EEE 415 ELECTRODYNAMICS & INSULATING MATERIALS

Matter is anything that occupies space and has weight.

Two main states

1. Physical

2. Chemical

1. Physical state

There are 3 types

i. Solid

ii. Liquid

iii. Gas

i.Solid

Has definite shape

Has definite volume

Has very strong intermolecular forces(compact)

Not compressible

Can change to liquid or gases

Examples: Rubies, all metals except wood, glass

ii.Liquids

Some have definite shape

Have weak intermolecular forces

Can change to liquid or solids

Example: mercury, paraffin

iii.Gases

Do not have definite shape

Have weak intermolecular forces

Can change to liquid or solid

Example: air, nitrogen, vapor

2.Chemical states

a) Elements

b) Compounds

c) Mixture

a.Element

This is a pure substance with same kind of atoms

Example: Hydrogen, Fluorine, Chlorine, Helium, Neon, Argon, Lithium, Sodium, Potassium,

Beryllium, Magnesium, Calcium, Boron, Aluminium, Carbon, Silicon, Nitrogen,

Phosphorus, Oxygen, Sulphur

b.Compounds

These are chemical combinations of two or more elements

Example: H20, CaCO3, NaCl

c.Mixture

Physical combination of two or more elements or compounds

Examples: Githeri, Milk, Colgate

Atom

The smallest unit of an element that cant be broken any further and retain its chemical

properties.

Molecule

The smallest particle of a compound that cant be broken down any further and retain its

chemical property.

Atomic theory

Neil Bohr atomic model

Proton

Positively charged particle (1.6*10^-19)

Neutron

Particle without charge

Electron

Negatively charged particle

Revolve around the nucleus in specified orbits

1.6*10^-19 (charge amount)

Nucleus

Innermost part of the atom that has protons and neutrons

Contributes to the weight of the atom

Orbit/quantum energy level

Pathway of electrons

Shell

A section of an atom where electrons can exist

It consists of subshells and electrons

Number of subshells is same as number of shells e.g. shell no. 4= 4ss

Number of electrons in a shell=2n^2, where n=shell number

For shell number 4

=2n^2=2*4^2=32 electrons

Subshell

Also a section of an atom where electrons can exist within a shell.

No of es=2+4(m_1)

Forbidden Band

A section of an atom where electrons cant exist.

Separates two shells

Theories

1st states that electrons will revolve in specified orbits without emitting or absorbing energy

(stable state)

2nd

states that if an electron transits from one energy level to the other, it will either emit or

absorb energy.

E=Ef-Ei= hf

h-planks constant

f-frequency

From a higher to lower energy level it will emit energy and absorb vice-versa

3rd

states: the momentum of an electron is given by

=n

=

Centrifugal energy and centripetal energy equal

Materials used in electrical and electronics eng

Historical perspective of use of materials

The earliest materials used by human beings were stones, wood, skin and horns, bones as tools to

make instruments and can be referred to as Stone Age.

It was followed by Bronze Age, then steel finally electronic age.

Electronic age started around 1940 by the invention of the PNj diode

Future trends

-MEMS and Nano technology (NEMS)

-Use of very light materials

-High temps

-Robots

-Smart materials

-adapting

-Mem resistor

-Biomaterials

Selection and classification of materials

Selection of materials

-Selection of materials depends on their properties and applications

-The selection can be done according to the following categories:

-Manufacturing

-Operational

-Functional

-Cost

Manufacturing

Deals with easy with which the material can be transformed into a product (molding)

For instance- Plasticity

-Ductility

-Machinability

-Finishing

Operational

Deals with environmental operating condition

They include:

-Temperature

-Humidity

-Pressure

-Friction

Functional

Deals with functionality of a material(product). Product is able to perform what it was intended

to perform effectively

They include:

-Strength

-Hardness

-Toughness

-Resistance to wear and corrosion

Cost

Deals with the monetary requirements

It includes cost from the time of acquiring raw materials up to the use of the product

-Raw materials

-Transport

-Storage

-Production

-Marketing

-Maintenance

-Taxes

Material properties to consider

There are several categories of material properties which include

-Electrical properties

-Mechanical properties

-Thermal properties

-Optical properties

-Magnetic properties

-Chemical properties

-Structural properties

Electrical properties

These are properties that indicate the ability of a material to allow current to pass through

They include- Conductance ( G)

-Resistance (R)

-Conductivity ()

-Resistivity ( )

Mechanical properties

Properties that indicate the strength of a material

Include- Tensile

-Impact strength

-Wear resistance

-Corrosion resistance

-Density

Optical properties

The behavior/operation of materials when exposed to light energy

They include

-Refraction index

-Reflectivity

-Absorption coefficient

Magnetic materials

Properties that enable materials to respond to magnetic energy

They include- cohesive

-permeability ()

Magnetic materials are classified as diamagnetic, paramagnetic, ferromagnetic,

antiferromagnetic e.g. iron, ferrite, nickel, cobalt

Chemical properties

They are the internal properties of a material which will determine its ability to react with other

materials.

They include:

-Atomic number

-Atomic weight

-Bonding

-Acidity

-Electronic structure

Material structure

It deals with the way a material is built, mostly the internal structure.

The different structures include:

-Macrostructure

-Microstructure

-Substructure

-Electronic structure

-Nuclear structure

Macrostructure deals with larger particles of a material. It requires a few levels of

magnification and use naked eye microscope.

Microstructure deals with the smaller sizes of a material studied using a microscope.

Substructure deals with much smaller sizes of a material studied using more powerful

magnifying equipment- x-ray.

Electronic structure: Deals with the smaller particles of an atom i.e. electrons studied using

electron spectroscope.

Nuclear structure: Deals with the nucleus of an atom studied using nucleus magnetic resonance

(NMR) (MR)

Material classification

Mostly materials are classified according to their properties and applications

Engineering materials are classified as follows:

-metals, semiconductors, ceramic, biomaterials, polymers, biological materials,

composites, advanced materials, smart materials, quantum dots.

Metals

They allow current and heat to pass through

There are 2 main types

-Ferrous

-Non-ferrous

Ferrous

Have Iron or a high % of the same in them. E.g. iron, ferrites

Non-ferrous

Dont have iron in them e.g. copper, aluminium

NB- most metals form alloys; brass

Ceramic

-They are materials that dont allow current to pass through

-They include most oxides for instance SiO2. Also burnt clay is a ceramic material

-They are commonly applied in insulators in high voltage. Can also be used to construct

capacitor

Polymers

-These are organic materials constructed by use of polymerization.

-They have complex structures of carbon, hydrogen and oxygen

-They are categorized into 2;

-Thermoplastics

-Thermosetting

-Thermoplastics: when exposed to heat they become soft and vice-versa. Include nylon,

polythene

-Thermosetting: they dont get soft when exposed to heat energy. Include PVC

Electrical properties of materials

Electrical property will deal with ability of a material to allow current to pass through.

The electrical property determine the classification of materials into 3 categories

1 .conductors

2. Insulators

3. Semiconductors

These properties include:

-Amount of free electronics

-Atomic number

-Bonding

-Forbidden band

-Energy gap

-Resistance

-Effect of changes in temperature

1) Amount of free electronics

-These are electrons found in the conduction band of a material.

-The more the number of electrons in the CB the better the conductivity of the material

2) Atomic number

This determines the position of a material (element) in the periodic table and therefore the group

it falls in also determines the number of electrons in the outer shell.

3) Bonding

This is the way atoms are combined together

This includes:

-covalent

-ionic

-metallic

4) Forbidden band

This is the section between the valence band and the conduction band

The size of FB determines classification of a material

5) Energy Gap (Eg)

This is the energy required to move an electron from the VB to the CB. Usually given in terms of

electron volts (ev)

6) Resistance (R)

-Opposition to flow of current in a material

-Measured in ohms

-It is given by R=

R-resistance

-resistivity

A-area

Resistance can also be given by ohms law

I=V/R

R=V/I

Another related property is conductance

G=I/R=

Measured in Siemens (s) Mhos ()

-Resistivity - ability of a material to oppose flow of current

Factors affecting resistivity

-Area, length & resistance

-additionally

-Amount of impurities

-Level of deformation- more

-Pressure and temperature

Conductivity (): ability of a material to allow current to flow through.

Measured in s/m, mhos/m

=1/

Conductivity of a semiconductor material is given by

Intrinsic

=nqn+ qp

Extrinsic conduction

N= qnND

N=donor concentration

N

x

p =qpNA

NA-Acceptor concentration

Dielectric-permittivity ()

The ability of a material to affect the electric field

= ro

- relative permittivity

Comparison of permittivity of a material to that of free space

-measured in F/m

-free space permittivity=8.854x10^-12

-C=

C-capacitance

d-distance

A-area (effective)

-D=E, =D/E

D- charge density

E-electric field

D=G/A

-F=

F-force

G1G2-Pt charges

r-radius, distance between point charges

=

-The dielectric material will be affected by:

-Dielectric constant ()

-Dielectric strength

-Dielectric loss

-According to the above electrical properties materials can be classified as conductors, insulators

and semi-conductors

Conductors

These are materials that allow current to pass through

Characteristics

-Have free electrons

-Have 1 to 3 electrons in the outer shell

-Have metallic bonding

-Have overlapping bonds

-Energy gap is very small or none at all Eg 0eV

-Very low resistance and low resistivity

10-4

-10-20

m

Thermal properties of insulating materials

-Insulating materials can be solids, liquids & gases

-Temp & heat affects these materials in one way or another

-At high temps solids can change either liquids or gases whole liquids usually change to gases

-If the materials are exposed to various temp, they undergo thermal deterioration

-Like solids are usually affected more by temp variations than liquids & gases.

-Liquids and gases sometimes can be used to cool down equipment i.e. oil in tx

-One of the main properties that makes them suitable in these apps is good heat conductivity

-Since some of the liquids being used are very expensive and have a high affinity to thieves who

want to sell it, theres a new technology which has been developed to allow the use of gas-air to

cool the transformers.

-One of the main disadvantages of these transformers in the technology used is very expensive

-This could be because its a new technology

-Only a few companies can afford them i.e. safaricom which has these transformers in their data

centers

-Change in temps can also affect electrical properties of insulating materials

-One needs to consider this when selecting an insulating material

Chemical properties of insulating materials

-Insulating materials should not react with the environ since this can affect their insulating

properties

-Solids should not undergo corrosion

-Since this can reduce their thickness therefore increasing their chances to electrical breakdown

-Liquids & gases should not conduct electricity

-They should not react with either elements in the environ

-Liquids & gases should not be combustible

Examples:

Solids: PVC, ceramic, glass, silica side, wood, concrete (limestone), micas

Liquids: oil, pure water

Gases: air, inert gases

Maxwells equations

These are equations used to analyze and study electromagnetic waves as they move from source

(transmitter) to the rx.

They were compiled by Maxwell after being developed by different scientists i.e. Gauss,

Amperes, Ohms, Faraday

Electromagnetic waves

These are signals produced by accelerating electrons

They have electrical and magnetic energy which travel in the same direction perpendicular to one

another

The energy changes from electrical to magnetic & vice-versa i.e. the signal moves from one

location to the other.

They can also be said to be transverse

electric

Time

magnetic

These waves are classified into the following categories

Gamma rays, X-rays, Ultra-violet, visible light, infrared, microwaves, radio waves

Maxwells equations in integral form

These equations are derived from different other equations like Gauss, Faradays and Amperes

1) Gauss of electricity

. da=q/ o

Surface integral of electrical field (enclosed) is equal to the charge

1) The enclosed area.

E-electric field intensity

A-area

Q-charge

o-free space permittivity (8.85x10-12

F/M )

2) Gausss law of magnetic

B-magnetic field intensity

Faradays law of induction

E s

S-length

B-magnetic intensity

t-time

Amperes law

s oi +

+

E

o-free space

permeability=4x10-n

A/M

i=current

C=speed

Differential form of Maxwells equations

Gauss law of electricity

/o

.D=

Gauss law of magnetism

.B= 0

Faradays law of induction

xE=

Amperes law

xH=Jc+JD

xH=Jc+

Where =divergence =

+

+

E electric field intensity which is a vector. Has magnitude and direction also given as E

D electric flux density, D=oE free space D is also a vector sometimes represented by D

B this is magnetic flux density. Also a vector B=oH

Where o=4x10-7

H-this is the magnetic field intensity. It is a vector represented by H

x this is a curl. Can also be represented by

ax ay az

Ex Ey Ez

= ax(

-

)- y (

-

) + az (

-

)

-represents the charge density

o-free space permittivity=8.85x10-12

F/M

JC-Conduction current density

JD-displacement current density both of which are given by JD=

Maxwells expressions in free space are given by:

Gauss law of electricity

0

Since =0

Gauss law of magnetism

0

Faradays law of induction

Amperes law

Since in free space JC=0

-The expressions are used to analyze static charge and constant current & moving charge as well

as varying current

-Static charge produces electric field but not magnetic field

E=F/q

E=q/4r2

-Constant current produces magnetic field but not electric field. This will be in surrounding of

a conductor

-Moving charge and varying current produces both electric and magnetic fields

-For Faradays law of induction with existence of E also there exists H

-Not one of them can exist without other

-The same case applies to amperes law where existence of varying H will give rise to E

Examples

Given an electric field of E=Emsin(wt+ Bz)y

Determine D, B and H in free space

D=E

=oy

D= o E

=oEmsin(wt-Bz) y

ii)xE=B/t

= ax ay az

0 Ey 0

= ax ay az

0 Ey 0

= x(

-

)- ay (

-

) + az (

-

)

=

x +

az

=

Emsin(wt+Bz) ax] +

Emsin(wt+Bz) az]

=

Emsin(wt-Bz) x]

= Emcos(wt-Bz) x

= Emcos(wt-Bz) x

Integrate both sides

= Emcos(wt-Bz)

-B =

Emsin(wt-Bz) x

B =

Emsin(wt-Bz) x T

iii) B=oH

H=B/o

H=1/o

Emsin(wt-Bz) x

Plot for E and H

x

H

z

E

y

Further analysis

xH=

ax ay az ax ay az

=

Hx Hy Hz Hx 0 0

= ax(0)- ay(

-

Hx) + az(

-

Hx)

= -

Hx ay = - ay

Hx

= - ay

[ 1/o

Emsin(wt-Bz)]

= 1/o

Emcos(wt-Bz) y =

=

(o Emsin(wt-Bz) y

=wo Emcos(wt-Bz) y

1/o

Emcos(wt-Bz) y= wo Emcos(wt-Bz) y

=1/o

= wo

o/o =w2/B

2

w/B=1/( oo)=C=3x10m/s

Proves that above is a wave moving in z direction

(Emsin(wt-Bz) y)

B/ow( Emsin(wt-Bz) x)

= ow/B

1/oo= w2/B

2

w/B= 1/( oo)

=o x1/( oo)

= o /( oo)

=( 2

o /oo)= (o /o) =377 impedance of free space

Example

Given the following

H=Hmej(wt+Bz)

y in free space determine B, D and E. prove that this is a wave moving in z

direction in free space.

Boundary conditions H2

1

H1

2

These are conditions that describe the relations between fields, transiting from one medium to

another.

Considering both electric fields and magnetic field relations can be summarized as follows

E

Et1=Et2

Dn1= Dn2

(D1-D2)n12= -s

Tan1 = r2 Tan2 r1

= 0 r

D-electric field density

s change surface density

t-tangential

n-normal

H/(1+t2) =Ht2

Bn1= Bn2

H-H2x n12

Tan1 = r2 Tan2 r1

o =4x10-17

H/m

Example

Given a field of H=4x +3 y- 6 z A/m

Transiting from a medium of r1=3 and r2=5 for x0 respectively. Determine B1, B2,

H2, 1 and 2

B=H

B1= 1H1= 0r1H1

B1 = 30 (4x +3y- 6z)

B1 = 0 (12x +9y- 18z)

Bn1= Bn2

The x value is the same, the rest will change according to the relative permeability

B1 = 0 (12x + r29y- r218z)

r1 r1

= 0 (12x +

x9y-

x18z)

= 0 (12x +15y- 30z)

H=B/

H2= B2/( 0 r2)

= 0 (12x +15y- 30z)

5x0

=2.4x +3y- 6z A/m

Magnitude of H2

H2= (2.4x +3y- 6z)

=7.125 A/m

Relating this to x value

Tan2 =Hx2/H2

2=tan-1

(Hx2/H2)

= tan-1

(2.4/7.125)

=18.62

2=90-2=90-18.62=71.38

=Tan1 = r2 Tan2 r1

Tan1 = r2 Tan2 r1

=

xtan71.38=

tan-1(4.947)=78.57

Poisson, Laplace and Helmholtz equations

Poissons equation

This is an equation that describes the relation between fields more so electric field. It is derived

from one of Maxwells equations.

.D=

D=E

. E =

. E =/

-.V=E

Gradient of potential difference

. (-.V)= /

2V=- / Poissons equation

=- /o for free space

For regions without charge the above expression becomes Laplace expression given by

2V=0 Laplace equation

Where 2 is the Laplace operator

Helmholtz equation is related to the above where it is given by

2A +K2A=0

Where

2- Laplace operator

K-Wave number

A-Amplitude

Laplace expression solution can be given in terms of Cartesian, cylindrical and spherical

Cartesian

2V=

Cylindrical solution

2V=

Spherical solution

2V=

s

Example

If the potential at r=0.1m is 0 and the potential at r=0.1m is 100v. Determine E and D for the

above conditions.

Taking the spherical co-ordinates solution, the potential does not depend on and . It depends

on r.

The solution will be given by

2V=

Integrating the above solution once

Integrate again

V=

+B

V= r1+B

For v=0 and v=0.1

0=

+B

For v=100 and r=2

100=

+B

0= 10A +B, B=10A

200=A +2B

200=A +20B

19A=200

A=10.53

B=10A=10x10.53=105.3v

V=

+B

V=

+10.53

-V=E

E=-V

E=- (

+10.53)

E=

E=10.53r2.r

D= E

D=0E for free space

D=10.53 0 r2.rC/m

2

Plane wave equations

Plane wave- this is an electromagnetic wave as it appears to an onlooker

It is called a plane wave since it comes as a plane i.e. two dimensions to the onlooker

The plane is x and y in most cases the electric field x in the x direction while magnetic field on

the y direction

H

y

E

x

The plane wave obeys Maxwells equations and therefore its equations are derived from

Maxwells equations

This is done as follows

D= E

B=H

J=E

In free space

=0 charge density

Taking these assumptions, Maxwells equations in terms of time varying quantities will be given

as follows

x H=Jc +JD =E+ jE= (+ j)E

xE=

= jH

x H=0

x E=0

Performing a curl of the curl of H will result to

xx H=(+ j)( x E)

(. H) 2H =(+ j)( x E)

=(+ j)( jH)

But (. H)=0

2H = j(+ j)H

2H=jw (+ j)H

Performing the curl of the curl of E, we obtain

xx E= j( x H)

(. E) 2E = j(+ j)E

2E = j(+ j)E

2E = j(+ j)E

2H =r2H

2E =r2E

rpropagation constant of wave

r+j

attenuation constant

wave number

r2= j(+ j)

r= ( j(+ j))

The and are obtained from the above expression using mathematical analysis

= (

( (1+ (

) 2

) 1)

= (

( (1+ (

) 2

) +1)

The solution to 2E =r2E is given in Cartesian coordinates from following expression

= r

2H

Since the wave is travelling in z direction then above results to

= r

2H

Which gives a solution of

H(zt)=Hoe+_rz

ejwt H

Solutions of different parameters can be determined considering 3 types of materials which

include partially conducting materials dielectrics and good conductors

The parameters will include; , , n, ,

Partially conducting materials

For the materials and need to be considered

= (

( (1+ (

) 2

) 1)

= (

( (1+ (

) 2

) +1)

=

=

=

=

=

Impedance

et=Ex/Hy=(

)

Assuming that

=0

=0

=

Impedance

=

dielectric

freespace=

=377

Good conductors

Assume that >> , I is not considered and

=

=

=

=

=

==

=

=

= skin depth

=1/

Impedance

=

=

< 45

Group velocity

Velocity of several waves moving together as a group is given by VP=

Phase velocity

Velocity of a wave is given by

VP=

Reflection, refraction, diffraction and scattering

The electromagnetic waves for H and E obey the laws of reflection, refraction, diffraction and

scattering as is the case with light energy.

Different media are considered with characteristics r and r

i =r

S ells law

S S

=

Reflection coefficient

= E

E =

Transmission coefficient

T= E

E

ANTENNA

It is a device that changes emf energy to electrical energy and vice versa. Used in transmission

and receiving system.

Antenna parameters

These are parameters used to analyze antennas

Also referred to as antenna property

Include: gain, directivity, radiation pattern, impedance matching, efficiency

Gain

This is the ratio of energy transmitted to that of the i/p energy

This parameter is important in determining the best antenna use for a given frequency. The

higher the better.

Directivity

This is the level of power transmitted in a given direction to the average power transmitted in all

directions

Radio transmitter antennas do not require to be directive while satellite antennas require.

Directive

Not directive

Radiation pattern

Given the distribution of electric on magnetic field with respect to a given angle.

Omni direction

Side lobe

Back lobe Main lobe

Efficiency

Ratio of output to power it can go up to 90%

Impedance matching

Process of making two impedances between two devices look same for max power transfer

Important in compiling different devices to avoid power reflection (loss)

NB: Add more info for instance equations

Examples of antennas: dipole, yagi uda, horn antenna, parabolic, patch antennas