Advanced Electromagnetics LN01_EM Fields, Matters, Wave Equation [email protected] 1 /47 بنام...

Advanced Electromagnetics LN01_EM Fields, Matters, Wave Equation [email protected] 1 /47

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Time-Varying and Time-Harmonic



Maxwell's Equations

Closed forms: Integral Form:


Boundary Conditions:

Finite Conductivity Media:

Infinite Conductivity Media:


Boundary Conditions


Power and Energy:

Using the combination of Maxwell's equations:

Integrating each of the terms:


Power and Energy

Using:

Conservation of power law:


Time-harmonic Electromagnetic Fields:

Boundary Conditions:

Time-harmonic Electromagnetic Fields

Surface Impedance:

LN01T1: Do Problems 1.1, 1.3, 1.7, 1.9, 1.12, 1.13 _ Due: 06.31


Electrical Properties of Matter


Bohr Models:

Moving of a electron from lower to higher energy level, absorbs energy.

Moving of a electron from higher to lower energy level, radiates energy.

Dielectrics (insulators): ideal Dielectrics don’t have Free Charges

Hydrogen

AluminumGermanium

Materials

Microwave Cooking Food:

[13, 18]


Nonpolar Material: Polar Material:

Bound surface charge is not permissible to separate positive and negative charges by an integration surface.

Therefore within a volume, an integral number of positive and negative pairs with an overall zero net charge must exist.

Hence bound surface charge should not be included to determine boundary conditions.

P is a result of the bound surface charge density.

Polar & Nonpolar Materials

electric susceptibility

is dielectric constant

or relative permittivity

static permittivity

Typical dipole moments: P=10−30 C-m

referred as: “electrets”


Magnetic Moments:

Apply a external field:

Magnetization Current Densities:

Modified Maxwell Equation:

Magnetic Materials

Orbiting electrons equivalent circular electric loop

equivalent square electric loop

no external field external field is applied

magnetic susceptibility

is static permeability

relative permeability


Classification of Materials:

Diamagnetic:

Net magnetization vector M is small in magnitude.

M opposes the applied magnetic field.

Paramagnetic & Antiferromagnetic:

M agrees the applied magnetic field and Relative permeability is greater than unity.

Ferromagnetic & Ferrimagnetic:

M agrees the applied magnetic field and Relative permeability is much greater than unity.

Magnetic Materials

1r

1r

1r


Conductors are material whose atomic outer shell electrons are not held very tightly and can migrate from one atom to another.

For copper: & For glass:

Applied field on a PEC:

Conductive Materials

J

0vq

0vq

/ rtt

rttv

tvv eqeqtq /

0)/(

0)(

relaxation time

static conductivity electron mobility

Conductivity varies as a function of frequency

But



Increasing temperature

Increasing lattice vibration

Increasing free electrons colliding

Decreasing conductivity

Temperature effect:

J


Conductivity becomes extremely large, almost infinity, at T=0K =-273°C.

Materials can be cooled to these temperatures with relatively inexpensive liquid hydrogen.

It was discovered in 1911 by Dutch physicist “H. Kamerlingh Onnes” .

It was observed experimentally in 1933 by “Meissner” and “Ochsenfeld”.

Aluminum becomes superconducting at a critical temperature of 1.2K.

Before 1986, it was accepted that if materials could become superconducting at temperatures of 25K or greater, there would be a major technological breakthrough.

In January 1986 a major breakthrough in superconductivity may have provided.

“Karl Alex Mueller” and “Johannes Georg Bednorz”, IBM Zurich Research Laboratory scientists, observed that a new class of “oxide materials” exhibited a superconductivity at a critical temperature much higher than.

This material was a “ceramic copper oxide” possessing critical temperature 35K which was substantially higher than 23K for “niobium-germanium”.

Many other groups have reported even higher superconductivities, Up to about 90K in a number of “ternary oxides” of rare earth elements.

“C. W. Chu” from University of Houston, found that by pressurizing a superconducting copper oxide, lanthanum, and barium he could observe critical temperatures of up to 70K.

In 1987 “Dr. Chu” also discovered that replacing lanthanum with yttrium resulted in even higher temperatures, up to about 95K.

In 1988 in Japan, a compound of bismuth, calcium, strontium-copper, and oxygen had achieved a critical temperature of 105K.

It is even reasonable to expect that superconductivity may be achieved at room temperature !!!!!!!!!!!!!.

Superconductors

Critical Temperature(T=-273°C)

Superconductor

H. Kamerlingh Onnes

Nobel prize in 1913

Therefore superconductivity at room temperature may be just around

the corner


Decades of 1990s and 2000s have introduced interest and excitement into field of electromagnetics, especially as they relate to integration of special types of “artificial dielectric materials”, coined “metamaterials”.

“meta”, is a Greek word that means “beyond” , and the term has been coined to represent materials that are artificially fabricated so that they have electromagnetic properties that go beyond those found readily in nature.

In fact, word has been used to represent materials which usually are constructed to exhibit “periodic formations” whose period is much smaller than wavelength.

Metamaterials can encompass:

Engineered Textured Surfaces,

Artificial Impedance Surfaces,

Artificial Magnetic Conductors (AMC),

Electromagnetic Band-Gap (EBG) structures,

Double Negative (DNG) Materials,

Frequency Selective Surfaces (FSS),

and even “Fractals” or “Chirals”.

Metamaterials


Types of Media:

Linear: Constitutive Parameters are not functions of the applied field.

For example, air is nearly linear for applied electric fields up to about 1MV/m and Beyond that, air breaks down and exhibits a high degree of nonlinearity.

Homogeneous: Constitutive Parameters are not functions of position.

All materials exhibit some degree of non-homogeneity.

Dispersive: Constitutive Parameters are function of frequency.

All materials used in practice display some form of dispersion.

Isotropic: Constitutive Parameters aren't function of the direction of field.

Many materials, especially crystals, exhibit a rather high degree of anisotropy.

Types of Media

Linear Isotropic Homogeny Non-Dispersive

?? ?

? ,,Constitutive Parameters

Permittivity Tensor:

Isotropic Parameters:


Complex Permittivity:

Total Current:

Effective conduction and displacement current densities:

Good Dielectrics:

Good Conductors:

Complex Permittivity

= Displacement Current Density

= Conduction Current Density

𝛿


Complex Permittivity

Electric polarization as a function of frequency:

In general:


Complex Permeability:

Magnetic Current:

Ferrites finds wide applications in design of nonreciprocal microwave components such as:

Isolators,

Hybrids,

Gyrators,

Phase shifters,

Ferrites become attractive for these applications because at microwave frequencies they exhibit strong magnetic effects that result in anisotropic properties and large resistances (good insulators).

These resistances limit current induced in them and in turn result in lower ohmic losses.

Complex Permeability

LN01T2: Do Problems 2.1, 2.3, 2.8, 2.15, 2.16due: 93.07.01


Wave Equation and its Solutions


Using Maxwell's Equations:

(1)(2)

From (1):

Using:

Using:

In a similar way:

For source-free regions:

and lossless media:

Wave Equations

Time-Harmonic:

Time-Harmonic:

Time-Harmonic:


Rectangular Coordinate System

Using the separation-of-variables method:

Solution to Wave Equations

General form:


Wave Types:



An example is rectangular pipe (waveguide)



Cylindrical Coordinate System:

& Using:

Laplacian of a scalar that in cylindrical coordinates takes the form of:


But:Using form of:

Coupled equation

Coupled equation

Un-Coupled equation


Cylindrical Coordinate System (cont.):

An coupled second-order PDE is the most difficult to solve but:

An uncoupled second-order PDE (will be most useful in construction of TEz and TMz) can be solved as:

Letting:

Classic Bessel differential equation:


Using:

is referred to as constraint (dispersion) equation

TW: Traveling Waves

TW

SW

or:

TW

SW

or:

SW

TW

or:

Appendix IV

SW: Standing Waves


An Example: Cylindrical waveguide of the circular cross section

Within the circular waveguide

For ρ=0, where Ym(βρρ) possesses a singularity and then:

Fields in the region outside the cylinder, like scattering by the cylinder:



Wave functions, zeroes, and infinities for radial wave functions:



Spherical Coordinate System:

Unfortunately, all PDE are coupled and would be most difficult to solve.

However, TEr & TMr can be formed that must satisfy the scalar wave equation of:



Spherical Coordinate System (cont.):

Dividing both sides by , multiplying by

Pmn(cosθ) & Qm

n (cosθ) are referred to Legendre functions of the first and second kind

Well-known Legendre differential equation (Appendix V):


Spherical Bessel functions first & second kind

Spherical Henkel function second kind



Wave functions, zeroes, and infinities for radial waves

Appropriate solution forms will depend on problem.

For example: a typical solution to represent the fields within a sphere:

For fields to be finite at r=0, where yn(βr) possesses a singularity, and for any value of θ, including θ=0,π where Qm

n(cosθ) possesses singularities and therefore:




To represent fields outside a sphere, like for the scattering:

whereby h(2)n(βr) has replaced first kind jn(βr) because outward traveling waves are formed outside.

Other spherical Bessel and Hankel functions that are most often encountered in boundary value EM poblems are those utilized by Schelkunoff.

These spherical Bessel and Hankel functions, denoted in general by:


LN01T3: Do Problems 3.1, 3.14 due: 93.07.12


Wave Propagation and Polarization


Polarization


Polarization

Poincare Sphere


H-RX

H-TX

V-TX

V-RX

H H

V

Processor

Processor

Polarizationاستخراج داده راداری

Polarimetric Radar

LN01T4: Do Problem 4.4 , 4.23 due: 93.07.12



Reflection and Transmission


Normal Incidence-lossless Media


Normal Incidence-lossless Media

Example:

A uniform plane wave traveling in free space is incident normally upon a flat semi-infinite lossless medium with a dielectric constant of 2.56 (polystyrene).

Determine the reflection and transmission coefficients and incident, reflected, and transmitted power densities.

Assume that the amplitude of the incident electric field at the interface is 1mV/m.

and


Oblique Incidence-lossless Media

Plane of incidence is formed by a unit vector Normal to reflecting interface and the vector in the direction of incidence

Perpendicular Polarization:

E is perpendicular to plane of incidence

Fresnel Refection and transmission coefficients:


Parallel Polarization:

Fresnel Refection and transmission coefficients:




Parallel Polarization:

R=0 T≠1 ?????

Surface Waves


Normal Incidence: Conductor-Conductor Interface:

Example:

A uniform plane wave, whose incident electric field has an x component with an amplitude at the interface of 0.001V/m, is traveling in a free-space medium and is normally incident upon a lossy flat earth, as shown.

Determine ‘‘skin depth’’ in the earth at frequency of 1MHz.

Lossy Media

and distributions in a lossy earth

material is a very good conductor


Multiple Interfaces

Normal Incidence, Lossless Media:

A single dielectric possesses

narrow characteristics around

the center frequency

To increase bandwidth:


Characteristics of such a response are very similar to band-stop of a single section filter or single section quarter-wavelength impedance transformer.

To increase BW, each layer with different dielectric constant, must be inserted between two semi-infinite media.

Multiple section dielectric layers can be used to design dielectric filters.

Coating radar targets with multilayer slabs can also be used to reduce or enhance their scattering characteristics.

Binomial (maximally flat) design:

An n-section transformer of:

…

Multiple Interfaces


At large observation distances the field radiated by a satellite antenna which is attempting to communicate with a submerged submarine is locally TEM as shown in the figure. Assuming E before it impinges on the water is 1mv/m and the submarine is directly below the satellite. Find at 1Mhz:

A Example: P-5.31

1. The intensity of the reflected E field2. SWR created in air3. Incident and reflected power densities4. Intensity of the transmitted E field5. Transmitted power density6. Skin depth “d” of the water.7. Velocity of the wave in water to that in air (v/v0).

Multiple Interfaces


12221.

o45e8.2

2)j1( 1e98.0

377e8.2

377e8.2 180

45

45

o

o

Nearly Total Reflection1901

1SWR

2.

mv98.0mv198.0EE ir

Strong Standing Wave

2

23

0

2

ii m/nW32.1

3772

10

2

ES

3. 2

23

0

2

rr m/nW29.1

3772

1098.0

2

ES

o453

0

e10152

4. m/v15EE it

5. 2i

2

ri

or2

tt m/nW02.0S)1(SS

2

ES

6. m5.0f

1

7. s/m1017.32

u 6

0u01.0

Multiple Interfaces

Example (cont.):

LN01T5: Do Problem 5.1 , 5.5 due: 93.07.12

Advanced Electromagnetics LN01_EM Fields, Matters, Wave Equation [email protected] 1 /47 بنام...

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