28538193-1-Vector-Identities-Triple-Products-♠

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_________________________________________________________________________ Microwave Engineering Dr.PVS&RSR

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1

Vector Identities:

Triple Products:

( ) ( ) ( ) A B C B C A C A B = = ( ) ( ) ( ) A B C B C A C A B = =

Product rules: ( ) ( ) ( ) fg f g g f = + ( ) ( ) ( ) ( ) ( ) A B A B B A A B B A = + + + ( ) ( ) ( ) fA f A A f = + ( ) A B = ( ) B A ( ) A B

( ) ( ) ( ) fA f A A f =

( ) ( ) ( ) ( ) ( ) A B B A A B A B B A = + Second derivatives:

( ) 0 A = ( ) ( ) 2 A A A = ( ) 2 f f = ( ) 0 f =

Fundamental theorems :

As is known, the volume is always enclosed by a closed surface and the surface isalways is enclosed by closed path. The path is a vector, is a directed curve, directionnormally being indicated with an arrow over the curve. The surface is also bydefinition a vector and, by definition, is always surrounded by a closed path. Thedirection of the surface at a point over it can be found by drawing a closed path,direction being same as that of the path surrounding the total surface, surroundingthat point. The direction pointed by the right hand thumb when wrapped by thefingers around the point gives the direction of the surface at that point. Theintegration of a vector over the closed surface can be related to the integration of the

same vector throughout the volume enclosed by the surface through the divergencetheorem. Similarly the integration of a vector along a closed path can be related tothe integration of the same vector over the surface enclosed by the path through Curltheorem.


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Gradient theorem

In word form it can be stated as 'the integral of the tangential component of

gradient of a scalar function along a path from ' a ' to' b ' is the difference of thefunction values at ' a ' and ' b ' i.e. ( ) f a and ( ) f b . '

( ) ( ) ( )b

a

f dl f b f a = Divergence theorem or Gauss theoremThis theorem connects surface integral with volume integral. In word form it can

be stated as 'the integral of the normal component of a vector over a closedsurface is equal to the integral of the divergence of the same vector through anyvolume enclosed by that surface.' Analytically

( ) A d A da =

Curl theorem or Stokes theorem

This theorem connects line integral with surface integral. In word form it can bestated as 'the integral of the tangential component of a vector around a closed

path is equal to the integral of the normal component of the curl of the samevector through any surface enclosed by the path.' Mathematically

( ) A da A dl = Significance:

These are useful in converting the Maxwell's equations from point formto integral form and vice versa.

They relate a surface integral to its corresponding volume integral andalso a line integral to its corresponding surface integral.

Operator Del i j k y z

+ +

It is a three dimensional, partial differential vector operator defined inCartesian system only. But it can be mapped into other co-ordinate

systems. Its units are ( ) 1mt . This operator can be applied over a scalar function to find its gradient,

over a vector function to find either its divergence or curl When applied to a position vector ' r ra ' joining origin with ( ), , y z

or ( )1 1 1, , y z with ( ), , x y z , then


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r r a =

2

1 r ar r

=

( ) ( )2 , 2n nr r a n r n = + ( )34 , 2r n = =

( ) 0n r r a =

Operator Laplacian2 2 2

22 2 2 y z

+ +

It is a three dimensional, second order, partial differential scalar operator defined in Cartesian system only. But it can be mapped into

other co-ordinate systems. Its units are ( ) 2mt . This operator can be applied over a scalar function as well as over a

vector function

Operator d'Alembertian2

2 20 0 2t

and Helmholtz operator ( )2k + :Both these operators are three dimensional partial differential operators. The first isnormally applied upon scalar functions.

Divergence. Curl And Gradient In Different Co-Ordinate Systems:

Let 1 2 3 dl h du u h dv v h dw w= + +


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1 2 3

1 1 1 V V V V u v wh u h v h w

= + +

( ) ( ) ( )2 3 3 1 1 21 2 3

1u v w D h h D h h D h h Dh h h u v w

= + +

1 2 3

1 2 3

1 2 3

1

u v w

h u h v h w

Dh h h u v w

h D h D h D

=

2 2 3 3 1 1 21 2 3 1 2 3

1 h h h h h hV V V V

h h h u h u v h v w h w

= + +

ELECTROMAGNETIC WAVE THEORY

Electromagnetic field theory or electromagnetic wave theory is the study of electricaland magnetic properties of the regions i.e. parts of the space.

The region surrounding the stationary charge distribution is called electric field or to be precise electrostatic field. The study of the electrostatic field is electrostatics.

The region surrounding the conductor carrying direct current (dc) distribution iscalled magnetic field or to be precise steady magnetic field. The study of the steady

magnetic field is magnetostatics.

The electrostatic fields and steady magnetic fields together are called static fields or dc fields. In static fields the field intensity can be function of position and independent of time.

The region surrounding the conductor carrying time varying or alternating current(ac) distribution is called time varying electromagnetic field. In these fields there exists bothelectric field intensity and magnetic field intensity which are related to each other. Thisrelation i.e. the relation between electric field and magnetic field in time-varying fields isgiven by Faradays law and Maxwells relation.

All the three types of fields are related because of the relation that exists in betweentheir respective sources. Stationary charge gives electrostatic field and charge moving withconstant velocity is the source of steady magnetic field whereas the charge moving withacceleration/deceleration gives rise to time-varying electromagnetic fields.


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All the fields whether dc or ac are reservoirs of the energy. And it is also possibleto add or subtract energy to the fields. One difference between electric and magnetic fieldsis: magnetic fields can do no work whereas electric fields can do.

Time varying fields exhibit one important property which is not possessed by staticfields. There exists travelling wave and consequently energy flow called radiation in timevarying electromagnetic fields. As the antennas which are critical components of wire-lesscommunication systems, functions based on radiation. So the study of the properties of thetime varying fields has become an important requirement for communication engineers.

The intensity of the electric field is a vector quantity indicated by E with unitsvolts/meter. The magnetic field intensity is also a vector quantity indicated by H with units.Electric field intensity is function of the medium properties where as the magnetic fieldintensity is independent of the medium of the field.

For static fields intensities are always inversely proportional to the square of thedistance from the source. And functions of position only where as time varying fields arefunctions of position and time also

A quantity D E = called electric displacement density can also be defined for electric fields with units coul/m 2. A quantity similar to this can be defined for magneticfields also. It is called magnetic flux density denoted by B with units of tesla or webers/m 2.It is related to the magnetic field intensity through B H = .

The electric displacement density is independent of the medium whereas themagnetic flux density is dependent upon the medium properties. For static fields these arefunctions of position only where as for time varying fields they are functions of time also.

Potential functions are also defined both for static and dynamic fields. For electric

field it is called electric scalar potential indicated by V with units of volts. With magneticfields the magnetic vector potential is defined indicated by A with units ----. For the staticsources the potentials are inversely proportional to the distance from the source. Potentialscan be related to their static sources as well their fields.

For dynamic fields the potential functions are called retarded potentials because of the retardation or delay is incorporated into the expressions. They can be related to thesource or the fields. In this case the two potentials can be related to each other also throughCoulombs gauze or Lorentz gauge

Electro statistics

Coulombs Law:- The force on a point charge Q due another point charge q is proportional

to the product of the charges,to the inverse of the distance between them andit is along the line joining these two charges,attractive for dissimilar and repulsive for similar charges,


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depends upon the medium in which charges are located

Mathematically,

214 r Q q F ar

=

( ) ( ) ( )

( ) ( ) ( )

1 2 1 2 1 23

2 2 2 21 2 1 2 1 2

4

x i y y j z z k Q q

x x y y z z

+ + = + +

( )1 23

21 2

4

r r Q q

r r

=

where 0 r =

By superposition, the force on Q due to n charges 1 2 3, , ,.... nq q q q is

21

14 k

nk

r k

Q q F a

r =

The force on the unit charge i.e. 1Q = is the electric field intensity

21

14 k

nk

r k

q F E a

Q r = =

For the continuous charge distributions

2

1

4r

dl E a

r

=

for the line charge distribution

2

14 r

ds E a

r

= for surface charge distribution

2

14 r

d E a

r

= for volume charge distribution

Field Intensities

For a point charge E 214 r Q E ar

=

For infinite line charge1 2

4 r E a

r

=


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For infinite sheet charge02

n E a

=

Electric Flux Density: D

It is a quantity proportional to the no. of flux lines crossing unit area equal to E i.e. D E = .

Gauss Law:

The net flux through any closed surface is equal to the net charge enclosed by thatsurface.

. enc D da Q= Integral form D = Differential form

Gauss law is useful to compute the field intensity when the charge distribution is highlysymmetrical i.e. plane symmetry, cylindrical symmetry or spherical symmetry.Depending upon the symmetry exhibited by the charge distributions, the Gaussian

surfaces (surfaces over which integration is performed) can be a pill box, coaxialcylinder or a concentric sphere.The application of the Gauss law to find the field intensity of charge distribution requiresthe prior knowledge of the field.The integration of the Gauss law becomes simpler only if the field is either normal or tangential to the Gaussian surface and when ever it is normal its value must remainconstant.

Scalar Potential: V

The scalar potential V of a point P in an electric field E is defined as the externalwork done to bring a unit positive charge from infinity to the point P

. P

V E dl

=

Potential of a point a with respect b is .a

abb

V E dl =

For a point charge1

4Q

V r

=

For the continuous charge distributions


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For line charge distribution1

4

dl V

r

=

For surface charge distribution1

4

daV r

=

For volume charge distribution1

4d

V r

= Field intensity E in terms of the potential V

.dV E dl = but .dV V dl = giving E V =

Considering divergence both sides and using Gauss Law 2 vV

= which is Poissons

Law.If the volume charge density 0

v = then 2 0V = which is Laplace Equation.

Divergence And Curl Of The Electrostatic Field:

( ) ( )21

' '4

r a E r r d r

= % where 'r r r = %

( )21

. . ' '4

r a E r d r

= % : But ( )3

2. 4r a r

r = %

%

Thus ( ) ( ) ( )31 1. 4 ' ' '4

E r r r d r

= = . v D =

CONDUCTORSBasics properties: 0 E = inside a conductor, E can be only perpendicular to the surface just

outside the conductor,0

E n

= , 0 = inside a conductor, the charge can reside only over

the surface, V is constant through out the conductor,

Polarization of materialsWhen a piece of dielectric material is placed in an electric field

The field will induce in each atom a tiny dipole moment pointing in the same directionas the field, if the material is made up of non-polar molecules.Each permanent dipole will experience a torque, tending to line it up along the fielddirection, of the material is made up of polar molecules.


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In either case, the result is a lot of little dipoles pointing along the direction of the field andthe dielectric is said to be Polarized . A convenient measure of this effect is polarization P=

Dipole moment per unit volume.Field of a polarized object:A single dipole P gives potential

20

14

r a P V r

= .

A differential volume d with dipole moment p P d = gives rise to a potential

20

14

r a P dV d r

= .

The total potential due to the entire object is 20

14

r

vol

P aV d

r

=

0

1 14 vol

P d r

=

0 0

1 14 4 sur vol

P da P d

r r

=

0 0

1 14 4

b b

sur vol

dad

r r

= where .b P n = and .b P =

It means the potential and hence also the field of a polarized object is the same asthat produced by a volume charge density b P = plus a surface charge density

b P n = Gauss law in the presence of dielectrics :With in the dielectric, the total charge density can be written as b f = + where b = aresult of polarization, bound charge density and f = which is not a result of polarization,free charge.

Now the Gauss law reads 0 b f f E P = = + = + where the E is total

field ( )0 0 f E P D E P + = + In terms of the D , the electric displacement, Gauss law reads f D = enc D da Q = Linear dielectrics:in linear dielectrics, the polarization is proportional to the field 0 e P E = . The

proportionality constant e is called electric susceptibility. So in linear media, we have

( )0 0 0 0 1e e D E P E E E = + = + = + . Now ( )0where 1 e D E = = + is called the


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permittivity of the material.0

1 e r k

= + = = . where k is called the dielectric constant of

the material. For lossy dielectric ( )0 ' '' ' 1 tanr r j j j

= = = & where

tan =loss tangent =0

''' r

=

Steady magnetic fields

Biot-savart law: the magnetic field intensity dH at an arbitrary point P due to a steady linecurrent element Idl is proportional to the current element Idl , inversely proportional to 2r ,r being the distance between the current element and the field point P and it is directed

perpendicular to both the current element and the distance

vector 2 21

4 4r r Idl a dl a I dH H

r r

= =

For surface currents 24r da a K H

r

= and

for volume currents 21

4r J a H d

r

=

Lorentz force law : the force on a moving charge Q with velocity v in a magnetic field B plus electric field E is ( )ele mag F F F QE Q v B= + = +

( )mag F I dl B= for line currents( )mag F K da B= for surface currents

( )mag F J B d = for volume currents.Magnetic forces can do no work .

Amperes law for steady currents : The magnetomotive force around a closed path is equal

to the net steady current through any surface enclosed by the path. integral formenc H dl I = and differentialformenc H I = Amperes law is useful in finding the field intensity when the current distribution exhibitssymmetry like infinite straight line, infinite plane, infinite solenoid and toroid.


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Magnetization of materials

When a magnetic field is applied to a material, a net alignment of the magnetic dipoles

inside the material occurs resulting inMagnetization of the material parallel to B ( ) paramagnets or Magnetization opposite to ( )diamagnets B or Retention of substantial magnetization indefinitely after the external field has

been removed ( )ferromagnetsA quantity used to describe the state of magnetic polarization of a material is Magnetization

= magnetic dipole moment per unit volume.Field of a magnetized object:

The vector potential of a single dipole m is0

24r m a

A r

= .

In the case of a magnetized object, the vector potential is 0 24r

vol

M a A d

r

=

0 0

4 4 sur vol

M da M d

r r

= +

0 0

4 4b b

sur vol

K da J d

r r

= + where b K M n= and b J M =

This relation says that the potential and hence also the field of a magnetized object is the

same as would be produced by a volume current density b J M through out thematerial plus a surface current b K M n= on the boundary.

Amperes law in magnetized materialsThe total current of the material can be expressed as b f J J J = + where b J is bound current,

a result of magnetization and J is free current, not a result of magnetization.

Amperes law is ( ) ( )0

1 f b f B J J J J M

= = + = +

0

B J

=

: let

0

B H

=

f H J = fencl H dl I = Linear media:For most substances the magnetization is proportional to the field 0 m H = , the

proportionality constant m is called the magnetic susceptibility.


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( ) ( )1o o m B H M H = + = + so B H = where ( )1o m = + which is called the permeability of the material.

Hence

Magnetism

The origin of magnetism lies in the orbital and spin motions of electrons and how theelectrons interact with one another. The magnetic behaviour of the materials can beclassified into the following five major groups; Diamagnetism Paramagnetism Ferromagnetism Ferrimagnetism

Antiferromagnetism

The first two groups of materials exhibit no collective magnetic interactions and are notmagnetically ordered where as the materials of the last three groups exhibit long rangemagnetic order below a certain critical temperature.

Ferromagnetic and Ferrimagnetic materials are strongly magnetic where as the other three are weakly magnetic.

Diamagnetism The substances are composed of atoms which have no net magneticmoments because all the orbital shells are filled with no unpaired electrons. It is usuallyweak with temperature independent negative susceptibility. Examples for the materials

that exhibit diamagnetism are quartz, calcite and water. Paramagnetism in this class of materials, some of the atoms or ions in the materials have

a net magnetic moment due to unpaired electrons in partially filled orbitals. One of themost important atoms with unpaired electrons is iron. However the individual magneticmoments do not interact magnetically and the magnetization is zero when the field isremoved. They have temperature dependant positive susceptibility.

Ferromagnetism the atomic moments in these materials exhibit very strong interactions produced by very large electronic exchange forces and result in parallel alignment of atomic moments. Two distinct characteristics of ferromagnetic materials are spontaneousmagnetization and the existence of magnetic ordering temperature. The elements Fe, Niand Co and many of their alloys are typical ferromagnetic materials. It is due to the

magnetic dipoles associated with the spins of unpaired electrons, each dipole likes to point in the same direction as its neighbour, the alignment occurs in relatively small patches called domains and they themselves are randomly oriented. For ferromagneticmaterials the susceptance is positive and is approximately 20 to 200 times that of

paramagnetic materials.


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Ferrimagnetism a ferrimagnetic material is one in which the magnetic moment of theatoms on different sub-lattices are opposed and unequal. This happens when the sub-lattices consists of different materials or ions. Ferrimagnetic materials have high

resistivity and external field induced anisotropic properties. Ferrimagnetism propertiesare similar to Ferromagnetism in that spontaneous magnetization, Curie temperature,hysterisis and remanence. But they have different magnetic ordering. Ferrimagnetism isexhibited by ferrites and magnetic garnets. The oldest known magnetic substancemagnetite is a ferrimagnet. Widely used ferrimagnetic materials are YIG and ferritescomposed of iron oxides and other elements such as aluminium, nickel, cobalt,manganese and zinc.

Antiferromagnetism an ferromagnetic material is one in which the magnetic momentof the atoms on different sub-lattices are opposed and equal resulting in a net moment of zero.

Electrodynamics

Faradays law : In 1831 Michael Faraday performed three important epoch makingexperiments.

Exp I: He pulled a loop of wire through a magnetic field resulting flow of currentthrough the loop.

Exp II: He moved a magnet moving its field holding the loop still resulting incurrent through the loop.

Exp III: With both the loop and the magnet at rest, he changed the strength of the field by varying the current in the coil resulting the flow of current in the

loop.In case of first experiment, which is an example of motional emf, it is the Lorentz force lawat work; the emf is magnetic. To explain the generation of emf in the last two experiments,Faraday assumed that 'a changing magnetic field induces an electric field' and this particular electric field caused the emf. So in the last two cases the emf is electric.

A time varying magnetic field produces an emf which may establish a current in a

suitable closed circuit. If the circuit is an N turn coil then emf d

N dt

= . A non-zero value

of d dt

may result due to a time changing flux linking a stationary closed path, relative

motion between a steady flux and a closed path or a combination of the above two.Lenzs law: The induced emf due to the time varying magnetic field is in such a direction asto produce a current whose flux, if added to the original flux would reduce the magnitude of the emf.


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Motional emf: it is due to the motion of a conductor is a magnetic field. The force on charge

Q located in the conductor ( ) ( ) m F

F Q v B v B E Q

= = =

Maxwell's correction:

In the case of steady magnetic fields H J = which is ampere's law. The relation thatholds for time varying magnetic fields must converge to this expression in case of no timevariations. With this aspect in consideration, let us suppose, for time varying fields

H J X = + where X is unknown to be determined. Now consider divergence both sides of this relation.

H J X = + The left hand side of this equation is always is zero. And the first term of the right hand side

is, according to equation of continuity, J t

=

. But the Gauss law says D = .

With this relation the equation continuity becomes D

J t

=

resulting in

0D

X t

= +

which gives

D X

t

=

Hence D

H J t

= +

in case of time varying fields.

Note the relations used in the above derivation, equation continuity and Gauss law both are valid for time varying fields. Therefore the resultant expression must also be validfor time varying fields.

According to this Maxwell's correction of Ampere's law 'a changing electric fieldinduces a magnetic field'. Maxwell called the term D t as displacement current.

Poynting Theorem

It states that the net power flowing out of a given volume v is equal to the time rateof decrease in the energy stored with in v minus the power dissipated plus the power output of the source. According to this theorem the vector product P E H = at any

point is a measure of the rate of energy flow per unit area at that point.

( ) 2 2 20 01

( ) ( . )2S V V V

d E H da E H dv E J dv E

dt = + +

It can also be stated as the work done on the charges by the electromagnetic force isequal to the decrease in the energy stored in the field less the energy that flowed outthrough the surface. In fact it is the work energy theorem of the electrodynamics.


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( )2 00 01 ( ).2V V S d dW

E Jdv E H dv E H dadt dt

= + =

Importance: This theorem gives the energy relations of the fields in any volume. It also gives the

net flow of the power out of given volume thorough its surface. The pointing vector is the power density on the surface of a volume. The direction of

the pointing vector is the direction of the flow of the power.

Complex Poynting Theorem

Total complex power fed into a volume is equal to the algebraic sum of

Active power dissipated as heat, plus Reactive power proportional to the difference between time-average magnetic andelectric energies stored in the volume, plus

Complex power transmitted across the surface enclosed by the volume.

( ) ( )201 1 122 2 2 m ev v v s E J dv E dv j w w dv P ds = + +

Maxwells Equations

Maxwells assumption a changing displacement density was equivalent to an electriccurrent density and as such would produce a magnetic field has had far-reaching effects.Faradays law indicates that a changing magnetic field produces electric field.

These two together lead to wave equations predicting the existence of electromagnetic wave propagation even thirty years before Hertzs experimentalverification.

The following four electromagnetic equations are known as Maxwells equations because of the contribution to their development and established them as a self-consistentset.

When equations are transformed from time varying form into phasor form twochanges take place: one is apparent t gets replaced by j , another is hidden fields andsources become independent of time. In the phasor form of equations the fields and sourcesare functions of just position only.

Equations in Integral FormWord form of equations:


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The magneto motive force around a closed path is equal to the conduction current

plus the time-derivative of electric displacement through any surface.

The electromotive force around a closed path is equal to the time derivative of themagnetic flux through any surface bounded by the path.

The net electric displacement through the surface enclosing a volume is equal to thetotal charge with in the volume.

The net magnetic flux emerging through any closed surface is equal to zero.

Mathematical form involving time varying quantities :

D H dl J ds

t

= +

B E dl ds

t

=

v D ds = 0 B ds =

Mathematical form involving phasor quantities:

( ) H dl J j E ds = + E dl j H ds =

v D ds = 0 B ds = Equations in Differential Form

Word form of equations:

The curl of the magnetic field at a point in a time varying field is the sum of theconduction and displacement current densities at that point.

The curl of the electric field at a point in a time varying field is equal to the negativetime-rate of change of the magnetic flux density at that point.

The divergence of the electric displacement density at a point in a time varying fieldis equal to the volume charge density at that point.


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The divergence of the magnetic flux density at a point in a time varying field is zero..

Mathematical form involving time varying quantities :

D H J

t

= +

B

E t

=

v D = 0 B =

Mathematical form involving phasor quantities:

H J j E = + E j H =

v D = 0 B =

Significance:o Electromagnetic phenomenon of any type i.e. any frequency ranging dc to

infinity, any amount of intensity can be explained interpreted and understoodusing the Maxwells equations.

o The time varying fields at a point, both electric and magnetic, obey Maxwells

equations the fact of which is used to compute the fields many times. In theabsence of the relations connecting the time varying fields with their sources, thisobservation is of very significant and useful.

o Maxwells equations lead to the development of wave equations. The fieldintensities of the time varying fields obey wave equations proving the existenceof the wave or energy flow in the time varying fields.

Boundary Conditions

The tangential component of E is continuous at the surface. It means E is same justoutside the surface as it is just inside.

The tangential component of H is continuous across a surface except at the surface of

a perfect conductor. The tangential component of H is discontinuous by an amountequal to the surface current per unit width at the surface of the perfect conductor.

The normal component of B is continuous at the surface of any discontinuity. The normal component of D is continuous except in the presence of surface charge

density. The normal component of D is discontinuous by an amount equal to thesurface charge density in the presence of surface charge density.


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Wave equations

In perfect dielectric

Applying curl on both sides of the Faradays law E H = &

But in free space, according to Amperes law with Maxwells correction H E = &

So E E = &&

The vector identity is2 E E E =

So2

E E E = &&

According to gauss law, as the medium is free space without any charge,

10 E D

= =

Hence2 E E = &&

Similarly it can be derived using the Maxwells equation2 H H = &&

In phasor form these two equations become2 2 E E = 2 2 H H =

These two equations are called vector Helmholtz equations .

In conducting media

Applying curl on both sides of the Faradays law E H = &

But in free space, according to Amperes law with Maxwells correction H J E E E = + = +& & as J E = is Ohms law.

So

E E E = & &&

The vector identity is

2 E E E = So

2 E E E E = & &&


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According to gauss law, as the medium is free space without any charge,1

0 E D

= =

Hence2 E E E = +& &&

This is the wave equation in E and similarly the wave equation for H 2 H H H = +& &&

can be derived using the Maxwells equationIn phasor form, these two equations become

( )( )

2 2

2

E j E

j j E E

=

= + =

( )( )

2 2

2

H j H

j j H H

=

= + =

Propagation constant The constant is known as propagation constant of the medium and in general a complexquantity having both the real and imaginary parts. The real part is called attenuationconstant and the imaginary part is called phase shift constant .

j = +

( ) j j = +

2

2 21 1

2

= +

2

2 21 12

= + +

For good dielectrics

2

2 21 12

= +

2

2 21 12 2

+

2

=


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2

2 21 12

= + +

2

2 21 12 2

+ +

2

2 21 8

= +

For good conductors

( ) j j = +

1j

j

= +

045 j = Therefore

2

= =

Vector Helmholtz equations

210r

r E k E

=

21 0r r

H k H

=

Scalar Helmholtz equations

21 0 z r z r

E k E

+ =

. Similarly for other components of the E .

21

0 z r z r

H k H

+ = . Similarly for other components of the H .


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Wave: A physical phenomenon that occurs at one place at a given time is reproduced at

other places at later times, the time delay being proportional to the space separation

from the first location then the group of phenomenon constitutes wave.Plane wave : It is a wave whose equi-phase surfaces are planes.

Uniform plane wave : it is a wave whose magnitude and phase, both are constant over a set of planes. In uniform plane wave E and H are independent of two dimensions and dependant

only on one dimension and time. These are transverse in nature i.e. E and H are perpendicular to the direction of

propagation of the wave. They i.e. E and H are perpendicular to each other. In fact E, H and direction of

propagation of the wave form RH vector system. The direction of propagation is given by H E . In fact E, H and direction of

propagation of the wave form RH vector system .

= y

x

H E

,

= x

y

H

E also

= H E

Classification of electromagnetic waves: where ever time varying fields exists, therethe wave exists and the converse is also true. The electromagnetic waves can be classifiedinto four categories.

Transverse electromagnetic (TEM) wave:

In this wave, also known as Principal wave, the electric vector E and magneticvector H both are entirely normal to the direction of the propagation of the wave.

In addition, the electric vector E , magnetic vector H and the direction of propagationall the three vectors form a right handed vector system.

The energy travels as TEM wave in free space and over parallel wire transmissionline. The coaxial lines can also hold this type of wave.

The phase velocity and group velocity is same for TEM wave. Neither one dependsupon the frequency. So the TEM wave is non-dispersive wave.

Transverse electric (TE) wave:In this wave, the electric vector is entirely normal to the direction of propagation and

hence no component in the direction of propagation. The magnetic vector has both thenormal and parallel components.

Transverse magnetic (TM) wave:In this wave, the magnetic vector is entirely normal to the direction of propagation

and hence no component in the direction of propagation. The electric vector has both thenormal and parallel components.


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Hybrid or mixed wave:It is a linear combination of TE and TM waves. In this wave, both the electric and

magnetic vectors posses both the components, normal and parallel to the direction of propagation of the wave. Non-TEM waves i.e. TE, TM and TE+TM waves exist in hallow pipe waveguides.

The phase velocity differs from group velocity in case of non-TEM waves. And both dependupon the frequency. So the non-TEM waves are always dispersive in nature.

The coaxial line, in addition to the TEM wave, can carry higher order forms of TM andTE waves with components of electric or magnetic field in the direction of the line axis.However for the usual coaxial lines the dimensions are small enough that the lines areoperating at frequencies far below cutoff for these modes.

Reflection And Refraction

Perfect conductor :

Normal incidence : The amplitude of the reflected electric field strength is equal to that of the incident electric field strength, but its phase is reversed on reflection i.e. r i E E = The electric field intensity in the standing wave pattern is ( ), 2 sin sinT i E x t E x t =% The magnetic field strength gets reflected without phase reversal i.e. r i H H = The magnetic field strength in the standing wave pattern is ( ), 2 cos cosT i H x t H x t =% In the reflected wave, the T E

% and T H % are 90 0 apart in time-phase. Also there exists a

surface current of ( )/ sec 0 s T J amp H x= = .Oblique incidence : The plane of incidence is the plane containing the incident ray and thenormal to the surface. Perpendicular polarization: 2 sin y j yT i z E jE z e

=


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where cos z = and sin y = .

Parallel polarization:2 cos y j yT i z H H z e

=

2 cos sin y j y y i z E j H z e =

2 sin cos y j y z i z E H z e =

In both the types of polarizations, thereexists standing wave distribution along the z axis i.e. cos z z or sin z z and

travelling wave along the y direction i.e. j ye

Perfect dielectric: Normal incidence : 1i i E H = , 1r r E H = and 2t t E H = Tangential components are continuous i r t E E E + = and i r t H H H + =

1 2

1 2

r

i

E E

=

+; 1

1 2

2t i

E E

=

+

Oblique incidence : energy conservation requires

Perpendicular polarization:

221 1

1

221 1

1

cos sin

cos sin

r

i

E E

=

+

1 1 2 2

1 1 2 2

cos cos

cos cos

=

+( )( )

2 1

2 1

sin

sin

=

+

Parallel polarization:

22 2

1 11 1

22 21 1

1 1

cos sin

cos sin

r

i

E E

=

+

2 2 1 1

2 2 1 1

cos cos

cos cos

=+

( )( )

1 2

1 2

tansin

=+


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Assuming the thickness of the conductor plate to be larger than the depth of penetration so that no reflection from the back surface of the conductor, the surface

current density becomes0

00 0

y

s

J

J J dy J e dy

= = =

Therefore, the surface impedance tan 00

s s

E J Z

J J

= = =

For conducting medium ( ) j j j = +

Hence 045 s j

Z

= = = .

The surface impedance is complex quantity and its real part is called surface resistance s R

whereas its imaginary part is called surface reactance s X . For a thick good conductor their

magnitudes are same.2 s

R

,

2 s X

Observations:

The surface impedance is equal to the intrinsic impedance for the conductingmedium.

It is also equal to the characteristic impedance of the thick plane conductor.

This is also input impedance of the thick plane conductor when viewed as atransmission line conducting energy into the interior of the metal.


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The surface resistance, with units of ohms, is same as the high frequency skineffect resistance per unit length of a flat conductor of unit width.

The surface resistance s R is related to the depth of penetration or skin depth in

a conductor through1

s R =

The surface resistance of a flat conductor at any frequency is equal to the dcresistance of a thickness of the same conductor. This means that the conductor having a thickness very much greater than and having exponential currentdistribution throughout its depth has the same resistance as would athickness of the conductor with the current distributed uniformly throughoutthis thickness.

The power loss per unit area of the plane conductor is 2 s eff s J R

Polarization of waveso Polarization of a radiated wave is defined as that property of an electromagnetic

wave describing the time-varying direction and relative magnitude of the electricfield vector. Specifically the figure traced as a function of time by the extremity of the vector at a fixed location in space and the sense in which it is traced as observedalong the direction of propagation. Basically polarization refers to the time-varying

behaviour/orientation of the E vector in an uniform plane wave at some fixed point.o Linear polarization: if the direction of the resultant E vector in the uniform plane

wave remains same with respect time then the wave is said to be linearly polarized . o Elliptical polarization: if the tip of the E vector of a travelling plane wave traces an

ellipse, then the wave is said to be elliptically polarized.o Circular polarization: if the tip of the E vector of a travelling plane wave traces a

circle, then the wave is said to be circularly polarized.o Consider a plane wave travelling in z direction. If the x and y components of

the E vector are in phase , then the wave is linearly polarized. If the x and y components are not in phase and/or unequal in magnitude then the wave iselliptically polarized. Circular polarization results when and y componentshave 90 0 phase difference and equal in magnitude.

o Sense of polarization: when the wave is receding , if the resultant E vector rotatesclockwise the wave is said to be clockwise or right circular/elliptical polarized wave.Anti-clock wise or left circular/elliptical polarized wave results when the E vector rotates in anti-clock wise direction.

o For a z-travelling waveo ( ) a y x E jaa E +=0 LCP or CCWo ) a y x E jaa E =0 RCP or CW


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o y x b jBa A E +=0 LEP or CCWo

y x b jBa A E =0 REP or CWo Reversal of the sense of rotation can be obtained by giving 180 0 phase shift to

either x or y component of the E field.o Polarization is specified by

Shape (axial ratio) Orientation Sense of polarization of the polarization ellipse.

o Other representations are Polarization ratio Stokes parameter Poincare sphere

o Axial ratio =ellipsetheof axismajor semi

ellipsetheof axisor semi

a

b min=

o Polarization ratio P = j x

y e P E

E =

Basically there are two types for representing polarization states; Wave polarization representation: ( ), AR it uses Axial ratio AR which

is ve+ for LH, and ve for RH polarizations, tilt angle which is theangle the major axis makes with the axis.

Electrical quantities representation: ( ), yo xo E E it uses the ratio yo xo E E , the angle between them .


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If ( )1cot AR = and ( )1tan yo xo E E = , then the state can be representedeither by the pair ( ), or ( ), .

AR varies from 1 for circle to for line, varies from 4 to 4 and varies from 0 to .

varies from 2 to 2 and from 0 to .

Poincare sphere provides a compact graphical representation of all the two typesand it also corresponds the above two representations. it is useful to find howclose two polarization states are or how much interaction takes place betweentwo states of polarization. Its salient features are

Equator 0, AR = = represents linear polarization. Longitude point0 = represents horizontal and 090 = vertical polarization.

North pole represents Left circular and South pole Right circular polarization Northern hemi-sphere represents left handed and Southern hemi-sphere

represents right handed polarization. Point is denoted by ( )2 , 2 . The xy plane or horizontal plane represents

0 = and z vertical plane represents 0 = . Matched states: when two states of polarization fall on the same point over the

Poincare sphere then they are said to be Matched states of polarization. And theycan interact maximum.

Orthogonal states: when the two states of polarization fall on radially opposite points on the Poincare sphere, then they are said to be orthogonal states. And nointeraction is possible between them.

Antennas And Polarization

Polarization of the antenna is defined in its transmitting mode. The polarization of anantenna in a given direction is defined as the polarization of the wave transmitted by theantenna .Polarization pattern of an antenna represents its polarization characteristics and it isdefined as the spatial distribution of the polarization of a field vector radiated by theantenna taken over its radiation sphere. At each point over the sphere the polarization isusually resolved into a pair of orthogonal components along and directions. Thesecomponents are called co-polarization and cross polarization.Polarization mismatch occurs due to the mismatch between the polarization of the receivingantenna and the polarization of the receiving wave. .Due to this mismatch the receivingantenna cannot extract maximum amount of power from the incoming wave.If the incoming wave is linearly polarized, the receiving antenna can be

Linearly polarized with its polarization aligned to the polarization of theincoming wave or


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Circularly polarized or Elliptically polarized with its major axis aligned to the polarization the

incoming wave.

If the incoming wave is circularly polarized, the receiving antenna can be Linearly polarized. Circularly polarized with the sense of rotation same as that of the

incoming wave. Elliptically polarized with the sense of rotation same as that of the

incoming wave.If the incoming wave is elliptically polarized, the receiving antenna can be

Linearly polarized with its polarization aligned to the major axis. Circularly polarized with the sense of rotation same as that of the

incoming wave. Elliptically polarized with the sense of rotation same as that of the

incoming wave, respective axes of the both pointing in the samedirection.The above observations are based on the fact that the power transfer efficiency between two

states represented by and ' on the Poincare sphere is given by 2'

cos2 pol

MOM

=

Transmission-Line Theory

Transmission lines :

These are metallic conductor systems involving two or more conductorsseparated by an insulator used to transfer low frequency electrical energy in TEMform from one point to another.

Types Balanced or differential Tr system:

Both the conductors in this line carry signal currents of equal magnitude wrt theelectrical ground but in opposite directions.

Any pair of wires can be operated in balanced mode provided that neither wireis at ground potential.

A balanced wire pair has the advantage of noise interference getting cancelled atthe load due to high CMRR of 40 to 70db.

Un-balanced or single ended Tr system: One conductor is at ground potential where as the other one is at signal potential. Because one wire being at ground potential, the probability of noise being

induced is more.


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The standard two conductor coaxial cable is an unbalanced line as the secondconductor or shield is generally connected to ground.

Parallel conductor tr. Lines: Suitable for low frequency applications At high frequencies they become useless as radiation and dielectric losses increase Susceptible for noise pick up. Ex. Open wire TL, Twin lead(ribbon) cable, Twisted pair cable etc.

Coaxial or concentric tr. lines; Extensively used for high frequency applications as they give low radiation and

dielectric losses. Also they give shield against external interference. Ex. Solid flexible (low losses), rigid air filled(relatively expensive) must be used in unbalanced mode and expensive.

Baluns : These are circuit devices used to connect a balanced TL to an unbalanced load like

antenna or unbalanced TL such as a coaxial cable. Ex. Transformer balun, bazooka balun

Non-resonant line: A line terminated in its characteristic impedance is called non-resonant or flat

or smooth line. The voltage and current over such a line are constantthroughout its length if it is loss-less and decreases exponentially if the line islossy.

Resonant line : It is loss-less line terminated over a short or open circuit.

Reflection factor k: It is defined as the ratio of the current actually flowing in the load to that

which might flow under image matched conditions.This ratio indicates the change in current of the load due to reflection at themismatched junction.

21

212

Z Z

Z Z k

+= where 1 Z and 2 Z impedances at the junction seen looking

towards both the sides.

Reflection loss:


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It is defined as the number of nepers or decibels by which the current in theload under image matched conditions would exceed the current actuallyflowing in the load.

nepers Z Z Z Z R L

21

21

2ln += db

Z Z Z Z

21

21

2log20 +=

Stub: A stub is a piece of transmission line whose input impedance is pure

reactance. Normally short circuited stubs are used as open circuited stubstend to radiate.

Infinite line:

It one whose length is infinite. It is characterized by the absence of reflected

wave and the current over the line depends only on its characteristicimpedance not on the termination.

Velocity factor: The velocity factor of a dielectric substance or a cable is the velocity

reduction ratio. It is given byr

f v 1

= where r is dielectric constant of the

medium.

Half wave transmission line:

Its input impedance is equal to the terminating impedance, this property is independent of characteristic impedance 0 Z but frequencydependant.

The short circuited 2 line can act as a band stop filter, can be used tomeasure velocity factor and dielectric constant of medium.

Half wave line is also used to measure the impedance that is notaccessible physically.

Quarter wave 4 transmission lines: Short circuited 4 line is equivalent to parallel LC circuit.

Open circuited 4 line is equivalent to series LC circuit. Short circuited line with length 4 > is equivalent to capacitor C . Open circuited line with length 4 > is equivalent to an inductor L . Short circuited line with length 4 < is equivalent to an inductor L . Open circuited line with length 4 < is equivalent to capacitor C .


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Applications of 4 line: Quarter wave transformer :

9 it is a loss-less uniform line of length 4 9 Its input impedance is inversely proportional to its terminating impedance.9 Provided the characteristic impedance is resistive, its input impedance is

inductive if the termination is capacitive and vice versa9 It acts as impedance transformer or inverter as it can step up or step down the

impedance. It is used for load matching purposes.9 It disadvantage is sensitivity to frequency change.

Opened out parallel wire 4 transmission line is used as wire radiator called half wavedipole.

Opened out parallel wire transmission line of length less than 4 is used as wire

parasitic radiator called director in Yagi-Uda array. So the director carries capacitivecurrents. In other words, an opened out line excited at a frequency less than resonant iscapacitive.

Opened out parallel wire transmission line of length more than 4 is used as wire parasitic radiator called reflector in Yagi-Uda array. So the reflectors carry inductivecurrents. In other words, an opened out line excited at a frequency more than resonant isinductive..

Equivalent Circuit Representation

Since each conductor has a certain length and diameter it must have resistance andconductance.

Since there arewires close to each other,there must be capacitance

between them. Dielectricmaterials, which cannot be

perfect in its insulation,separate the wires: the

leakage through it can be represented by a short conductance.

All the quantities R,L,Gand C are proportional to the


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length of the line and unless measured and quoted per out length they are meaningless.These are distributed through out the length of line. Under no circumstances can they beassumed to be limped at any one point.

Losses in TR Lines:

Losses in TR lines occur in types a) Radiation b) Conductor heating c) Dielectricheating.

Radiation losses arise because a TR line may act as an antenna if the separation of theconductors is an appreciable fraction of wavelength.

Conductor heating or I 2 R loss is proportional to current and there for inversely

proportional to characteristic impedance. It also increases with frequency because of theskin effect.

Dialectic heating is proportional to the Voltage across the dielectric and inversely proportional to the characteristic impedance. It increases with frequency.

The transmission line is characterized by primary constants and secondary constants.

Secondary constants : Characteristic Impedance 0 Z and propagation constant are calledthe secondary constants of the transmission line.

Characteristic Impedance 0 Z :- The characteristic impedance of a transmission line,0 Z is the impedance measured at the input of the line when its length is infinite : It can

also be defined as the input impedance of a transmission line when it is terminated on itscharacteristic impedance.

S

S

I V

Z =0 when the length of line is infinite or when the line is terminated over Z o. it is

related to the primary constants of the line through)()(

0 C jG L j R

Z

++

=

Propagation Constant : It is defined as the natural

logarithm of the ratio of the input to the output current. R

S e I

I log= , where

S I and R I are at a unit distance apart on the line of infinite length. It is a


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complex quantity and is represented by j+= where is known as theAttenuation constant and is known as the phase shift constant. It is related

to the primary constants through ZY = loss- less line or RF line G R == 0

Consider a small section dx or a transmission line. The series impedance of this section will be dx L j R )( + and the shunt admittance will be( )dxC jG +

Then( )dx L j R I V dV V +=+ )( , ( )dxC jGV I dI I +=+ )(

and I L j RdxdV

)( += I C jGdxdI

)( +=

dxdI

Z dx

V d =2

2

and =22

dx I d

dxdV

Y L j R Z +=Q , C jGY +=

V ZY dx

V d =2

2

and ZYI dx

I d =2

2

22

2

d V V

dx = and

22

2

d I I

dx =

where = Propagation Constant= ZY = ( )( )C jG L j R ++ )

The general solution to the above equations can be expressed in either of the two forms:In terms of exponential functions.


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x x ebeaV += : x x ed ec I +=

In terms of hyperbolic functions. x B x AV sinhcosh += ; x D xC I sinhcosh +=

Out of the four constants Dand C B A ,, only two are independent. The last two equationscan be written as

x B x AV sinhcosh += ; )sinhcosh(1

0

x A x B Z

I +=

In terms of sending end voltage and currents x Z I xV V S S sinhcosh 0=

x Z V

x I I s

s sinhcosh0=

In terms of receiving end voltage and currents

)(sinh)(cosh 0 xl Z I xl V V R R += ; )(sinh)(cosh0

xl Z V

xl I I R R +=

In terms of sending end impedance s Z and 0 Z

( ) ( )[ ] z s z s s e Z Z e Z Z I V 002 ++=

( ) ( )[ ]

z

s

z

s

s e Z Z e Z Z Z

I I

0002

+=

In terms of receiving end impedance l Z and 0 Z

( ) ( )[ ]d l d l l e Z Z e Z Z I V ++= 002

( ) ( )[ ]d l d l l e Z Z e Z Z Z I

I += 0002


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Waves on LinesThe reflected wave is generated at the load as a result of reflection of the incident wave

by the load impedance. This reflection is of such a character as simultaneously to meet

the following conditions. If ' '' E E E = + and ' '' I I I = + theno The voltage and current of the incident wave at the load must satisfy

0' ' E I Z = o The voltage and current of the incident wave at the load must satisfy

0'' '' E I Z = o The load voltage L E is the sum of the voltages of the incident and

reflected waves at the load, that is 1 2 L E E E = + o The load voltage L I is the sum of the currents of the incident and reflected

waves at the load, that is 1 2 L I I I = +

o The vector ratio of L L L E I Z = must equal the load impedance L Z

Pure travelling wave is one whose SWR is unity indicating no reflections from theload.

o In this wave V and I are in-phase.o It occurs when the line is terminated with its matched impedance.o The ratio of V to I is constant 0 Z , Characteristic impedance of the line.o In pure travelling wave the phase varies continuously along the length of the

line but not the amplitude. Pure standing wave is one whose SWR is infinity indicating the total reflection of the

incident wave by the load.o In this wave V and I are 90 0 out of phase.o It occurs when the line is terminated with oc or sc or pure reactance.o The ratio of V to I is function of the position along the length of the line.o At all points between a pair of successive voltage nulls i.e. in one half cycle,

the voltage is in phase. All the points in the next half cycle exhibit 180 0 phasedifference with the points of previous half cycle. Similar is the case with thecurrent wave form.

Impure standing wave is a combination of pure travelling wave and pure standingwave.

o In this case V and I are not 900 out of phase and in general it varies along thelength of the line.

o It occurs when the termination is different from oc, sc, or pure reactance or matched termination.


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Reflection coefficient The vector ratio '' ' E E of the voltage of the reflected wave to the voltage of the

incident wave at a distance l from the load is defined as Reflection coefficient at the point l and denoted usually by . It has both magnitude and phase so is a complex quantity.

If the reflection coefficient is considered at the load then it is called reflection

coefficient of the load and equals to( )( )

02

1 0

1

1 L

load L

Z Z E E Z Z

= =

+

In case of zero loss line, the reflection coefficient has everywhere has the samemagnitude and equals the reflection coefficient of the load. In case of lossy line the reflectedwave becomes smaller and the incident wave larger with increasing distance from the loadcausing to decrease correspondingly.

The relation between the load voltage and current and the voltages of the incidentand reflected waves at the load can be deduced as

01 1 2

L L L E I Z E E +

= = +

02 1 1 2

L L L E I Z E E E

= = = +

Standing Waves

The distance between two successive nodes or anti-nodes of voltage (or current) isalways 2 g . And it is 4 g between voltage node to current anti-node or voltageanti-node to current node.

When the termination is open circuit, the current gets reflected with 180 0 phase shiftwhere as the voltage gets reflected without any phase shift. It results in current nodeand voltage anti-node over the open circuit termination.

When the termination is short circuit, the voltage gets reflected with 180 0 phase shiftwhere as the current gets reflected without any phase shift. It results in voltage nodeand current anti-node over the short circuit termination

When the termination load is either open circuit or short circuit or pure reactance thetotal incident wave gets reflected, as the load cannot dissipate any power. In suchcase the amplitude of the reflected wave is same as that of the incident wave

resulting in perfect cancellation at the nodes. Consequently SWR becomes zero. When the termination is pureresistance, then voltage nodeand current anti-node occursover the termination for

0 R R L < . For the case 0 R R L >


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38

voltage anti-node and current node occurs over the termination. The ratio of voltage to current at a point over the line is the impedance of the line at

that point. The impedance at two points A and B are equal if they are separated by a

distance equal to integer multiples of 2 g . The impedances have inverse proportionality if the distance is odd multiples of 4 g

The line impedance is capacitive in a distance of 4 g right side of voltage node andit is inductive in a distance of 4 g to its left side.

Standing-wave Ratio

The ratio of the maximum amplitude to minimum amplitude possessed by thevoltage or current distribution is defined as standing-wave ratio and denoted by .

The character of the voltage or current distribution on a transmission line can beconveniently described by SWR.

Standing-wave ratio is a measure of amplitude ratio of reflected to incident waves.Thus a SWR of unity denotes the absence of a reflected wave, while a very high SWR indicates that the reflected wave is as large as the incident wave.

Theoretically, for the case of zero attenuation, the SWR will be infinite when theload is either open- or short-circuited or is a lossless reactance.

The SWR is one means of expressing the magnitude of the reflection coefficient : the exact relation between the two is

1

1

+ =

or

11

=

+

Significance of SWR:

The importance of the standing wave ratio arises from the fact that it can be veryeasily measured experimentally.

The SWR indicates directly the extent to which reflected waves exist on a system.In addition, Standing wave measurements provide an important means of measuring

impedance.


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39

Line Impedance In terms of sending end impedance s Z and 0 Z

z Z Z z Z Z Z Z

s

s

tanhtanh

0

00

=

In terms of receiving end impedance l Z and 0 Z

d Z Z d Z Z

Z Z l

l

tanhtanh

0

00 +

+=

Input Impedance

l Z Z l Z Z

Z Z l

l in

tanhtanh

0

00 ++

=


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40

Advantages of Load Matching

Maximum power can be transferred over the line

Maximum efficiency in power transfer can be achieved. Load power is independent of l With Lesser Peak voltage over the line, power can be transferred. Eliminates modulation distortion Prevents the frequency shift in the source Minimize errors in measurement systems

Single-stub matching

Stub position0

1tan2 Z

Z l L=

Stub length0

001' tan2 Z Z

Z Z

Z Z

l L

L

OS =

Line Distortion:

The deviation of the waveform at the output of the line from that at its inputis called line distortion .

It is due to the fact that all frequencies in the waveform do not have sameattenuation and same delay during the propagation.

The characteristic impedance being function of the frequency, attenuation being function of the frequency and velocity of propagation on the line beingfunction of frequency are causes of distortion.

Types of distortion: frequency distortion and delay distortion. Frequency distortion is due to various frequency components of the signal

undergoing different amounts of attenuation when the attenuation

constant is function of frequency. Toeliminate this distortion the attenuation constant must be made independentof frequency.

Phase or delay distortion is due to different frequency components of thesignal undergoing different amounts of phase delays while reaching thedestination thus spoiling the original phase relation between them. To


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41

eliminate this phase shift constant must be made proportional to angular frequency .

Frequency distortion can be reduced by cascading the lines with networksknown as equalizers . Equalizer is network whose attenuation versusfrequency characteristic is just opposite to that of the line. Delay distortioncan also be reduced with equalizers, but it must be designed in such a waythat the for the total circuit is proportional to . For audio transmissionfrequency distortion is serious problem whereas for video transmission bothare serious.

Distortion-less line: it is a line which transmits the input signal without anydistortion. This occurs when the primary constants the line are related

through the relationC G

L R

= . With this interrelation among the primary

constants of the line the attenuation constant LC L R

= becomes

independent of the frequency, the phase shift constant LC = is proportional to angular frequency making the velocity of propagationindependent of the frequency thus eliminating both types of distortion.

Loading: in the actual linesC G

L R

>> . To make the line distortion-less L R

is

decreased by increasing the inductance L . This is affected by either bychanging the line configuration or by using high inductance coils. Themethod of reducing the distortion by increasing the inductance of the line iscalled loading . It is of two types.

Continuous loading: the tape of steel or some other magnetic materials suchas perm alloy or mumetal is wound around the conductor to be loaded. Itincreases the permeability of the surrounding medium and thereby increasingthe inductance. It is costly and used in sub-marine cablesonly.

Lumped loading: in this method inductance coils are introduced at definiteand uniform intervals along the length of the line to increase its inductance.

For telephone cable R Z = and C jY = .CR

vCR

2

;2

=== as

and v are functions of frequency distortions of both types take place.


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42

Smith Chart

Smith chart developed by P.Smith in 1939, is the best known and widely usedgraphical aid in solving transmission line problems.

The real utility of the smith chart lies in the fact that it can be used to convert thereflection coefficients to normalized impedances or admittances.

It is a polar plot of complex reflection coefficient with the normalized impedance or admittance in a unity circle.

0

11

l l l

l

Z z

Z +

= =

11

r i

r i

jr jx

j+ +

+ =

( )

2 2

22

1

1

r i

r i

r

= +

and( )

22

2

1

i

r i

x

= +

. These can

be rearranged as2

2 11 1r i

r r r

+ =

+ + it represents

a family of constant resistance circles with radius( )1 1 r + and centre at ( )1r r + along the real axis.

( )2 2

2 1 11r i x

+ =

it represents a family of

constant reactance circles with radius 1 x and centre at1

1,r i x = = .

It consists of two sets of circles or arcs of circles:o The constant resistance r circles whose centres lie on the straight line of the

chart. They represent the normalized resistance along the transmission line.o The constant reactance x loci, arcs of the circles lie on both sides of the

horizontal line. They represent various values of the normalized reactance of the transmission line.

o The circles and arcs are orthogonalo Upper half of the chart represents inductive reactance/susceptance where as

the lower half represents the capacitive reactance/susceptance.o Smith chart describes the line of half-wavelength onlyo It can be used only with normalized impedances or admittances.o The movement towards the generator corresponds to clockwise motion on the

chart and towards the load corresponds anti-clockwise motion. The constant resistance r and constant reactance x loci form two families of

circles and all of them pass through the point 0,1 = = ir


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Multiple choice questions

1. The Smith chart is [D ]

a) A Polar plot b) represents complex reflection coefficientc) Inscribed in a unity circle d) all

2. The complete circles in the smith chart represent [C ] a) Normalized resistance

b) Normalized conductancec) Both d) none

3. The arcs in the smith chart represent [C ] a) Normalized reactance

b) Normalized susceptancec) Both d) none

4. The circles and arcs over the smith chart are [A ] a) Orthogonal b) opposite to each other c) At 45 0 d) None

5. The upper half of the smith chart represents [C ] a) Inductive reactance b) inductive susceptancec) Both d) None

6. The lower half of the smith chart represents [C ]

a) Capacitive reactance b) capacitive susceptancec) Both d) None7. The radius of the constant SWR circle is [C ]

a) Voltage SWR b) current SWR c) Both d) None

8. The centre of the constant SWR circle is [C ] a) 1of horizontal line b) centre of the chartc) Both d) None

9. The smith chart represents the line for a length of [ ] a) 2 g b) 2

c) d) None

10. In the left half of the chart, the resistance [ ] and reactance values are

a) Less than 1 b) more than 1


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c) Both d) None

11. In the right half of the chart, the resistance and [ ] reactance values are

a) Less than 1 b) more than 1c) Both d) None

12. The left most point of the chart represents [ ] a) ( )0,0 b) ( ), c) Either one d) None

13. The right most point of the chart represents [ ] a) ( )0,0 b) ( ), c) Either one d) None

14. The top most point of the chart represents [ ] a) ( )1,1 b) ( )1, 1 c) Either one d) None

15. The bottom most point of the chart represents [ ] a) ( )1,1 b) ( )1, 1 c) Either one d) None

16. Smith chart is always used with [B ] a) Normalized impedances

b) Normalized admittancesc) Both d) None

17. Smith chart is useful to analyse [ ] a) Loss-less lines b) lossy-linesc) Both d) None

18. The horizontal line left of the centre represents [ ] a) max min,V I b) min max,V I

c) Both d) None

19. The horizontal line right of the centre represents [ ] a) max min,V I b) min max,V I c) Both d) None

20. Towards load over the line corresponds [ ]


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a) Clock-wise rotation over the smith chart b) Anti-clockwise rotation over the smith chartc) Both d) none

21. Towards source over the line corresponds [ ] a) Clock-wise rotation over the smith chart b) Anti-clockwise rotation over the smith chartc) Both d) none

22. The point over constant SWR circle diametrically [ ] opposite to load impedance point representsa) Load admittance b) load impedance

c) either one d) None

23. The point over constant SWR circle diametrically [ ]

opposite to load admittance point isa) Load admittance b) load impedancec) Either one d) None

24. Travel of length 2 g over the line corresponds [ ] rotation over the chart

a) 0180 b) 0360 c) 090 c) None

25. If the load is pure reactance, then the load point [ ] over the smith chart staysa) At the periphery b) over the horizontal linec) In the lower half d) in the upper half

26. If the load is pure resistance, then the load point [ ] over the smith chart staysa) At the periphery b) over the h

28538193-1-Vector-Identities-Triple-Products-♠

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