Electromagnetic waves, Circuits and Applications

8/22/2019 Electromagnetic waves, Circuits and Applications

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2

ELECTROMAGNETIC WAVES,

CIRCUITS AND APPLICATIONS

2.1 Electromagnetism Introduction

Electromagnetism is a branch of Physics that describes the interactions involving electric

charge. Classical electromagnetism, summarized by Maxwells equations, includes the phenomena

of electricity, magnetism, electromagnetic induction (electric generators) and electromagnetic

radiation (including all of classical optics). In this chapter, we shall discuss the fundamentals of

electromagnetism along with Maxwells equation and propagation of electromagnetic waves in

free space.

2.1.1 Electrostatics

If a stick of sealing wax is rubbed with cats fur, both bodies are put into a peculiar

condition in which light bodies in their neighbourhood are set in motion. We say that by rubbing,

bodies become electrified and that they carry an electric charge. In the light of modern views,charge is a fundamental property of elementary particles, which make up matter. It is evident from

these definitions of charge that it is always associated with mass. It is also found experimentally

that charge can be transferred from one body to the other by contact. The unit of charge in SI

system is coulomb.

Two charged bodies exert a force upon one another. This force can be used to measure the

charge as for example by means of an electrometer. From results, which have been obtained in

such measurements, existence of charges in two different kinds called positive and negative has

been concluded. These charges when in combination add algebraically i.e.the charge is a scalar

quantity.

Faradays laws of electrolysis and Millikans oil drop experiment have shown that the

smallest charge that exists in nature is the charge of an electron and that charge of any other

electrified body is an integral multiple of this electronic charge. This all in turn means that charge

is quantisedi.e. it appears in discrete units.

Further, since charge is a fundamental property of the ultimate particles making up matter,

the total charge of a closed system cannot change i.e. net charge is conserved in an isolated

system. The law of conservation of charge itself beautifully illustrated by nature in pair production

or annihilation in which equal quantities of charges of each sign (positive and negative) appear or

disappear.


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Physics for Technologists

Electrostatics is the branch of Physics, which deals with the behaviours of stationary

electric charges. Now, we shall discuss the fundamental definitions in electrostatics.

Coulombs Inverse Square Law

Coulombs inverse square law gives the force between the two charges. According to this

law, the force (F) between two electrostatic point charges (q1 and q2) is proportional to the productof the charges and inversely proportional to the square of the distance (r) separating the charges.

i.e. Fq1 q2

F2

1

r

(or)2

21

r

qqKF =

where K is proportionality constant which depends on the nature of the medium. This force acts

along the line joining the charges. For a dielectric medium of relative permittivity r, the value ofK

is given by,

4

1

4

1

0

==r

K

where = permittivity of the medium.

0 = permittivity of free space = 8.854 1012 F m 1

Fig. 2.1 Coulomb Inverse Square Law

For air medium, r = 1

In the scalar form, the force between the electric charges is given by,

2

21

04

1

r

qqF

=

where229

0

1094

1 = CNm

Electric field

Electric charges affect the space around them. The space around the charge within which

its effect is felt or experienced is called electric field.

Electric field Intensity (or) Strength of the electric field (E)

The electrostatic field intensity E due to a point charge qa at a given point is defined as

the force per unit charge exerted on a test charge qb placed at that point in the field.

1-1-

20

CN(or)mvolt4

r

rq

q

FE aa

b

baa

==

Electrostatic Potential (V)

2.2

q 1 q 2

m e d i u m r


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Electromagnetism and Microwaves

As in the study of mechanics, it is useful to think in terms of the work done by electrical

forces and the potential energy in electric charges to understand the behaviour of electric charges.

Just as the heat flows from a higher temperature to lower temperature, water flows from higher

level to lower level and airflows from higher pressure to lower pressure, electric charge flows from

a body where electrical level is more to a body where it is less. This electrical level is called

electric potential.

The electric potential is defined as the amount of work done in moving unit positive

charge from infinity to the given point of the field of the given charge against the electrical force.

Unit: volt (or) joule / coulomb

The electric potential at any point is equal to the work done in moving the unit positive

charge from infinity to that point.

Potential dxEVr

.==

dx

x

qr

2

04

=

r

q

r

qV

00 4

11

4 =

=

The electric field intensity (E) and the potential (V) are related in differential form as,

dr

dVE

=

In vector notation,

E = V= negative gradient of the potentialwhere

zk

yj

xi

+

+

= . = Gradient operator

Electric lines of force

An electric field may be described in terms of lines of force in much the same way as a

magnetic field. The electric field around a charged body is represented by imaginary lines called

electric lines of force.

The direction of lines of force at any point is the direction along which a unit positive

charge (+1C) placed at that point would move or tend to move.

Properties of electric lines of force

1. Every line originates from a positive charge and terminates on a negative charge.

2. Lines of force never intersect.

3. The tangent to line of force at any point gives the direction of the electric field Eat

that point.

2.3


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Fig. 2.2 Electric lines of force

4. The number of lines of force per unit area at right angles to the lines is proportional to

the magnitude ofE.

5. Each unit positive charge gives rise to0

1

lines of force in free space.

Electric flux

The electric flux is defined as the number of lines of force that pass through a surface

placed in the electric field.

The electric flux (d) through an elementary area ds isdefined as the product of the area and the component of

electric field strength normal to the area.

The electric flux normal to the area ds = d= E . ds

Fig.2.3 Flux of the electric field

d =E ds cos= (Ecos) . ds

= (Component ofEalong the direction of the normal area)

The flux over the entire surface = = S

d ds.cosES

= Unit: Nm2 C 1

Gauss theorem (or) Gauss law

This law relates the flux through any closed surface and the net charge enclosed within the

surface. The electric flux () through a closed surface is equal to the

0

1

times the net charge q

enclosed by the surface.

q

=

0

1

(or) =

= cos0

dsEq

Dielectric materials

Dielectric materials are also called as insulators. In dielectric materials, all the electronsare tightly bound to their parent molecules and there are no free charges. In addition, theforbidden energy band gap for dielectric materials is more than 3eV. Therefore, it is not possiblefor the electrons in the valence band to excite to the conduction band, by crossing the energy gap,even with normal voltage or thermal energy. Because of this, no electrical conduction takes place.Generally, dielectrics are non-metallic materials of high specific resistance and negativetemperature coefficient of resistance.

Electric flux density or Electric displacement vector (D)

2.4

E

ds


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It is defined as the number of electric lines of force passing normally through an unit area

of cross section in the field. It is given by,

AD

= Unit : coulomb / m2

where, = total electric flux (coulomb) and

A = Area of cross section (m2).

Permittivity ()Permittivity is defined as the ratio of electric displacement vector (D) in a dielectric

medium to the applied electric field strength (E). Mathematically it is given by,

E

D= Unit : farad /metre

The permittivity indicates the degree to which the medium can resist the flow of electriccharge and is always greater than unity. The permittivity () can also be given as,

= o . r

where o = permittivity of free space or vacuum and r = permittivity or dielectric constant of themedium.

Dielectric constant ( r)The dielectric constant or relative permittivity ( r) of a material determines its dielectric

characteristics. It is the ratio of the permittivity of the medium ( ) and the permittivity of freespace (0) and is given by,

0

=r

Since it is a ratio of same quantity, r has no unit. It is a measure of polarization in thedielectric material.

2.1.2 Magnetostatics

A stationary electric charge always produces a static electric field. The behaviour of thestationary charges has been discussed in the previous section. The electric current in a circuit isdue to the movement of electric charges i.e. electrons. Oersterd and Ampere proved experimentallythat the current carrying conductor produces a magnetic field around it. Hence, the origin ofmagnetism is linked with current and magnetic quantities are measured in terms of current.

The Coulombs inverse square law of magnetism gives the force of attraction between two

individual magnetic poles. The interaction between the magnets can be explained on the basis ofinverse square law similar to that in electricity. But this does not imply that there are magnetic freecharges as there are in electricity. The magnetic poles are analogous to the polarization charges ininsulators, the smallest entity being a dipole and a simple pole i.e. no isolated monopole exists.The magnetostatics deals with the behaviour of stationary magnetic fields. We shall discuss thefundamentals of magnetism.

Magnetic dipole

Any two opposite magnetic poles separated by a distance dconstitute a magnetic dipole.

Magnetic dipole moment (m)Ifm is the magnetic pole strength and lis the length of the magnet, then its dipole moment is

given by,

2.5


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m= ml

It can also be defined as follows:

When an electric current of i amperes flows through a circular wire of 1 turn having an area

of cross section a m2, then it is said to have a magnetic moment of,

m = ia Unit: ampere (metre)2

Fig.2.4 Magnetic moment

Dipole moment is a vector quantity. Its direction is normal to the plane of the loop to theright, if the current is clockwise.

Magnetic flux ( )It is defined as the total number of magnetic lines of force passing perpendicular through a

given area.

Unit: weber.

It can also be defined as the total number of lines of force emanated from North Pole.

Magnetic flux density (or) Magnetic induction (B)

It is defined as the number of magnetic lines of force passing through an unit area of cross

section. It is given by,

2Magnetic Flux weber/m (o )Unit Area

B r tesla

A= =

It is also defined as the magnetic force (F) experienced by an unit north pole placed at the

given point in a magnetic field.

Force experienced

Polestrength

FB

m= =

Magnetic field strength (or) Magnetic field intensity (H)

Magnetic field intensity or magnetic field strength at any point in a magnetic field is equal

to

1times the force acting on a unit north pole placed at the point.

metre/turns.ampereB

m

FH.e.i

=

= 1

where = permeability of the medium in which the magnetic field is situated.

Magnetization (or) Intensity of Magnetization (M)

2.6

A

mi


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The term magnetization is the process of converting a non-magnetic material into a

magnetic material. It measures the magnetization of the magnetized specimen. Intensity of

magnetization (M) is defined as the magnetic moment per unit volume.

It is expressed in ampere/metre.

2.7


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Magnetic susceptibility ()It is the measure of the ease with which the specimen can be magnetized by the magnetizing

force. It is defined as the ratio of magnetization produced in a sample to the magnetic field

intensity. i.e. magnetization per unit field intensity.

unit)(noHM=

Magnetic permeability ()It is the measure of degree at which the lines of force can penetrate through the material. It

is defined as the ratio of magnetic flux density in the sample to the applied magnetic field

intensity.

H

B.e.i r == 0

where 0 = permeability of free space = 4 10 7

H m 1

r = relative permeability of the medium

Relative permeability ( r)It is the ratio of permeability of the medium to the permeability of free space.

i.e. r =0

(No unit)

Relation between rand When a magnetic material is kept in a magnetic field (H), then two types of lines of

induction passes through the material.. One is due to the magnetic field (H) and the other one is

due to self-magnetization of the material itself. Therefore, total flux density (B) in a solid can be

given as,

B = 0 (H+M) (1)

We know that

HB)or(H

B == (2)

Equating (1) and (2), we get,

H = 0(H+M)

H = 0H+0M

0rH = 0H +0M ][ 0 r =

0 0

0 0

r

H M

H H

= +

1rM

H = +

+= 1r.e.i

2.8


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Bohr Magneton (B)Bohr magneton is the magnetic moment produced by one unpaired electron in an atom.

It is the fundamental quantum of magnetic moment.

1 Bohr magneton m

ehh

m

e

42.2 ==

1B = 9.27 1024 ampere metre 2

Current density (J)

Current density is defined as the ratio of the current to the surface area whose plane is

normal to the direction of charge motion. It is denoted by Jand is a vector having direction of

charge motion.

Consider a surface ds whose normal is parallel to the motion of electrons. The current

density is given by,

ds

dI=J (or) dI =J . ds

Therefore the net current flowing through the conductor = =S

J.dsI

Conduction Current Density ( J1)

The current density due to the conduction electrons in a conductor is known as the

conduction current density.

By ohms law, the potential difference across a conductor having resistance R and currentI

is,V = IR (1)

For a length land potential difference V,

V=El (2)

whereE= electric field intensity.

From equations (1) and (2),

IR = El (3)

But R =

=

A

l

A

l

1

(4)

where and are the electrical resistivity and conductivity respectively.

Using equation (4) in (3), lEA

lI .=

(or) EA

IE

A

I

=

= (or) (or) EJ =1 (5)

J 1 may be referred to as conduction current density, which is directly proportional to the

electric field intensity.

Displacement Current Density )J( 2

2.9


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There is no direct current in a circuit containing capacitor while an alternating current can

flow in it. The conduction current due to the motion of electrons cannot pass through a capacitor as

its plates are separated by a dielectric. As the current does not pass through the capacitor so we

have to conclude that in a capacitor a certain process closes the conduction current, i.e. it enables

in someway the charge exchange between the capacitor plates without actually transporting a

charge between the plates. The current associated with this process is called as displacement

current. The displacement current per unit area is known as displacement current density.

In a capacitor, the current is given by,

dt

dV.C

dt

)CV(d

dt

dQIc ===

(1)

where Q, Cand Vrepresents charge across the plates, capacity and potential difference across the

plates of the capacitor respectively.

In a parallel plate capacitor, the capacitance is given by,

C=d

A(2)

where , A and drepresents electric permittivity, area of the plates of the dielectric filled capacitor

and distance between the plates of the capacitor respectively.

Using equation (2) in (1)

=CIdt

dV

dA

Ior

dt

dV

d

A C .)(.

=

J2 = Displacement current density =dt

Ed

dt

dE

d

V

dt

d )( ==

dt

DdJ =2 [since ED = = electric displacement vector]

This is not a current, which directly passes through a capacitor, and is only an apparent

current representing the rate at which the flow of charge takes place from electrode to electrode in

the external circuit. Hence the displacement is justified.

In the presence of magnetic fields in free space due to time varying electric fields, the net

current density =J = J1 + J2

dt

DdEJ +=

Biot Savart Law

Biot Savart law is used to calculate the magnetic field due

to a current carrying conductor. According to this law, the

magnitude of the magnetic field at any point P due to a small current

element I.dl ( I = current through the element, dl = length of the

element) is

1. directly proportional to the current (I)

directly proportional to the length of the current element (dl)Fig. 2.5 Biot Savart law

2. directly proportional to the sine of the angle () between the direction of the current andthe line joining the current element to the point P and

2.10

P

dB

X

r

Y

I

A

B

I.dl C


11/46


3. inversely proportional to the square of the distance between (r) of the point P from the

current element

i.e.2

sin

r

IdldB

(or)

2

0 sin.4 r

IdldB

=

In vector notation,3

0 .4 r

ridldB

=

The direction of vectordB is the direction of the vector ridl (i.e.) perpendicular to the

plane of the paper and inwards.

Amperes circuital law

It states that the line integral of the magnetic field (vectorB) around any closed path orcircuit is equal to 0 (permeability of free

space) times the total current (I) threading

through the closed circuit. Mathematically,

IdlB 0. =

It may be noted that the magnitude

of the magnetic field at a point on the

circular path changes with the change in

radius of the circular path but the line

integral of vectorB over any closed path

will be independent of its radius i.e. equalto 0times the current threading the circle.

Fig. 2.6 Amperes circuital law

Faradays Law of electromagnetic induction

Michael Faraday found that whenever there is a change in magnetic flux linked with a

circuit, an emf is induced resulting a flow of current in the circuit. The magnitude of the induced

emf is directly proportional to the rate of change of magnetic flux. Lenzs rule gives the direction

of the induced emf which states that the induced current produced in a circuit always in such a

direction that it opposes the change or the cause that produces it. By combining Lenzs rule with

Faradays law of electromagnetic induction, the induced emf can be written as,

dt

de

=)(emfinduced

where dis the change magnetic flux linked with a circuit in a time dtsecond.

2.1.3 Electromagnetic waves

According to Faradays laws of electromagnetic induction, a time varying magnetic field

behaves as a source of electric field. The principle of generating electric field by changing

magnetic fields is employed in transformers, inductances etc. According to Maxwells

modification of Amperes law, a changing electric field gives rise to a magnetic field. It means that

when either of the field (magnetic or electric) changes with time, the other field is induced in thespace. This leads to the generation of electromagnetic disturbance comprising of time varying

2.11

X

Y

I

O a

P

B

Q

dlPQ=


12/46

Y

X

Z

E E

H

H

H

E


electric and magnetic fields. Such a disturbance can be propagated through space even in the

absence of any material medium. These disturbances have the properties of a wave and are called

electromagnetic waves.

Fig. 2.7 Electromagnetic waves

The variations of electric intensity and magnetic intensity are transverse in nature. The

variations ofE and H are perpendicular to each other and also to the directions of wave

propagation. The wave patterns ofE and H for a travelling electromagnetic wave obey the

fundamental equations, called Maxwells equations. These equations are mathematical abstractions

of experimental results.

Electromagnetic waves cover a wide range of frequencies and they travel with the same

velocity as that of light i.e. 3 10 8 m s1. The electromagnetic waves include radio frequency

waves, microwaves, infrared waves, visible light, ultraviolet rays, X-rays and gamma rays. The

classification electromagnetic wave is done according to their main source. However, different

sources may used to produce waves in overlapping range of frequencies.

The history of evolution of electromagnetic waves is summarized as:

1. James Clerk Maxwell (1831 1879) unified all previous known results, experimental and

theoretical on electromagnetic waves in four equations and predicted the existence of

electromagnetic waves.

2. Heinrich Rudolf Hertz (1857 1937) experimentally confirmed Maxwells prediction.

3. Guglielmo Marconi (1854 1937) transmitted information on an experimental basis at

microwave frequencies.

4. George C. Southworth (1930) really carried out Marconis experiments on a commercial

basis.

5. During World War - II (1945) based on the previous developments; radarwas invented

and was exploited for military applications.

2.1.4 Del, Divergence, Curl and Gradient Operations in Vector calculus

(i) Del (nabla) Operator ():

The del operator is defined through the partial derivatives of the with respect to space

variables. In Cartesian coordinates, the del operator is written as,

zk

yj

xi

+

+

=

2.12


13/46

-

PP P

(a) positivedivergence

(b) negativedivergence

(c) zero divergence


It is a vector operator and it may be applied on scalars, vectors or tensors. The del operator

is important since it provides a number of indications as to how a vector or scalar functions vary

with position. It shows up in the gradient, curl, divergence and Laplacian.

(ii) Divergence

The divergence of a vectorV written as div V represents the scalar quantity.

div V = V =z

V

y

V

x

V zyx

+

+

Fig. 2.8 Divergence

Physically the divergence of a vector quantity represents the rate of change of the field

strength in the direction of the field.

Fig. 2.9 Example for divergence

If the divergence of the vector field is positive at a point then something is diverging froma small volume surrounding that point and that point is acting as a source. If it negative, then

something is converging into the small volume surrounding that point is acting as sink. However,

if the divergence at a point is zero then the rate at which something entering a small volume

surrounding that point is equal to the rate at which it is leaving that volume. The vector field

whose divergence is zero is called solenoidal.

(iii) Curl

The curlofV is written as curl V represents a vector quantity.

2.13

Div is positive

Div = 0

Div is negative


14/46

Curl


curl V =

zyx VVV

zyx

kji

V

=

Physically, the curl of a vector field represents the rate of change of the field strength in a

direction at right angles to the field and is a measure of rotation of something in a small volume

surrounding a particular point. For streamline motions and conservative fields, the curl is zero

while it is maximum near the whirlpools.

(i) No rotation of the (ii) Rotation of the (iii) direction of curlpaddle wheel means paddle wheel means

the curl of the field is where the curl of the

zero field exists.

Fig.2.10 Example for curl

For vector fields whose curl is zero there is no rotation of the paddle wheel when it is

placed in the field. Such fields are called irrotational.

(iv) The Gradient

The gradient of a scalar function is a vector whose cartesian components are

zy,

x

and (i.e.)z

ky

jx

++

==

igrad

The magnitude of this vector gives the maximum rate of change of the scalar field and its

direction is the direction in which this maximum change occurs.

For example, the electric field intensity at any point is given by,

E= grad V= negative gradient of potentialThe negative sign implies that the direction ofE opposite to the direction in which V

increases.

Some Important Vector Results and Theorems

In electromagnetism, we shall use the following vector results:

1. E)E()E( 2=

curl curlE= grad divE E2

2. div gradS= S2

2.14


15/46


S)S(2=

3. div (SV) =Sdiv V+ VgradS)S(V)V(S)VS(

+=

4. curl grad = 00)( =

5. Gauss Divergence Theorem

It relates the volume integral of the divergence of a vectorVto the surface integral of thevector itself. According to this theorem, if a closed S bounds a volume , then

(div V) d= s V ds (or) = Sd dsVV )(6. Stokes Theorem

It relates the surface integral of the curl of a vector to the line integral of the vector itself.

According to this theorem, if a closed path C bounds a surface S,

s (curl V) ds = C V dls (V) ds = C V dl

2.1.5 Maxwell Equations

Maxwells equations combine the fundamental laws of electricity and magnetism and are

of profound importance in the analysis of most electromagnetic wave problems. The behaviour of

electromagnetic fields is studied with the help of a set of equations given by Maxwell and hence

called Maxwells equations. These equations are the mathematical abstractions of certain

experimentally observed facts and find their application to all sorts of problem in

electromagnetism. Maxwells equations are derived from Amperes law, Faradays law and Gauss

law. They are listed in the Table 2.1.

Table 2.1 Maxwells Equations

Maxwells Law Differential form Integral form

First law:

(Based on Gauss law of electrostatics)

=D. =vs

dvds.D

Second Law:

(Based on Gauss law of magnetostatics)

0=B. 0=

sds.B

Third law:

(Based on the Faradays law of

electromagnetism)t

BE

=

=s

d.t

Bdl.E

Fourth Law:

(Based on the Amperes circuital law or Biot

Savart law)

t

DEH

+=

+=

slt

DE(dl.H

where D = electric displacement vector (C m2

)

2.15


16/46


= volume charge density (C m 3)

B = magnetic induction (Wb m2)

E = electric field intensity (V m 1 )

H = magnetic field intensity (A m1)

Maxwells equations: Derivation

Maxwells First Law

Suppose the charge is distributed over a volume V. Let be the volume density of the

charge, then the charge q is given by,

q = v

dv

The integral form of Gauss law is,

==v

sdv

dsE

0

1(1)

According to Gauss divergence theorem,

=vs

dv)E(dsE (2)


=vv

dv

)dvE(0

1

(3)

Since, this is true for any volume V, integral must be equal.

0

= E

(4)

div0

=E (5)

But electric displacement vector, ED 0= (6)

(5) 0

0

0

0 div

=E

(or) div ( =)E0

(or) div =)D(

2.16


17/46


= )D( (7)

This is the differential form of Maxwells I law.

From (1), =v

sdvdsE0

=

vs

dvdsD (8)

This is the integral form of Maxwells I law.

Maxwells Second Law

From Biot - Savart law of electromagnetism, the magnetic induction at any point due to a

current element,

dB =24 r

sinidlo

In vector notation,

)ridl(r

dB

=3

0

4

= )ridl(

r

^

2

0

4

Therefore, the total induction B = )rdl.r(

i ^ 2

0 1

4

This is Biot - Savart law.

If we replace the current iby the current densityJthe current per unit area,A

iJ= then,

dvrJr

B )..(1

4

^

2

0 =

[ i =J . A andI . dl = J(A . dl) =

J . dv]

Taking divergence on both sides,

dvrJr

Bv

).1

(4

^

2

0 =

If the current density is assumed to be constant, then 0= J

0=

B

This is the differential form of Maxwells second equation.

Experiments to date have shown that magnetic monopoles do not exist. Hence, the

number of magnetic lines of force entering any arbitrary closed surface is exactly the same leavingit. Therefore the flux of magnetic induction B across a closed surface is zero.

2.17


18/46


By Gauss divergence theorem,

v 0 ==

s

B.dsdv)B(

This is the integral form of Maxwells second law.

Maxwells Third Law

By Faradays law of electromagnetic induction,

dt

de

=

Now, let us consider work done on a charge, moving it through a distance dl.

W= dl.E which is a line integral

If the work is done along a closed path, emf = dl.E

The magnetic flux linked with closed area Sdue to the inductionB = =s

ds.B

[ ]=== s dsBdtd

dt

de .emf

=

sds

dt

Bd

Hence, ds.dt

Bddl.E

s =

This is the Maxwells third equation in integral form.

Using Stokes theorem, the line integral of a vector function along a closed pathdl.E can be converted to the surface integral of the normal component, the vector E

of the enclosed surface.

(i.e) dl.E = s

ds).E(

=s s

ds.dt

Bdds).E(

Hence,t

B)E(

=

This is the Maxwells third equation in differential form.

Maxwells Fourth Law

By Amperes circuital law,

=

idl.B 0

But,H

B=0 (or)B = 0 H

2.18


19/46


Therefore, =i

idl.H

But i = s

ds.J

Hence, =

ds.Jdl.Hs

Butt

DEJ

+=

+= ss

ds.t

Dds.Edl.H

This is Maxwells fourth equation in integral form.

Using Stokes theorem,

ds).H(dl.H

s

=

Hence,

+=s ss

ds.t

Dds.Eds).H(

ds.t

DEds).H(

s

+=

(or)t

DEHH

+==

curl

This is Maxwells fourth equation in integral form.

2.1.6 Maxwells equations in free space

In free space, the volume charge density () = 0 and conduction current density (J1) = 0

(since = 0 ) and therefore, the Maxwells Equations becomes,

D = 0 (1)

B = 0 (2)

E =t

B

=t

(0 H) =

H0

(3)

H =t

D

=t

E

0

(4)

Differentiating (4) with respect to time,

t

22

2

0

2

t

E

t

D)H(

=

= [SinceD = 0E]

2.19


20/46


2

2

0t

E

t

H)H(

=

=

(5)

Taking curl on both sides of (3),

)H()E(= 0

(6)

But,

E)E()E( 2= = 2E [since E= 0]

(7)

Using equation (7) in (6),

2E= )H(

0

2

2

0002

tE)H(E

==

(8)

This is called free space electromagnetic equation.

In one dimension,

2

2

002

2

2

2

002

2

.1

)(x

E

t

Eor

t

E

x

E

=

=

(9)

Comparing this with standard mechanical wave equation,

22

22

2

x

yC

t

y

=

(10)

We get,8

0000

2103

11===

C)or(C m/s.

The velocity of electromagnetic wave in free space

C=00

1

(11)

Similarly, the wave equation in terms ofHcan be written as,

2H =2

2

00t

H

(12)

In a medium of magnetic permeability and electric permittivity , the wave equation becomes,

2H =2

2

t

H

(13)

2E =2

2

t

E

(14)

The above equations (13) and (14) are known asHelmholtzs wave equations.

2.20


21/46


The velocity of electromagnetic wave in any medium is,

C=

1(15)

Worked Example 2.1: An electromagnetic wave of frequency f = 3.0 MHz passes from vacuum

into a non magnetic medium with relative permittivity 4. Calculate theincrement in its wavelength. Assume that for a non-magnetic medium

r=1.

Frequency of the electromagnetic wave =f= 3.0 MHz = 3 10 6 Hz

Relative permittivity of the non-magnetic medium = r = 4

Relative permeability of the non-magnetic medium =r= 1

Velocity of em wave in vacuum = C =00

1

Wavelength of the em wave in vacuum = =00

1.

1

ff

C=

Velocity of em wave in non-magnetic medium =

rr

C 00

11==

Wavelength of the em wave in non-magnetic medium =

rrff

C

00

1.

1=

=

Therefore the change in wavelength =

= 1

11.

1

00 rrf

= m5014

1

103

1036

8

=

i.e. the wavelength decreased by 50 m.

2.1.7 Characteristic Impedance

The solution of the equation for the electric component in the electromagnetic wave is,

Ey =Eo sin

2(ct x) (1)

For magnetic component,

Hz= HOsin

2(ct x) (2)

Differentiating equation (1) with respect to time,

)xCt(C

E

t

Ey

=

2cos

20 (3)

2.21


22/46


But,

zyx HHH

zyx

kji

H

=

H=xH

yH

xHk

zH

xHj

zH

yHi zxyxzyz

=

+

(4)

SinceHvaries only in the Z direction and wave travelling along X-axis, the component

ofHother thanx

Hz

becomes zero.

But from the fourth law of free space Maxwells equation,

t

EH

y

= 0 (5)


t

Ey

x

Hz

=

0 (6)

Substituting equation (3) in (6),

)xCt(C

Ex

Hz

=

2cos

200

(7)

Integrating with respect tox,

Hz=

2

2sin

200 )xCt(

CE

(8)

Hz= C0E0 sin

2(Ct x)

Hz=00

1

0E0 sin2 (Ct x)

Hz=0

0

E0 sin

2(Ct x)

Hz=0

0

.Ey (9)

(or) ZH

E

z

y

== 00

= Characteristic Impedance of the medium (10)

2.22


23/46

Y

X

Z

Ey

Hz

P


For free space,Z=0

0

= 376.8

For any medium,Z=

ohm

Worked Example 2.3: Electromagnetic radiation propagating in free space has the values of

electric and magnetic fields 86.6 V m 1 and 0.23 A m 1 respectively.

Calculate the characteristic impedance.

Electric field intensity =E =86.6 V m 1

Magnetic field intensity =H= 0.23 A m 1

Characteristic impedance =Z= ==23.0

6.86

H

E376.52 ohm

Z = 376. 52 ohm

2.1.8 Poynting vector(P)

The rate of energy flow per unit area in a plane electromagnetic wave is defined by a

vector )(P called the poynting vector.

HEBEP ==0

1

The direction of )(P gives the direction in which the energy is transferred. Unit: W/m2

Taking the divergence of poynting vector in free space,

H)E.(E)H.(H).(E =

=t

DE

t

BH

.. =

+

t

DE

t

BH ..

=

+

t

HH

t

EE .. 00 =

+

tH

Ht

E

E ).2(2

1

).2(2

1

00

=

+

t

H

t

E2

0

2

0

)(.

2

1)(.

2

1 =

+

202

02

1

2

1HE

t

2.23


24/46


Fig.2.11 Poynting vector

Considering the surface Sbounds a volume Vand integrating the above equation over the

volume V, we get

+

= VVdVHE

tHE

2

0

2

0 2

1

2

1).(

On applying divergence theorem to the LHS term of the above relation, we get

+

=VS

dVHEt

dSHE2

02

02

1

2

1).(

The term on the RHS within the integral of the above equation represents the sum of

energies of electric and magnetic fields. Hence the RHS of the above equation represents the

amount of energy transferred over the volume Vin one second i.e. it represents the rate of flow of

energy over the volume V.

Energy associated with the electric field2

20EUE

= and that with the magnetic field

0

220

22

BHUm == . As

( )Em UE

CB

HU ==== 20

0

22

0

2

1

22

[as C

B

E= and

00

1

=C ], which shows that instantaneous energy density associated with electric field i.e.

energy is equally shared by the two fields.

The vector HEP = is interpreted as representing the amount of field energy

passing through the unit area of surface in unit time normally to the direction of flow of energy.

This statement is termed as Poyntings theorem and the vectorP is called Poynting Vector. The

direction of flow of energy is perpendicular to vectors EandH i.e. in the direction of the vector

HE .

2.1.9 Skin Depth or Penetration Depth:

It can be proved that the amplitude of the electromagnetic wave propagating through a

conducting medium is damped, i.e. in a good conductor, the wave is attenuated as it progresses.

At higher frequencies, the rate of attenuation is very large, and the wave may penetrate only a very

short distance before being reduced to a small value. This effect is called skin effect. The reason

for the rapid attenuation of electromagnetic waves in a conducting medium is the conversion ofelectromagnetic energy into joules heat energy.

The skin depth or penetration depth () is defined as that depth in which the amplitude of the

electric field of the wave has been attenuated to

e

1or approximately 37% of its original value.

The penetration depth is given by,

1=

where is the attenuation constant.

From the Maxwells Equations, the attenuation constant can be derived as,

2.24


25/46


+= 11

2 22

2

where = angular frequency of the wave, = permeability of the medium, = permittivity of

the medium, = electrical conductivity of the medium.

For good conductors,

>>1

Hence,2

The penetration depth is given by,

21==

The above equation shows that at high frequencies, the current will flow only on the

surface of the conductor.

2.2 Waveguides

For transmitting electromagnetic energy from one place to another, transmission lines can

be used. At frequencies below microwaves, coaxial cable is the primary means of carrying radio

signals. But at microwave frequencies, this kind of transmission line is less effective. At

frequencies higher than 3 GHz, transmission of electromagnetic energy along the transmission

lines and cables becomes difficult mainly due to the losses that occur both in the solid dielectric

needed to support the conductor and in the conductors themselves. A metallic tube can be used to

transmit electromagnetic wave at these frequencies.

A hollow metallic tube of uniform cross section for transmitting electromagnetic waves by

successive reflections from the inner walls of the tube is called waveguide.

Waveguides may be used to carry energy between pieces of equipment or over longer

distances to carry transmitter power to an antenna or microwave signals from an antenna to a

receiver. Waveguides are made from copper, aluminium or brass. These metals are extruded into

long rectangular or circular pipes. Often the insides of these waveguides are plated with silver to

reduce their resistance to a very low level.

An electromagnetic energy to be carried by a waveguide is

injected into one end of the waveguide. Thus is done with antenna like

device which creates an electromagnetic wave that propagates through

the waveguide. The electric and magnetic fields associated with the

signal bounce off the inside walls back and forth as it progresses down

the waveguide. The waveguide completely contains the signal so that

none escapes by radiation.

In order to determine the EM field configuration within the

waveguide and to know how these waves are transmitted along the tube,

Maxwells equations should be solved subject to appropriate boundary

conditions at the walls of the guide. Such solutions give rise to a

2.25

O

X

Y

Z

Ex

, Hx

E z , H z

E

y

,

H

y

Fig.2.12 Components of

electric and magneticfield intensities in an EM

wave


26/46


number of field configurations. Each configuration is known as a mode. The following are the

different modes possible in a waveguide system:

Transverse Electro Magnetic (TEM) wave: Here both electric and magnetic fields are Z directed

components. (i.e.) E z = 0 and Hz = 0.

1. Transverse Electric (TE) wave: Here only the electric field is purely transverse to the

direction of propagation and the magnetic field is not purely transverse. (i.e.) E z = 0, Hz

0.

2. Transverse Magnetic (TM) wave: Here only magnetic field is transverse to the direction of

propagation and the electric field is not purely transverse. (i.e.) E z 0, Hz = 0.

3. Hybrid (HE) wave: Here neither electric nor magnetic fields are purely transverse to the

direction of propagation. (i.e.) E z 0, Hz 0.

2.2.1 Rectangular and Circular waveguides

Any shape of cross section of a waveguide can support electromagnetic waves. But since

irregular shapes are difficult to fabricate, analyze and are rarely used, rectangular and circular

waveguides have become more common. A waveguide having rectangular cross section is known

as rectangular waveguide and that having circular cross section is known as circular waveguide

Most waveguides are of the rectangular variety.

It is the size of the waveguide that determines its

operating frequency range. Consider a rectangular pipe

with width a and height b as shown in Fig.2.13. The

frequency of operation is determined by the a

dimension. This dimension is usually made equal to one

half the wavelength at the lowest frequency ofoperation. This frequency is known as the waveguide

cutoff frequency. At the cutoff frequency and below, the

waveguide will not transmit energy. At frequencies

above the cutoff frequency, the waveguide will

propagate energy. Normally, the height of the

waveguide is made equal to approximately one half

the a dimensions.

When a probe launches energy into the waveguide, the electromagnetic fields bounce off

the side walls of the waveguide as shown in Fig.2.14. The angles of incidence and reflectiondepend upon the operating frequency. At high frequencies, the angles are large and therefore, the

path between the opposite walls is relatively long as shown in Fig.2.14 (a). As the operating

frequency gets lower, the angles decrease and the path between the sides shortens. When the

operating frequency is reaches the cutoff frequency of the waveguide, the signal simply bounces

back and forth directly between the side walls of the waveguide and has no forward motion. At the

cut off frequency and below, no energy is propagated.

Rectangular waveguides usually come in a variety of standard sizes. The exact size is

selected based on the desired operating frequency. The size of the waveguide is chosen so that its

rectangular width is greater than one half the wavelength but less than the one wavelength at the

operating frequency. This gives a cutoff frequency that is below the operating frequency, thereby

ensuring that the signal will be propagated down the line.

2.26

ab

Fig. 2.13 Dimensions of the waveguide

determining the operating frequency

range


27/46

Angle of incidence(A) Angle of reflection (B)

(A = B)


By solving Maxwells equations for a rectangular waveguides, it can be proved the TEM

waves cannot exists inside a waveguide. The electric and magnetic field configurations in

waveguides are represented by subscripts. The general symbol will be TE m, n or TM m, n where the

subscript m indicates the number of half wave variations of the electric field intensity along the

b( wide) dimension of the waveguide. The second subscript n indicates the number of half wave

variations of the electric field in the a (narrow) dimension of the guide. The TE 1, 0 mode has the

longest operating wavelength and is designated as the dominant mode. It is the mode for the lowest

frequency that can be propagated in a waveguide.

(a) at high frequency

(b) at medium frequency

(c) at low frequency

(d) at cutoff frequency

Fig. 2.14 Wave paths in a waveguide at various frequencies

For a standard rectangular waveguide, the cutoff wavelength are given by,

22

2

+

=

b

n

a

mc

where a and b are measured in centimetres.

The circular waveguide is used in many special applications in microwave techniques. The

circular guide has the advantage of greater power handling capacity and lower attenuation for a

given cutoff wavelength, but it has the disadvantage of somewhat greater size and weight. Also,

the polarization of the transmitted wave can be altered due to the minor irregularities of the wall

surface of the circular guide, whereas the rectangular cross section definitely fixes the polarization.

The wave of lowest frequency or the dominant mode in the circular waveguide is the

TE11 mode. The subscripts which describe the modes in the circular waveguide are different than

for the rectangular waveguide. For the circular waveguide, the first subscript m indicates the

number of full wave variations of the radial component of the electric field around the

circumference of the waveguide. The second subscript n indicates the number of half wave

variations across a diameter. Also, the second subscript indicates the number of diameters that can

be drawn perpendicular to all electric field lines and in the case of TE 0 n waves, it indicates the half

wave variations of the electric field across a radius of the guide.

2.27


28/46


The cutoff wavelength for dominant mode of propagation TE11 in circular waveguide of

radius a is given by

1.814

2 ac =

The cutoff wavelength for dominant mode of propagation TM01 in circular waveguide ofradius a is given by

2.405

2 ac =

where a is measured in centimetres.

The applications of circular waveguides include

1. Rotating joints in radars to connect the horn antenna feeding a paraboloid reflector

(which must rotate for tracking)

2. TE01 mode suitable for long distance waveguide transmission above 10 GHz.

3. Short and medium distance broad band communication (could replace / share coaxial and

microwave links)

Worked Example 2.4: The dimensions of the waveguide are 2.5 cm 1 cm. The frequency is8.6 GHz. Find (i) possible modes and (ii) cut off frequency for TE

waves.

Given a = 2.5 cm , b = 1 cm andf = 8.6 GHz

Free space wavelength = cm488.3

108

1039

10

0 =

==

f

C

The condition for the wave to propagate is thatC> 0

For TE01 mode, cm212222

22222====

+= b

a

ab

anbm

abC

SinceC< 0, TE01 does not propagate.

For TE10 mode,C= 2a = 2 2.5 = 5 cm

SinceC> 0, TE10 mode is a possible mode.

Cut off frequency = GHz65

103 10

=

== CC

C

f

Cut-off wavelength for TE11 mode =

cm856.1)1()5.2(

15.222

2222=

+

=

+ba

ab

AsC for TE11 < 0 , TE11 is not possible.

From the above analysis, it is concluded that only the possible mode

is TE10 mode.

The cut off frequency = 6 GHz

2.28


29/46


2.3 Microwaves

Microwaves are electromagnetic waves whose frequencies range from about 300 MHz

300 GHz (1 MHz = 10 6 Hz and 1 GHz = 10 9 Hz) or wavelengths in air ranging from 100 cm 1

mm. The word microwave means very short wave. Microwave region is the shortest wavelength

region of the radio spectrum, which is a part of the electromagnetic spectrum. Its lower edge

actually overlaps with the infra red region. Microwaves are becoming more and more importantdue to the rapid development of various branches of science and engineering such as radar,

telecontrol and telemetry, telecommunications, television, industrial electronics, basic research and

medicine etc.

2.3.1 Properties of Microwaves

1. Microwave is an electromagnetic radiation of short wavelength.

2. They can be reflected by conducting surfaces just like optical waves since they travel

in straight line.

3. Microwave currents flow through a thin outer layer of an ordinary cable.

4. Microwaves are easily attenuated within short distances.5. They are not reflected by ionosphere but penetrate it and pass into outer space.

2.3.2 Advantages and Limitations

In communications, there are some unique advantages of microwaves over the low

frequency signals:

1. Increased bandwidth availability:

Microwaves have large bandwidths compared to the common bands like short waves

(SW), ultrahigh frequency (UHF) waves, etc. To explain this more clearly, let us consider the

microwaves extending from = 1 cm - = 10 cm (i.e) from 30,000 MHz 3000 MHz. Thisregion has a bandwidth of 27,000 MHz. If this region is used for communication and if 6 MHz is

allotted for each station ( as in TV), 13,500 different microwave broadcasting stations can be

accommodated in this range of bandwidth. Since sound transmission requires a bandwidth of only

4 Hz, each station can use a large number of channels for sound transmission.

It is the current trend to use microwaves more and more in various long distance

communication applications such as telephone networks, TV networks, space communication,

telemetry, defence, railways etc. Frequency modulation and present day digital modulation

schemes also require higher bandwidth.

2. Improved directive properties:

The second advantage of microwaves is their ability to use high gain directive antennas.

Any EM wave can be focused in a specified direction (Just as the focusing of light rays with lenses

or reflectors) by making the radiating antennas several wavelengths wide. Since the wavelength of

microwaves is in the order of centimeter, it is possible to make high gain directive antennas. The

wider the aperture of the antenna in terms of wavelength, the narrower the beam and higher the

gain of the antenna.

3. Fading effect and reliability:

Fading effect due to the variation in the transmission medium is more effective at low

frequency. Due to the Line Of Sight (LOS) propagation and high frequencies, there is less fading

effect and hence microwave communication is more reliable.

2.29


30/46


4. Power requirements:

Transmitter / receiver power requirements are pretty low at microwave frequencies

compared to that at short wave band.

5. Transparency property of microwaves:

Microwave frequency band ranging from 300 MHz 10 GHz are capable of freely

propagating through the atmosphere. The presence of such a transparent window in a microwave

band facilitates the study of microwave radiation from the sun and stars in radio astronomical

research of space. It also makes it possible for duplex communication and exchange of

information between ground stations and space vehicles.

In spite of these advantages, microwaves cannot replace radio frequency waves for

round - the - world communication because these high frequency waves penetrate through the

ionosphere and are lost into the outer space. It must be mentioned here that round - the - world

communication using radio frequency waves is made possible by the reflections of these waves

from the ionosphere surrounding the earth.

2.3.3 Generation of microwaves

The microwaves can be generated by using vacuum tubes and semiconductor diodes. High

power microwaves can be generated using tubes and solid state devices are used to produce low

power microwaves. The following are the some of the examples for devices which are used to

produce the microwaves:

1. Magnetron oscillator

2. Klystron oscillator

3. Travelling wave tube

4. Gunn diode

5. IMPATT diode etc.

Basically, a diode can be used to generate electromagnetic waves at microwave

frequencies. But the conventional triode valve can not be used to generate microwaves even

though they can be used to generate RF waves. The various problems involved in the construction

of a microwave oscillator using triode valve are listed below:

(i) Any pair of electrode in a triode acts as a capacitor and this inter electrode capacitance

becomes important at microwave frequencies.

(ii) The distributed circuit elements [(i.e) the circuit elements whose dimensions

become comparable to the wavelength], begin to radiate and so there is large power

loss at microwave frequencies, where it is not so at radio frequencies.

(iii) The transit time (i.e) the time taken by an electron to travel from cathode to anode

becomes more important at microwave frequencies. In the conventional triode valve

oscillators, the feedback circuits are designed on the assumption that the transition

time is negligible compared to the period of oscillations of the waves generated by the

oscillator. But, the period of oscillation of the microwave is in the order of transit time

and hence the usual type of feedback circuits does not work at microwave frequencies.

2.30


31/46


In order to overcome the problem of transit time, inter electrode capacitance, distributed

reactances etc., various structures and ingenious circuits have been invented. In this section, we

shall discuss the production of microwaves using vacuum tubes.

2.31


32/46

Copper Anode BlockCathode

RF out

CavitySlot

InteractionSpace

Coaxial lineoutput

Resonant (or)anode cavity


2.3.4 Magnetron oscillator

The magnetron was first invented by Hull in 1921 and an improved high power magnetron

was developed by Randall and Boot around 1939. Magnetrons provide microwave oscillations of

very high peak power.

There are three types of magnetrons.

1. Negative resistance type

2. Cyclotron frequency type

3. Cavity type

Negative resistance Magnetrons make use of negative resistance between two anode

segments but have low efficiency and are useful only at low frequencies (< 500 MHz).

Cyclotron frequency Magnetrons depend upon synchronization between an alternating

component of electric and periodic oscillation of electrons in a direction parallel to this field. They

are useful only for frequencies greater than 100 MHz.

Cavity Magnetrons depend upon the interaction of electrons with a rotatingelectromagnetic field of constant angular velocity. These provide oscillations of very high peak

power and hence are useful in radar applications. This being the most useful one, we shall study

this in detail.

Cavity Magnetrons

Cavity magnetron is simply a vacuum tube with two elements, a cathode and an anode.

The anode is a hollow cylindrical block made of copper. The cathode is a rod heated along the axis

of the anode. The space between the anode and cathode is called the interaction space. The

anode has several cavities that open into the interaction space. One of the cavities in the anode is

connected to a co axial line or waveguide for extracting the output. The major elements in the

magnetron oscillator are shown in Fig.2.15 (i) and its cross sectional view of the anode assembly isshown in Fig.2.15 (ii).

Fig.2.15 (i) Major elements in the Magnetron oscillator Fig. 2.15 (ii) Cross sectional view of the

Anode assembly

Each cavity in the anode acts as an inductor having only one turn. The slot connecting the

cavity and the interaction space acts as a capacitor. These two elements together form a parallel

resonant circuit, and its resonant frequency depend on the value ofL of the cavity and the Cof the

slot, (i.e) on the dimensions of the slot and the cavity. The frequency of the microwaves generated

2.32

Magnet pole

piece

Magnet polepiece

Anode

assembly

Cathode

Waveguide

Co-axial

output system


33/46

a

b

c

d

CathodeAnode

InteractionSpace


by the magnetron oscillator depends on the frequency of the RF oscillations existing in the

resonant cavities.

Magnetron is a cross field device as the electric field between the anode and the cathode is

radial whereas the magnetic field produced by a permanent magnet is axial. A high DC potential

can be applied between the cathode and anode which produces the radial electric field. Thepermanent magnet is placed such that the magnetic lines pass parallel to the axis of the cylindrical

anode. Depending on the relative strengths of the electric and magnetic fields, the electrons

emitted from the cathode and moving towards the anode will traverse through the interaction space

as shown in Fig.2.15 (iii).

In the absence of magnetic field (B = 0), the electron travel straight from the cathode to the

anode due to the radial electric field force acting on it [indicated by the trajectory a in Fig. (iii)].

If the magnetic field strength is increased slightly (i.e) for moderate value ofB, it will exert a

lateral force bending the path of the electron as shown by the path b in Fig. (iii). The radius of

the path is given by,eBmvR = , that varies directly with electron velocity and inversely as the

magnetic field strength. If the strength of the magnetic field is made sufficiently high so as to

prevent the electrons from reaching the anode (as shown by the path c and those inside in

Fig. 2.15(iii)), the anode current becomes zero.

Fig. 2.15 (iii) Electron trajectories in the presence of crossed electric and magnetic fields (a) no

magnetic field (b) small magnetic field (c) Magnetic field = Bc (d) Excessive magnetic field

The magnetic field required to return electrons back to the cathode just grazing the surface

of the anode is called the critical magnetic field (Bc), the cut off magnetic field. If the magnetic

field is larger than the critical field (B > Bc), the electron experiences a greater rotational force and

may return back to the cathode quite faster. All such electrons may cause back heating of the

cathode. This can be avoided by switching off the heater supply after commencement of

oscillation. This is done to avoid fall in the emitting efficiency of the cathode.

All the above explanation is for a static case in the absence of the RF field in the cavity of

magnetron. In order to understand the working of magnetron oscillator, it must be assumed that RF

oscillations are induced in the cavities. Oscillations of transient nature produced when the HT is

switched on, are sufficient to produce the oscillations in the cavities. Now, it has to be shown that

these oscillations are maintained in the cavities reentrant feedback which results in the production

of microwaves. Reentrant feedback takes place as a result of interaction of the electrons(circulating through the interaction space) with the electric field of the RF oscillations existing in

2.33


34/46

Electricfield

a

b

Magnetic field

(perpendicular to paper)


the cavities. The cavity oscillations produce electric fields which fringe out into the interaction

space from the slots in the anode structure, as shown in Fig.2.15 (iv). Energy is transferred from

the radial dc field to the RF field by the interaction of the electrons with the fringing RF field.

Due to the oscillations in the cavities, the either sides of the slots (which acts as a

capacitor) becomes alternatively positive and negative. Hence the directions of the electric fieldacross the slot also reverse its sign alternatively. Also adjacent sides of the slot are always at

opposite polarity. At any instant, if that part of the anode close to the spiralling electron goes

positive (due to the RF oscillations in the cavity), the electrons gets retarded. This is because; the

electron has to move in the RF field, existing close to the slot, from positive side to the negative

side of the slot. In this process, the electron loses energy and transfer an equal amount of energy to

the RF field which retard the spiralling electron. Now, the electron moves to a smaller orbit

momentarily but the anode pulls it back to the previous orbit. On its return to the previous orbit the

electron may reach the adjacent section or a section farther away and transfer energy to the RF

field if that part of the anode goes positive at that instant. This electron travels in a longest path

from cathode to the anode as indicated by ain Fig. 2.15 (iv). Such electrons which participate intransferring the energy to the RF field are called as favoured electrons and are responsible for

bunching effect. These electrons give up most of its energy before it finally terminates on the

anode surface. An electron b is accelerated by the RF field and instead of imparting energy to the

oscillations, takes energy from oscillations resulting in increased velocity. Hence bends more

sharply, spends a very little time in the interaction space and is returned back to the cathode. Such

electrons are called unfavoured electrons which do not participate in the bunching process rather

they are harmful in the sense they cause back heating.

Fig.2.15 (iv) Possible trajectory of electrons from cathode to anode in an eight cavity magnetron

operating in mode

Every time an electron approaches the anode in phase with the RF signal, it completes a

cycle. This corresponds to a phase shift 2. For a dominant mode, the adjacent poles have a phase

difference of radians. This called the - mode.

The anode must have any convenient even number of cavities. At any particular instant,

one set of alternate poles goes positive and the remaining set of alternate poles goes negative dueto the RF oscillations in the cavities. As the electron approaches the anode, one set of alternate

2.34


35/46


poles accelerates the electrons and turns back the electrons quickly to the cathode. The other set

alternate poles retard the electrons, thereby transferring the energy from electrons to the RF signal.

This process results in the bunching of electrons in certain regions. The mechanism by which

electron bunches are formed and by which electrons are kept in synchronism with the RF field is

called phase focusing effect. The number of bunches depends on the number of cavities in the

magnetron and the mode of oscillations. In an eight cavity magnetron oscillating with

- mode, the electrons are bunched in four groups as shown in Fig. 2.15 (v).

Two identical resonant cavities will resonate at two frequencies when they are coupled

together. The two resonant frequencies lie above and below the resonant frequency of the

individual resonators. This is due to the effect of mutual coupling. A common means of separating

the pi mode from adjacent modes is by a method called strapping. The straps consists of either

circular or rectangular cross section connected to alternate segments of the anode block.

Fig.2.15. (v) Bunching of electrons in multicavity magnetron

Performance Characteristics

1. Power output: In excess of 250 kW ( Pulsed Mode), 10 mW (UHF band), 2 mW

(X band), 8 kW (at 95 GHz)

2. Frequency: 500 MHz 12 GHz

3. Duty cycle: 0.1 %

4. Efficiency: 40 % - 70 %

Applications of Magnetron

1. Pulsed radar is the single most important application with large pulse powers.

2. Voltage tunable magnetrons are used in sweep oscillators in telemetry and in missile

applications.

3. Fixed frequency, CW magnetrons are used for industrial heating and microwave ovens.

2.3.5 Klystron Oscillator

A klystron is a vacuum tube that can be used either as a generator or as an amplifier of

power at microwave frequencies. This was invented by Russel H. Varian at Stanford University in

1939 in association with his brother S.P. Varian.

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Reflex Klystrons

The reflex klystron has been the most used source of microwave power in laboratory

applications. A reflex klystron consists of an electron gun, a cavity with a pair of grids and repeller

plate as shown in Fig.2.16. In this klystron, a single pair of grids does the functions of both the

buncher and the catcher grids. The cathode emits electrons which are accelerated forward by anaccelerating grid with a positive voltage on it and focused into a narrow beam. The electrons pass

through the cavity and undergo velocity modulation, which produces electron bunching. Then this

beam is repelled back by a repeller plate kept at a negative potential with respect to the cathode.

On its return, the electron beam once again enters the same grids which act as a buncher. Thus the

same pair of grids acts simultaneously as a buncher for the forward moving electron and as a

catcher for the returning beam.

The feedback necessary for electrical oscillations is developed by reflecting the electron

beam so that it passes through the resonator a second time. The velocity modulated electron beam

does not actually reach the repeller plate, but is repelled back by the negative voltage. The point at

which the electron beam is turned back can be varied by adjusting the repeller voltage. Thus the

repeller voltage is so adjusted that complete bunching of the electrons takes place at the catcher

grids. The distance between the repeller and the cavity is chosen such that the repeller electron

bunches will reach the cavity at proper time to be in synchronization. Because of this, they deliver

energy to the cavity. The result is the oscillation at the cavity frequency. The cavity itself is made

positive so that the electrons are ultimately attracted by the cavity and cause direct current flow in

the external circuit. A coupling loop in the cavity removes the RF energy.

Performance Characteristics

1. Frequency: 4 200 GHz2. Power: 1 mW 2.5 W

3. Theoretical efficiency : 22.78 %

4. Practical efficiency : 10 % - 20 %

5. Tuning range : 5 GHz at 2 W 30 GHz at 10 Mw

Applications

The reflex klystrons are used in

1. radar receivers

2. local oscillator in microwave receivers

3. signal source in microwave generator of variable frequency

4. portable microwave links

5. pump oscillator in parametric amplifier

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Filament

Coaxinput

Coaxoutput

HTDC

+

-

Collectorplate

HelixGlass tube Accelerating

anode

Cathode


Output

Cavity

Loop

Repeller plate

Focussingelements

Cathode emitselectrons

Filamentheats cathode

+-

+

-

+DC

+

-

Direction ofelectron beam

Deceleratinggrid

Fig. 2.16. Reflex Klystrons used as an oscillator

2.3.6 Travelling Wave Tube

One of the most versatile microwave RF power amplifiers is the Travelling Wave Tube

(TWT). The main virtue of the TWT is its extremely wide band width of operation. TWT was

designed by Pierce and others in 1946.

Fig.2.17 shows the basic structure of a TWT. It consists of a cathode and filament heater

plus an anode that is biased positively to accelerate the electron beam forward and to focus it into a

narrow beam. The electrons are attracted by a positive plate called the collector to which is applied

a high dc voltage. The length of the tube can be anywhere from approximately one ft. to several

feet. In any case, the length of the tube is usually many wavelengths at the operating frequency.

Surrounding the tube are either permanent magnets or electromagnets that keep the electrons

tightly focused into a narrow beam.

Fig.2.17. Basic structure of a travelling wave tube (TWT)

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The unique feature of the TWT is a helix or coil that surrounds the length of the tube. The

electron beam passes through the centre or axis of the helix. The microwave signal to be amplified

is applied to the end of the helix near the cathode and the output is taken from the end of the helix

near the collector. The purpose of the helix is to provide a path for RF signal that will slow down

its propagation. The propagation of the RF signal along the helix is made approximately equal to

the velocity of the electron beam from the cathode to the collector. The structure of the helix is

such that the wave travelling along it is slightly slower than that of the electron beam.

The passage of the microwave signal down the helix produces electric and magnetic fields

that will interact with the electron beam. The effect on the electron beam is similar to that in a

klystron. The electromagnetic field produced by the helix causes the electrons to be speeded up

and slowed down. This produces velocity modulation of the beam which produces density

modulation. Density modulation causes bunches of electrons to group together one wavelength

apart. These bunch of electrons travel down the length of the tube toward the collector. Since the

density modulated electron beam is essentially in step with the electromagnetic wave travelling

down the helix, the electron bunches induce voltages into the helix which reinforce the voltage

already present there. The result is that the strength of the electromagnetic field on the helix

increases as the wave travels down the tube towards the collector. At the end of the helix, the

signal is considerably amplifier. Coaxial cable or waveguide structures are used to extract the

energy from the helix.

The primary benefit of the TWT is its extremely wide bandwidth. Tubes can be made to

amplify signals from UHF to hundreds of gigahertz. Most TWTs have a frequency range of

approximately 2:1 in the desired segment of the microwave region to be amplified. The TWTs can

be used in both continuous and pulsed modes of operation with power levels up to several

thousands watts.

Performance characteristics

1. Frequency of operation : 0.5 GHz 95 GHz

2. Power outputs: 5 mW (10 40 GHz low power TWT)

250 kW (CW) at 3 GHz (high power TWT)

10 MW (pulsed) at 3 GHz

3. Efficiency : 5 20 % ( 30 % with depressed collector)

(A depressed collector is set of collector rings maintained at successively at higher

potentials to effectively slow down the electron beam that still possesses a considerable

kinetic energy)

Applications of TWT

1. Low noise RF amplifier in broad band microwave receivers.

2. Repeater amplifier in wide band communication links and long distance telephony.

3. Due to long tube life (50,000 hours against th for other types), TWT is power output

tube in communication satellite.

4. Continuous wave high power TWTs are used in troposcatter links (due to larger power

and larger bandwidths).

5. Used in Air borne and ship borne pulsed high power radars.

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2.3.7 Biological effects of microwaves

As a result of technical and industrial development, an increasing number of man made

electromagnetic fields have appeared in the human environment. Electromagnetic radiation in the

1 mm to 1 m wavelength range (300 MHz to 300 GHz) is referred to as microwave radiation, and

is part of what is known as radiofrequency (RF) radiation. The latter covers the 0.5 MHz to 300

GHz range and is considered in the context of adverse biological effects.

Fig.2.18 Ionizing and non ionizing radiations electromagnetic energy

The electromagnetic radiations can be classified as ionizing and non ionizing

radiation. The ionizing and non-ionizing electromagnetic radiations are presented in Fig.2.18.

Ionization is a process by which electrons are stripped from atoms and molecules. This process can

produce molecular changes that can lead to damage in biological tissue, including effects on DNA,

the genetic material. This process requires interaction with high levels of electromagnetic energy.Those types of electromagnetic radiation with enough energy to ionize biological material include

X-radiation and gamma radiation. Therefore, X-rays and gamma rays are examples of ionizing

radiation.

The energy levels associated with RF and microwave radiation, on the other hand, are not

great enough to cause the ionization of atoms and molecules and RF energy is, therefore, is a type

of non-ionizing radiation. Other types of non-ionizing radiation include visible light, infrared

radiation and other forms of electromagnetic radiation with relatively low frequencies. Often the

term "radiation" is used to apply to ionizing radiation such as that associated with nuclear power

plants. Ionizing radiation should not be confused with the lower-energy, non-ionizing, radiation

with respect to possible biological effects, since the mechanisms of action are quite different.

Microwave energy is non-ionizing electromagnetic radiation. Ionizing radiation messes up

molecules, non-ionizing radiation merely heats them. This means that, in general, it does not have

sufficient energy to kick an electron off an atom thus producing charged particle in a body and

cause biological damage. The only proven harmful effect from exposure to microwave (or RF)

radiation is thermal. RF radiation can enter deep into the body and heat human organs. Thus, the

depth of penetration and the level absorption of radiation in the body are relevant. The blood

vessels are dilating and the blood flow increases substantially as the thermoregulatory mechanism

is activated in order to keep the body temperature constant. With rising body temperature the

metabolic rate rises also, what may lead to Stress-Adaptation-Fatigue Syndrome. The following

are the effects produced by the electromagnetic waves at the different frequency level:

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Above 10 GHz (3 cm wavelength or less) heating occurs mainly in the outer skin surface.

From 3 GHz to 10 GHz (10 cm to 3 cm) the penetration is deeper and heating higher.

From 150 MHz to about 1 GHz (200 cm to 25 cm wavelength), penetration is even deeper

and because of high absorption, deep body heating can occur.

Any part of the body that cannot dissipate heat efficiently or is heat sensitive may be

damaged by microwave radiation of sufficient power. The lens of the eye and testes are respective

examples.

The microwave energy exposure is measured in terms of SAR (Specific Absorption Rate)

or PD (Power Density). SAR is the energy which is absorbed in a unit of mass or volume of the

body per unit time. The standards that limit microwave exposure were set at 0.4 W/kg SAR for

occupational and 0.08W/Kg for public exposure. The averaging time for determination of SAR

was 6 minutes. Power density is the energy absorbed per unit area in unit time. A safety factor of

10 applied to the solar radiation level has been widely adopted for RF radiation, the standard is 10

mW/cm2 maximum. The high power microwaves definitely cause some adverse effects in the

human system. The Table 2.2 shows the effects of exposure to certain power levels, without time

limit.

Table 2. 2 Effects of exposure of microwave energy

Power level

(mW / cm2)

Long-term effect on

human body

Remarks

0.01 Nothing -

0.1 Nothing -

1 Nothing -

5 Nothing Accepted standard for microwave oven leakage

10 Nothing Accepted standard for maximum continuous

exposure to radiated emissions (cell phones, etc.)

30 You can feel heat -

100 Cataracts can be

produced

Summer sunlight is at this level.

1000 Pain is induced -

5000 Cooking commences -

2.3.8 Applications of Microwaves

Microwaves have a wide range of applications in modern technology. Broadly the

applications can be in the areas listed below:

1. Telecommunication: Intercontinental Telephone and TV, space communication (Earth - to

- space and space - to - Earth), telemetry communication link for railways etc.

2. Radars: Radar is an acronym forRAdio Detection And Ranging. It can be used to detect

and locate a target. It is also used to determine the velocity of moving objects and find

many characteristics of the targets such as size, shape etc.

Radar works on the principle of radio echoes. It has a transmitter which radiates high power

radio pulses into space. When these electrical pulses are incident on any distant moving orstationary target, they get scattered in all directions. A part of the scattered energy travels back to

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the receiver (for transmission as well as reception of waves, a single device called duplexer is

used). By measuring the time taken by the pulse to reach the target and back to the receiver, the

range or the distance of the target can be determined. In order to find the direction of the target,

directional antennas are used.

The essential components of a simple radar system are shown in Fig. 2.19.

R e c e i v e r( s u p e r h e t )

T r a n s m i t t e r( R F o s c i l l a t o r )

S i g n a lp r o c e s s o r

W a v e f o r mg e n e r a t o r

T od i s p l a y

D a t a

p r o c e s s o r

T r a n s m i t t e d p u l s eR a d i a t i n ga n t e n n a

R e f l e c t e d o re c h o p u l s e

T R s w i t c ho rD u p l e x e r

R

T

Fig. 2.19. Functional Block diagram of a RADAR

The radar signal waveform as generated by the waveform generator modulates a high

frequency carrier and the modulated signal is raised by to the desired power level in the transmitter

portion. The transmitter could be a power amplifier which employs microwave tube amplifiers

such as Klystron, Travelling Wave Tube etc. The transmitter is turned on and off with a periodic

high power pulse i.e. the transmitter is turned on whenever a pulse arrives and turned off whenever

the pulse is absent. Thus the transmitter generates periodic pulses of very short duration. These

pulses are fed to the antenna which radiates them into the space. The antenna is highly directional.

In most of the cases, a single antenna is used for both transmission and reception. This is

achieved with the use of a TR (Transmitter Receiver) switch. This is also called as duplexer.

This connects the antenna to the transmitter during transmission and to the receiver duringreception. This also isolates the sensitive receiver from the damaging effects of high power

transmitter.

The echo signal received by the antenna is directed to the receiver input. The receiver is

usually of the superheterodyne type. The purpose of signal processing is to reject the undesired

signal such clutter and enhance the desired signals due to the targets. Data Processing refers to the

processing done after the detection decision has been made. Functions like automatic tracking,

target recognition are examples of data processing in a radar system. The displays put the

processing information in a form usable by radar operators and others wanting to use the

information such as Air Traffic Controllers, Weapon System Operators etc.

The distance of a target (target range) can be calculated using the following relation:

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2

CtR =

whereR = Target range, t= round trip propagation time and C= velocity of propagation.

Radar Range Equation:

The radar range equation relates the various parameters of a radar such as transmitted

power, transmit antenna gain, radar cross section of the target, receiver antenna aperture, minimum

detectable power at the receiver input and various loss factors.

The maximum range of radar may be defined as the distance of the target which can be

just detected by the radar.

The radar range equation is given by,

( ) ( )

4

1

min3

22

max 4

= r

rt

P

S

Electromagnetic waves, Circuits and Applications

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