omain, how shown the superficial density charge.

The continuity condition of electric and magnetic field on the interface between two next equations:

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________________________________________________________________________________ ɵ ɵ ɵ ɵ 1 2 1 2 1 2 1 2 1 2 1 2 ( ) 0, ( ) 0, ( ) 0, ( ) 0 r r r r n E E n H H n E E n H H

ε

µ

× − = × − = ⋅ − = ⋅ − = (1.3)

where

n

ɵ

is the normal vector at the interface. It can be noted that the tangential components of electric and magnetic field are continuous, while the normal components show a discontinuity at the interface between the two materials.

From the first two equations of (1.1), we can obtain the “curl – curl” form of the vector Helmotz equations: 2 2 1 0 1 0 r r r r E k E H k H ε µ µ ε             ∇ × ∇ × − = ∇ × ∇ × − = (1.4)

where

k

2

=

ω µ ε

2 _{0 0} and it is called wavenumber of the medium. This equation can take several forms, if we consider two or three dimensional problem.

1.2 Volumetric Equivalence Principle

In the previous paragraph, we have shown a way for electromagnetic scattering analysis, using differential equation. The alternative at this formulation is given by integral equations. To obtain that solution, we have to replace the inhomogeneous dielectric and magnetic material existent in the problem by equivalent induced polarization currents and charges. In this way, Maxwell’s equations can be rewritten to produce

0 0 , , ( ) , ( ) , o e o m E j H M H j E J E H ωµ ωε ε ρ µ ρ ∇ × = − − ∇ × = − ∇ ⋅ = ∇ ⋅ = (1.5) where

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________________________________________________________________________________ ________________________________________________________________________________ 0 0 0 0 ( 1) , ( 1) , 1 , 1 . r r e r r m r r M j H J j E E H ωµ µ ωε ε ρ ε ε ε ρ µ µ µ = − = −   = ⋅∇       = ⋅ ∇     (1.6)

The equations (1.5) define the same field described in equations (1.1) but involve a homogeneous space characterized by permittivity

ε

o and permeability

µ

o instead of the original heterogeneous environment. The difference between the two set of equations is compensated from the source terms. Since the two set of equations are equivalent, we refer to the procedure of replacing the dielectric or magnetic material by induced sources as a volumetric

equivalence principle. The new equivalent sources of equations (1.6) radiate in free space. The task of finding electromagnetic fields in free space is much more simple than the original burden of solving equations (1.1) directly in the inhomogeneous environment.

1.3 Brief description of a scattering problem

The scatterings problems are related to the scattered field, which is due to the presentence of an object in the referent space. To explain this concept, we suppose the scatter of Figure 1.1 is illuminated by a field produced by a source, called primary source, placed somewhere outside the scatterer. The fields produced by this source in absence of scatterer are called incident fields, Einc and Hinc. But, since the object can be replaced by equivalent induced sources radiating in free space, we have a secondary induced source, which produce the scattered fields,

E

scat and Hscat. The total field can be written as:

, , inc scat inc scat E E E H H H = + = + (1.7)

The incident fields in the immediate vicinity of the scatterer, away from its source, satisfy the vector Helmholtz equations:

2 2 2 2

0 ,

0 ;

inc inc inc inc

E

k E

H

k H

∇

+

=

∇

+

=

(1.8)

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________________________________________________________________________________ 2 2 0 0 2 2 0 0 , , scat scat scat scat J E k E j J M j J H k H J j M j ωµ ωε ωε ωµ ∇∇ ⋅ ∇ + = − + ∇ × ∇∇ ⋅ ∇ + = −∇ × + − (1.9)

where J and M denote the equivalent sources defined in the equation (1.6). In a source free homogeneous medium, we can use the vector Laplacian:

(

)

2

E E E

∇ = ∇ ∇ ⋅ − ∇ × ∇ × (1.10)

In a two-dimensional problem, radiation conditions have the form of Sommerfeld radiation conditions: lim lim scat scat z z scat scat z z E jkE H jkH ρ ρ

ρ

→∞ →∞ ∂ = − ∂ ∂ = − ∂ (1.11)

for the TE and TM polarizations, respectively, where ρ is the radial variable in cylindrical coordinates. Whereas, in three-dimensional problem, the form is more complicate. Regarding r how the conventional spherical coordinate variable, radiation conditions are form:

lim lim scat scat r scat scat r r E jk E r H jk H →∞ →∞ × = × = ɵ ɵ (1.12)

These radiation conditions ensure that the field satisfy Helmholtz equation.

1.4 Potentials in theory of the radiation

To solve equations in homogenous infinite space (Helmholts) we can use several approach. The most used approach is to express the field in term of the magnetic and electric vector potential, respectively, A and F, as:

2 0 2 0 , . scat scat A k A E F j F k F H A j ωε ωµ ∇∇ ⋅ + = − ∇ × ∇∇ ⋅ + = ∇ × + (1.13)

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________________________________________________________________________________ ________________________________________________________________________________ 2 2 2 2 , . A k A J F k F M ∇ + = − ∇ + = − (1.14) A solution, for a homogeneous medium, to these equations, satisfying the radiation condition, can be written in the form

' ' ' ' ' ' ( ) ( ) , 4 ( ) ( ) , 4 jk r r jk r r e A r J r d r J G r r e F r M r d r M G r r π π − − − − = = ⊗ − = = ⊗ −

∫∫∫

(1.15)

where, the scalar function is the three-dimensional Green’s function:

. 4 jk r e G r π − = (1.16)

The electrical and magnetic fields can be obtained by equations (1.13), which involve differentiations.

An alternative formulation to previous one, which uses a mixed formalism, is shown below:

0 0 , , scat e scat m E j A F H A j F

ωµ

ωε

= − − ∇Φ − ∇ × = ∇ × − − ∇Φ (1.17)

where

Φ

e and

Φ

m are the scalar potential functions. These are given by:

0 0 , . e e m m G G ρ ε ρ µ Φ = ⊗ Φ = ⊗ (1.18)

A and F can be still shown to be the identical convolution expressions appearing in equations (1.15). This particular choice of scalar and vector potentials results in a complete decoupling of the contribution to the field from the electric current density, magnetic current density, electric charge density, and magnetic charge density. After the scalar and vector potentials are determinate, equations (1.15) require only a single differentiation to obtain the

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________________________________________________________________________________

1.5 Duality relationships

We consider electrical sources (J,

ρ

_e) radiate in a linear, homogeneous and isotropic medium, with constant ε1 and µ1. The electromagnetic fields produced by this system of sources can be obtained by Maxwell’s equations:

1 ₁ 1 1 ₁ 1 1 1 , , ( ) , ( ) 0. e E j H H j E J E H ωµ ωε ε ρ µ ∇ × = − ∇ × = − ∇ ⋅ = ∇ ⋅ = (1.19)

Now, we consider a fictitious system of magnetic sources (M,

ρ

_m) in a medium with a same permittivity ε2 and permeability µ2. These sources generate electromagnetic fields given by:

2 ₂ 2 2 ₂ 2 2 2 , , ( ) 0, ( ) _m. E j H M H j E E H ωµ ωε ε µ ρ ∇ × = − − ∇ × = ∇ ⋅ = ∇ ⋅ = (1.20)

Substituting the quantities that appear in the first system (1.19) with the corresponding amount in the second column of the TABLE 1, we get a system identical to (1.20).

The idea of duality is a useful aid to generating new formulas.

TABLE 1 Matches in the principle of duality

J

M M

−

J

e

ρ

m m

ρ

e 1

ε

µ

2 1

µ

ε

2 1

E

H

₂ 1

H

−

E

₂ 1

A

F

₂ 1

F

−

A

₂

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________________________________________________________________________________

1.6 Reciprocity Theorem

The reciprocity theorem is an important theorem very used in the problems solution of transmitting and receiving properties of radiating system. We assume to be in a linear, isotropic medium, there exists two set of sources J1, M1 and J2, M2, that are allowed to

radiate simultaneously or individually inside the same medium at the same frequencies and produce fields

E H

1, 1 and

E H

2, 2, respectively. For both the Maxwell’s equations have to be

satisfied: 1 ₁ 1 1 1 ₁ 1 1 , , E j H M H j E J

ωµ

ωε

∇ × = − − ∇ × = + (1.21) 2 ₂ 2 2 2 ₂ 2 2 , . E j H M H j E J

ωµ

ωε

∇ × = − − ∇ × = + (1.22)

Multiplying the first of equations (1.21) by H2and the second of equation (1.22) by E1 and after some mathematical operation, we can obtain:

(

E1 H2 E2 H1

)

E1 J2 H2 M1 E2 J1 H1 M2

−∇ ⋅ × − × = ⋅ + ⋅ − ⋅ − ⋅

(1.23) which is called the Lorentz reciprocity theorem in differential form. Integrating on an entire volume both sides of (1.23) and using the divergence theorem on the left side, we obtain the Lorentz reciprocity theorem in integral form:

(

1 2 2 1

)

'

(

1 2 2 1 2 1 1 2

)

'

S E H E H ds V E J H M E J H M dv

−

∫∫

× − × ⋅ =

∫∫∫

⋅ + ⋅ − ⋅ − ⋅ (1.24)

For a source-free region ( J1=J2 =M1=M2=0 ) , equations (1-23) and (1-24) reduce, respectively, to

(

E1 H2 E2 H1

)

0 ∇ ⋅ × − × = (1.25)

(

1 2 2 1

)

' 0 S⋅ E ×H −E ×H ⋅ds =

∫∫

(1.26)

If we consider the fields E E H H1, 2, 1, ₂ and the sources J J1, 2,M M1, 2 are within a medium that is enclosed by a sphere of infinite radius and assume that sources are positioned within a finite region, while the fields are observed in far fields, the equation (1.24) became:

(

1 2 2 1 2 1 1 2

)

' 0

V E ⋅J +H ⋅M −E ⋅J −H ⋅M dv =

∫∫∫

(1.27)

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________________________________________________________________________________

(

1 2 1 2

)

'

(

2 1 2 1

)

'

V E ⋅J −H ⋅M dv = V E ⋅J −H ⋅M dv

∫∫∫

(1.28)

1.7 Reaction theorem

The equation (1.28) can be interpreted as a coupling between a set of fields and a set of sources. This coupling is defined reaction and each of the integrals in (1.28) can be shown as:

(

)

(

)

1 2 1 2 2 1 2 1 1, 2 ' 2,1 ' V V E J H M dv E J H M dv = ⋅ − ⋅ = ⋅ − ⋅

∫∫∫

(1.29)

The relation 1, 2 connects the coupling (reaction) of the fields E H1, 1 (produced by sources 1

1,

J M ) to the sources J2,M2 (produce the fields E H2, 2). The same procedure can be followed for the relation 2,1 .

1.8 Surface Equivalence Principle

The surface equivalence principle (Huygens’s principle) is a very useful principle for the study of radiation of antenna and, in general, transmitter. The idea is the sources can be replaced by equivalent sources. The fictitious sources are said to be equivalent within a region because they produce within that region the same field as the actual sources.

To explain this we can take account the system shown in Figure 1.3, where the radiating sources are represented electrically by current density J1 and M1. This sources radiate the fields E1 and H1. We can chose a closed surface S, which encloses the current densities J1 and M1. S splits the whole volume in two volumes: an inner volume

V

₁ and an external volume

V

2. The first step is replace the original sources with equivalents that will yield the

same fields E1 and H1 in the volume

V

₂. The equivalent problem is displayed in figure , where the original current densities J1 and M1 are removed. We assume that exist the fields

E and H inside the surface S, and the fields E1 and H1 outside the same surface. These fields must satisfy the boundary condition on the tangential electrical and magnetic field

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_________________________________________________________________________

________________________________________________________________________________ which radiate into a unbounded space

Figure 1.3 Problem model: a) original sources; b) equivalent sources

The equivalent sources produce the original fields

E and Hare different from the originals and exist only in

external region, therefore we can assumed internal field are zero. (1.30) became:

s

J n H H n H

M n E E n E

that equation takes is known as Love’s equivalence principle. Since the internal fields are zero, we can change the propriety of the medium without disturb the internal fields.

1.9 Surface integral equation

In the precedent paragraph is shown the relation b density. Assuming that a surface S

conductor (PEC), using the first equation of (1.2) the equation (1.31) becomes:

_________________________________________________________________________ ________________________________________________________________________________ ɵ

₍

₎

ɵ

₍

₎

1 1 s s J n H H M n E E = × − = − × − space

Problem model: a) original sources; b) equivalent sources

The equivalent sources produce the original fields E1 and H1in the volume

V

₂, while the fields are different from the originals and exist only in

V

1. We are interested at the fields in

external region, therefore we can assumed internal field are zero. In this way the equation

ɵ

₍

₎

ɵ ɵ

₍

₎

ɵ 1 1 0 1 1 0 s H s E J n H H n H M n E E n E = = = × − = × = − × − = − ×

that equation takes is known as Love’s equivalence principle. Since the internal fields are zero, we can change the propriety of the medium without disturb the internal fields.

Surface integral equation

In the precedent paragraph is shown the relation between electromagnetic fields and current

S encloses a scatter and this scatter is made of perfect electric conductor (PEC), using the first equation of (1.2) the equation (1.31) becomes:

________________________________________________________________________________

________________________________________________________________________________ (1.30)

2, while the fields

. We are interested at the fields in In this way the equation

(1.31)

that equation takes is known as Love’s equivalence principle. Since the internal fields are zero,

etween electromagnetic fields and current encloses a scatter and this scatter is made of perfect electric

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________________________________________________________________________________ ɵ ɵ 1 1 0 s s J n H M n E = × = − × = (1.32)

Considering an incident field

E

incand

H

inc on the PEC surface, the equations (1.13) can be combined with the equation (1.32), to produce:

2 0 , inc inc A k A E E j H H A ωε ∇∇ ⋅ + = − = − ∇ × (1.33)

The equations (1.33) provide a definition for the fields E and H in presence of a PEC medium. This means the total field is given by superimposing the incident field with the scattered field. Imposing the first boundary condition of (1.2) on the surface we obtain:

ɵ ɵ 2 0 inc S A k A n E n jωε _{∇∇ ⋅ +}  × = − ×     (1.34)

This equation is called Electric Field Integral Equation (EFIE), whereas imposing the second equation of (1.2) on the surface, the result is:

ɵ inc ɵ

s

S

n H× =J −_n×∇×A_ (1.35)

that is Magnetic Field Integral Equation (MFIE). Using a combination of EFIE and MFIE, we obtain the Combined Field Integral Equation (CFIE). In both equation (1.34) and (1.35) the unknowns is the electric current densityJS.

Either equation (1.34) or (1.35) can be solved to find the unknown Js, from which E and H can be determinate everywhere in space using equation (1.33).

In the case of infinitesimally thin open PEC surface, as plate, strip or shell, the surface equivalent principle can be applied alike of a solid structure. We can assume the surface S collapse on the thin plate, but the equation previously present are unable to distinguish between the two equivalent sources. We have to work with a single equivalent source that represents both equivalent sources of two sides. The boundary condition for PEC is valid for an infinitesimally thin structures, whereupon it possible to use EFIE for this type of scattering problems. Instead, the second equation in (1.2) is not valid for this type of geometry, hence we cannot use MFIE to describe scattering problems.

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_________________________________________________________________________

________________________________________________________________________________

1.9.1 Aperture in a conducting

It is now presented the scattering problem of an aperture on an infinite PEC plane. In reference to the Figure 1.4, the sources are defined on the region 1.

Figure

Using the equivalence principle, we define a surface plane and introduce equivalent sources (as equation (1.31)):

These sources will replicate the original fields in region 1 and create a null field in region 2. The source current M1exists only over the aperture, because the tangential electric field vanish on the PEC part of plane, while J

paragraph 1.8, being interested only in the field in the region 1, we can change the material on the region 2. Hence, we can introduce an additional PEC plate under the origina

aperture and create a uniform PEC plate. In this way, the equivalent sources irradiate in presence of a PEC, then we can apply the method of images, as shown in

method says that we can remove the plane and place on this an imaginary current. The electric current have a negative image on PEC, then

mirror image, therefore the two components are added. These sources produce a non zero field in region 2, but it is not the same of the original problem, therefore the equivalence only holds for the region 1.

_________________________________________________________________________

________________________________________________________________________________

conducting plane

It is now presented the scattering problem of an aperture on an infinite PEC plane. In reference , the sources are defined on the region 1.

Figure 1.4 Aperture in a ground plane

Using the equivalence principle, we define a surface S on the region 1 side of the conducting plane and introduce equivalent sources (as equation (1.31)):

ɵ ɵ 1 1 1 1 J n H M E n = × = ×

These sources will replicate the original fields in region 1 and create a null field in region 2. exists only over the aperture, because the tangential electric field vanish

1

J is non-zero over the entire surface S. How we said in the paragraph 1.8, being interested only in the field in the region 1, we can change the material on the region 2. Hence, we can introduce an additional PEC plate under the origina

aperture and create a uniform PEC plate. In this way, the equivalent sources irradiate in presence of a PEC, then we can apply the method of images, as shown in Figure

method says that we can remove the plane and place on this an imaginary current. The electric current have a negative image on PEC, then J1 are deleted, while the magnetic current have a mirror image, therefore the two components are added. These sources produce a non zero field in region 2, but it is not the same of the original problem, therefore the equivalence only holds ________________________________________________________________________________

________________________________________________________________________________ It is now presented the scattering problem of an aperture on an infinite PEC plane. In reference

on the region 1 side of the conducting

(1.36) These sources will replicate the original fields in region 1 and create a null field in region 2. exists only over the aperture, because the tangential electric field vanish . How we said in the paragraph 1.8, being interested only in the field in the region 1, we can change the material on the region 2. Hence, we can introduce an additional PEC plate under the original, close to the aperture and create a uniform PEC plate. In this way, the equivalent sources irradiate in Figure 1.5. The method says that we can remove the plane and place on this an imaginary current. The electric are deleted, while the magnetic current have a mirror image, therefore the two components are added. These sources produce a non zero field in region 2, but it is not the same of the original problem, therefore the equivalence only holds

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_________________________________________________________________________

Figure 1.5 Method of image for an aperture in a ground plane for region 1

In a similar manner we can solve the problem in the region 2. In this case, the equivalent sources are:

Referring to the Figure 1.6, using the method of images, the current magnetic are replaced by 2 2

M that irradiate in free space. This source is confined to the original aperture and replicates the fields in the region 2.

Figure 1.6 Method of image for an aperture in a ground plane for region 2

Imposing the continuity of the original tangent electric field through the aperture, we can assume

and then we just need to know only

_________________________________________________________________________

Method of image for an aperture in a ground plane for region 1

In a similar manner we can solve the problem in the region 2. In this case, the equivalent

ɵ ɵ 2 2 2 2 ( ) J n H M E n = − × = × −

, using the method of images, the current magnetic are replaced by 2 that irradiate in free space. This source is confined to the original aperture and replicates

Imposing the continuity of the original tangent electric field through the aperture, we can

1 2

M = −M

and then we just need to know only M1.

________________________________________________________________________________

In a similar manner we can solve the problem in the region 2. In this case, the equivalent

(1.37) , using the method of images, the current magnetic are replaced by 2 that irradiate in free space. This source is confined to the original aperture and replicates

Imposing the continuity of the original tangent electric field through the aperture, we can

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________________________________________________________________________________

1.10 Scattering Cross Section and Radar Cross Section

When a plane wave impresses a target in a specific direction, an amount of power will be scattered in other direction. The quantity of power scattered is proportional to the apparent size, shape and composition of the target in that specified direction. It also depends on the frequency and nature of the incident wave. To characterize this quantity we can use a bistatic scattering cross section [6], which has the measure of an area. We can also define a radar cross section, how the power that is returned in a given direction, normalized with respect to the power density of the incident field. The normalization is due to the necessity of independence by distance between the antenna radiating and the target. Generally, the receiver of the scattered energy is assumed to be located far enough from the target so that the target is essentially a point scatterer. This implies that the distance of the target is much longer than any significant target dimension. This “point scatterer” radiates energy isotropically. The definition of the radar cross section is based upon this concept of isotropic scattering, assuming that the target is illuminated by a plane wave. For a three dimensional geometry, the radar cross section can be expressed as:

(

)

(

)

2 2 2 , ( , , , ) lim 4 0, 0 s inc inc r inc E r E θ φ σ θ φ θ φ π →∞ = (1.39)

The limit is presented because it is supposed to have a plane wave and the target is assumed to be a point scatterer. When an object is illuminated by an electromagnetic wave, energy is dispersed in all directions. For a given object, the radar cross section is a function of the angle of incident wave and of the angle to the observation point. The convention adopted is that the angles inc

θ and

φ

inc define the direction of propagation of the incident field and the angles

θ

, φ define the direction of propagation of the scattered field of interest.