microwaves components

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_________________________________________________________________________Microwave Engineering Dr.PVS&RSR __________________________________________________________________________

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:

♠ ( ) 0A∇ • ∇× =

♠ ( ) ( ) 2A A A∇× ∇× = ∇ ∇ • − ∇

♠ ( ) 2f f∇ • ∇ = ∇

♠ ( ) 0f∇× ∇ =

Fundamental theorems:

As is known, the volume is always enclosed by a closed surface and the surface is always is enclosed by closed path. The path is a vector, is a directed curve, direction normally being indicated with an arrow over the curve. The surface is also by definition a vector and, by definition, is always surrounded by a closed path. The direction 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, surrounding that point. The direction pointed by the right hand thumb when wrapped by the fingers around the point gives the direction of the surface at that point. The integration 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 divergence theorem. Similarly the integration of a vector along a closed path can be related to the integration of the same vector over the surface enclosed by the path through Curl theorem.

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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 the function values at ' a ' and 'b ' i.e. ( )f a and ( )f b . '

( ) ( ) ( )b

a

f dl f b f a∇ • = −∫

Divergence theorem or Gauss theorem This 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 closed surface is equal to the integral of the divergence of the same vector through any volume enclosed by that surface.' Analytically

( )A d A daτ∇ • = •∫ ∫

Curl theorem or Stokes theorem This theorem connects line integral with surface integral. In word form it can be stated 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 same vector through any surface enclosed by the path.' Mathematically

( )A da A dl∇× • = •∫ ∫

Significance: • These are useful in converting the Maxwell's equations from point form

to integral form and vice versa. • They relate a surface integral to its corresponding volume integral and

also a line integral to its corresponding surface integral.

Operator Del i j kx y z

∂ ∂ ∂∇ + +∂ ∂ ∂

• It is a three dimensional, partial differential vector operator defined in Cartesian 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 ' rra ' joining origin with ( ), ,x y z

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

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rr a∇ =

2

1 rar r

∇ = −

( ) ( )2 , 2n nrr a n r n∇ • = + ≠ −

( )34 , 2r nπ δ= = −

( ) 0nrr a∇× =

Operator Laplacian2 2 2

22 2 2x 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'Alembertian 2

2 20 0 2t

µ ε ∂∇ −∂

and Helmholtz operator ( )2k∇ • ∇ + :

Both these operators are three dimensional partial differential operators. The first is normally applied upon scalar functions.

Divergence. Curl And Gradient In Different Co-Ordinate Systems: Let 1 2 3ˆ ˆ ˆdl h du u h dv v h dw w= + +

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♠ 1 2 3

1 1 1ˆ ˆ ˆV V VV u v wh u h v h w

∂ ∂ ∂∇ = + +∂ ∂ ∂

♠ ( ) ( ) ( )2 3 3 1 1 21 2 3

1u v wD h h D h h D h h D

h h h u v w ∂ ∂ ∂ ∇ • = + + ∂ ∂ ∂

♠

1 2 3

1 2 3

1 2 3

ˆ ˆ ˆ1

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 2

1 2 3 1 2 3

1 h h h h h hV V VVh 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 electrical

and 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 is called 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 both electric field intensity and magnetic field intensity which are related to each other. This relation i.e. the relation between electric field and magnetic field in time-varying fields is given by Faraday’s law and Maxwell’s relation.

All the three types of fields are related because of the relation that exists in between their respective sources. Stationary charge gives electrostatic field and charge moving with constant velocity is the source of steady magnetic field whereas the charge moving with acceleration/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 possible to add or subtract energy to the fields. One difference between electric and magnetic fields is: magnetic fields can do no work whereas electric fields can do.

Time varying fields exhibit one important property which is not possessed by static

fields. There exists travelling wave and consequently energy flow called radiation in time varying electromagnetic fields. As the antennas which are critical components of wire-less communication systems, functions based on radiation. So the study of the properties of the time varying fields has become an important requirement for communication engineers.

The intensity of the electric field is a vector quantity indicated by E with units

volts/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 field intensity is independent of the medium of the field.

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

A quantity D Eε= called electric displacement density can also be defined for electric fields with units coul/m2. A quantity similar to this can be defined for magnetic fields also. It is called magnetic flux density denoted by B with units of tesla or webers/m2. It is related to the magnetic field intensity through B Hµ= .

The electric displacement density is independent of the medium whereas the magnetic flux density is dependent upon the medium properties. For static fields these are functions 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 magnetic fields the magnetic vector potential is defined indicated by A with units ----. For the static sources the potentials are inversely proportional to the distance from the source. Potentials can 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 the source or the fields. In this case the two potentials can be related to each other also through Coulombs gauze or Lorentz gauge

Electro statistics

Coulomb’s 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 and it 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,

2

14 r

Q qF arπε

=

( ) ( ) ( )

( ) ( ) ( )1 2 1 2 1 2

32 2 2 2

1 2 1 2 1 24

x x i y y j z z kQ q

x x y y z zπε − + − + − = − + − + −

( )1 23

21 2

4r rQ q

r rπε−

=−

where 0 rε ε ε= By superposition, the force on Q due to n charges 1 2 3, , ,.... nq q q q is

21

14 k

nk

rk

Q qF arπε

= ∑

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

21

14 k

nk

rk

qFE aQ rπε

= = ∑

For the continuous charge distributions

2

14 r

dlE a

rλ

πε= ∫ for the line charge distribution

2

14 r

dsE a

rσ

πε= ∫ for surface charge distribution

2

14 r

dE a

rρ τ

πε= ∫ for volume charge distribution

Field Intensities

For a point charge Eε 2

14 r

QE arπε

=

For infinite line charge 1 24 rE a

rλ

πε=

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For infinite sheet charge 02 nE aσ

ε=

Electric Flux Density: D It is a quantity proportional to the no. of flux lines crossing unit area equal to Eε i.e. D Eε= . Gauss Law:

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

. encD da Q=∫ Integral form

D ρ∇ • = Differential form Gauss law is useful to compute the field intensity when the charge distribution is highly

symmetrical 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, coaxial cylinder or a concentric sphere.

The application of the Gauss law to find the field intensity of charge distribution requires the 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 remain constant.

Scalar Potential: V

The scalar potential V of a point P in an electric field E is defined as the external work 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 charge 14

QVrπε

=

For the continuous charge distributions

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For line charge distribution 14

dlV

rλ

πε= ∫

For surface charge distribution 14

daV

rσ

πε= ∫

For volume charge distribution 14

dVr

ρ τπε

= ∫

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 ρε

∇ = − which is Poisson’s

Law. If the volume charge density 0vρ = then 2 0V∇ = which is Laplace Equation. Divergence And Curl Of The Electrostatic Field:

( ) ( )2

1 ' '4

raE r r dr

ρ τπε

= ∫ % where 'r r r= −%

( )2

1. . ' '4

raE r dr

ρ τπε

∇ = ∇∫ %: But ( )3

2. 4ra rr

πδ∇ = %%

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

E r r r d rπδ ρ τ ρπε ε

∇ = − =∫

. vD ρ∇ = CONDUCTORS

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

outside the conductor,0

ˆE nσε

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

the surface, V is constant through out the conductor,

Polarization of materials

When 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 direction

as 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 field

direction, 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 and the 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

ra PVrπε•= .

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

20

14

ra PdV dr

τπε

• = .

The total potential due to the entire object is 20

14

r

vol

P aV dr

τπε

•= ∫

0

1 14 vol

P dr

τπε

= • ∇ ∫

0 0

1 14 4sur vol

P da P dr r

τπε πε

• ∇ •= −∫ ∫

0 0

1 14 4

b b

sur vol

da dr r

σ ρ τπε πε

= −∫ ∫

where ˆ.b P nσ = and .b Pρ = −∇ It means the potential and hence also the field of a polarized object is the same as

that produced by a volume charge density b Pρ = −∇ • plus a surface charge density ˆb P nσ = •

Gauss law in the presence of dielectrics: With in the dielectric, the total charge density can be written as b fρ ρ ρ= + where bρ = a result of polarization, bound charge density and fρ = which is not a result of polarization, free charge. Now the Gauss law reads 0 b f fE Pε ρ ρ ρ ρ∇ • = = + = −∇ • + where the E is total

field ( )0 0fE P D E Pε ρ ε→ ∇ • + = → ≡ +

In terms of the D , the electric displacement, Gauss law reads fD ρ∇ • = encD da Q→ • =∫ Linear dielectrics: in linear dielectrics, the polarization is proportional to the field 0 eP Eε χ= . The proportionality constant eχ is called electric susceptibility. So in linear media, we have

( )0 0 0 0 1e eD E P E E Eε ε ε χ ε χ= + = + = + . Now ( )0where 1 eD Eε ε ε χ= = + is called the

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permittivity of the material.0

1 e rk εχ εε

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

the material. For lossy dielectric ( )0

' '' ' 1 tanr r j j jσε ε ε ε ε θωε

= − = − = −& where

tanδ =loss tangent =0

''' r

ε σε ωε ε

=

Steady magnetic fields

Biot-savart law: the magnetic field intensity dH at an arbitrary point P due to a steady line current 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 2

14 4

r rIdl a dl aIdH Hr rπ π× ×= → = ∫

For surface currents 24rda aKH

rπ×= ∫ and

for volume currents 2

14

rJ aH dr

τπ

×= ∫

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

( )magF I dl B= ×∫ for line currents

( )magF K da B= ×∫ for surface currents

( )magF J B dτ= ×∫ for volume currents. Magnetic forces can do no work.

Ampere’s 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 formencH dl I• = →∫ and differential formencH I∇× = → Ampere’s law is useful in finding the field intensity when the current distribution exhibits symmetry 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 in

Magnetization of the material parallel to B ( )paramagnets or

Magnetization opposite to ( )diamagnetsB or Retention of substantial magnetization indefinitely after the external field has

been removed ( )ferromagnets A quantity used to describe the state of magnetic polarization of a material is Magnetization M = magnetic dipole moment per unit volume.

Field of a magnetized object: The vector potential of a single dipole m is

024

rm aAr

µπ

×= .

In the case of a magnetized object, the vector potential is 024

r

vol

M aA dr

µ τπ

×= ∫

0 0

4 4sur vol

M da M dr r

µ µ τπ π

× ∇×= +∫ ∫

0 0

4 4b b

sur vol

K da J dr r

µ µ τπ π

= +∫ ∫ where ˆbK M n= × and bJ 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 bJ M≡ ∇× through out the material plus a surface current ˆbK M n= × on the boundary.

Amperes law in magnetized materials The total current of the material can be expressed as b fJ J J= + where bJ is bound current, a result of magnetization and fJ is free current, not a result of magnetization.

Amperes law is ( ) ( )0

1f b fB J J J J M

µ∇× = = + = + ∇×

0f

B M Jµ

→ ∇× − =

: let

0

B M Hµ

− =

fH J→ ∇× = fenclH dl I→ • =∫ Linear media: For most substances the magnetization is proportional to the field 0 mM Hµ χ= , the proportionality constant mχ is called the magnetic susceptibility.

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( ) ( )1o o mB H M Hµ µ χ= + = + so

B Hµ= where ( )1o mµ µ χ= + 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 the electrons interact with one another. The magnetic behaviour of the materials can be classified into the following five major groups; • Diamagnetism • Paramagnetism • Ferromagnetism • Ferrimagnetism • Antiferromagnetism

The first two groups of materials exhibit no collective magnetic interactions and are not

magnetically ordered where as the materials of the last three groups exhibit long range magnetic 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 magnetic

moments because all the orbital shells are filled with no unpaired electrons. It is usually weak 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 the most important atoms with unpaired electrons is iron. However the individual magnetic moments do not interact magnetically and the magnetization is zero when the field is removed. 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 spontaneous magnetization and the existence of magnetic ordering temperature. The elements Fe, Ni and 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 ferromagnetic materials 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 the atoms 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 properties are similar to Ferromagnetism in that spontaneous magnetization, Curie temperature, hysterisis and remanence. But they have different magnetic ordering. Ferrimagnetism is exhibited by ferrites and magnetic garnets. The oldest known magnetic substance magnetite is a ferrimagnet. Widely used ferrimagnetic materials are YIG and ferrites composed of iron oxides and other elements such as aluminium, nickel, cobalt, manganese and zinc.

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

Electrodynamics Faraday’s law: In 1831 Michael Faraday performed three important epoch making experiments.

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

• Exp II: He moved a magnet moving its field holding the loop still resulting in current 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 law at 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 dNdtφ= − . A non-zero value

of ddtφ 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. Lenz’s law: The induced emf due to the time varying magnetic field is in such a direction as to 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 ( ) ( ) mFF Q v B v B EQ

= × → = × =

Maxwell's correction: In the case of steady magnetic fields H J∇× = which is ampere's law. The relation that holds for time varying magnetic fields must converge to this expression in case of no time variations. 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, Jtρ∂∇ • = −

∂. But the Gauss law says D ρ∇ • = .

With this relation the equation continuity becomes DJt

∂∇ • = −∇ •∂

resulting in

0 D Xt

∂= −∇ • + ∇ •∂

which gives DXt

∂=∂

Hence DH Jt

∂∇× = +∂

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 valid for time varying fields.

According to this Maxwell's correction of Ampere's law 'a changing electric field induces 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 rate

of 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 0

1 ( ) ( . )2S V V V

dE H da E H dv E J dv Edt

ε µ σ× • = − + + −∫ ∫ ∫ ∫

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

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( )2 00 0

1 ( ).2V V S

d dWE Jdv E H dv E H dadt dt

ε µ• = − + − × =∫ ∫ ∫

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 and

electric energies stored in the volume, plus • Complex power transmitted across the surface enclosed by the volume.

( ) ( )20

1 1 122 2 2 m e

v v v s

E J dv E dv j w w dv P dsσ ω∗− • = + − + •∫ ∫ ∫ ∫

Maxwell’s Equations

Maxwell’s assumption a changing displacement density was equivalent to an electric current density and as such would produce a magnetic field has had far-reaching effects. Faraday’s 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 Hertz’s experimental verification.

The following four electromagnetic equations are known as Maxwell’s equations because of the contribution to their development and established them as a self-consistent set.

When equations are transformed from time varying form into phasor form two changes take place: one is apparent t∂ ∂ gets replaced by jω , another is hidden fields and sources become independent of time. In the phasor form of equations the fields and sources are functions of just position only.

Equations in Integral Form Word 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 the magnetic flux through any surface bounded by the path.

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

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

Mathematical form involving time varying quantities:

DH dl J dst

∂ • = + • ∂ ∫ ∫ BE dl dst

∂• = − •∂∫ ∫

vD ds ρ• =∫ 0B ds• =∫ Mathematical form involving phasor quantities:

( )H dl J j E dsωε• = + •∫ ∫ E dl j H dsωµ• = − •∫ ∫

vD ds ρ• =∫ 0B 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 the conduction 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 negative

time-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 field

is 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:

DH Jt

∂∇× = +∂

BEt

∂∇× = −∂

vD ρ∇ • = 0B∇ • =

Mathematical form involving phasor quantities:

H J j Eωε∇× = + E j Hωµ∇× = −

vD ρ∇ • = 0B∇ • =

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

infinity, any amount of intensity can be explained interpreted and understood using the Maxwell’s equations.

o The time varying fields at a point, both electric and magnetic, obey Maxwell’s equations the fact of which is used to compute the fields many times. In the absence of the relations connecting the time varying fields with their sources, this observation is of very significant and useful.

o Maxwell’s equations lead to the development of wave equations. The field intensities of the time varying fields obey wave equations proving the existence of 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 just

outside 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 amount equal 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 the surface 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 Hµ∇×∇× = − ∇× & But in free space, according to Amperes law with Maxwell’s correction

H Eε∇× = & So

E Eµε∇×∇× = − && The vector identity is

2E E E∇× ∇× = ∇∇ • − ∇ So

2E E Eµε∇∇ • − ∇ = − && According to gauss law, as the medium is free space without any charge,

1 0E Dε

∇ • = ∇ • =

Hence 2E Eµε∇ = &&

Similarly it can be derived using the Maxwell’s equation 2H Hµε∇ = &&

In phasor form these two equations become 2 2E Eω µε∇ = − 2 2H Hω µε∇ = −

These two equations are called vector Helmholtz equations. In conducting media Applying curl on both sides of the Faradays law

E Hµ∇×∇× = − ∇× & But in free space, according to Amperes law with Maxwell’s correction

H J E E Eε σ ε∇× = + = +& & as J Eσ= is Ohms law. So

E E Eµσ µε∇×∇× = − −& && The vector identity is

2E E E∇× ∇× = ∇∇ • − ∇ So

2E E E Eµσ µε∇∇ • − ∇ = − − & &&

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19

According to gauss law, as the medium is free space without any charge, 1 0E Dε

∇ • = ∇ • =

Hence 2E E Eµσ µε∇ = +& &&

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

can be derived using the Maxwell’s equation In phasor form, these two equations become

( )( )

2 2

2

E j E

j j E E

ωµσ ω µε

ωµ σ ωε γ

∇ = −

= + =

( )( )

2 2

2

H j H

j j H H

ωµσ ω µε

ωµ σ ωε γ

∇ = −

= + =

Propagation constantγ The constant γ is known as propagation constant of the medium and in general a complex quantity having both the real and imaginary β parts. The real part is called attenuation constantσ and the imaginary part is called phase shift constant β .

jγ α β= +

( )j jγ ωµ σ ωε= +

2

2 21 12

µε σα ωω ε

= + −

2

2 21 12

µε σβ ωω ε

= + +

For good dielectrics

2

2 21 12

µε σα ωω ε

= + −

2

2 21 12 2

µε σωω ε

≅ + −

2σ µ

ε=

_________________________________________________________________________


20

2

2 21 12

µε σβ ωω ε

= + +

2

2 21 12 2

µε σωω ε

≅ + +

2

2 218

σω µεω ε

= +

For good conductors

( )j jγ ωµ σ ωε= +

1 jj ωεωµσσ

= +

045jωµσ ωµσ≅ = ∠ Therefore

2ωµσα β= =

Vector Helmholtz equations

21 0rr

E k Eεµ

∇× ∇× − =

21 0r

r

H k Hµε

∇× ∇× − =

Scalar Helmholtz equations

21 0z r zr

E k Eεµ

∇ • ∇ + =

. Similarly for other components of the E .

21 0z r zr

H k Hµε

∇ • ∇ + =

. Similarly for other components of the H .

_________________________________________________________________________


21

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 HE × . In fact E, H and direction of

propagation of the wave form RH vector system.

• εµ=

y

x

HE ,

εµ−=

x

y

HE

also εµ=

HE

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

• Transverse electromagnetic (TEM) wave: In this wave, also known as Principal wave, the electric vector E and magnetic

vector 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 propagation

all the three vectors form a right handed vector system. The energy travels as TEM wave in free space and over parallel wire transmission

line. The coaxial lines can also hold this type of wave. The phase velocity and group velocity is same for TEM wave. Neither one depends

upon 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 the normal 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 the normal and parallel components.

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22

• 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 depend upon 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 and

TE 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 are operating 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 iE E= − The electric field intensity in the standing wave pattern is ( ), 2 sin sinT iE x t E x tβ ω=% ♠ The magnetic field strength gets reflected without phase reversal i.e. r iH H= The magnetic field strength in the standing wave pattern is ( ), 2 cos cosT iH x t H x tβ ω=%

♠ In the reflected wave, the TE% and TH% are 900 apart in time-phase. Also there exists a surface current of ( )/ sec 0s TJ amp H x= = .

Oblique incidence: The plane of incidence is the plane containing the incident ray and the normal to the surface. ♠ Perpendicular polarization: 2 sin yj y

T i zE jE z e ββ −=

_________________________________________________________________________


23

where coszβ β θ= and sinyβ β θ= . ♠ Parallel polarization:

2 cos yj yT i zH H z e ββ −=

2 cos sin yj yy i zE j H z e βη θ β −=

2 sin cos yj yz i zE H z e βη θ β −=

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

travelling wave along the y −direction i.e. yj ye β− Perfect dielectric: ♠ Normal incidence: 1i iE Hη= , 1r rE Hη= − and 2t tE Hη= Tangential components are continuous i r tE E E+ = and i r tH H H+ =

1 2

1 2

r

i

EE

ε εε ε

−=

+; 1

1 2

2t

i

EE

εε ε

=+

♠ Oblique incidence: energy conservation requires ♠ Perpendicular polarization:

221 1

1

221 1

1

cos sin

cos sin

r

i

EE

εθ θε

εθ θε

− −

=

+ −

1 1 2 2

1 1 2 2

cos coscos cos

ε θ ε θε θ ε θ

−=

+( )( )

2 1

2 1

sinsin

θ θθ θ

−=

+

♠ Parallel polarization:

22 21 1

1 1

22 21 1

1 1

cos sin

cos sin

r

i

EE

ε εθ θε ε

ε εθ θε ε

− −

=

+ −

2 2 1 1

2 2 1 1

cos coscos cos

ε θ ε θε θ ε θ

−=

+( )( )

1 2

1 2

tansin

θ θθ θ

−=

+

_________________________________________________________________________


24

♠ Brewster angle: it is the angle of incidence at which there is no reflected wave when the

incident wave is parallel or vertical polarized. It is 1 21

1

tan εθε

−= .

♠ Total internal reflection: it takes place if the medium 1 is denser than medium2 and 1θ

is large enough to satisfy 1 21

1

sin εθε

−>

♠ Fields across media: If 1θ and 2θ are the angles made by the fields with normals in the

two different media then 1 1

2 2

tantan

θ εθ ε

= and 1 1

2 2

tantan

θ µθ µ

=

Skin depth or depth of penetration: when the medium is conductive, the wave gets attenuated as it progresses into the medium. Skin depth or depth of penetrationδ is defined as the depth in which the wave gets attenuated to 1 e or approximately 37 percent of its original value.

As the amplitude of the wave decreases exponentially with depth 1αδ =

2

2 2

1 1 2

1 12

δα ωµσµε σω

ω ε

= = ≈

+ −

for a good conductor.

Surface impedance: At high frequencies, when wave falls over the conductor the current flows which is confined almost entirely to a very thin sheet at the surface of the conductor.

Surface impedance is defined as the ratio of tangential electric field strength at the surface of the conductor to the surface current density that flows as a result of the incident wave.

tans

s

EZJ

=

If the conductor is flat plate with its surface at 0y = plane, then the current distribution in the y −direction will be 0

yJ J e γ−= where 0J is the current density at the surface which is related to the tangential electric field through 0 tanJ Eσ= .

_________________________________________________________________________


25

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 becomes 00

0 0

ys

JJ J dy J e dyγ

γ

∞ ∞−= = =∫ ∫

Therefore, the surface impedance tan 0

0s

s

E JZJ J

γ γσ σ

= = =

For conducting medium ( )j j jγ ωµ σ ωε ωµσ= + ≈

Hence 045sjZ γ ωµ ωµ

σ σ σ= = = ∠ .

The surface impedance is complex quantity and its real part is called surface resistance sR whereas its imaginary part is called surface reactance sX . For a thick good conductor their

magnitudes are same. 2sR ωµσ

≈ , 2sX ωµσ

≈

Observations:

• The surface impedance is equal to the intrinsic impedance η for the conducting medium.

• 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 a transmission line conducting energy into the interior of the metal.

_________________________________________________________________________


26

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

• The surface resistance sR is related to the depth of penetration or skin depth δ in

a conductor through 1sR

σδ=

• The surface resistance of a flat conductor at any frequency is equal to the dc resistance of a thickness δ of the same conductor. This means that the conductor having a thickness very much greater thanδ and having exponential current distribution throughout its depth has the same resistance as would a thicknessδ of the conductor with the current distributed uniformly throughout this thickness.

• The power loss per unit area of the plane conductor is 2seff sJ R

Polarization of waves o Polarization of a radiated wave is defined as “that property of an electromagnetic

wave describing the time-varying direction and relative magnitude of the electric field 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 observed along 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 is elliptically polarized. Circular polarization results when x − and y − components have 900 phase difference and equal in magnitude.

o Sense of polarization: when the wave is receding, if the resultant E vector rotates clockwise 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 wave o ( ) ayx EjaaE +=0 LCP or CCW o ( ) ayx EjaaE −=0 RCP or CW

_________________________________________________________________________


27

o yx bjBaAE +=0 LEP or CCW o yx bjBaAE −=0 REP or CW o Reversal of the sense of rotation can be obtained by giving 1800 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 = ellipsetheofaxismajorsemiellipsetheofaxisorsemi

ab min=

o Polarization ratio P = δj

x

y ePEE

=

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

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

• Electrical quantities representation: ( ),yo xoE E φ it uses the ratio

yo xoE E , the angle between them φ .

_________________________________________________________________________


28

If ( )1cot ARε −= ± and ( )1tan yo xoE Eγ −= , then the state can be represented

either by the pair ( ),ε τ or ( ),γ φ . • AR varies from 1 for circle to ∞ for line, ε varies from 4π− to 4π and

τ varies from 0 to π . • φvaries from 2π− to 2π and γ from 0 to π .

Poincare sphere provides a compact graphical representation of all the two types

and it also corresponds the above two representations. it is useful to find how close two polarization states are or how much interaction takes place between two states of polarization. Its salient features are

• Equator 0, ARε = = ∞ represents linear polarization. Longitude point 0τ = represents horizontal and 090τ = 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 , 2ε τ . The xy plane or horizontal plane represents 0ε = and xz vertical plane represents 0τ = .

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 they can 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 no interaction is possible between them.

Antennas And Polarization Polarization of the antenna is defined in its transmitting mode. The polarization of an antenna in a given direction is defined as “ the polarization of the wave transmitted by the antenna “. Polarization pattern of an antenna represents it’s polarization characteristics and it is defined as “ the spatial distribution of the polarization of a field vector radiated by the antenna taken over its radiation sphere. At each point over the sphere the polarization is usually resolved into a pair of orthogonal components along θ and φ directions. These components are called co-polarization and cross polarization. Polarization mismatch occurs due to the mismatch between the polarization of the receiving antenna and the polarization of the receiving wave. .Due to this mismatch the receiving antenna 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 the incoming wave or

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29

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 same direction.

The above observations are based on the fact that the power transfer efficiency between two

states represented by M and 'M on the Poincare sphere is given by 2 'cos2pol

MOMη ∠ =

Transmission-Line Theory

Transmission lines:

• These are metallic conductor systems involving two or more conductors separated by an insulator used to transfer low frequency electrical energy in TEM form from one point to another.

Types Balanced or differential Tr system:

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

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

• A balanced wire pair has the advantage of noise interference getting cancelled at the 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.

_________________________________________________________________________


30

• The standard two conductor coaxial cable is an unbalanced line as the second conductor 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 constant throughout its length if it is loss-less and decreases exponentially if the line is lossy.

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 the mismatched junction.

21

212ZZZZ

k+

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

towards both the sides.

Reflection loss:

_________________________________________________________________________


31

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

nepersZZZZ

RL21

21

2ln

+= db

ZZZZ

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 stubs tend 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 characteristic impedance not on the termination.

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

reduction ratio. It is given by r

fvε1= where rε 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 0Z but frequency dependant.

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

• Half wave line is also used to measure the impedance that is not accessible physically.

Quarter wave 4λ transmission lines: • Short circuited 4λ line is equivalent to parallel LC circuit. • Open circuited 4λ line is equivalent to series LC circuit. • Short circuited line with length 4λ> is equivalent to capacitorC . • Open circuited line with length 4λ> is equivalent to an inductor L . • Short circuited line with length 4λ< is equivalent to an inductor L . • Open circuited line with length 4λ< is equivalent to capacitorC .

_________________________________________________________________________


32

Applications of 4λ line: • Quarter wave transformer:

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

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

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

• Opened out parallel wire 4λ transmission line is used as wire radiator called ‘half wave

dipole’. • Opened out parallel wire transmission line of length less than 4λ is used as wire

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

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

Equivalent Circuit Representation Since each conductor has a certain length and diameter it must have resistance and conductance.

Since there are wires close to each other, there must be capacitance between them. Dielectric materials, 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,G and C are proportional to the

_________________________________________________________________________


33

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 be assumed to be limped at any one point. Losses in TR Lines: Losses in TR lines occur in types a) Radiation b) Conductor heating c) Dielectric heating. Radiation losses arise because a TR line may act as an antenna if the separation of the conductors is an appreciable fraction of wavelength. Conductor heating or I2 R loss is proportional to current and there for inversely proportional to characteristic impedance. It also increases with frequency because of the skin 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 0Z and propagation constant γ are called the secondary constants of the transmission line.

Characteristic Impedance 0Z :- The characteristic impedance of a transmission line,

0Z 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 its characteristic impedance.

S

S

IV

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

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

0 CjGLjRZ

ωω

++=

• Propagation Constant γ : It is defined as the natural

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

Se I

Ilog=γ , where

SI and RI are at a unit distance apart on the line of infinite length. It is a

_________________________________________________________________________


34

complex quantity and is represented by βαγ j+= where α is known as the Attenuation constant and β is known as the phase shift constant. It is related

to the primary constants through ZY=γ • loss- less line or RF line GR == 0

• Consider a small section dx or a transmission line. The series impedance of

this section will be dxLjR )( ω+ and the shunt admittance will be ( )dxCjG ω+

Then

( )dxLjRIVdVV ω+=+− )( , ( )dxCjGVIdII ω+=+− )(

⇒ − andILjRdxdV )( ω+= ICjG

dxdI )( ω+=−

⇒dxdIZ

dxVd −=2

2

and =2

2

dxId

dxdVY− LjRZ ω+=Q , CjGY ω+=

⇒ VZYdx

Vd =2

2

and ZYIdx

Id =2

2

2

22

d V Vdx

γ= and 2

22

d I Idx

γ=

where γ = Propagation Constant= ZY = ( )( )CjGLjR ωω ++ ) The general solution to the above equations can be expressed in either of the two forms: In terms of exponential functions.

_________________________________________________________________________


35

xx ebeaV γγ −+= : xx edecI γγ −+=

In terms of hyperbolic functions. xBxAV γγ sinhcosh += ; xDxCI γγ sinhcosh += Out of the four constants DandCBA ,, only two are independent. The last two equations can be written as

xBxAV γγ sinhcosh += ; )sinhcosh(1

0

xAxBZ

I γγ +−=

• In terms of sending end voltage and currents xZIxVV SS γγ sinhcosh 0−=

xZV

xII ss γγ sinhcosh

0

−=

• In terms of receiving end voltage and currents

)(sinh)(cosh 0 xlZIxlVV RR −+−= γγ ; )(sinh)(cosh0

xlZV

xlII RR −+−= γγ

• In terms of sending end impedance sZ and 0Z

( ) ( )[ ]zs

zs

s eZZeZZI

V γγ002

−++= −

( ) ( )[ ]zs

zs

s eZZeZZZI

I γγ00

02−−+= −

• In terms of receiving end impedance lZ and 0Z

( ) ( )[ ]dl

dl

l eZZeZZI

V γγ −−++= 002

( ) ( )[ ]dl

dl

l eZZeZZZI

I γγ −−−+= 0002

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36

Waves on Lines The 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= + then

o 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 LE is the sum of the voltages of the incident and reflected waves at the load, that is 1 2LE E E= +

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

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

• Pure travelling wave is one whose SWR is unity indicating no reflections from the

load. 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 0Z , 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 900 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 1800 phase difference with the points of previous half cycle. Similar is the case with the current wave form.

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

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

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

_________________________________________________________________________


37

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 Γ . 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

11

Lload

L

Z ZEE Z Z

−Γ = =

+

In case of zero loss line, the reflection coefficient has everywhere has the same magnitude and equals the reflection coefficient of the load. In case of lossy line the reflected wave becomes smaller and the incident wave larger with increasing distance from the load causing Γ to decrease correspondingly. The relation between the load voltage and current and the voltages of the incident and reflected waves at the load can be deduced as

01 1 2

L LL E I ZEE + = = + Γ

02 1 1 2

L LL E I ZEE E −Γ = Γ = = + Γ

Standing Waves

• The distance between two successive nodes or anti-nodes of voltage (or current) is

always 2gλ . And it is 4gλ between voltage node to current anti-node or voltage anti-node to current node.

• When the termination is open circuit, the current gets reflected with 1800 phase shift where as the voltage gets reflected without any phase shift. It results in current node and voltage anti-node over the open circuit termination.

• When the termination is short circuit, the voltage gets reflected with 1800 phase shift where as the current gets reflected without any phase shift. It results in voltage node and current anti-node over the short circuit termination

• When the termination load is either open circuit or short circuit or pure reactance the total incident wave gets reflected, as the load cannot dissipate any power. In such case 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 pure resistance, then voltage node and current anti-node occurs over the termination for

0RRL < . For the case 0RRL >

_________________________________________________________________________


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 2gλ . The impedances have inverse proportionality if the distance is odd multiples of 4gλ

• The line impedance is capacitive in a distance of 4gλ right side of voltage node and it is inductive in a distance of 4gλ to its left side.

Standing-wave Ratio

The ratio of the maximum amplitude to minimum amplitude possessed by the voltage or current distribution is defined as standing-wave ratio and denoted by’ ρ ’. The character of the voltage or current distribution on a transmission line can be conveniently 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 the load is either open- or short-circuited or is a lossless reactance. The SWR is one means of expressing the magnitude of the reflection coefficient ‘ Γ ’: the exact relation between the two is

11

ρ+ Γ

=− Γ

or 11

ρρ

−Γ =+

Significance of SWR: The importance of the standing wave ratio arises from the fact that it can be very easily 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 sZ and 0Z

zZZzZZ

ZZs

s

γγ

tanhtanh

0

00 −

−=

• In terms of receiving end impedance lZ and 0Z

dZZdZZ

ZZl

l

γγ

tanhtanh

0

00 +

+=

Input Impedance

lZZlZZ

ZZl

lin γ

γtanhtanh

0

00 +

+=

_________________________________________________________________________


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 lβ • 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 position 0

1tan2 Z

Zl L−=

πλ

Stub length 0

001' tan2 ZZ

ZZZZ

lL

L

OS −= −

πλ

Line Distortion:

• The deviation of the waveform at the output of the line from that at its input

is called line distortion. • It is due to the fact that all frequencies in the waveform do not have same

attenuation 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 being function 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 constantα is function of frequency. To eliminate this distortion the attenuation constant α must be made independent of frequency.

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

_________________________________________________________________________


41

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

• Frequency distortion can be reduced by cascading the lines with networks known as ‘equalizers’. Equalizer is network whose attenuation versus frequency characteristic is just opposite to that of the line. Delay distortion can also be reduced with equalizers, but it must be designed in such a way that the β for the total circuit is proportional to ω . For audio transmission frequency distortion is serious problem whereas for video transmission both are serious.

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

through the relationCG

LR = . With this interrelation among the primary

constants of the line the attenuation constant LCLR=α becomes

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

• Loading: in the actual linesCG

LR >> . To make the line distortion-less

LR is

decreased by increasing the inductance L . This is affected by either by changing the line configuration or by using high inductance coils. The method of reducing the distortion by increasing the inductance of the line is called ‘loading’. It is of two types.

• Continuous loading: the tape of steel or some other magnetic materials such as ‘perm alloy’ or ’mumetal’ is wound around the conductor to be loaded. It increases the permeability of the surrounding medium and thereby increasing the inductance. It is costly and used in sub-marine cables only.

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

• For telephone cable RZ = and CjY ω= . CR

vCR ωωβα 2;2

=== as α

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 used graphical aid in solving transmission line problems.

• The real utility of the smith chart lies in the fact that it can be used to convert the reflection 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 ll

l

ZzZ

+ Γ= =− Γ

11

r i

r i

jr jxj

+ Γ + Γ→ + =− Γ − Γ

( )2 2

2 2

11

r i

r i

r − Γ − Γ→ =− Γ + Γ

and( )2 2

21

i

r i

x Γ→ =− Γ + Γ

. These can

be rearranged as 2

2 11 1r i

rr r

Γ − + Γ = + + 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 x Γ − + Γ − =

it represents a family of

constant reactance circles with radius 1 x and centre at 11,r i xΓ = Γ = .

• 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 orthogonal o 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 only o 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 =Γ=Γ ir

_________________________________________________________________________


43

• The constant SWR circle is drawn on the chart with SWR as radius and ‘1’ of the horizontal line as centre.

o The impedances of represented by various points over the SWR circle denote the impedances of the line in a distance of 2gλ .

o Distance along the line is represented by angular distance around the chart, total circumference or 3600 corresponding to the line length of 2gλ

o The point at which the constant SWR circle intersects the horizontal straight line corresponds the SWR the load

• The upper half of the diagram represents inductive or positive reactance and lower

half represents capacitive or negative reactance. • There exists minV on the line at a point of min 1z ρ= and maxV exists where

ρ=maxz on the line. • The horizontal radius to the right of the chart centre corresponds to

max min max, , andV I z ρ and left of the chart centre corresponds to

min max min, , and 1V I z ρ • Circles of constant SWR ρ are concentric with the centre of the chart. These circles

intersect the zero reactance or susceptance axis at points ρ0z and ρ/0z • Radial lines represent loci of the constant line angle zβ . In the chart wavelength

scales corresponding to the line angle are included around the outside edge of the chart.

• For a lossy line not terminated in its characteristic impedance the path of travel on the chart from the load to the generator is a decreasing logarithmic spiral.

Single Stub Matching

• Plot the normalized impedance and draw the constant SWR circle on Smith chart. • Move a distance of 4λ along the constant SWR circle to locate load admittance.

Let it be 1P • On the SWR circle nearest to the load admittance point locate a point, which

represents admittance jb±1 . This is the point of intersection of constant SWR circle and 1=r circle. Let it be 2P

o Read the distance between 1P and 2P using the scale provided at the circumference of the chart. This gives the distance in wavelengths where the stub has to be placed from the load.

_________________________________________________________________________


44

o Starting from the point ( ∞∞ j, ) find the distance of the point at which the susceptance is jb± . This gives the length of the short-circuited stub in wavelengths to be connected for matching.

Double-Stub Matching

• Single stub matching is impractical when the stub is not being able to be placed physically in the ideal location. Particularly it is very difficult to place the stub at the exact required location in the case of coaxial lines.

• Plot the normalized

impedance and draw the constant SWR circle on Smith chart. Move a distance of 4λ along the constant SWR circle to locate load admittance. Let it be 1P

• From the point 1P move a distance equal to that of the first stub from the load along the constant SWR circle towards the generator and locate its admittance. Let it be 1dy

• Matching is not possible if 1dy is within 2=g circle for stub spacing is 8λ or 83λ and 1=g circle for stub spacing of 4λ

• Draw the spacing circle, which is the constant conductance unity circle, rotated

counter clockwise by λλd×

4360 degrees, λd is the distance between the stubs in

wavelengths. • Move from 1dy along constant −g circle to intersect spacing circle at two points

representing 11y and '11y . And then find the lengths of the SC stubs which can

neutralize the reactive parts of 11y and '11y

• If the load admittance is inside the forbidden region, move out of it clockwise along constant SWR circle. The distance thus travelled gives the position of the stub from the load.

_________________________________________________________________________


45

Multiple choice questions

1. The Smith chart is [D ] a) A Polar plot b) represents complex reflection coefficient

c) Inscribed in a unity circle d) all

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

c) Both d) none

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

c) Both d) none

4. The circles and arcs over the smith chart are [A ] a) Orthogonal b) opposite to each other c) At 450 d) None

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

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

a) Capacitive reactance b) capacitive susceptance c) Both d) None

7. 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) ‘1’of horizontal line b) centre of the chart c) Both d) None

9. The smith chart represents the line for a length of [ ]

a) 2gλ b) 2λ c) λ d) None

10. In the left half of the chart, the resistance [ ]

and reactance values are

a) Less than 1 b) more than 1

_________________________________________________________________________


46

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 1 c) 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 admittances c) Both d) None

17. Smith chart is useful to analyse [ ]

a) Loss-less lines b) lossy-lines c) 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 [ ]

_________________________________________________________________________


47

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

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

c) either one d) None 23. The point over constant SWR circle diametrically [ ] opposite to load admittance point is

a) Load admittance b) load impedance c) Either one d) None

24. Travel of length 2gλ 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 stays a) At the periphery b) over the horizontal line c) In the lower half d) in the upper half

26. If the load is pure resistance, then the load point [ ]

over the smith chart stays a) At the periphery b) over the horizontal line c) In the lower half d) in the upper half

27. The centre of the arcs of the smith chart is [C ] a) ( )1 ,0r r+ b) ( )1,1 x

c) ( )( )1 ,1r r+ d) ( )1, 1r r+ 28. The centre of the circles of the smith chart is [C ]

a) ( )1 ,0r r+ b) ( )1,1 x

c) ( )( )1 ,1r r+ d) ( )1, 1r r+ 29. The load impedance is pure resistance equal to [ ] characteristic impedance of line, then the

_________________________________________________________________________


48

load point over the smith chart will be a) At the centre b) at the periphery c) in the upper half d) in the lower half

30. In wave meter the pointer is [B ]

a) Oscillatory b) Rotary c) Stationary d) None

31. In the bench the source is always followed by [B ]

a) Attenuator b) Isolator c) Wave meter d) Detector

32. In the bench the source is modulated by a frequency [A ]

a) 1 KHz b) 10 KHz c) 100 KHz d) None

33. Tunable probe exists over / in [B ]

a) VSWR meter b) Slotted section c) Attenuator d) None

34. The method used to measure high VSWR is [B ]

a) Slotted line method b) Double minimum method c) Both d) None

35. Low VSWR method can be used to measure VSWR up to [A ]

a) ten b) five c) three d) None

36. In low VSWR method, in VSWR meter the pattern maximum [A ]

is set to

a) One b) zero db c) Both d) None

37. In high VSWR method, in VSWR meter the pattern minimum [C ] is set to a) One b) zero db c) Three db d) None

38. While measuring guide wave length, the termination of the bench [A ] must be a) Short b) Matched termination c) Open d) None

_________________________________________________________________________


49

39. While measuring guide wave length, always minimum points are [C ] considered because

a) Minima are sharp b) minima are easily locatable c) Both d) None

40. Guide wave length does not depend upon [A ] a) Termination b) frequency

c) Mode of the wave d) none 41. While measuring attenuation, the termination of the bench must be [B ]

a) Short b) Matched termination c) Open d) None

42. Short is required at a distance from diode in wave guide detector [B ]

a) 4λ b)

4gλ

c) 2λ d) None

43. Gyrator gives a phase shift to a wave travelling from output [B ]

to input a) 180o b) 0o c) 90o d) None

44. Calibration standards are [C ]

a) Rotary type phase shifter b) Rotary type attenuator c) Both d) None

45. When the depth of penetration is > λ /4, the susceptance

of the probe is [ ] a) Inductive b) Capacitive c) Nil d) None

46. The range of VSWR in db indicated VSWR meter is [B ] a) 0-2db b) 0-10db c) 0-5db d) 0-100db

38. For low power measurement the technique used is [A ]

a) Bolometer technique b) Calorimeter technique c) Reflect meter technique d) Calorimetric wave meter technique

39. The temperature coefficient of the Thermisters is [ ]

a) Positive b) negative c) Zero d) None

_________________________________________________________________________


50

40. The ‘Q’ of Cavity with single opening can be measured [ ] a) Transmission method b) Double minimum method c) Both d) None

41. Barretters have [ ] a. Positive temp efficient of resistance b. Negative temp coefficient of resistance c. Both d) None

42. The magic T can be used for the measurement of [B ] a) Frequency b) Impedance c) Q factor d) None

43. Impedance of the line to the left of voltage minimum is [ ]

a) Inductive b) capacitive c) High resistance d) low resistance 44. Impedance of the line to the right of voltage minimum is [ ]

a) Inductive b) capacitive c) High resistance d) low resistance 45. Impedance of the line at the voltage minimum is [ ]

a) Inductive b) capacitive c) High resistance d) low resistance 46. Impedance of the line at the voltage maximum is [ ]

a) Inductive b) capacitive c) High resistance d) low resistance 47. Thermo couple is a junction of [B ]

a) Two similar metals b) Two dissimilar metals c) Identical but different lengths d) None

48. The relation between VSWR (S) & reflection coefficient (P) is [ ]

a) S = PP

+−

11 b) S =

PP

−+

11

c) S = P

P+1

d) None

49. If rP is reflected power, iP is the incident power, [B ]

the reflection coefficient ‘P’ is a) r iP P b) r iP P

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51

c) i rP P d) i rP P

50. Q – factor is measured using [ ]

a) Reflect meter method b) Transmission method c) Power ratio method d) None

51. The bolometer that is having a negative temperature coefficient [ ] of resistivity that is called a) Barrater b) Varistor c) Thermisters d) Calorimeter

52. In laboratory experiments the output from Reflex [ ]

Klystrons are modulated by square waves because a) It is easy generative a square wave b) It prevents frequency modulation c) Detector circuit is easy to design d) The termination is less complicated

53. In microwave power measurement using bolometer the principle of working is the

variation of a) Inductance with absorption of power b) Resistance with absorption of

power c) Capacitance with absorption of power d) All

54. We use two 20db directional couplers along with two [ ]

detectors in which technique of impedance measurement a) Slotted line b) Reflecto-meter c) Heterodyne technique d) None

55. We use two matched detector in which

technique of q factor measurement [ ] a) Slotted line b) Reflectometers c) Heterodyne technique d) None

56. .The technique used to measure the dielectric constant is [ ] a) Slotted line b) Waveguide method

c) Reflect meter method d) Wave meter method 57. for impedance measurement the following oscillator is used [ ] a) Reflex klystron tube oscillator b) Gunn oscillator

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52

c) a and b d) a or b

Logarithmic units

Two voltages, or currents, differ by one neper when one of them is ‘ e ’ times as large as the other.

1 1

2 2

nepers ln lnV INV I

= = .

The ratio of input to output power can be expressed as 21

2

NP eP

=

The bel is defined as the logarithm of a power ratio,

1

2

number of bels=log PP

As it is more convenient to handle with a smaller unit, decibel abbreviated ‘db’ which is one tenth of a bel has been defined.

1

2

db=10log PP

In terms of voltages and currents 1 1

2 2

20 log 20logV IdbV I

= =

Nepers and dbs are related through 2 1010N dbe = giving 1neper 8.686db=

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53

MICROWAVE ENGINEERING

Microwaves: • The term microwaves refer to ac signals of small or tiny wavelengths. Even though the

limits of the microwave frequency range are not defined, the frequencies which fall in

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54

between 0.3 GHz to 300GHz with corresponding wavelength between 1m to 1mm respectively are considered as microwave region..

• IEEE RADAR BAND DESIGNATIONS

HF 3-30 MHz 100-10 mts VHF 30-300 MHz 10-1 mts UHF 0.3-1 GHz 100-30 cm. L-BAND 1-2 GHz 30-15 cm. S-BAND 2-4 GHz 15-7.5 cm. C-BAND 4-8 GHz 7.5-3.75 cm. X-BAND 8-12 GHz 3.75-2.5 cm. Ku-BAND 12-18 GHz 2.5-1.67 cm. K-BAND 18-26 GHz 1.67-1.0 cm. Ka-BAND 26-40 GHz 10-7.5 mm. MILLIMETER WAVES 40-300 GHz 7.5-1 mm. SUB-MILLIMETER WAVES

>300 GHz < 0.1 cm.

Salient Features

• Possibility of larger bandwidths: More band width (information-carrying capacity) is possible

• Possibility of smaller sized systems: Miniaturized communication systems possible because high gains are possible with a given physical size of antenna at microwave frequencies.

• Frequency reusability: Satellite and terrestrial communication links with high capacities are possible with frequency reuse at minimally distant locations because wave signals travel by line of sight.

• Achievability of larger antenna gains: More radar cross section and more antenna gain at microwave frequencies make these frequencies preferred for radar systems.

• Various molecular, atomic and nuclear resonance occur at microwave frequencies giving rise to applications in the areas of remote sensing, medical diagnostics and treatment and heating methods.

Major Applications

• Radar: these are the systems used for detecting and locating air, ground or sea going targets and also for air-traffic control systems, missile tracking radars, automobile

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55

collision-avoidance systems, whether prediction, motion detectors, and a wide variety of remote sensing systems.

• Communication systems: a large fraction of the wire less communication systems is micro wave communication systems that include long haul trunk telephone, data, and television transmissions. Direct broadcast satellite (DBS) television, cellular video(CV) systems and global positioning satellite systems(GPS) also use the microwave technology.

• Microwave Heating: Microwave oven is one application which is used not only for heating food but also in industrial and medical applications. The source used is in general a Magnetron tube operating at 12.5GHz with power out of either 500 or 1500W. When compared to conventional cooking, microwave cooking generally gives faster and more uniform heating of food more cooking efficiency.

• Energy Transfer: It is still in a stage of conception or infancy. Electrical power transmission lines are very efficient and convenient to transfer energy but there are cases where it is inconvenient or impossible to use such power lines. In such cases it is conceivable that electrical power can be transmitted with out wire by a well focussed microwave beam. One example is power transmission from solar satellite power station to earth. Another concept is transmission of electrical power from earth to a vehicle such a small helicopter or airplane.

• Electronic Warfare: Crippling the military radar and communication systems by deliberate means such as interference, jamming and other counter measures, is known as electronic warfare which can be divided into three major heads:

o Electronic support measures (ESM): these are the methods to detect the presence of a search or tracking radar, or the presence of a jamming signal using a receiver placed on aircraft, ships or ground vehicles.

o Electronic countermeasures (ECM): these are the methods used to either confuse or deceive a radar or communication system.

o Electronic counter countermeasures (ECCM): the aim of ECM is to make the radar or communication system ineffective. The purpose of the ECCM is to make the crippling too costly to achieve.

Hallow-Pipe Wave-Guides

• George C. Southworth and W.L Barrow invented wave-guides independently in 1930s. Earlier in 1893 Heaviside considered the possibility of the propagation of the wave inside closed hallow tube and Lord Raleigh later proved mathematically the possibility of propagation in wave-guide.

• Hallow-pipe wave-guides are single-conductor systems; usually made with highly conducting metal walls, which can support TE or TM waves. Wave guides have dimensions that are convenient in the frequency range of 3 to 100 GHz only.

_________________________________________________________________________


56

• If the cross-section is uniform then it is called cylindrical wave-guide, in case of rectangular cross-section then the guide is called rectangular wave guide and if the cross-section is circular, the guide is called circular waveguide.

• Wave guides have the advantages of o Higher power handling capability and low loss o Mechanical simplicity, simpler to manufacture o Flash-over less likely o Higher max. Operating frequency (325 GHz.)

But they are bulky and expensive. • Coaxial lines have

o very large band width and are convenient for test applications o But low max. operating frequency(18GHz.) and also it is difficult to fabricate

complex microwave components with coaxial lines • The wave travels through the dielectric filling the wave guide after getting reflected from the

walls instead of conduction along them. So it is the hallow region through which the wave actually travels and the purpose of the walls is to confine the wave.

• Even though the behaviour of the wave is same, the CWGs have geometry, mode designation and applications different from that of RWGs.

• CWGs are easy to join, easy to manufacture but their cross sectional area is bigger than RWGs for a given frequency and also rotation of the plane of polarization occurs in them.

• A serious disadvantage with CWG is that there is only a very narrow range between the cut-off wavelength of the dominant mode and the cut-off wavelength of the next higher mode. Thus the frequency range over which pure mode operation is assured is relatively limited.

• RWGs have the plane of the polarization of the wave uniquely defined with E across the narrow dimension but CWG is the most common form of a dual polarization transmission line.

• Guide wave length is an important parameter of waveguides which can be defined as the axial length corresponding to one cycle of variation of the field configuration in the axial direction. It also represents the distance that a wave travels down the guide when undergoing a phase shift of 2π radians.

Wave-guides compared with transmission lines:

• Wave guides are single conductor systems where as transmission lines are multi-conductor systems.

_________________________________________________________________________


57

• Energy travels in the form of TE or TM or hybrid mode wave in wave guides whereas in transmission lines it travels always in TEM wave.

• The frequency of the wave must be larger than certain value known as cut off frequency in order to be propagated through the wave guide whereas the wave of any frequency can be transmitted along the transmission line.

• Modal propagation is an important and special feature of the energy transfer through the waveguides where as the field distribution in transmission lines is non-modal in nature.

Analysis

In this section we find the field-distribution in the hallow pipe waveguides of both the rectangular as well as circular geometries. As the rectangular waveguide involves rectangular symmetry, Cartesian co-ordination is used in their analysis where as cylindrical coordinate system is used for circular waveguides as they involve cylindrical symmetry.

The procedure to be followed is same in the both the cases. Using Maxwell’s curl equations the transverse components are expressed in terms of longitudinal components in order to reduce the volume of the problem to one third: now the no. of unknowns is two. These two unknowns are then found by solving their respective wave equations.

Fields in Rectangular waveguides:

Let us suppose the wave-guide with inner dimensions a and b , ba ≥ is lying along z −axis carrying a travelling wave in the positive z direction. The walls of the waveguide are made with perfect conductor i.e. conductivity σ of the walls is ∞ and that hallow region is a perfect dielectric i.e. its conductivityσ is zero.

Let us also suppose the time variations of the field quantities are exponential i.e. j te ω . If the time variations are exponential then the fields must vary in the same manner i.e. exponentially along the direction of the propagation of the wave, according to transmission line theory. So the fields must vary with z as zeγ where γ is known as the propagation constant which in general is a complex quantity jα β+ .

The propagation constantγ is an important parameter describing the behaviour of the medium with respect to the wave. The real part α is known as the attenuation constant representing the attenuation offered to the wave by the medium. Te imaginary part β is known as the phase shift constant representing the phase change in the wave motion. If the medium offers no attenuation to the wave motion, then attenuation

_________________________________________________________________________


58

constant 0α = making the propagation constant γ a pure imaginary quantity i.e. γ jβ= . If the medium does not allow the wave into it behaving like a pure attenuator then 0β = making the propagation constant γ a pure real quantity i.e. γ α= . In general, when the wave is travelling through the medium, some amount of attenuation is offered by the medium; for such case the propagation constant is complex jγ α β= + .

In the present case, where the walls of the waveguide are made with perfect conductor and hallow region is a perfect dielectric, the power of the can not be absorbed by any one of them. Because perfect conductor walls are equivalent to a short circuit load and perfect dielectric region is equivalent to a open circuit load of circuit theory. The net result is unattenuated transmission of the wave with the propagation constant a pure imaginary quantity i.e.γ jβ= .

If E and H are electric and magnetic fields at an arbitrary point P in the hallow region,

they must be related through the Maxwell’s curl equations. tDH

∂∂=×∇ ;

tBE

∂∂−=×∇ .

Here the fields are function of , , andx y z t . Apart from this the fields and also their individual components separately obey wave equations.

In the analysis we aim to find the fields E and H which are vectors and so in general each one must have three components.. We find all the six components

xE , yE , zE , xH , yH and zH of the fields first by expressing the transverse components

xE , yE , xH and yH in terms of the longitudinal ones, zE and zH and then finding the longitudinal components by solving their wave equations.

The longitudinal components of the fields can be expressed in terms of the transverse components.

=xH 2zH

h xγ ∂−

∂

yE

hj z

∂∂+ 2

ωε : =yH 2zH

h yγ ∂−

∂ xE

hj z

∂∂− 2

ωε

=xE 2zE

h xγ ∂−

∂

yH

hj z

∂∂− 2

ωµ : =yE 2zE

h yγ ∂−

∂ xH

hj z

∂∂+ 2

ωµ

where 2 2 2h γ ω µε= + is called the characteristic equation. The constant h is also denoted frequently as ck and called cut-off wave number. In the above relations all the field quantities are functions of x and y only.

Critical Observations that can be made are: With 0=zE , 0=zH simultaneously all the field components become zero indicating TEM cannot exist inside the wave-guide. With 0≠zE , 0=zH all the components are not zero indicating the possibility of TM wave in the waveguide. With 0=zE , 0≠zH there exist non-zero field components indicating

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59

possibility of TE wave. With 0≠zE , 0≠zH the wave can exist in the guide, as there are non-zero field components. This wave called hybrid or mixed wave.

We find zE and zH by solving the wave equations which are partial differential

equations and hence their particular solution requires boundary conditions.

The boundary conditions to be used in the present context while solving the wave equation are derived from the fact that the tangential component of the electric field at the surface of the perfect conductor is zero. At 0=y and by = lies inner surfaces of the broader walls =xE zE=0 . At 0=x and ax= lies the inner surfaces of the narrow walls

=yE zE=0 Transverse Magnetic(TM) wave. 0=zH Wave equation for zE is

2

22

tEE z

z ∂∂=∇ µε .

With exponential time-variations as well as z -variations i.e. ( ) ( ), , , , z j t

z zE x y z t E x y e eγ ω−= the wave equation becomes

2 22 2

2 2z z

z zE E E Ex y

γ ω µε∂ ∂+ + = −∂ ∂

.

( )2 2

2 22 2 0z z

zE E Ex y

γ ω µε∂ ∂+ + + =∂ ∂

2 22

2 2 0z zz

E E h Ex y

∂ ∂+ + =∂ ∂

This equation, known as Helmholtz equation, is a second order, two dimensional partial differential equation in which zE is function of the x and y only i.e. it doesn’t involve z or t .

This equation can be solved using method of variable separation. According to this method, let zE XY= where X is function of variable x alone and Y is a function of variable y alone. With this assumption the above wave equation can be decomposed into two second order ordinary differential equations for which solutions are readily available.

2 22

2 2

1 1 0d X d Y hX dx Y dy

+ + =

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60

Let us suppose 2

22

1 d X AX dx

= − where A is an arbitrary constant. They the above equation becomes

22 2

2

1 0d YA hY dy

− + + =

( )2

2 2 22

1 d Y h A BY dy

= − − = − where B is another arbitrary constant. Here 2 2 2h A B= + .

The general solutions to the above equations are 1 2cos sinX c Ax c Ax= + and

3 4cos sinY c By c By= + making

( )( )1 2 3 4 cos sin cos sinzE XY c Ax c Ax c By c By= = + + where 1 2 3 4, , andc c c c are arbitrary constants whose values can be fixed with the boundary conditions. Now applying the boundary conditions 0 at 0, 0zE x y= = = give 1 30, 0c c= = whereas the

conditions 0 at ,zE x a y b= = = give andm nA Ba bπ π= = . Now if the product

2 4 , another arbitraryconstantc c C= , then

sin sin j zz

m nE C x y ea b

βπ π − =

and 22

2

+

=

bn

amh ππ

With these values of zH and zE other components can be computed using the relations connecting transverse components to longitudinal ones. After including the exponential time variations and z −variations, the complete set of the field components in the wave guide are

0=zH ; sin sin j z j tz

m nE C x y e ea b

β ωπ π − =

2 cos sin j z j tx

j m nE B C x y e eh a b

β ωβ π π − − =

2 sin cos j z j ty

j m nE A C x y e eh a b

β ωβ π π − − =

2 sin cos j z j tx

j m nH A C x y e eh a b

β ωωε π π − =

2 cos sin j z j ty

j m nH B C x y e eh a b

β ωωε π π −− =

Transverse Electric(TE) type wave:. 0=zE

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61

Wave equation for zH is

2

22

tHH z

z ∂∂=∇ µε .

When the time and z -variations are exponential i.e. ( ) ( ), , , , z j tz zH x y z t H x y e eγ ω−=

the wave equation becomes 2 2

2 22 2

z zz z

H H H Hx y

γ ω µε∂ ∂+ + = −∂ ∂

.

( )2 2

2 22 2 0z z

zH H Hx y

γ ω µε∂ ∂+ + + =∂ ∂

2 22

2 2 0z zz

H H h Hx y

∂ ∂+ + =∂ ∂

In this equation zH is function of the x and y only. Solving this equation using method of variables separation, the general solution will be

( )( )1 2 3 4 cos sin cos sinzH XY c Ax c Ax c By c By= = + + where 1 2 3 4, , andc c c c are arbitrary constants whose values can be fixed with the boundary conditions. As the boundary conditions are not available on zH and available on xE and yE , find these transverse components of the electric field using the available expressions for zH and zE . Then apply the boundary conditions to fix the values of the arbitrary constants.

0 at 0 and 0 at 0y xE x E y= = = = give 2 40, 0c c= = whereas the conditions

0 at and 0 aty xE x a E y b= = = = give andm nA Ba bπ π= = . Now if the product

'1 3 , another arbitraryconstantc c C= , then

22

2

+

=

bn

amh ππ and

' cos cos j zz

m nH C x y ea b

βπ π − =

As the boundary conditions are not available on zH , with the general solution for zH obtain xE , yE using the relations connecting transverse components to lateral components and then fix the values for the constants with the available boundary conditions. With these values of zH and zE other components can be computed using the relations connecting transverse components to longitudinal ones.

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62

0=zE ; 'cos cos j z j tz

m nH C x y e ea b

β ωπ π − =

2 'cos sin j z j tx

j m nE A C x y e eh a b

β ωωµ π π − =

2 'sin cos j z j ty

j m nE B C x y e eh a b

β ωωµ π π −− =

2 'sin cos j z j tx

j m nH B C x y e eh a b

β ωβ π π − =

2 'cos sin j z j ty

j m nH A C x y e eh a b

β ωβ π π − =

Filter characteristics of the rectangular waveguides: The hallow pipe waveguides behave like high-pass filters. They admit and allow the wave to propagate through them only if the frequency of the wave is more than certain value known as the cut off frequency whose value depends upon the dimensions of the guide and the mode of the wave.

Cut-off frequency cf : It is the frequency above which the frequency of the wave should be in order to get entry into the wave guide for propagating through it. It depends upon the dimensions of the guide and as well on the mode of the wave.

Cut-off wavelength cλ : The wavelength corresponding to cut-off frequency is called cut-off wavelength. It can be defined as the wavelength below which the wavelength of the wave should be in order to get entry into the waveguide for propagating through it. Its value is related to the dimensions of the guide and mode numbers of the wave through

2 2

2cm na b

λ = +

.

Proof:

We know the propagation constant γ can be related to the frequency ω of the wave through

2 2 2hγ ω µε= − or 2 2hγ ω µε= − Depending upon the relation between 2h and 2ω µε the propagation constant γ can be pure real or imaginary.

When 2 2h ω µε> the propagation constant is pure real quantity indicating that the wave guide is acting as a pure attenuator without wave motion refusing entry to the wave.

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63

When 2 2h ω µε< the propagation constant is pure imaginary quantity indicating that the wave guide is acting as a pure transmission line without any attenuation to the wave. This must be the case to which the waveguide under consideration belongs because the wave is already in and loss less condition is assumed.

The change over in the behaviour of the waveguide from pure attenuator to pure

transmission line takes place as the frequency is increased from low to high when 2 2h ω µε= . The frequency of the wave which satisfies this relation is called cut off

frequency cω . Hence

2 2c hω µε = . But

222

+

=

bn

amh ππ

Solving for cf we get 22

21

+

=

bn

amfc µε

The corresponding wavelength, cut off wave length can be found from 2 21 2c

c c

v m na bf f

λµε

= = = +

Cut-off frequency cf and Cut-off wavelength cλ are same for both the types of the wave i.e. TE or TM in rectangular wave guides.

Modal propagation characteristics of rectangular waveguides:

The electromagnetic energy propagation along a wave guide in the form of some definite field patterns known as ‘modes’ is an important and special feature of the energy propagation through the wave guides. The mode subscripts in RWG are denoted with m and n in that order as mnTE or mnTM . • Subscripts m and n indicate the number of half period variations of the fields along

x and y directions respectively. For TM wave m and n can assume any integer value from 1, 2, 3 etc. For TE wave m and n can assume any integer value from 1, 2, 3 etc. but either m or n only one at a time can assume zero value.

• Dominant mode of a wave-guide is one, which has the lowest cut-off frequency. It is 10TE for RWG. The significance of the dominant mode is smaller wave-guide is

sufficient to transmit a given frequency dominant modes. Another advantage is as the frequency can be transmitted in dominant mode in a guide that is too small to hold

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64

higher order modes the energy loss through the generation of spurious modes is prevented.

• For this mode the cut-off wavelength is twice the inner distance between side walls:: 0 2aλ = .

• The field components when the wave is in dominant mode are

2

2

0 ; 'cos

0 ; 'sin

'sin ; 0

j z j tz z

j z j tx x

j z j ty y

E H C x e ea

jE H B C x e eh a

jE B C x e e Hh a

β ω

β ω

β ω

π

β π

ωµ π

−

−

−

= =

= =

− = =

• We can observe the electric field is entirely in y − direction whereas the magnetic field

is devoid of a component in y − direction. • Regarding the field distribution in y − direction, the electric field is maximum at the

middle of the guide becoming nil at the ends. • The magnetic field has both x and z components; so it can only be in the xz −plane • Any plane containing E vector and parallel to the narrow walls of the waveguide is

called E − plane where as the plane containing H vector and parallel to the broader walls is called H − plane.

• Higher order modes i.e. modes with large m and n values can be used to transmit several

signals simultaneously through one wave-guide. These modes, as they require the wave guides of larger dimensions, are also capable of transmitting higher frequencies. But they are difficult to excite and also require wave-guides of larger dimensions to transmit a given frequency.

• Degenerate modes are different modes of wave-guides having the same cut-off frequency. Ex. mnTE and mnTM .

• Evanescent modes are modes which are beyond cut off i.e. wavelength more than cut-off value and so cannot propagate. They represent localized field distribution i.e. induction fields, that introduce reactive effects but do not carry energy away from the point of origin as does the dominant mode.

• Suppression of unwanted modes: It is preferred to operate waveguides so that only single pure mode is present because coupling systems and terminations can be designed on the basis of a definitely known type of field pattern. Waveguide carrying more than one mode is called ‘overmoded’

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65

o In most of the cases, the dominant mode is preferred because the guide then has the smallest possible dimensions and the undesired modes can be very simply eliminated.

• By proportioning the guide so that it is large enough to transmit the dominant mode while too small to permit propagation of any other mode, the higher modes do not travel down the guide, but rather are confined to the region where they are generated.

• In rectangular guides, when the guide is so proportioned that 2a b = there is a two to one frequency range over which only the dominant mode propagates. In contrast if the guide were square, the 01TE mode would have the same cut off wavelength as the 10TE mode, and there would be no frequency range over which only a single mode would propagate. Because of the considerations of this type, rectangular guides are practically always proportioned so that 2a b = , as this ratio gives the best mode separation of all possible proportions.

• If the ratio 2a b < then the range frequencies over which single mode propagation is possible gets reduced. If the ratio 2a b > then the power handling capability of the waveguide decreases. So the optimum ratio is 2a b =

o Dimensions versus frequency: Single mode propagation exists in rectangular wave guide proportioned so that 2a b = from a frequency of 2c a to c a . If lower frequencies are to be transmitted then 2c a should be low thus requiring the waveguide of larger dimensions. To be used for transmitting higher frequencies, c a should be large thus requiring waveguide of smaller dimensions.

Bandwidth of rectangular waveguides:

The bandwidth of a rectangular waveguide is for all practical purposes, less than a 2:1 frequency range because the mode 20TE begins to propagate at a frequency equal to twice the cut off frequency of the 10TE mode.

The cut off frequency of the dominant mode can be lowered and consequently the bandwidth increased by loading the waveguide with a conducting ridge on the top and/or bottom walls.

Apart from increasing the bandwidth, the loading gives better impedance characteristics making the guide suitable for impedance matching purposes.

The presence of the ridge however reduces the power handling capacity of the waveguide.

Excitation of the rectangular waveguides:

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66

The mode of propagation of the wave is determined by the type and location of the excitation. Although either probes or loops may be used as excitation sources, the probes are normally preferred for their simplicity.

The guide is closed at one end by a conducting wall and an appropriate exciting probe is inserted through the end or side of the guide. The end of the guide serves as a reflector and if the distance between the probe and the wall is properly adjusted, the reflected waves arrive at the probe in phase with the emitted wave, and the two propagate down the guide as one wave.

The probes should coincide with the positions of maximum electric field intensity in the modes they are intended to excite, with attention being given to the proper phasing of the potentials supplied to the probes in accordance with the phasing of the fields to be excited. If the loops are used for excitation, the plane of the loop will be made normal to the magnetic field and the loop will be located at a point of maximum magnetic field intensity.

The sources excite not only the desired modes but also higher order unwanted modes. But the choosing the guide dimensions appropriately, it is possible to have only the

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67

desired wave above cutoff frequency, the other waves then being attenuated and not propagated.

Fields in Circular Waveguides

Let us suppose the circular wave-guide of inner radius ‘ a ’ is lying along z −axis carrying travelling wave in positive z direction. The walls of the waveguide are made with perfect conductor i.e. conductivity σ of the walls is ∞ and that hallow region is a perfect dielectric i.e. its conductivityσ is zero.

Let us also suppose the time variations of the field quantities are exponential i.e. j te ω . If the time variations of the field quantities are exponential then the fields must vary in the same manner i.e. exponentially along the direction of the propagation of the wave, according to transmission line theory. So let us assume the fields vary with z as zeγ . Here γ is the propagation constant along the direction of propagation i.e. z − direction, which is in general a complex quantity α β+ .

o α is the attenuation constant along z − direction representing the attenuation to

the wave. o β is known as the phase shift constant

along z − direction representing the phase shift in the wave motion.

o Pure real γ indicates no wave motion, only attenuation.

o Pure imaginaryγ indicates wave motion with out attenuation.

o Complex quantityγ indicates wave motion with attenuation.

According to Maxwell where ever time varying fields exist there electromagnetic field exists and the converse is also true. As the wave exists inside the waveguide there fields must exist. If E and H are fields at an arbitrary point P in the hallow region, they must be related through the Maxwell’s curl equations. As we consider this problem in cylindrical coordinate system, the fields are functions of , , and .z tρ φ

The longitudinal components of the fields can be expressed in terms of the transverse components.

=ρH ρ

γ∂

∂− zHh2

φρωε

∂∂+ zE

hj

2 : =φHφρ

γ∂

∂− zHh2 ρ

ωε∂∂− zE

hj

2

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68

=ρE ρ

γ∂∂− zE

h2 φρ

ωµ∂

∂− zHhj

2 : =φEφρ

γ∂

∂− zEh2 ρ

ωµ∂

∂+ zHhj

2

where µεωγ 222 +=h . The constant h is also denoted frequently as ck and called cut-off wave number.

o The boundary conditions to be used while solving the wave equation are derived based on the fact that the tangential component of the electric field at the surface of the perfect conductor is zero. At a=ρ the inner surface of the CWG exists so the tangential component of the electric field must be zero i.e. =φE zE=0 at a=ρ .

Transverse Magnetic wave 0=zH

Wave equation for zE is

2

22

tEE z

z ∂∂=∇ µε .

When time variations and z -variations are exponential i.e. ( ) ( ) tjz

zz eeEtzE ωγφρφρ −= ,,,, then the wave equation becomes

zz

zzz EEEEE µεω

ρργ

φρρ22

22

2

2

2

−=∂

∂++∂

∂+∂∂ .

In this equation zE is function of the ρ and φ only. This equation can be solved using method of variable separation. According to this method, let zE PQ= where P is function of variable ρ alone and Q is a function of variable φ alone. With this substitution the above wave equation can be decomposed into two second order differential equations for which solutions are readily available.

22

2

1 d Q nQ dx

= −

2 22

2 2 0P P nh Pρ ρ ρ ρ

∂ ∂+ + − = ∂ ∂ where n is an arbitrary integer constant.

The first one is standard second order differential equation for which solution is available. cos sinn nQ A n B nφ φ= +

The second equation is known as the Bessel equation. Its solutions which can represent the physical fields are known as Bessel functions ( )nJ hρ . These are called Bessel functions of first kind and n order. Now

( )cos sin ( )z n n nE PQ A n B n J hφ φ ρ= = + .

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69

This is general solution and to fix the arbitrary constants we use the boundary conditions. Applying 0 atzE aρ= = gives

( )nm

nm ha

hah == where ( )nmha are the roots of the equation ( ) 0nJ ha = in the

increasing order of their magnitude. Hence ( ) ( )cos sin j z

z n n n nmE A n B n J h e βφ φ ρ −= + It is also possible to orient the reference to φ axis so that 0nB = . Therefore

( )cos j zz n n nmE A n J h e βφ ρ −= With these values of zH and zE other components can be computed using the relations

connecting transverse components to longitudinal ones. And attaching the time-variations they become

( ) ( )cos j z j tz n n nmE A J h n e eβ ωρ φ −=

( ) ( )2 sin ;j z j tn n nm

nm

jH nA J h n e e E Hh

β ωρ φ ρ

ωε βρ φρ ωε

−= − = −

( ) ( )' cos ;j z j tn n nm

nm

jH A J h n e e E Hh

β ωφ ρ φ

ωε βρ φωε

−= − =

Transverse Electric(TE) wave 0=zE Wave equation for zE is

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70

2

22

tHH z

z ∂∂=∇ µε .

When both the time and z -variations are exponential i.e. ( ) ( ) tjzzz eeHtzH ωγφρφρ −= ,,,,

the wave equation becomes

zz

zzz HHHHH µεω

ρργ

φρρ22

22

2

2

2

−=∂

∂++∂

∂+∂

∂ .

( )2 2

2 2 22 2 2

z z zz z

H H H H h Hω µε γρ ρ φ ρ ρ

∂ ∂ ∂+ + = − + = −∂ ∂ ∂

Note in this equation zH is function of the ρ and φ only. o Solving this equation using method of variable separation and with the initial conditions

mentioned above gives ( ) ''

nmnm h

ahah == and; =zH '

nA ( )'nmn hJ ρ cos ( )φn zje β−

o As the boundary conditions are not available on zH , with the general solution for zH obtain xE , yE using the relations connecting transverse components to lateral components and then fix the values for the constants with the available boundary conditions

o With these values of zH and zE other components can be computed using the relations connecting transverse components to longitudinal ones.

( ) ( )' ' cos j zz n n nmH A J h n e βρ φ −=

( ) ( )

( ) ( )

' ''

' '' 2

cos ;

sin ;

j zn n nm

nm

j zn n nm

nm

jH A J h n e E Hh

j nH A J h n e E Hh

βρ ρ φ

βφ φ ρ

β ωµρ φβ

β ωµρ φρ β

−

−

− ′= =

= = −

Filter characteristics of the circular waveguide: Cut-off frequency cf and Cut-off wavelength cλ : The frequency of the wave should be more than a certain value, known as cut-off frequency, in order to get admitted into the waveguide to propagate further. Its value depends upon the dimensions of the guide as well on the mode of the wave.

µεπ2nm

chf = for TM and

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71

µεπ2

'nm

chf = for TE wave.

The wave length corresponding to the cut off frequency is cut off wavelength and it can be formally defined as the wavelength below which the wavelength of the wave should be in order to get admitted into the guide to propagate further. Its value depends upon the dimensions of the guide as well as on the mode of the wave. It is

( )nmc ha

aπλ 2= for TM and

( ),

2

nmc ha

aπλ = for TE wave.

Proof:

We know the propagation constant γ can be related to the frequency ω of the wave through

2 2 2hγ ω µε= − or 2 2hγ ω µε= − Depending upon the relation between 2h and 2ω µε the propagation constant γ can be pure real or imaginary.

When 2 2h ω µε> the propagation constant is pure real quantity indicating that the wave guide is acting as a pure attenuator without wave motion refusing entry to the wave.

When 2 2h ω µε< the propagation constant is pure imaginary quantity indicating that the

wave guide is acting as a pure transmission line without any attenuation to the wave. This must be the case to which the waveguide under consideration belongs because the wave is already in and loss less condition is assumed.

The change over in the behaviour of the waveguide from pure attenuator to pure

transmission line takes place as the frequency is increased from low to high when 2 2h ω µε= . The frequency of the wave which satisfies this relation is called cut off

frequency cω . Hence 2 2c hω µε = . But 2 2

nmh h= in case of TM wave and it is 2

nmh′= in case of TE wave. Solving for cf we get

µεπ2nm

chf = for TM and

'

2nmh

π µε= for TE wave.

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72

The corresponding wavelength, cut off wave length can be found from

( )nmc ha

aπλ 2= for TM wave and

( ),

2

nm

aha

π= for TE wave.

In case of the circular wave guides, both the cut-off frequency and cut-off wavelength depends upon the type of the wave i.e. their values are different for TE and TM waves in circular wave guides.

Modal propagation characteristics in the circular waveguides:

o The mode subscripts in CWG are denoted with n and m in that order as nmTE or

nmTM . o ‘ n ’ indicates the order of the Bessel function and m indicates the roots of the equation

( ) 0=xJn or ( ) 0' =xJn in the order of the magnitude. o The subscript ‘ n ’ can assume any integer value 0, 1, 2, 3 etc. but for the subscript ‘ m ’

zero value is forbidden and so it can assume any integer value 1, 2, 3 etc. Note 10TE and

00TM are not possible. o Dominant mode of a wave-guide is one, which has the lowest cut-off frequency. It is

11TE for CWG. o Degenerate modes are different modes of wave-guides having the same cut-off

frequency. Ex. mTE0 and mTM1 . Dispersive characteristics of the waveguides:

The media in which the velocity of the wave depends upon its frequency are called dispersive media. Otherwise they are called non-dispersive media. The velocities of the wave, both phase as well as the group velocity, in the wave guide vary with the frequency and hence hallow pipe wave guide is a dispersive medium. Also the TE/TM waves carried by these media are dispersive waves.

Dispersion of the signal spoils the original phase relation between different frequency components as it travels down the guide leading to signal distortion.

o Guide wavelength gλ : It is the distance between two consecutive equi-phase planes in

the wave-guide.

It is related to cut-off wavelength through the relation ( )21g cλ λ λ λ= −

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73

o Phase/wave velocity pv : It is the velocity at which the phase of the wave changes along the length of the guide.. It is more than or equal to the velocity of wave in free space. It is by

definition equal to ( ) ( )21p cv vω β λ λ= = −

o Group velocity gv : It is the velocity with which a narrow band signal travels in the guide.

It is always less than velocity of wave in free space. ( ) ( )1 21g cv d d vβ ω λ λ

−= = −

Proof: We know for the hallow pipe waveguide, either rectangular or circular, the propagation constant is 2 2hγ ω µε= − In the present case, it is given that the waveguide is lossless and the wave is inside, so the propagation constantγ must be pure imaginary i.e.

2 2h jγ ω µε β= − = And hence the phase shift constant becomes

2 2hβ ω µε= − But we have 2 2

ch ω µε= . So the phase shift constant becomes 2 2

cβ ω µε ω µε= −

( )21 cω µε ω ω= − ( )21 cf fω µε= − ( )21 cω µε λ λ= − To be precise, γ is the propagation constant along the z −direction i.e. along the length of the waveguide. So β must be the phase shift constant along the length of the waveguide. By definition phase shift constant is phase shift per unit length and in this case the length has to be considered along the z −direction. Hence

2

g

πβλ

=

where gλ is the wavelength along the z −direction called guide wavelength. Now

2g

πλβ

=

( )2

2

1 c

π

ω µε λ λ=

− ( )2

2

2 1 c

f

f

π λ

π λ λ=

− ( )21 c

λ

λ λ=

−.

Here we used the relation 1v f λ µε= = for the velocity of the wave. The phase velocity of the wave, from its basic definition, is

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74

pv ωβ

=

Substituting

( )21 cβ ω µε λ λ= − we get

( )21p cv v λ λ= − Similarly from its basic definition, the group velocity of the wave can be related to free space wavelength and guide wavelength by

( ) ( )1 21g cv d d vβ ω λ λ

−= = −

As the frequency of the wave is increased from the cut-off value to infinity, guide wavelength and phase velocity vary from infinity to their free space value where as the group velocity varies from zero to its free space value.

The phase velocity and group velocity are same for TEM wave. None of these two depends upon the frequency and so TEM wave is non dispersive. Impedances of the waveguides:

Oliver Heavyside first coined the term impedance in nineteenth century to describe the complex ratio V I in AC circuits. Schelkunoff extended this concept to electromagnetic fields in a systematic way and noted that impedance should be considered as a characteristic of the type of field as well as the medium.

o Wave impedance zZ of a wave-guide is defined as the ratio of transverse electric field strength to transverse magnetic field strength.

In case of rectangular waveguides 2 2

2 2

x ytransz

trans x y

E EEZH H H

+= =

+

( )21 for TE mode wavecη λ λ= −

( )21 for TM mode wavecη λ λ= −

In case of circular waveguides 2 2

2 2trans

ztrans

E EEZH H H

ρ φ

ρ φ

+= =

+

( )21 for TE mode wavecη λ λ= −

( )21 for TM mode wavecη λ λ= −

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75

Proof: Let us first consider the rectangular waveguides. For TE mode

=xE y

Hhj z

∂∂− 2

ωµ : =yEx

Hhj z

∂∂+ 2

ωµ

=xH 2zH

h xγ ∂−

∂ : =yH 2

zHh yγ ∂−

∂

Substituting these expressions with jγ β= in the basic defining relation we get 2

2g

z

fZ

π µλωµβ π

= =

( ) ( ) ( )2 2 21 1 1c c c

f vλµ µ µ

λ λ λ λ µε λ λ= = =

− − −

( ) ( )2 21 1c cµ λ λ η λ λε

= − = −

For TM mode

=xE 2zE

h xγ ∂−

∂ : =yE 2

zEh yγ ∂−

∂

=xH y

Ehj z

∂∂+ 2

ωε : =yHx

Ehj z

∂∂− 2

ωε

Substituting these expressions with jγ β= in the basic defining relation we get 2

2zg

Zf

β πωε π ελ

= =

( ) ( ) ( )2 2 21 1 1c c c

f vλ λ λ λ µε λ λ

λε ε ε− − −

= = =

( ) ( )2 21 1c cµ λ λ η λ λε

= − = −

The relations for circular waveguide can be derived by following the procedure similar to that for rectangular waveguides.

In general the wave impedance is a characteristic of the particular type of wave TEM, TE, and TM which may depend upon the type of line or guide, the material and the operating frequency.

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76

The wave impedance of the waveguide medium can be likened conceptually to the intrinsic impedance of the free space medium.

o Characteristic impedance 0Z : The concept of the Characteristic impedance has been

borrowed into wave guides from the transmission line theory. It has been defined in several different ways for a finite length line in transmission line theory in terms of the voltages and currents over the line and power through the line in the following forms.

Voltage-current formula ( )IVIVZ =,0 ,

Power-current formula ( ) ∗=II

PIPZ 2,0 ,

Power-voltage formula ( )P

VVVPZ2

,0

∗

=

where V and I are voltage and current and P is the power flowing over the line when extended to infinity, all represent peak phasors. o All the above formulae give same value for low frequency line carrying TEM

wave, but different values for different modes of wave-guide, which carry TE and TM waves.

o In case of rectangular waveguides, for dominant wave

Voltage-current formula gives ( ) TEZabIVZ

2,0

π= ,

Power-current formula ( ) ( )IVZIPZ .4

, 00π= ,

Power-voltage formula ( ) ( )IVZVPZ ,4, 00 π=

o In case of circular waveguides, for dominant wave

Voltage-current formula gives ( )0 , 520 gZ V Iλλ

= ,

Power-current formula ( )0 , 354 gZ P Iλλ

= ,

Power-voltage formula ( )0 , 764 gZ P Vλλ

=

o Since voltages and impedances are not uniquely defined for TE and TM waves

their characteristic impedance cannot be unique and for such waves it can be defined in several ways.

To match a wave-guide to an uniquely defined impedance, the usual approach is to use the definition that gives best agreement between theory and experimental data.

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77

Attenuation of the waveguides:

o Attenuation of the wave-guides can be divided into two categories. Reflective attenuation and dissipative attenuation. .

o Reflective attenuation When the frequency of the wave is less than the cut-off frequency ( cff < ) it cannot enter into the wave-guide. This behaviour of the wave-guide is described mathematically by ascribing large amount of attenuation to the wave-guide known as the ‘Reflective attenuation’

o When a waveguide is excited at a wavelength greater than cut-off, the electric and magnetic fields decay exponentially with distance at a very much more rapid rate than is accounted for by the dissipative losses.

o More over the rate of attenuation depends only on the ratio cλ λ of the free space wavelength to the cut-off wavelength; unlike the wave shorter than the cut-off wavelength, the attenuation is independent of the material of the guide walls.

o The exact law of attenuation per unit length in db is ( )254.6 1 cc

α λ λλ

= −

When the actual wavelength is much greater than cut-off then 54.6

c

αλ

≈

These relations apply to all modes of propagation in all types of waveguides. An important observation from these relations is when cλ λ large, the attenuation is substantially independent of frequency.

o Waveguides operated at wavelengths greater than cut-off are called waveguide attenuators and are often used as attenuators in signal generators.

o Dissipative attenuation. The energy of wave when it is travelling through the wave-guide ( cff > ) gets absorbed by the walls due to their finite conductivity and also by the hallow region due to its non-zero conductivity resulting in ‘Dissipative attenuation.’

• The attenuation due to the dielectric loss

dα mNpk /2tan2

βδ= .

Proof: Consider a perfect dielectric, then its permittivity

0 rε ε ε= .

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78

For an imperfect dielectric material the permittivity

( )0 0 00

tanr r rj jσε ε ε ε ε ε ε ε δωε

= = − = −

&

Here tanδ is known as loss tangent of the dielectric and it is equal to

tano r

σδωε ε

=

If the wave guide hollow region is filled with perfect dielectric material, then

2 2 2 20 0 rj h hγ β ω µε ω µ ε ε= = − = − assuming 1rµ =

If the wave guide hollow region is filled with imperfect dielectric material, then

2 2 2 2 20 0 0 0 0 0

0

tanr r rh j h jσγ ω µ ε ε ω µ ε ε ω µ ε ε δωε

= − − = − +

assuming 1rµ = .

2 2 2 2 20 0 0 0Let and tanr ra h x jω µ ε ε ω µ ε ε δ= − =

22 2 1But for

2xa x a x aa

+ ≈ +

.

Using this relation 2 2

2 2 0 0 0 00 0 2 2

0 0

tan tan22

r rr

r

h j jh

ω µ ε ε δ ω µ ε ε δγ ω µ ε ε ββω µ ε ε

= − + = +−

.

From this 2

0 0 tan2

rd

ω µ ε ε δαβ

= Np/meter for TE or TM wave.

In case of TEM wave this quantity becomes

0 0tan

2d rδα ω µ ε ε= Np/meter.

It can be observed that the dielectric loss becomes zero when the conductivity or loss tangent of the dielectric is zero. The above relations for attenuation constant due to the dielectric loss can be used for both the rectangular as well as circular wave guides.

• The attenuation due to conductor loss

./2

/ mNpguidewavethedownflowpower

lengthunitdissipatedpowerc −×

=α

Proof: Let us suppose the voltage and current phasors along the line of infinite length are

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79

0 0andz j z z j zV V e e I I e eα β α β− − − −= =

Then the average power transmitted is { } { } 20 0

1 1Re Re2 2

zavP VI V I e α∗ ∗ −= =

The rate of decrease of transmitted power along the line will be 2avav

P Pz

α∂− = +∂

The decrease of transmitted power per unit length of line is 2av avP Pα−∆ =

And this is the power dissipated per unit length Power lost per unit length 2Power transmitted

α= leading

to Power lost per unit length2 Power transmitted

α =×

For rectangular waveguides, o Power flow down the wave-guide

= dydxHZ transz

2

21

∫

= daaHE z∫∫ •× ∗Re21 = ( ) dydxHEHE xyyx∫∫ ∗∗ −Re

21

o Power dissipated/unit length

= daHRs 2

tan2 ∫

= [ ]∫∫ ∫∫ ==×+×× dzdxHadzdyHaR

yyxxs 2

0

2

022

For circular waveguides,

o Power flow down the wave-guide

= φρρ ddHZ transz

2

21

∫

= daaHE z∫∫ •× ∗Re21 = ( ) φρρρφφρ ddHEHE∫∫ ∗∗ −Re

21

o Power dissipated/unit length

= daHRs 2

tan2 ∫

= [ ]∫∫ =× dzdHaR

as φρ

ρρ2

2

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80

.

Micro-strip Transmission lines:

It is one of the most popular types of planar transmission lines. It can be easily integrated with other passive and active micro-wave devices. It can be fabricated by photolithographic processes. It can be viewed as a two wire line consisting of two flat strip conductors of width ‘ w ’

separated by a distance ‘ 2d ’ in the absence of the dielectric. The micro-strip line can not support a pure TEM wave. In most practical applications,

the dielectric substrate is electrically very thin i.e. d λ and so the fields are quasi-TEM. The phase velocity p ev c ε=

The propagation constant 0 0 eβ ω µ ε ε=

1 1 12 2 1 12

r re d w

ε εε + −= ++

Micro strip has most of its field lines in the dielectric region and some fraction in the air region above substrate.

It is an asymmetrical type strip transmission line.

Strip line Transmission lines: Planar type transmission line widely used in microwave integrated circuitry.

It is a sort of ‘flattened out’ coaxial line. It has two conductors and a homogenous dielectric. It supports TEM wave

and this is the usual mode of operation. Strip-line can also support higher order TM and TE modes but these are

suppressed with shorting screws between the ground planes and restricting ‘b ’ to less than 4λ

The analysis of the strip-line is quite complex and difficult process. A reasonable approximation to the exact results with the application of Laplace equation gives.

Cavity Resonators

These are tuned circuits at the highest frequencies. Their behaviour is identical to a LC tuned circuit. Theoretically a given resonator has an infinite number of resonant modes and each mode corresponds to a definite resonant frequency. So each cavity resonator has an

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81

infinite number of resonant frequencies. The mode having the lowest resonant frequency is known as the dominant mode.

In practice the rectangular-cavity resonators, circular-cavity resonators and re-entrant-cavity resonators are commonly used in many microwave applications. Types: Regular shaped resonators like spheres, cylinder or rectangular prisms. But their various resonant frequencies are harmonically related which is a defect. Irregular shaped resonators known as re-entrant cavities are also useful as tuned circuits as well as they can be easily integrated into the structure of the microwave device. It is also convenient to couple the signal to and take the signal form the re-entrant cavity.

A re-entrant cavity is one in which the metallic boundaries extend into the interior of the cavity. The examples are coaxial cavity, radial cavity, butterfly cavity etc. They are designed to for use in klystrons and microwave triodes. One of the commonly used re-entrant cavities is the coaxial cavity. Tuning: With adjustable screws or posts, by introducing solid dielectric material or ferrites, by moving a wall in and out slightly with screw are some of the commonly used methods of cavity tuning. Coupling: Power can be coupled using slots, loops and probes. But they load the cavity and also change its resonating frequency. Beam coupling is another method of power coupling of the cavity which is widely used in microwave tubes. Applications: Used as input and output circuits of amplifiers as well as in oscillators. These are also used in filters, with mixers. Another major application of the cavity resonator is as cavity wave meter Analysis of the regular cavities:

Consider a piece of rectangular wave guide lying along z −axis carrying a travelling wave in positive z direction.

When the opening at the output side is closed with a shorting plate, made with perfect conductor, the forward travelling or positive z travelling wave hits the shorting plate and gets reflected. Now there comes into being another wave travelling in negative z direction. Due to the interference between the two waves travelling in opposite directions but over the same path, the standing wave pattern comes into being.

When the opening at the inlet side is also closed with another shorting plate, the input to the forward wave is cut off but its place is taken over by the wave resulted due to reflection of the negative z travelling wave. The net result is the trapping of a travelling

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82

wave which keep on travelling in between the shorting plates for ever in case of loss free conditions.

The frequency at which the wave hits the walls is called resonant frequency and in general it depends upon the dimensions of the wave guide as well as the mode of the wave. Consider the case Rectangular Cavity Resonator. Structurally it is a piece of waveguide closed at both the ends. So we start with a piece of rectangular wave guide of length ‘ d ’lying along z direction. When both the openings are remained opened and a wave is travelling in positive z direction then the fields in phasor form at an arbitrary point in the waveguide can be expressed as

ixoix EE = zje β− and ixoix HH = zje β−

iyoiy EE = zje β− and iyoiy HH = zje β−

izoiz EE = zje β− and izoiz HH = zje β−

When the opening at the outlet is closed, the forward travelling wave gets reflected and the fields of the reverse or –z travelling wave

rxorx EE = zje β and rxorx HH = zje β

ryory EE = zje β and ryory HH = zje β

rzorz EE = zje β and rzorz HH = zje β Due the combining or interference of these two waves standing wave pattern comes into being. The fields of the standing wave pattern are

ixosx EE = zje β− rxoE+ zje β ; ixosx HH = zje β− rxoH+ zje β

iyosy EE = zje β− ryoE+ zje β ; iyosy HH = zje β− ryoH+ zje β

izosz EE = zje β− rzoE+ zje β ; izosz HH = zje β− rzoH+ zje β It is possible to relate the amplitudes of the reflected waves with those of the incident waves. Magnitude wise: As the waveguide walls are made with perfect conductor and hallow region is made up of perfect dielectric the resonator under consideration is a loss-less system. So the magnitudes of the reflected waves remain same as that of the incident waves. And phase wise: The tangential component of the E fields and normal components of the H fields suffer180 o phase shift and the normal components of the E fields and tangential components of the H fields suffer no phase shifts on reflection at the surface of the perfect conductor resulting in

ixoE rxoE−= izoE rzoE=

iyoE ryoE−= ixoH rxoH=

izoH rzoH−= =iyoH ryoH With these relations the fields in the standing wave become

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83

2 sinsx ixoE jE zβ= − 2 cossx ixoH H zβ=

2 sinsy iyoE jE zβ= − 2 cossy iyoH H zβ=

2 cossz izoE E zβ= 2 sinsz izoH jH zβ= −

o For TM mode fields are.

0izoH = ; sin sinizom nE C x ya bπ π =

2 cos sinixoj m nE B C x y

h a bβ π π − =

2 sin cosiyoj m nE A C x y

h a bβ π π − =

2 sin cosixoj m nH A C x yh a bωε π π =

2 cos siniyoj m nH B C x yh a bωε π π− ==

o For TE mode fields are

0izoE = ; 'cos cosizom nH C x ya bπ π =

2 'cos sinixoj m nE A C x yh a bωµ π π =

2 'sin cosiyoj m nE B C x yh a bωµ π π− =

2 'sin cosixoj m nH B C x yh a bβ π π =

2 'cos siniyoj m nH A C x yh a bβ π π =

gλπβ 2= and 2

0

1

−

=

λλ

λλg

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84

When the both ends are closed, the wave gets trapped inside the waveguide and keep on travelling from one end to another end with a certain frequency know as resonant frequency. Certain field components like xE , yE and zH get reflected with phase reversal and as a result there exists minima or nodes occur at both the ends in their standing wave pattern. Other field components like xH , yH and zE get reflected with out any phase reversal resulting in maxima or anti-nodes at both the ends in their standing wave pattern. From the properties of the standing waves we know the distance between two nodes or anti-nodes and hence the distance p between the ends must be an integral no. of half (guide) wave lengths. i.e.

dppd g πβ

λ=⇒=

2

With both the openings closed , the fields inside the resonator become

, 2 sinres x ixopE jE zdπ = −

, 2 cosres x ixo

pH H zdπ =

, 2 sinres y iyopE jE zdπ = −

, 2 cosres y iyo

pH H zdπ =

, 2 cosres z izopE E zdπ =

, 2 sinres z izo

pH jH zdπ = −

Resonant frequency: We know that for a rectangular wave guide

2 2hβ ω µε= − . In the case of the resonator, this relation becomes 2 2r

phdπβ ω µε= − = .

Solving this equation for rf with 22

2

+

=

bn

amh ππ leads to the expression for the

resonating frequency 222

21

+

+

=

dp

bn

amfr

πππµεπ

The physical significance of the mode subscripts nm , and p depends upon the fact that they represents the number of half-wave periodicity in yx, and z directions respectively. The allowed integral values for them are

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85

For TM mode:: ,.....3,2,1=m ,.....3,2,1=n

.....,2,1,0=p For TE mode:: 0,1,2,3,.....m =

,.....3,2,1,0=n .....4,3,2,1=p

Dominant mode of the resonator is one having the lowest resonant frequency. It is TE101 for dba <> in the case of rectangular cavity resonator..

Consider the case Circular Cavity Resonator. We start with a piece of rectangular wave guide Fields of the forward or +z travelling wave

j zi i oE E e βρ ρ

−= j zi i oH H e βρ ρ

−=

j zi i oE E e βφ φ

−= j zi i oH H e βφ φ

−= j z

iz izoE E e β−= j ziz izoH H e β−=

Fields of the reverse or –z travelling wave j z

r r oE E e βρ ρ= j z

r r oH H e βρ ρ= zje β

j zr r oE E e βφ φ= j z

r r oH H e βφ φ=

j zrz rzoE E e β= j z

rz rzoH H e β=

Fields of the standing wave j z j z

s i o r oE E e E eβ βρ ρ ρ

−= + j z j zs i o r oH H e H eβ βρ ρ ρ

−= + j z j z

s i o r oE E e E eβ βφ φ φ

−= + j z j zs i o r oH H e H eβ βφ φ φ

−= + j z j z

sz izo rzoE E e E eβ β−= + j z j zsz izo rzoH H e H eβ β−= +

But the tangential component of the electric fields and normal components of the magnetic fields suffer 180 o phase shift and the normal components of the electric fields and tangential components of the magnetic fields suffer no phase shifts on reflection at the surface of the perfect conductor resulting in

i o r oE Eρ ρ= − izo rzoE E=

i o r oE Eφ φ= − i o r oH Hρ ρ=

izo rzoH H= − i o r oH Hφ φ= And hence the fields in the standing wave become

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86

jEE ois 2ρρ −= zβsin 2ois HH ρρ = zβcos

jEE ois 2φφ −= zβsin 2ois HH φφ = zβcos

2izosz EE = zβcos jHH izosz 2−= zβsin

TM mode fields. 0=izoH ; =izoE nA ( )nmn hJ ρ cos ( )φn zje β−

=oiH ρ 2nmh

njρωε− nA ( )nmn hJ ρ sin ( )φn zje β−

=oiH φnmh

jωε− nA ( )nmn hJ ρ' cos ( )φn zje β−

=oiE ρ φωεβ H

=oiE φ ωεβ− oiH ρ

o TE mode fields. 0=izoE ; =izoH 'nA ( )'

nmn hJ ρ cos ( )φn zje β−

=oiH ρ 'nmhjβ− '

nA ( )''nmn hJ ρ cos ( )φn zje β−

=oiH φ 2'nmhnj

ρβ '

nA ( )'nmn hJ ρ sin ( )φn zje β−

=oiE ρ oiH φβωµ

=oiE φ β

ωµ− oiH ρ

gλπβ 2= and

( )2

01g

λλλ λ

=−

When the both ends are closed, the distance between the ends must correspond to an integer no. of half(guide) wave lengths.

i.e. dppd g πβ

λ=⇒=

2

Resonant frequency:

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87

We know that for a circular wave guide 2 2hβ ω µε= − . In the case of the resonator, this relation becomes 2 2r

phdπβ ω µε= − = .

Solving this equation for rf with 22

2

+

=

bn

amh ππ leads to the expression for the

resonating frequency 222

21

+

+

=

dp

bn

amfr

πππµεπ

leading to the resonating frequency of

( ) 22

21

+

=

dq

ahaf nm

rπ

µεπ for TM

( ) 22'

21

+

=

dq

aha nm π

µεπfor TE

• For TM mode ,...2,1,0=n , .....3,2,1=m , ,....2,1,0=q • For TE mode ,...2,1,0=n , .....3,2,1=m , ....3,2,1=q • n … indicates the periodicity in the φdirection • m ….indicates the number of the zeros of the field in the radial direction. • q …the number of half-waves in the axial direction

If ad < the dominant mode is 110TM and it is 111TE when The ‘Q’ factor of a cavity resonator: Q is a measure of the frequency selectivity of a resonant or anti-resonant

circuit. maximum energy stored2energy dissipated per cycle

Q π= WP

ω= where 2 2

2 2v v

W E d H dε µτ τ= =∫ ∫

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88

2

2s

ts

RP H da= ∫ here E and H are peak values of the field intensities and tH is the peak

value of tangential magnetic field intensity.

2

2v

s ts

H dQ

R H da

ωµ τ=

∫

∫( )

( )2 s

volumeR surface area

ωµ≈


1. The dominant TE mode in rectangular wave guide is [ D ]

a) TE01 b) TE11

c) TE20 d) TE10

2. Wave guides can carry [ D ] a) TE mode b) TM mode c) Mixed mode d) All

3. Cut – off wave length of Circular Wave-guide in TM mode is [ B ]

a) ) 22

2

+

bn

am

b) nmha

a)(

2 π

c) ( )'

2

nmhaaπ d) None

4. Cut off wave length of Rectangular wave guide is [ A ]

a) 22

2

+

bn

am

b) 22

2

+

an

bm

c) both d) None

5. Guide wave length of cylindrical wave guide is [ C ]

a) 2

01 λλ λ

−

b) 2

01 ff

λ −

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89

c) both d) None

6. Wave impedance of wave-guide in TE mode can be [ A ]

a) 2

01 λη λ

−

b) η 20 )/(1 λλ−

c) both d) None

7. Dominant modes in a cylindrical wave guide is [ C ] a) TE11 b) TE10 c) Either d) None

8. Dominant modes in regular cavity resonators can be [ A ]

a) TE111 b) TE110 c) Either d) None

9. Non existent modes in circular wave guides are [ B ] a) TE10 b) TE00 c) both d) None

10. Resonant frequency of Rectangular c resonator is [ A ]

a) 222

21

+

+

dp

bn

am

µε b) 1

2m n pa b dµε

+ +

c) 222

21

+

+

dp

bn

am

µεπ d) None

11. Attenuation constant due to dielectric loss is in Nepers/mt [ D ]

a) β

δβ2tan

2

b) 2

2 tanβ δβ

c) 2

2tanω δ

β d)

20 0

2r tanω µ ε ε δ

β

12. Attenuation constant due to conductor loss is [ A ]

a) powerflowx

lengthunitdissipatedpower2

/ b) lengthunitpowerDissipated

flowpowerx/

2

c) flowPower

lengthunitpowerdissix /2 c) None

13. Reflective attenuation comes into being when the frequency of the wave is [ A ]

a) < cut off frequency b) > cut off frequency

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90

c) both d) None

14. In RWG, the mode subscripts m and n indicate [ A ] a) no of half wave patterns b) No. of full wave patterns c) no of the zeros of the field d) None

15. In RWG, for dominant mode, the cut off- wave length is [ A ]

a) 2a b) 2b c) a d) None

16. The wave whose frequency is 1.5 GHz falls in the band [ A ] a) L b) S c) C d) None

17. The wave whose frequency is 2.5 GHz falls in the band [ B ]

a) L b) S c) C d) None

18. The wave whose frequency is 7 GHz falls in the band [ C ] a) L b) 5 c) C d) None

19. The wave whose frequency is 9 GHz falls in the band [ B ] a) L b) X c) C d) None

20. Degenerate modes in circular wave-guides are [ A ]

a) TE01 & TM11 b) TE22 & TM22 c) Both d) None

21. At infinite frequency, the guide wave length is [ B ]

a) Infinite b) Free space wave length c) cut-off wavelength d) None

22. An air filled rectangular waveguide has dimensions of 6 X 4cm. [ A ] Its cut-off frequency for TE10 mode is

a) 2.5GHz b) 25GHz c) 25MHz d) 5GHz

23. The phase velocity of the guided wave at a frequency

of 3.0GHz in TE10 for the above problem(probno:1) is [ ] a) 0.1m/s b) 5.42×108 m/s c) 5.4×106 m/s d) 3.78×108 m/s

24. In problem1 the group velocity is [ ]

a) 1.659×108 m/s b) 5.42×108 m/s c) 0.185×108 m/s d) 3.78×108 m/s

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91

25. In problem1, the wave impedance in the waveguide at 3GHz is [ ] a) 120π b) 681.72 c) 300 d) 600

26. In problem1, the phase constant in the waveguide is [ ]

a) 69 rad/m b) 100 rad/m c) 34.5 rad/m d) 50rad/m

27. The cut-off frequency of a waveguide depends on [ D ]

a). dimensions of the waveguide b). the dielectric property of the medium in the waveguide c). wave mode d) all

28. In hollow rectangular waveguides [ C ]

a). the phase velocity is greater than the group velocity b). The phase velocity is greater than the velocity of light in free space

c) both d) none

29. In a waveguide, the suffix m, n of the modes TE/TM denote [ B ]

a). half wavelength of E field and full wavelength of H field b) half wavelengths of E and H fields c) full wavelength of E field and half wavelength of H field d) half wavelengths of H and E fields

30. The waves in a waveguide [ B ]

a). travel along the border walls of the waveguide b).are reflected from side walls but do not travel along them c). travel through the dielectric without touching the walls d). travel along the all the four walls

31. Which of the following modes is not supported by RWG [ D ] a) TE10 b) TE11

c) TM11 d) TM10

32. For a wave of finite frequency in an air filled RWG [ D ]

a) guide-wavelength is never less than the free space wavelength

b) Wave impedance is never equal to the free space impedance

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92

c) TEM is possible d) all

33. Dominant mode in circular wave guide is [ B ]

a) TE10 b) TE11 c) TE01 d) TE12

34. In TE20 of RWG, the no of half waves in x − direction [ A ] a) 2 b) 1 c) 4 d) 0

35. In cylindrical waveguide TEZ is [ B ]

a) βωµ

b) ωµβ

c) ωβµ

d) ωµβ

36. Theoretically no. of modes that can exist in cylindrical waveguides [ D ]

a) Zero b) One c) 2 d) Infinite

37. The primary mode in a rectangular resonant cavity [ B ] a) TE111 b) TE101 c) TE100 d) TE001

38. Real power transmitted in a rectangular waveguide is [ D ]

a) E×H* b) H×E*

c) ½ Re( H×E*) d) ½Re( E×H* ) 39. A disadvantage of microstrip compared with strip-line is [ ] a) Does not readily lend itself to printed circuit technique b) More likely to radiate c) Bulkier d) complex and expensive 40. The transmission system using two ground planes is [ ] a) Microstrip b) Rectangular waveguide

c) Circular waveguide d) Strip line 41. A disadvantage of strip line wave guide is [ ] a) Smaller bulk b) Greater bandwidth

c) Higher power handling capability d) Greater compatibility with solid state devices

_________________________________________________________________________


93

42. A disadvantage of strip line over microstrip is its [ ] a) Easier integration with semiconductor devices b) Lesser tendency to radiate c) Higher isolation between circuits d) Higher ‘Q’

Attenuators Attenuation in db of a device is ten times logarithmic ratio of power flowing into the device to the power flowing out of the device when both the input and output circuits are matched.

Attenuation in db 10log i

o

PP

=

Of the input circuit is not matched to the device then the iP is equal to the power incident minus the power reflected. If the output circuit is not matched then the oP becomes equal to the power consumed in the output circuit plus the power reflected into the circuit.

o Resistive-card attenuator: This type can provide either fixed amount or a variable amount of attenuation.. • In the fixed version, the resistance card tapered at both ends is bonded in place. The

tapering of the card helps in maintaining low SWR at the input as well as at the output ports over the useful wave-guide band.

• Maximum attenuation per unit guide length can be achieved by placing the card parallel to the electric field and at the centre of the wave guide where the field is maximum for the dominant mode.

• The amount of attenuation provided is a function of frequency, a disadvantage. It in general increases with frequency.

• In the variable version, called Flap attenuator, the resistance card enters into the wave guide through the slot provided in the broader wall thereby intercepting and absorbing a

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94

portion of the wave. A hinge arrangement is used to change the depth of penetration of the resistance card, there by changing the amount of attenuation from 0db to typically30db .

• The biggest disadvantage with Flap attenuators is their attenuation is frequency sensitive and also the phase of the output signal is function of attenuation.

o Rotary-vane attenuator: The essential parts of this device are; two fixed and one rotary

wave-guide sections. It also includes input and output transition sections to provide low SWR connections to rectangular wave-guides.

Structure: The two fixed circular waveguide sections are identical in all respects; each

attached to a transition and each consists of a piece circular wave guide with a lossy dielectric plate lying horizontal in it. In middle exists a rotatable circular waveguide section with a dielectric plate which can be placed at any angle by rotating the waveguide section. The plates are normally thin with 1rε > , 1rµ = and conductivityσ a finite nonzero value.

The plates attenuates the wave travelling, the amount of attenuation being dependant upon the properties of the material from which the plate is cut, the dimensions of the slab and also the angle between the plane of the plate and the E vector of the wave.

When the E vector of the wave is normal to it, the plate does not attenuate the wave in any significant manner, whereas it attenuates the wave in good amount when the E vector is parallel. In the present case, the lengths of the plates are selected in such a way that after travelling past the plates with its E vector parallel, the wave amplitude becomes insignificant.

Analysis: It can be shown that the wave undergoes an amount of attenuation in db

( )410log 1 cos mA θ= when the rotatable section is rotated by an angle equal to mθ from horizontal..

When the wave with its E vector vertical falls over and crosses the input fixed section in which plate is horizontal, it does so without any attenuation.

The unattenuated wave at the input of the rotatable section can be resolved into two components, one parallel to the rotatable plate and another normal to it. The parallel component gets absorbed and attenuated almost completely by the plate whereas the normal component crosses without any significant attenuation.

Now it is the only the normal to rotatable plate component that exists at the input of the

fixed output section. This component can be resolved into two one horizontal and the other vertical. The horizontal component is parallel to the fixed section plate and hence gets absorbed whereas the vertical one comes out unattenuated which is 2cos mE θ .

If the amplitude of the input field is E , then the output field strength will be mE θ2cos .

Hence the attenuation provided by the device in db is ( )410log 1 cos mA θ= .

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95

The attenuation is controlled by the rotation of the centre-section, minimum attenuation at 0=mθ and maximum at 090=mθ .

The attenuation provided by this device depends only on the rotation angle mθ and not upon the frequency. This device is very accurate and hence being used as a calibration standard. Its accuracy is limited only by imperfect matching and by mis-alignment of the resistance cards.

Phase shifters These devices find wide applications in test and measurement systems, but most

significant use is in phased array antennas where antenna beam is steered in space by

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96

electronically controlled phase shifters. The phase shifters which use ferrites in their construction are non-reciprocal where as others in general are reciprocal.

The phase shift that can be introduced into the wave by a waveguide section of length

‘ l ’ is given by 2 gl lβ π λ= where ( )22g r aλ λ ε λ= − . From this relation we can

observe that the phase of the wave can be controlled either by varying rε or the guide width

‘ a ’ thus changing the guide wavelength

Dielectric phase-shifters: The variable type dielectric phase shifters employ the a low loss dielectric insertion into the air filled guide at a point of max electric field to increase its effective dielectric constant thereby causing the guide wavelength gλ to decrease. Thus the insertion of the dielectric increases the phase shift in the wave passing through the fixed length wave guide section. Tapering of the dielectric slab is resorted to reduce the reflections. In another version, a pair of thin rods is used to move the dielectric slab from a region of low electric field intensity to one of the high intensity to increase the effective dielectric constant.

Squeeze type phase-shifters: It is a length of waveguide whose broader walls contain long non-radiating slots. A clamping arrangement is used to reduce the guide width a thus increasing the guide wavelength gλ resulting in a decreased phase shift in the wave through the wave guide section. It is also called line stretcher.

Rotary phase-shifters: The essential parts of this phase shifter are three wave guide

sections, two fixed and one rotary. The fixed sections consist of quarter wave plates and the rotary section consists of half wave plate, all the plates are of dielectric type.

Structure: The two fixed quarter wave sections identical in all respects and the rotatable half wave section is just the double of a quarter wave section. Each of the two fixed sections, attached to a transition, consists of a piece circular wave guide with a dielectric plate making an angle of 045 with the horizontal. The dielectric plate is normally thin with 1rε > , 1rµ =

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97

and 0σ ≈ . When the E vector of the wave is normal to it, the plate does not effect the wave in any way, whereas it adds an additional phase lag when the E vector is parallel. The additional phase lag depends upon the properties of the material from which the slab is cut and the dimensions of the slab. The length of the plate is selected in such a way that this additional phase lag is 090 in case of quarter wave plate and 0180 in case of half wave plate. As same materials are used to make half and quarter wave plates, the length of one becomes the double of the other.

Analysis: It can be shown that the output wave experiences an additional phase delay of 2 mθ when the half-wave plate is rotated by an angle equal to mθ .

When the wave with its E vector vertical falls and crosses over the quarter wave plate which is making an angle of 045 with the horizontal, the component of the wave parallel to the plate undergoes a phase shift of 090 in addition to the regular phase shift of lβ where as the component normal to the plate undergoes only the regular phase shift of lβ .

The above two components having phase shift of 090 can be resolved into two components each making total of four, one pair parallel to the half wave plate and another pair normal to the plate. The resultant of the pair normal to the half wave plate will have a lagging phase angle of mlβ θ+ where as the pair parallel to the half wave plate results in a lagging phase angle of 090 mlβ θ+ + .

The two components one normal and the other parallel to the half wave plate while crossing undergoes a phase change 2 lβ and 02 180lβ + resulting in a net phase lag of 3 mlβ θ+ and 03 270 mlβ θ+ + respectively. These two components which are available at the output of the half wave plate, can now be resolved into two components each , one along the quarter wave plate and the other normal to it. The resultant of the two components normal to the plate will have a phase lag equal to 3 2 mlβ θ+ whereas the component parallel posses

03 270 2 mlβ θ+ + . These two components, one is normal and the other is parallel to the quarter wave plate,

while travelling through the output quarter wave plate undergoes phase delays lβ and 090lβ + resulting in a net phase lag of 4 2 mlβ θ+ and 04 360 2 4 2m ml lβ θ β θ+ + = +

respectively. These two equi-phase components whose magnitudes are 2E , can be combined into one equal to 4 2 mE lβ θ∠ + .

In the absence of the plates the magnitude and phase of the out put would have been 4E lβ∠ .The presence of the plates makes the output to have an additional phase equal to

2 mθ when the half-wave plate is rotated by an angle equal to mθ .

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98

The output remains vertically polarized, which means that the phase shifter is loss less and reflection less for any position of the rotary section.

It is used as calibration standard because of its high accuracy.

Hybrid phase-shifters: The usefulness of the rotary phase shifter is limited to power levels of a few watts or even less. For higher power applications, the hybrid-type phase shifter is often used. These consist of a 3db short slot coupler and a pair of moving shorts. The shorts are coupled mechanically so that they can move as a unit. Moving the two shorts as a unit can vary the phase of the output wave. Moving them back a distance d causes the output wave to be delayed by an additional 2d , since the round the trip path of the wave is 2d . Thus the phase change in a hybrid phase shifter is ( )2 4 gd dθ β π λ∆ = = . Even though the phase change is showing a linear relationship with shorts movement, in practice the θ∆ versus d curve exhibits some deviation from linearity due to the imperfect operation of the 3db coupler.

Ferrite Phase shifters: They are two-port devices which can provide variable phase shift with the change of the bias field. One of the most useful designs is the latching or remnant non-reciprocal phase shifter employing ferrite toroid in a rectangular wave-guide.

Fixed phase-shift sections: To achieve a differential phase shift, the guide wavelength

can be altered by changing the guide width. It gives a differential phase shift of

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99

'

1 12g g

lθ πλ λ

∆ = −

. Another method of introducing differential phase change is by

inserting reactive elements into wave guide. Fixed phase sections are used in microwave bridge circuits which require a wave guide section in which the phase delay differs from that associated with wave propagation thro a standard RWG of equal length.

Wave Guide Windows These are used for impedance matching purpose at microwave frequencies.

• Inductive windows: The conducting diaphragms extending into the wave guide from either one or both of the sidewalls produce the effect of adding an inductive susceptance across the wave guide at the point at which the diaphragm is placed. These are called inductive diaphragms.

The amount of normalized susceptance added by the window depends upon the window insertion distance. The susceptance increases with the depth.

If the insertion is from both the side walls with two diaphragms then the resultant window is called symmetrical one. If the insertion is from either one wall only then it is called un-symmetrical window. The choice between symmetrical and unsymmetrical type is governed by mechanical considerations such as ease of machining and installation of pressurized windows.

• Capacitive windows: The conducting diaphragms extending into a rectangular wave

guide either from top or bottom or both walls produce the effect of adding capacitive susceptance shunted across the wave guide at that point. They are therefore called capacitive windows.

The amount of normalized susceptance due to the window depends upon the window insertion depth, in general increasing with the depth. These are not used extensively because of the lowering of the breakdown voltage and the consequent reduction in the maximum power that can be transferred through the wave guide.

• Resonant windows: A conducting diaphragm with a rectangular opening inside gives the

effect of a parallel LC circuit shunted across the guide at that point. This window is called resonant window.

It can give zero susceptance at a chosen frequency whose value depends upon the dimensions of diaphragm opening. It acts as a band-pass filter centred around this frequency, giving inductive susceptance on side and capacitive susceptance on the other side.

Obtainable Q values are of the order of 10 and decrease as the size of the aperture is increased.

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• Limitations: the windows suffer with two drawbacks; one is they cannot be made readily adjustable and provide only fixed amount of susceptance and the second one is; difficulty in maintaining the perfect contact between the diaphragm and walls of the wave guide.

Tuning Screws And Posts These are also used for impedance matching purposes. • Screws: A screw inserted into the top or bottom

of the wave guide walls, parallel to the E field lines can give variable amount of susceptance.

A screw of length less than 4λ produces capacitive susceptance whose value increases with depth of penetration. When the depth of penetration is 4λ the screw is in series resonance and further insertion causes the susceptance to be inductive. The most direct method of impedance matching with a matched screw is to use a single screw adjustable both in length and position along the wave guide. But it requires a slot in the wave guide. An alternative arrangement is to use double or triple screw units spaced at 8gλ or

4gλ . • Posts: A metal post or screw extending completely across the wave guide, parallel to E

field adds an inductive susceptance in parallel with the wave guide. A post extending across the wave guide at right angles to the E field

produces an effective capacitive susceptance in shunt with the wave guide at the position of the post.

Coupling Probes And Loops:

Probes and loops are used to couple coaxial line to wave guide or resonator.

• Probes: They consist of an extension of the centre conductor of the coaxial line at the mid point of one of the broader walls of the guide where E field is maximum and normal to the wall.

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Usually the wave guide is terminated in a short and the probe is placed approximately 4gλ from the termination. To minimize the reflections at the junction, the probe must be matched to the wave guide by proper choice of the length and position of the probe relative to the closed end of the wave guide.

The centre conductor of the coaxial line may extend completely across the wave guide or it may project an appreciable distance into the wave guide. In that case the magnetic as well as electric coupling is effective. For matching over an appreciable frequency band one or more of the following methods may be adopted:

The centre conductor may be flared at the point at which it enters the wave guide.

Height of the terminating section of the wave guide can be increased. A tapered section or some other type of impedance transformer can be used.

To excite a particular mode, the probe or probes should be placed parallel to the E

field at a position where the field has its largest value. When several probes are used, then they must be excited with appropriate phasing relation.

• Loops: Loop coupling is principally magnetic, so the loop must be placed at or near a

point of high H field strength and turned in such a way that its plane is normal to the flux lines.

Loops can be mounted in the end wall of a shorted wave guide or in the middle of the top or bottom wall at a distance of integral 2gλ from the shorted end. The plane of the loop should be normal to the H -field lines for maximum coupling. The amount of coupling obtainable with the loop depends upon its size and shape and in general increases with the area of the loop. • Comparison: The choice between loop and probe coupling is dictated partly by

mechanical and partly by electrical considerations. The important factors are • Likelihood of voltage breakdown in the vicinity of voltage anti-node. • Ease in adjusting the coupling • Constancy of coupling when mechanical changes are made. • Avoidance of interference with electron streams.

In microwave oscillators loops rather than probes are usually used because a probe in proper position for adequate coupling may interfere with electron movement with in the tube. Bends These are used to alter the direction of propagation in a wave-guide system. If the bending of the wave-guide is in the E − plane then the resultant structure is called E − plane Bend. H − plane Bend results when the bending is in the H − plane.

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102

• The reflection due to the bend is a function of its radius: the larger the radius, the lower the SWR.

• When the space available is limited, a double-mitred bend can be used. It gives a low VSWR when the spacing between the joints is 4gλ .

Twists • These are used to change

the plane of polarization of the propagating wave. The Gradual twist changes the plane of polarization in a continuous manner. It gives a SWR of less than 1.05 when the twist length is greater than few wavelengths. The step twist is used

when the space available in the propagation direction is limited. It contains a rectangular guide section that is oriented 450 with respect to the input and output guides. Microwave Junctions Microwave junctions are devices used to split or combine µwave power.

The important parts of microwave junctions are ports, arms and junction regions. These are used to describe the structures of the junctions. Ports are openings to which the source or load is connected. H.A.Wheeler introduced the term ‘port’ in 1950s to replace the less descriptive and more cumbersome phrase, ‘two terminal pair’. Arms are pieces of the transmission lines or waveguides with which the junction device is fabricated and Junction region is the common space where all the arms of the device meet each other

A port is said to be perfectly matched to the junction if nothing out of the power incident at the port is reflected back to the port by the junction. Two ports are said to be perfectly isolated if nothing out of the power incident at one port appears at the other port.

Three-port junctions: E-plane tee and H-plane tee are examples for three port junctions. As they are in the shape of English capital letter ‘T’ these are called ‘tees’.

Reciprocal three port junctions suffer with one drawback i.e. lack of isolation between the output ports resulting in dependence of the power consumed at one port on the termination at the other out-put port. This lack of isolation between the output ports limits the usefulness of the three port junctions, particularly in power monitoring and divider applications.

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plane teeH −

• As the side arm port is in the H − plane, it is called H − plane tee. It is also called, current junction, shunt junction or parallel junction

• The two arms which are in line are called coplanar arms whereas the other arm is called side arm or H − arm or shunt arm.

• Port 3 is perfectly matched to the junction.

• Ports1 and 2 are electrically symmetrical with respect to port3 when the collinear arm lengths are same

• For an ideal tee i.e. loss-less reciprocal junction the S-matrix is

−−

02121212121212121

• The transmission line equivalent circuit is As power divider:

• If the amplitude of the input wave at port 3 is A , then the amplitude of the waves at port 1 and 2 are same and equal to 2A . They are in-phase when its collinear arm-lengths are same

• When the power incident at port 3 is P then the powers that appear at ports 1 and 2 is 2P each. That is why it is called 3db splitter.

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104

• If the power incident at ports 1 or 2 is P , then the power out of ports 1 and 2 is 4P each and at port3 it is 2P

As power combiner:

• When equal input signals are given at both the collinear ports then the output signal appears at the side arm port whose power is sum of the powers of the input signals provided the collinear arm lengths same and sources are in phase

• The output power is zero and SW formation takes place in the collinear arms preventing the power entering into the junction when the sources are equal, out of phase and collinear arms lengths are same.

plane teeE −

• As the side arm port is in the E − plane, it is called E − plane tee. It is also called Series junction or voltage junction

• The two arms which are in line are called coplanar arms whereas the other arm is called side arm or E − arm or series arm.

• Port 3is perfectly matched to the junction.

• Ports 1 and 2 are electrically anti -symmetrical with respect to port3 when its collinear arm lengths are same.

• For an ideal tee i.e. loss-less reciprocal junction the S-matrix is

−−

02121212121

212121

• The transmission line equivalent circuit is

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105

As power divider

• If the amplitude of the input wave at port 3 is A , then the amplitude of the waves at port1 and 2are same and equal to 2A . They are out of phase when its collinear arms lengths are same

• When the power incident at port 3 is P then the powers that appear at ports1 and 2 is 2P each. That is why it is called 3db splitter.

• If the power incident at ports 1 or 2 is P , then the power out of ports 1 and 2 is 4P each and at port3 it is 2P

As power combiner

• When equal input signals are given at both the collinear ports then the output signal appears at the side arm port whose power is sum of the powers of the input signals provided the collinear arm lengths are same and sources are out of phase.

• The output power is zero and standing wave formation takes place in the collinear arms preventing the power entering into the junction when the sources are equal, in phase and collinear arms lengths are same.

Theorems of the tee junctions:- o A short circuit may always be placed in one of the arms of a three port junction

in such a way that no power can be transferred through the other two arms o If the junction is symmetric about one of its arms, a short circuit can always be

placed in that arm so that no reflection occurs in power transmission between the other two arms.

o It is impossible for a general three port junction of arbitrary symmetry to present matched impedances at all three arms

Applications:- RWG tees are used as

o Tuners by placing a short circuit in the symmetrical arm o Power dividers and adders o In the duplexer assemblies of the radar installations.

Four-port junctions: Magic tee and Directional couplers are examples for four port hybrid junction devices..

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106

Magic tee:

o It is formed by attaching sidewalls to the slots cut in the narrow wall and broad wall of a piece of wave-guide. Structurally. It is a combination of E-plane Tee and H-plane Tee.

o It is a hybrid in which the power is divided equally between the out put ports. The outputs can exhibit either 00 or 1800 phase difference

o It is also known as anti-symmetric coupler, 3db hybrid and 3db coupler.

o One of the main advantages of magic tee, in fact for any hybrid, is that the power delivered to one port is independent of the termination at the other output port provided the other port is match terminated.

Its properties are

All the ports are perfectly matched to junction and the &E H arm ports are decoupled, as are the coplanar arm ports.

A signal into a coplanar arm splits equally between &E H arms. For each output signal 2out inP P= and 2out inA A=

A signal into H arm splits equally between the coplanar arms, the outputs being in phase, equidistant from the junction.

A signal into E arm splits equally between the coplanar arms, the outputs being out of phase, equidistant from the junction.

For signals into both coplanar arms. 1. The signal output from the E-arm equals 1/ 2 times the

phasor difference of the input signals. (Difference arm). 2. The signal output from the H-arm equals 1/ 2 times the

phasor sum of the input signals. (Sum arm).

The S-matrix for an ideal Magic T is

[ ]

−

−=

001100111100

1100

21S

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107

Its important applications are in the ‘measurement of impedance’, as ‘duplexer’, as ‘tuner’ and also as ‘mixer’.

Hybrid Ring or Rat race Circuit:

Structure: The hybrid rings consists of an annular waveguide of proper electrical length to sustain standing waves, to which four arms are connected at proper intervals by means of series or parallel junctions. The arrangement shown consists of a piece of rectangular wave guide bent in the E − plane to form a complete loop whose median circumference is1.5 gλ . It has four openings from each of which a waveguide emerges forming parallel junctions. If there are no reflections from the terminations in any of the arms then any one arm is coupled to two others but not to the fourth. Hybrid rings can also be constructed by bending them in the H − plan also with the connection of series junctions. Functioning: The hybrid ring has the characteristics similar to those of hybrid tee. The wave fed into port 1 can not appear at port3 because the difference of phase shifts for the waves travelling in the clockwise and counter clockwise directions is 1800 thus cancelling each other. Similarly the waves fed into port 2 can not emerge at port4 and waves fed into port 3 can not emerge at port1. But the perfect cancellation takes place only in ideal hybrid rings and at the designated frequency. In actual hybrids there exists always a small amount of leakage wave resulting in non-zero wave where it is supposed to be nil. The S matrix for an ideal hybrid ring can appear as

[ ]12 14

21 23

32 34

41 43

0 00 0

0 00 0

s ss s

Ss s

s s

=

Hybrid ring vs hybrid tee:

The rat race and magic tee may be used interchangeably but o The hybrid tee is less bulky but requires internal matching which

doesn’t require for hybrid ring if the thickness is properly chosen. o Hybrid ring seems preferable at higher frequencies since its

dimensions are less critical.

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Directional couplers:

• It is a (any) reciprocal, loss-less and matched 4-port network. DC is a 4-port

network in which portions of the forward and reverse traveling waves on a line are separately coupled to two of the ports.

• DC is also called symmetric coupler and quadrature type hybrid

(i) In an ideal DC all the four ports are perfectly matched and also ports 1,2and 3,4 are perfectly isolated

(ii) A portion of the wave travelling from the port 1 to 4 is coupled to port 3 but not to port 2. Similarly a portion of the wave travelling from the port 4 to 1 is coupled to port 2but not to port 3.

(iii) A portion of the wave travelling from the port 2 to 3 is coupled to port 4 but not to port 1. Similarly a portion of the wave travelling from the port 3to 2is coupled to port 1 but not to port 4.

(iv) The coupling between port 1 and port 3is same as that between port 2 and 4. Similarly the coupling between port 1 and 4is same as that between port 2 and 3.

(v) The outputs are always at phase quadrature i.e. exhibit a phase difference of 090 . For this reason DC is called quadrature type hybrid.

(vi) For an ideal DC the scattering matrix is

[ ]

=

0000

0000

21

pjqjqp

pjqjqp

S p is equal to coupling coefficient ck related to

coupling factor cK , through 1c

c

kK

=

Two hole directional coupler:

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109

Structure:

• A two-hole directional coupler is formed by placing one piece of RWG over another and

cutting two holes at a distance of ( )4

12 gnλ

+ in the common broader wall. The size,

shape and location of the holes decide the amount of coupling.

• The waveguide system into which input is given is called primary or main wave guide system whereas the waveguide system from which the coupled output is extracted is called secondary or auxiliary waveguide system.

• Any one of the four ports can be the input port. If the port1 is input port, then the port2 which is opposite to the input port becomes through port or output port, the port3 which is just below the input port is called decoupled port and the port4 which is below the through port is called coupled port. With matched terminations on all the output ports,

Functioning: a fraction of the wave energy entered into port1 passes through the holes and is radiated into the secondary guide as the holes act as slot antennas. The forward waves in the secondary guide are in the same phase, regardless of the hole space and are added at port4. The backward waves in the secondary guide are out of phase resulting in cancellation at port3. Two hole directional coupler:

Structure: It has two versions. One is parallel guide coupler and the second one is skewed guide coupler. In parallel guide coupler version the two guides are parallel, one lying over the broad wall of the other with a small hole aperture in the common broad wall whose offset s from the side wall of the guide controls the coupling. In the skewed guide coupler version, one guide is over the other at an angleθ which controls the amplitude of the coupled waves. The geometry of the skewed Bethe hole coupler is often a disadvantage in terms of fabrication and application. Also both coupler designs operate properly only at the design frequency.; deviation from this frequency will alter the coupling level and the directivity. Functioning In this coupler, one waveguide is coupled to another through a single small hole in the common broad wall between the two guides. According to small-aperture coupling theory, an aperture can be replaced with equivalent sources consisting of electric and magnetic dipole moments. The normal electric dipole moment and the axial magnetic dipole moment radiate with even symmetry in the coupled guide, while the transverse

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110

magnetic dipole moment radiates with odd symmetry. Thus by adjusting the relative strengths of these two equivalent sources, we can cancel the radiation in the direction of the isolated port, while enhancing the radiation in the in the direction of the coupled port In case of parallel guide coupler, the coupling is controlled by the aperture offset s from the side wall where as the angleθ between the guides controls the coupling in case of skewed wave guide coupler. • Performance:- The performance of DC is described in terms of its coupling and

directivity. o Coupling is a measure of the power being sampled from the incident wave.

Coupling in db = C=c

i

PPlog10 = cKlog10 where CK is known as the

Coupling factor = c

i

PpowercoupledPpowerincident

o Directivity is a measure of how well the coupler distinguishes between forward

and reverse traveling waves.

Directivity in db = D= dd

c Kpp log10log10 = where is known as the

Directivity factor = d

c

PpowercoupleddePpowercoupled

−

o Isolation is also a performance index of directional coupler used.

Isolation in db 10 log i

d

PIP

= . The three are related through I C D= +

o For an ideal DC the directivity is infinite. But in practice D >30db.Loose coupling means C>20db

Uses: -Extensively used in systems that measure the amplitude and phase of travelling waves. The major applications are

• In power monitors and • In reflectometers

Hybrids: These are directional couplers where the coupling factor is 3db. There are two types of hybrids: One is Magic-T hybrid or rat-race hybrid which has a 1800 phase difference between the outputs and second one is quadrature hybrid which has 900 phase difference between the outputs

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111

Ferrites: Microwave isolators, gyrators and circulators use non-reciprocal transmission

materials such as ferrimagnetic and ferromagnetic materials. A ferrites or ferrimagnetic materials are non-metallic insulators but with magnetic

properties similar those of ferrous metals. Commonly used ferrites are manganese ferrite 2 3MnFe O and zinc ferrite 2 3MnFe O . Apart from these compounds one widely used Ferromagnetic material is Yttrium-Iron-Garnet 3 2 4 3[Y Fe (FeO ) ] or YIG in short.

The magnetic anisotropy of a ferrimagnetic material is exhibited only upon the application of a DC magnetic bias field. This field aligns the magnetic dipoles in the ferrite to produce a net non-zero magnetic dipole moment and causes these dipoles to precess at a frequency which depends upon the strength of the bias field. A microwave signal circularly polarized in the same direction as this precession interact strongly, while an oppositely polarized field interact lesser with the dipole moments. Since, for a given direction of rotation, the sense of polarization changes with the direction of propagation, a microwave signal propagate through ferrite differently in different directions. This effect is utilized in the fabrication of directional devices such as isolators, circulators and gyrators.

Another useful characteristic is that the interaction with the applied microwave signal can be controlled by adjusting the strength of the bias magnetic field. This property is used in the design of phase shifters, switches, tunable resonators and filters.

The ferrite is non-linear material and its permeability is an asymmetric tensor given

by ( )ˆˆ 1o mµ µ χ= + where 0

ˆ 00 0 0

m

m m

jjχ κ

χ κ χ =

which is tensor magnetic susceptibility.

In case of the ferrite biased in z − direction

( ) ( ) 00 0 0 2 2

0

1 1 1 mxx yy

ω ωµ µ χ µ χ µω ω

= + = + = + −

00 0 0 2 2

0

mxy yxj j ω ωκ µ χ µ χ µ

ω ω= − = =

−

[ ]0

0ˆ0 ___ bias

0 0

jj zµ κ

µ κ µµ

= − −

A material having a permeability tensor of this form is called 'Gyrotropic'. Here x or y component of H gives rise to both x and y components to B with a phase shift of 090 in between them.

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112

[ ]0 0 0

ˆ0 ___ bias0

j xj

µµ µ κ

κ µ

= − −

[ ] 0

0ˆ0 0 ___ bias

0

jy

j

µ κµ µ

κ µ

− = −

The two properties of the ferrites which are important and relevant to microwave

engineer are Faraday rotation and gyromagnetic resonance. Faraday rotation: Consider the linearly polarized plane wave propagation through the

ferrite in the direction of bias. The linearly polarized wave can be considered as sum of an RHCP and an LHCP wave. Due to creation of a preferred direction for magnetic dipole precession by the bias field, one sense of circular polarization causes the precession in this preferred direction where as the other causes the precession in the opposite direction. For the RHCP wave the ferrite material offers an effective permeability of µ κ+ and it is µ κ− for the LHCP wave. So these waves travel through the ferrite medium with different propagation constants. At any point in the ferrite medium, the total wave, sum of RHCP and LHCP waves, is still linearly polarized but with the polarization rotated. This phenomenon in which a linearly polarized wave undergoes a change in its direction of polarization is called Faraday effect. It can be shown a wave that travels from one end to other and back to the first in the ferrite rod undergoes a total polarization direction change of 2φ where φ is polarization change when travelled from one end to another. So Faraday rotation is a non-reciprocal effect.

Gyromagnetic resonance: It occurs when the forces precession frequency is equal to the free precession frequency making the elements of permeability tensors infinite. In the absence of loss the loss may be unbounded. The most important parameters of Ferrites are

o Line width which is the range of magnetic field strengths over which absorption will take place. It is defined between the half-power points for absorption. A wide line width indicates the wide band properties.

o Curie temperature is one at which a magnetic material loses its magnetic properties. It is up to 6000C for ferrites. For YIG it is 2800C. Curie temperature places a limitation on the maximum temperature at which a ferrite may be operated and therefore the power that it can handle.

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113

o Maximum frequency of operation: for devices using resonant absorption it depends upon the maximum field strength that can be generated. At present it is more than 220GHz.

Isolator • It is a two-port non-reciprocal lossy device having unidirectional transmission

characteristics. • A common application uses isolator between a high power source and a load to

prevent possible reflections from damaging the source.

• For an ideal isolator the scattering matrix is [ ]

=

0110

S

• The ferrite isolators, which have practical importance, are field displacement isolator, Faraday rotation isolator and resonance absorption isolator.

• Isolators can be fabricated with one 045 twist and one 045 ferrite rod. If one gives rotation in clockwise the other must be selected to give the rotation in the opposite direction i.e. anti-clockwise direction..

Field displacement isolator: Field displacement isolators are based on the fact that the electric field distributions

of the forward and reverse waves in a ferrite slab-loaded wave-guide are quite different. The electric field for a forward wave can be made to vanish at the side of the ferrite slab where the electric field of the reverse wave can be quire large. Then if a thin resistive sheet is placed in this position the forward wave will be unaffected while the reverse wave will be attenuated.

Its bandwidth is around 10%, high values of isolation can be obtained with this relatively compact device. Another advantage is relatively much smaller bias field is required Faraday rotation isolator:

It consists of a piece of circular wave guide capable of carrying wave in the dominant mode

11TE with transitions to a standard rectangular guide which can carry

10TE at both the ends. The end-transitions are twisted through 045 and the input transition carries a resistive plate attenuator whose plane is parallel to horizontal. A thin ferrite rod is placed inside the

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114

circular waveguide supported by polyfoam and the waveguide is surrounded by a permanent magnet which produce dc magnetic field in the ferrite rod. The forward travelling wave while travelling through the waveguide gets its plane of polarization rotated by 045 in clock wise direction by the ferrite and comes out of the rectangular wave guide transition without any attenuation because it is already twisted by the same angle. When the wave is travelling in the reverse direction, the polarization of the wave gets rotated by 045 and also in clock wise direction same as that of the forward travelling wave. This when the wave emerges into the input transition, not only it gets absorbed by the resistive vane but also it cannot propagate in the input rectangular wave guide because of its dimensions.

The ferrite isolators based on Faraday rotation are useful only for powers up to a few hundred watts where as those based on resonant absorption they can handle higher powers. This type of isolator is limited in its power handling capability to about 2kW. It has wide range of applications in the low-power field mostly in low power microwave amplifiers and oscillators. Resonant absorption isolator:

Resonance isolators are based on the principle that a circularly polarized plane wave rotating in the same direction as the precessing magnetic dipoles of a ferrite medium will have a strong interaction with the material, while circularly polarized wave rotating in the opposite direction will have weaker interaction. It results large attenuation of the wave near the gyro magnetic resonance of the ferrite, while the attenuation of the wave travelling in the opposite direction is very low.

It consists of a piece of RWG capable of carrying the wave in dominant mode 10TE with a piece of longitudinal ferrite material placed about a quarter of the way from one side of the waveguide and half way between its ends. A permanent magnet is placed around to generate the required strong magnetic field. The ferrite is place where the magnetic field is strong and circularly polarized. This polarization is clockwise in one direction and anti-clockwise in the opposite direction. When the wave is travelling in one direction, resonant absorption takes place while travelling in the opposite direction it gets unaffected.

These are commonly used for high powers and they handle powers up to 30MW peak. The maximum power handling ability of resonance isolators is limited by temperature rise and consequently by Curie temperature. Its bandwidth is less than 2%

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115

Gyrator

• It is an important canonical non-reciprocal two-port component having 1800 differential phase shift.

• For an ideal gyrator i.e. loss-less, matched and non-reciprocal one the scattering

matrix is [ ] 0 11 0

S

= − .

• Using the gyrator as a basic nonreciprocal building block in combination with reciprocal dividers and couplers can lead to useful equivalent circuits for non reciprocal components such as isolators and circulators.

• Gyrators can be fabricated with one 090 twist and one 090 ferrite rod. Both must rotate the E vector in the same sense i.e. either in clockwise or anti-clockwise.

Circulator

• It is an n-port, loss-less and non-reciprocal device matched at all the ports in which power flow occurs from ports 1to 2, 2 to 3…n to 1but not in the reverse direction.

• By transposing the port indices, the opposite circularity can be obtained. In

practice, changing the polarity of the bias field produces this result. • A 3-circulator can be used as an isolator by

terminating one of the ports with a matched load

• For an ideal three-port circulator the

scattering matrix is [ ]

=

010001100

S

Faraday rotation circulator: It consists of a piece of

circular wave guide capable of carrying wave in the dominant mode 11TE with transitions to a standard rectangular guide which can carry 10TE at both the ends. The end-transition ports ‘1’ , ‘2’ and two

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116

rectangular side ports‘3’ and ‘4’ placed with their broader walls along the length of the waveguide are twisted through 045 . A thin ferrite rod is placed inside the circular waveguide supported by polyfoam and the waveguide is surrounded by a permanent magnet which produce dc magnetic field in the ferrite rod. The wave travelling from port ‘1’ to ‘2’ passes port ‘3’ unaffected as its electric field is not cut significantly, gets rotated 045 by the ferrite rod, continues past the port ‘4’ unaffected reaching and emerging from the port ‘2’. Power fed into port ‘2’ travels past the port‘4’ unaffected, gets rotated 045 by the ferrite and reaches the port ‘3’ to emerge from it. In this case the wave cannot come out port ‘1’ because of its dimensions. Similarly, port ‘3’ is coupled only to port ‘4’ and port‘4’ only to port ‘1’. This type of circulator is power limited and so eminently suitable for low power applications. It is bulkier restricting its use to highest frequencies, in the millimetre range and above.

s-Parameters

• The concept of s-parameters was first popularized around the time Kaneyuka Kurokawa of Bell labs wrote his 1965 IEEE article ‘Power Waves and Scattering Matrix’ even though a good work of E.M. Mathews, Jr. titled ‘The use of Scattering Matrices in Microwave Circuits’ appeared a decade earlier that is in 1950’s and Robert Collins text book 'Field Theory Of Guided Waves’ published in 1960 has a brief discussion on scattering matrix. Scattered waves refer to both the reflected and transmitted waves.

• The S-parameters relate the amplitudes and phases of the traveling waves that are

incident on, transmitted through or reflected from a network port. The s -parameters are based on the concepts of (a) traveling waves and (b) matched terminations. These are useful to characterize linear networks at microwave frequencies.

• Formulation: - Let a ’s andb ’s are the incident waves into and scattered waves out of

various ports of an n − port network respectively. And in general these are complex quantities.

The scattered waves at various ports b ’s are dependant, in general, on the incident waves a ’s at various ports. That is, 1b depends on 1 2, ,...... na a a and so it can be expressed as

1 11 1 12 2 13 3 1 1.... ...r r n nb s a s a s a s a s a= + + + + + +

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117

The scattered wave at port '2' which is 2b also depends on 1 2, ,...... na a a and its dependence can be expressed as

2 21 1 22 2 23 3 2 2.... ...r r n nb s a s a s a s a s a= + + + + + + The scattered wave rb at a port ‘ r ’ depends upon the incident waves a ’s at various ports i.e. 1 1 2 2 3 3 .... ...r r r r rr r rn nb s a s a s a s a s a= + + + + + + Similarly for the scattered wave from port ' n ' that is nb depends on

1 2, ,...... na a a through

1 1 2 2 3 3 .... ...n n n n nr r nn nb s a s a s a s a s a= + + + + + +

.All these relations can be put in matrix form as

1 11 12 1 1

2 21 22 2 2

1 2

.. ..

.. .... .. .. .. .. .. .... .. .. .. .. .. ..

.. ..

n

n

n n n nn n

b s s s ab s s s a

b s s s a

=

or [ ] [ ][ ]asb =

• The matrix [ ]s relates scattered wavesb ’s with the incident waves a ’s of various

ports. Hence it is called scattering matrix of the circuit being modelled. The elements of the matrix are called scattering parameters or s -parameters. The s-parameters of a network are properties only of the network itself and are defined under the condition that all ports are match terminated.

• Chain s -parameters: - Another formulation that is very much useful when the circuit is in cascaded configuration uses Chain s − parameters. These are a set of parameters defined relating the incident wave and scattered wave at a port to the waves at an another port. For example the waves at port '1' can be related to the waves at port '2' through

=

2

2

2221

1211

1

1

ab

TTTT

ba

The parameters which relate the waves at one port to the waves at other port in the manner shown above are called ' Chain s -parameters'. These parameters are useful to model circuits which are in the form of cascaded sections. For cascaded networks [T]=[T’][T’’]

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118

• Features:

o all ports except port jmatch terminated

iij

j

bsa

= and the s -parameters are in general complex

quantities. o When the circuit is connected to transmission lines of unequal characteristic

impedances then 0allportsexcept port jmatch terminated0

jiij

j i

zbsa z

=

o When port i is perfectly matched to the junction then 0=iis o When ports i and j are perfectly isolated then 0== jiij ss o The diagonal elements of the matrix represent reflection coefficient. s -

parameter iis is reflection coefficient at port i with matched termination at all other ports.

• For some components and circuits, the scattering parameters can be calculated using network analysis techniques. Otherwise they can be measured directly with a vector network analyzer.

• The s-matrix is always a square one. For a n − port network, the S-matrix is a nn ×

square matrix • For reciprocal networks i.e. networks consisting of passive linear bilateral

impedances, the S-matrix is symmetric jiforss jiij ≠= . In addition if the ports i and j are electrically symmetrical with respect to port r then jrirjjii ssandss == , and

if the ports i and j are electrically anti-symmetrical with respect to port r then

jrirjjii ssandss −== • For loss-less passive networks i.e. with no resistive components the S matrix is

Unitary [ ] [ ] [ ]ISS t =∗

o Zero property of unitary matrix: - The sum of products of each term of any row multiplied by the complex conjugate of the corresponding terms of any other row is zero. Mathematically it can be stated as

10

N

pj qjj

s s p q∗

=

= ≠∑ . Similarly the sum of products of each term of

any column multiplied by the complex conjugate of the corresponding terms

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119

of any other column is zero. Mathematically it can be stated as

10 ;

N

ip iqi

s s p q∗

=

= ≠∑

o Unity property of unitary matrix: - The sum of the products of each term of

any one row multiplied by its complex conjugate is unity. Mathematically it can be stated as

2*

1 11

N N

ij ij ijj j

s s s= =

= =∑ ∑

Similarly the sum of the products of each term of any one column multiplied by its complex conjugate is unity. Mathematically

2*

1 11

N N

ij ij iji i

s s s= =

= =∑ ∑

• Shifting of reference plane:- The location of reference plane has a bearing over the exact phase of the s -parameter. In general

1

2

11 12 1 11 12 1

21 22 2 21 22 2

1 2

.. .. .. ..0 .. .. 0

.. .. .. ..0 .. .. 0.. .. .. .. .. .. .. .. .. .... .. .. .. .... .. .. .. .. .. .. .. ... .. .. .. ..

.. .. 0 0 .. .. n

j ln n

j ln n

j ln n nn

s s s s s ses s s s s se

s s s e

β

β

β

±

±

±

′ ′ ′ ′ ′ ′ = ′ ′ ′

1

2

1 2

0 .. .. 00 .. .. 0.. .. .. .. ..

. .. .. .. .. .. .... .. 0 0 .. .. n

j l

j l

j ln n nn

ee

s s s e

β

β

β

±

±

±

or in short hand form [ ] [ ]' n nj l j lS diag e S diag eβ β± ± = where +/- sign is if the shift is towards/away from the junction. We can observe that for diagonal elements, 2 ij l

ii iis s e β−′ = whereas for the off-diagonal

elements, ( )i jj l lij ijs s e β− +′ =

• Uses: o They provide a means by which complete characterization of a network at

microwave frequencies is possible. o They make the requirement of open and short circuits completely

unnecessary. o They are defined with matched loads for termination avoiding the possibility

of serious reflections back towards the source.

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120

S-matrix of an ideal E-plane Tee:

Consider the E- plane tee with the corresponding port designations as shown.

The E-plane tee is a three port junction. So its s-matrix must be a 3 3× square matrix. Let it be

[ ]11 12 13

21 22 23

31 32 33

s s ss s s s

s s s

=

.

As the junction is ideal it is reciprocal. So the matrix is symmetrical i.e. jiforss jiij ≠= . In addition port 3is perfectly matched to the junction so

33 0s = . Incorporating these two aspects, [ ]11 12 13

12 22 23

13 23 0

s s ss s s s

s s

=

.

In E-plane tee, the port3 is electrically anti-symmetrical with respect to the

ports1 and 2 so 11 22s s= and 23 13s s= − resulting in [ ]11 12 13

12 11 13

13 13 0

s s ss s s s

s s

= − −

.

As the junction is ideal it must be lossless and its s-matrix is unitary obeying unit property and also zero property. Applying unit property to column one C1

2 213 13 1s s+ = resulting 13 1 2s = . Now adjusting the reference plane at

either port 1 or 3 the phase of 13s can be made zero. So 13 1 2s = resulting

in [ ]11 12

12 11

1 2

1 2

1 2 1 2 0

s s

s s s

= −

−

.

Now applying the zero property to R1 and R3, 11 121 1 02 2

s s− = resulting in

11 12s s= . Now apply the unit property to row one R1 2

2 211 11

1 12

s s + + =

giving 2 211 11

12

s s+ = leading to 1112

s = . But 12 1112

s s= = . Now the reference

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121

planes at ports 1 and/or 2can be adjusted to make the phases of 11s and 12s are

zeros. So 11 1212

s s= = .

Incorporating all these findings into the matrix it becomes

[ ]1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 0

s

= −

−

.

S-matrix of an ideal H-plane Tee:

Consider the H-plane tee with the corresponding port designations as shown.

The H-plane tee is a three port junction. So its s-matrix must be a 3 3× square matrix. Let it be

[ ]11 12 13

21 22 23

31 32 33

s s ss s s s

s s s

=

.

As the junction is ideal it is reciprocal. So the matrix is symmetrical i.e. jiforss jiij ≠= . In addition port 3is perfectly matched to the junction so

33 0s = . Incorporating these two aspects, [ ]11 12 13

12 22 23

13 23 0

s s ss s s s

s s

=

.

In E-plane tee, the port3 is electrically symmetrical with respect to the ports1

and 2 so 11 22s s= and 23 13s s= resulting in [ ]11 12 13

12 11 13

13 13 0

s s ss s s s

s s

=

.

As the junction is ideal it must be lossless and its s-matrix is unitary obeying unit property and also zero property. Applying unit property to column one C1

2 213 13 1s s+ = resulting 13 1 2s = . Now adjusting the reference plane at


in [ ]11 12

12 11

1 2

1 2

1 2 1 2 0

s s

s s s

=

.

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122

Now applying the zero property to R1 and R3, 11 121 1 02 2

s s+ = resulting in

11 12s s= − . Now apply the unit property to row one R1 2

2 211 11

1 12

s s + + =

giving 2 211 11

12

s s+ = leading to 1112

s = . But 12 1112

s s= − = − . Now the

reference planes at ports 1 and/or 2can be adjusted to make the phases of 11s

and 12s are zeros. So 12 1112

s s= − = −

Incorporating all these findings into the matrix it becomes

[ ]1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 0

s

−

= −

.

S-matrix of an ideal Magic Tee:

Consider the magic tee with the port designations as shown in the diagram.

The Magic tee is a four port junction. So its s-matrix must be a 4 4× square matrix. Let it be

[ ]11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

s s s ss s s s

ss s s ss s s s

=

.

In Magic tee all the ports12 3and 4 are perfectly matched to the junction so 11 22 33 44 0s s s s= = = = and ports12 and 34 are perfectly isolated i.e.

12 21 34 43 0s s s s= = = = . Incorporating these two aspects,

[ ]13 14

23 24

31 32

41 42

0 00 0

0 00 0

s ss s

ss ss s

=

.

As the junction is ideal it is reciprocal. So the matrix is symmetrical i.e. jiforss jiij ≠= . Also in Magic tee, the E arm port3 is electrically anti-

symmetrical with respect to the ports1 and 2 so 11 22s s= and 23 13s s= − , the H

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123

arm port4 is electrically anti-symmetrical with respect to the ports1 and 2 so

11 22s s= and 24 14s s= resulting in [ ]13 14

13 14

13 13

14 14

0 00 0

0 00 0

s ss s

ss ss s

− = −

. As the junction is ideal it must be lossless and its s-matrix is unitary obeying unit

property and also zero property. Applying unit property to column three C3 2 2

13 13 1s s+ = resulting 13 1 2s = . Now adjusting the reference plane at either

port 1 or 3 the phase of 13s can be made zero. So 13 1 2s = resulting in

[ ]

0 0 1 2 1 2

0 0 1 2 1 2

1 2 1 2 0 0

1 2 1 2 0 0

s

− =

−

. Now applying unit property to column

three C4 2 2

14 14 1s s+ = resulting 14 1 2s = . Now adjusting the reference plane at


in [ ]14

14

14 14

0 0 1 2

0 0 1 2

1 2 1 2 0 00 0

s

ss

s s

− = −

S-matrix of an ideal Directional coupler:

Consider the directional coupler with the port designations as shown in the diagram.

The directional coupler is a four port junction. So its s-matrix must be a 4 4× square matrix. Let it be

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124

[ ]11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

s s s ss s s s

ss s s ss s s s

=

.

In the directional coupler all the ports12 3and 4 are perfectly matched to the junction so 11 22 33 44 0s s s s= = = = and ports12 and 34 are perfectly isolated i.e. 12 21 34 43 0s s s s= = = = . Incorporating these two aspects,

[ ]13 14

23 24

31 32

41 42

0 00 0

0 00 0

s ss s

ss ss s

=

.

As the coupler is ideal it is reciprocal. So the matrix is symmetrical i.e.

jiforss jiij ≠= . So [ ]13 14

23 24

13 23

14 24

0 00 0

0 00 0

s ss s

ss ss s

=

. In ideal directional coupler the coupling between ports 13and 24 is same

making 13 24s s= . the coupling between ports 14and 23is also must be same making 14 23s s= . Incorporating these changes the matrix becomes

[ ]13 14

14 13

13 14

14 13

0 00 0

0 00 0

s ss s

ss ss s

=

As the junction is ideal it must be lossless and its s-matrix is unitary obeying unit property and also zero property. Applying zero property to row one and row two R1R2

13 14 14 13 0s s s s∗ ∗+ = . Now if 14s is real quantity equal to p then this equation

becomes ( )13 13 0p s s∗+ = . To satisfy this equation the 13s must be pure imaginary and let it be jq . Incorporating these changes the matrix becomes

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125

[ ]

0 00 0

0 00 0

jq pp jq

sjq pp jq

=

. One observation from this matrix is outputs of the coupler

exhibit quadrature phase difference.

Any loss-less, reciprocal three port microwave junction cannot be matched at all the three ports.

Let us suppose the junction can be matched at all the three ports. Then , when

reciprocal, its s-matrix becomes [ ]12 13

12 23

13 23

00

0

s ss s s

s s

=

.

As the junction is loss-less its s-matrix is unitary obeying zero property and unity property. Applying zero property to rows one and two R1 R2

13 23 0s s∗ = . So either 13s or 23s or both must be zero. Now apply unity property to all the rows to have

2 212 13

2 223 12

2 213 23

1

1

1

s s

s s

s s

= −

= −

= −

. From these relations we can see if 13s is zero 12s is one, 23s is zero

and the last relation gives 13s as one which is contradictory to that with which we started i.e. 13s is zero. This contradiction also exist even when 23s or both are zero.

So the junction cannot be matched at all the three junctions Carlin’s theorem: Any loss-less, matched and non-reciprocal three port microwave

junction can be a perfect three port circulator. The s-matrix of a perfectly matched three port junction is

[ ]12 13

21 23

31 32

00

0

s ss s s

s s

=

. As the junction is non-reciprocal, the matrix is not

symmetrical. But as it is loss-less, its s-matrix is unitary exhibiting unity and zero laws. So

2 212 13 13 23

2 221 23 12 32

2 231 32 21 31

1; 0

1; 0

1; 0

s s s s

s s s s

s s s s

∗

∗

∗

+ = =

+ = =

+ = =

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126

From the above equations it can be seen if 21 0s ≠ then 31 0s = leading to 32 1s = .

If 32 0s ≠ then 12 0s = leading to 13 1s = . If 13 0s ≠ then 23 0s = leading to 21 1s = . So there exists perfect transmission form port 1 to 2 as 21 1s = , from port 2to3 as 32 1s = and from port 3 to 1 as 13 1s = . And zero transmission in any other direction defining a perfect three port circulator.

Its scattering matrix becomes [ ]13

21

32

0 00 0

0 0

ss s

s

=

.

UNITARY PROPERTY: For a loss-less junction, the scattering matrix is an unitary

matrix i.e. [ ] [ ] [ ]tS S U∗ = .

Let us suppose nV + , nI + represent incident wave voltage and current and nV − , nI − represent scattered wave voltage and current respectively at port ‘n’. For a loss-less junction, the total

power leaving its N-ports must be equal to the total incident power i.e. 2 2

1 1

N N

n nV V− +=∑ ∑

which can be expressed as t t

V V V V∗ ∗− − + + = . But

[ ]( ) [ ]( ) [ ] [ ]tt ttV V S V S V V S S V

∗∗ ∗∗− − + + + + = = . So

[ ] [ ]tt tV V V S S V

∗ ∗∗+ + + + = . It results in

[ ] [ ] [ ]( ) 0tt

V U S S V∗∗+ + − = giving [ ] [ ] [ ]t

S S U∗ = .

SYMMETRY PROPERTY: For a reciprocal junction, the scattering matrix is symmetrical i.e. nm mns s= provided the equivalent voltages have been chosen so that

power into port ‘n’ is given by 21

2 nV + for all modes.

With the necessary normalization, the total voltage and current at port ‘n’ can be expressed as andn n n n n n n nV V V I I I V V+ − + − + −= + = − = − , [ ] [ ] [ ]V Z I= . Now

[ ] [ ]V V Z V Z V+ − + − + = −

[ ] [ ]( ) [ ] [ ]( )Z U V Z U V− + + = − so [ ] [ ] [ ]( ) [ ] [ ]( )1S Z U Z U

−= + −

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127

Alternatively [ ] [ ]( ) [ ] [ ]( )[ ]1 12 2

V V I Z U I+ = + = + and

[ ] [ ]( ) [ ] [ ]( )[ ]1 1 12 2

V V I Z U I− = − = − = [ ] [ ]( ) [ ] [ ]( ) 11V Z U Z U V−− + = − + after

substituting [ ]I from the expression for V + . From this [ ] [ ] [ ]( ) [ ] [ ]( ) 1S Z U Z U

−= − + and

considering transpose both sides [ ] [ ] [ ]( ) [ ] [ ]( )1

t t tS Z U Z U

−= + − . but in this expression the

matrices in the parentheses are symmetrical so they are equal to their transposes. Therefore

[ ] [ ] [ ]( ) [ ] [ ]( )1

tS Z U Z U

−= + − . Comparing both the underlined equations [ ] [ ]t

S S= and

hence [ ]S is a symmetrical matrix. Multiple choice questions

1. When arm lengths are same in a Magic T, the outputs can be [C ]

a) In phase b) out of phase c) Either d) None

2. When arm lengths are same in a Magic T, the outputs can be [B ]

a) Coupled b) Decoupled c) Either d) None

3. When the co-planar arm lengths are not same, then outputs in H plane T can be [C ]

a) In phase b) Out of phase c) With phase difference d) either

4. When the co-planar arm lengths are not same, then outputs in E plane T can be [C ]

a) In phase b) Out of phase c) With phase difference d) either

5. The outputs of E-plane T are [A ] a) Coupled b) Decoupled c) Either d) None

6. The outputs of H-plane T are [A ]

a) Coupled b) Decoupled c) Either d) None 7. The outputs of directional coupler have a phase difference of [A ]

a) 90o b) 45o c) 180o d) None 8. Co planner arm ports in 3-port junctions can be [A ]

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128

a) Matched to junction b) mismatched c) either d) None

9. For its S-matrix to be unitary, the circuit must be [B ]

a) Reciprocal b) loss-less c) Both d) None

10. Chain S-parameters are similar to [C ]

a) ABCB parameters b) Transmission line parameter c) Both d) None

11. E plane T divides the power in the ratio of [A ]

a) 1:1 b) 1:3 c) 2:3 d) None

12. In hybrids the outputs are decoupled [A ]

a) When third output port in match terminated b) Always c) When all ports match terminated d) None

13. Symmetric coupler is [B ]

a) Magic Tee b) Directional Coupler c) Both d) None

14. Unsymmetrical coupler is [A ]

a) Magic Tee b) Directional Coupler c) Both d) None

15. The power division in a Tee when the input is at a coplanar arm is

[D ] a) 2:2:1 b) 3:2:1 c) 1:1:2 d) None

16. Unitary matrix is one satisfying [A ]

a) [A]t [A]* = [I] b) [ ] *tA = [ ] 1−A

c) Both d) None

17. E plane Tee is [C ] a) Voltage junction b) Series Junction c) Both d) None

18. H-Plane Tee is [B ]

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129

a) Voltage junction b) Shunt Junction c) Both d) None

19. If the circuit is symmetric, then its scattering matrix must be [D ]

a) Reciprocal b) unitary c) Symmetric d) None

20. Microwave Junctions are basically [C ]

a) Power dividers b) Combiners c) Both d) None

20. A ferrite is [A ] a) A non conductor with magnetic properties

b) An inter-metallic compound with particularly good conductivity c) An insulator which heavily attenuates magnetic fields d) A microwave semiconductor invented by Faraday

21. Manganese ferrite (MnFe2O3) may be used in [C ] a) Circulator b) Isolator c) Both d) none

22. The maximum power handled by a ferrite is limited by [A ]

a) Curie temperature b) Saturation magnetization c) Line-Width d) Gyro-magnetic resonance

1. A microwave circulator is a [C ]

a) 4 port µw junction b) 3 port µw junction c) Multi port Uni-directional coupler d) 3db µw coupler

2. An isolator or Uniline is a [A ]

a) two- port µw passive device b) single port µw passive device c) Two port µw active device d) Multi port active device

3. The degree of rotation of ferrite rod depends upon [C ]

a) The length of the rod b) The length of rod and diameter of the rod c) The length of the road and diameter of the rod and applied ‘dc’ magnetic field d) None

4. With circular wave guides the type of attenuator used is [A ]

a) Rotary type b) Flap type c) Either a or b d) neither a nor b

5. With rectangular wave guides the type of attenuator used is [B ]

a) Rotary type b) Flap type

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130

c) Either a or b d) neither a nor b 6. The gyrator is a two port passive device [B ]

a) From port 1 to 2 it offers 00 phase shift b) From port 1 to 2 it offers 1800 phase shift c) From port 2 to 1 it offers 00 phase shift d) From port 2 to 1 it offers 1800 phase shift

Basics of the microwave sources Fundamental principles of working of microwave tubes:

o During interaction between particle and field energy transfer takes place. o When the field favours the particle motion, energy transfer takes place from

the field to particle. o If the field opposes the particle motion, the particle loses and field gains the

energy. o The gain of the energy by one is equal to the loss by the other. o The amount of energy transferred is proportional to

The charge on the particle Intensity of the field. Length or duration of the interaction.

o When the gap voltage is sinusoidal time varying and the charge is distributed If the particle crossing at positive peak effects maximum transfers of

energy to field, the crossing of particle at negative peak effects maximum energy transfer to particle

the distributed charge must be compressed into a thin sheet or bunch, where it requires lesser amount time to cross the gap for effecting maximum amount of energy transfer.

o When the gap voltage is sinusoidal time varying and bunch-crossing is at a

constant rate: For maximum unidirectional flow of energy there is only one instant,

either at positive peak or negative peak where the bunch has to cross the gap. So the bunch crossing must be once per cycle of the gap voltage.

In case of bunch-crossing at a uniform rate f , maximum energy transfer can takes place only to a component of grid gap field whose frequency is also f .

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131

the components in the frequency range from ( )4 3 f to ( )4 5 f take different amounts of energy(but not maximum) from the bunches.

Other components of the grid gap voltage like 2 f , 4 f , 8 f etc. don’t involve in the energy transfer, where as the components 3 f , 5 f , 6 f etc. and 2f , 3f , 4f etc. they take/ give negligible amount of energy

• Conventional tubes are less useful at frequencies above 1GHz because o Lead inductance and inter-electrode capacitance o Transit-angle effects o Gain-bandwidth product limitations

• Disadvantages of solid-state µ wave devices are o Low efficiency at frequencies below 10GHz. o Small tuning range o Large dependence of frequency on temperature. o Higher noise.

• Advantages of tubes are o For generation of very high powers (10kw to 1Mw) o Generation of higher millimetre wave frequencies (100GHz and higher)

• Types of tubes o Linear type or ‘O’ type tubes: the electron beam traverses the length of the

tube and is parallel to the electric field. Ex. Klystrons, TWTs o Crossed-field or ‘M’ type tubes: the focusing field is perpendicular to the

accelerating electric field. Ex. Magnetron • Klystrons

o Varian brothers invented klystron amplifier in 1939. o The major disadvantage is their narrow bandwidth, which is a result of the

high-Q cavities required for electron bunching. o They have very low AM and FM noise levels.

• TWTs o Kompfner invented the helix TWT in 1944. o It has the highest bandwidth of any amplifier tube ranging from 30 to 120%. o Its power rating can be increased to several kilowatts by using an interaction

region consisting of a set of coupled cavities. o The efficiency is relatively small ranging from 20% to 40%

• Magnetron o Hull invented the magnetron in 1921. o It is the first high power microwave source. And now they are capable of

very high power outputs on the order of several kilowatts. o Their efficiency is 80% or more.

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132

o Their disadvantage is they are very noisy and cannot maintain frequency or phase coherency when operated in pulsed mode

o The application of magnetrons is now primarily for microwave cooking. • Difference between the Klystron and TWT

o The interaction of electron beam and RF field in the TWT is continuous over the entire length of the circuit, but the interaction in the klystron occurs only at the gaps of a few resonant cavities.

o The wave in the TWT is a propagating wave where as in the klystron it is not a travelling one.

o In the coupled cavity TWT there is coupling effect between the cavities, whereas each cavity in the klystron operates independently.

• Assumptions in the analysis of Klystrons

o The electron beam is with uniform density in the cross section of the beam. o Space charge effects are negligible o The magnitude of the input signal is assumed much smaller than the dc

accelerating voltage.

TWO-CAVITY KLYSTRON AMPLIFIER

• It is a µ wave amplifier with two cavities: one is input cavity known as

buncher cavity and the other is output cavity known as catcher cavity.

• The region in between the cavities is called drift region, which is a field-free

space.

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133

• At one end exists an electron gun and at the other end electron collector, a

metal plate grounded.

WORKING • An uniform electron beam is formed from the divergent electrons emitted by

the cathode using a focusing grid and accelerated to high velocities by an

accelerating voltage.

• All the electrons enter the grid gap with one velocity meV

v 00

2= . In the grid

gap they encounter the signal voltage, which we assume a single sinusoid

tVVs ωsin1= of frequency ω resulting in ‘velocity modulation’ given

by ( ) )]2

sin(2

1[ 10

101

gi tVV

vtvθ

ωβ−+= .

o All the electrons which cross during positive half cycle of the gap

voltage gets accelerated and that cross during the negative half cycle

get decelerated.

o Different electrons that enter the gap at different instants during the

positive/negative half cycle get different amounts of acceleration..

o As the amount of acceleration is equal to the deceleration the average

acceleration or deceleration of the electrons is nil i.e. the average

velocity of the electrons with which they leave the grid gap is same as

that with which they entered.

• While travelling through the drift region the beam undergo ‘density

modulation’. All the electrons that cross the gap during the gap voltage

changes from negative peak to positive peak come together forming ‘bunch’,

a thin dense electron cloud.

• One bunch per cycle forms and the length of the drift region is selected so

that bunch formation completes by the time it reaches the grid gap.

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134

• While crossing the grid gap, the bunch gives energy to the cavity to be taken

out using a loop.

ANALYSIS

• Let us suppose the device is along the z-axis with its first grid walls at z=0 and z=d.

If 0v is the velocity of the electrons with which they enter the cavity gap then. It is

given by meV

v 00

2=

• If the gap voltage due to the input signal is tVVs ωsin1= , then the average gap

voltage during the transit of the reference electron is )2

(sin 01g

iS tVVθ

ωβ +=

where

o Iβ is known as Beam coupling coefficient. It is the ratio of the alternating

current induced in the resonator to the alternating component of beam current

that produces it. For the present case it is

2

2sin

g

g

I θ

θ

β =

o gθ is the average gap transit angle for the electrons ωτ= and τ is average

gap transit time for the electrons = 010

ttvd −≈

• Velocity with which the electrons leave the buncher

gap ( ) )]2

sin(2

1[ 10

101

gi tVV

vtvθ

ωβ−+= . This is the equation of velocity modulation,

which describes the variation of velocity of electrons in the drift space. The quantity

0

1

VViβ is known as the depth of velocity modulation.

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135

• If T is the transit time of the electrons in drift space then 12 ttT −= =

−−=

2sin

21

)( 10

10

1

gi tVV

TtvL θ

ωβ

• The

transit

angle

of the

electron in drift space is Tωθ = =

−−

2sin

21 1

0

10

gi tVV θ

ωβθ

or

−−=

2sin 10

gtXθ

ωθθ where

o 0

0 vLωθ = is the transit angle of the dc electron and

o X is Bunching parameter of the two cavity klystron = 00

10

0

1

22 vL

VV

VV ii ωβ

θβ

= which

determines the degree of bunching and the waveform of the density –modulated

beam

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136

• Optimum length of the drift space 1

000

6.3VVv

Liβω

=

o Let i2 (t2) be the current at catcher cavity and from the principle of the charge

conservation 0022 dtIdti =

0

2

02

dtdtI

i =

o

+−++=

2sin

21 0

0

1002

gi tVV

Tttθ

ωβτ giving

00002 )2

cos( dttTdtdt gθωω +−=

)2

cos(1)2

cos(1 02

0

0

0

0

2

0

gg tX

I

tX

I

dtdtI

θθω

θω −−−

=+−

=

o As i2 (t2) is a periodic current, it can be expanded in Fourier series

( ) [ ]∑∞

=

−−+=1

020022 )(cos)(2n

gn tnnxJIIti θθω . Out of these infinite number of

components only the primary or fundamental component that is useful to amplify the

signal. The amplitude of the fundamental component I2 = 2I0 J1 (x) which assumes is

maximum value when X = 1.841 = 00

1

2 vL

VVi ωβ

leading to the optimum length of the

drift space 1

000

6.3VVv

Liβω

=

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137

• Equivalent circuit

o It consists of a parallel combination of wall resistance of the catcher cavity, beam loading resistance and external load resistance

• Output power

o I2 ind = Amplitude of fundamental component of induced current in catcher cavity

)(2 10020 XJII ββ ==

o Pout = shshout RXJIVI

RI

P2

)](2[22

)( 2100220

220 βββ

===

• Electronic efficiency = η = 00

220

2 VIVI

PP

in

out β=

If the coupling is perfect β0 = I1 , max I2 is 2I0 (0.58), and V2 = V0 , then the max

electronic efficiency η = 58%, in practice it is 15 to 30%.

• Voltage gain Av

Av ≅ 1

2

VV

= vsh

sh ARR

XXJR

VI ==

0

10

20

1

20 )(θββ

0

00 I

VR = dc beam resistance.

_________________________________________________________________________


138

• Mutual conductance mG

mG ≅ voltageInput

currentoutputinducedV

i ind =1

2 = 1

100 )(2V

XJIβ

∴ But V1 = XVV

iβ12 from X = 0

0

1

2θ

β

V

Vi so

mG = mGRX

XJ=

0

10

20

1.)(θβ

Av and Gm are related thro Av= Gm Rsh

• PERFORMANCE

• Efficiency about 40% (58% max)

• Power output: CW power up to 500 kW

Pulsed power up to 30 MW

Power gain: about 30 db

• Beam Loading: The production of an electronic admittance between two grids when an

initially un-modulated beam of electrons is shot across the gap between them is called

“Beam Loading”.

• If the buncher cavity gap width is more, the gap transit angle will be large and the

buncher cavity supplies energy to the beam to bunch. The power PB required to produce

bunching action is given as PB = V12 GB where GB is known as beam loading

conductance equal to

−

2cos

2 020

0 gG θββ .

• The Conductive component of beam loading admittance is bG ≅

−

151

24

22

0

0 gg

VI θθ

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139

• Positive bG indicates * Power is being delivered to the beam by the source of

alternating gap voltage. If the voltage source is resonator reduction in its Q

broadening its tuning.

• Negative bG indicates * Beam is delivering power to the source of alternating

gap voltage. If the source of alternating gap voltage is resonator, it may go into

oscillations.

• The susceptive component is bB ≅

−

203

112

2

0

0 gg

VI θθ

• It affects the resonance frequency of a resonator connected across the gap and

therefore cause the frequency of an oscillator to change with beam current and

voltage.

MULTICAVITY KLYSTRON:

• It gives more power gain than two-cavity Klystron. In a multi-cavity klystron each of the

intermediate cavities, placed at a distance of the bunching parameter X of 1.841 away

from the previous cavity acts as a buncher with the passing electron beam inducing a

more enhanced RF voltage than the previous cavity, which in turn sets up an increased

velocity modulation. The spacing between the consecutive cavities would therefore

distinguish [∴ From Lopt = 1

006.3VV

w iβϑ , L ∝

1

1V

]

• The multi-cavity klystrons are often operated with their cavities stagger tuned to obtain

greater bandwidth.

Power gain: 40 to 50 dB

Bandwidth: several percent

Frequency: 0.5 GHz to 14 GHz

Power range: 25 kW to 40 MW

_________________________________________________________________________


140

• These are noisy because the bunching is never complete and so the electrons arrive at

random at catcher cavity. So it is too noisy for use in receivers.

REFLEX KLYSTRON

• Reflex Klystron is a low power, low efficiency µ -wave oscillator; used as signal source

in Microwave generators, as local oscillator in microwave receivers,

as pump oscillator in parametric amplifiers, as frequency modulated oscillators in

portable microwave links. Its power output ranges from 10 mW to 3W and frequency

from 4 to 200 GHz

• Its basic parts are one reentrant cavity, beam emitter, accelerator and repeller.

• The electron beam emitted is accelerated to high velocities by the accelerating voltage

and while passing through the grid gap the beam gets velocity modulated.

• The velocity-modulated beam enters the repeller region to face repulsive field of the

repeller region.

• All the electrons, which cross the grid gap during the period from the positive peak to negative peak, come together forming a bunch after spending different amounts of time in the repeller region.

• The thin dense electron cloud i.e. bunch crosses the gap giving energy to the gap. For maximum transfer of energy the gap voltage must be large and opposing the bunch movement.

• Repeller protection:

o The voltage to repeller is always applied before the cathode and o A cathode resistor is often used to ensure that the repeller can never be more

positive than the cathode.

• Tuning:

o Frequency can be adjusted by adjustable screw, bellows or dielectric insert (mechanical methods)

_________________________________________________________________________


141

o Frequency can also be varied by adjusting the repeller voltage (electronic tuning). It is an important feature of the reflex klystron because it provides a means of obtaining fine tuning and also a means of introducing frequency modulation. In a typical case, a total frequency variation of the order of 1 percent can be obtained.

ANALYSIS

• Suppose the oscillator is lying along the z-axis with it grid walls at z=0 and z=d Let

0t is the instant at which the reference electron enters the cavity gap at z = 0, 1t is the

instant at which the electron leaves cavity gap at z = d and 2t is the instant at which

the same electron returned back to the gap by retarding field at z = d and collected

by the walls of cavity

• Let 0v is the velocity with which the electrons enter the cavity gap after getting

accelerated by potential 0V . Then it must be equal to meV02 . So the velocity with

which the electron enters the cavity gap at t = t0 and z = 0 is meV02 .

• The electrons while traversing the grid gap undergo a process known as ‘velocity

modulation’ with the following features. Assuming the presence of single frequency

component in the grid gap,

o The electrons that cross the gap during the positive half cycle of the gap

voltage get accelerated and those that cross during the negative half cycle

undergoes deceleration.

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142

o Different electrons that enter the gap at different instants during a half cycle

undergo different amounts of acceleration/deceleration.

o The amount of acceleration is equal to the amount of deceleration during one

cycle of the gap voltage as the beam is uniform. So the average amount of

acceleration/deceleration is zero.

o The average velocity of the electrons is same as the velocity with which they

enter the gap i.e. 0v

o Different electrons leave the gap with different velocities Let ( )1tv is the

velocity with which the reference electron leaves the cavity gap at 1tt = and

dz = . Then it is equal to ( )

−+=

2sin

21 1

0

101

gi tVVvtv

θωβ . This is known as

the equation of velocity modulation.

• The electrons crossing the gap enter into the repeller region in the face of retarding

force and spend different amounts of time before returning back to the gap..

Electrons accelerated during the gap transit enter deep into the repeller region and

hence require more time to come back to gap when compared to the electrons that

got decelerated.

o E = Retarding electron field in the repeller space =

LVV

LwtVVV rr 010 )sin( +≅++

o The force equation for an electron in the repeller region is

LVVeeE

dtzdm or +−=−=2

2

. Integrating this equation twice and applying the

conditions ( )1tvdtdz = and dz = at 1tt = we have

dtttvttmL

VVez r +−+−+−= )()()(2

)(11

21

0

_________________________________________________________________________


143

• After the round trip the electron returns to the gap at t = t2, so at t = t2 , z = d giving

dtttvttmL

VVedz r +−+−+−== )()()(2

)(121

212

0 . The transit time of the electron in the

repeller region is 1

0

112 )(

)(2 TVVetvmLtt

r

=+

=− . Form this

−+=

2sin

21 1

0

1'0

' gi tVVTT

θωβ

where 10T =

)(2

0

0

VVemLv

r +. This equation for the repeller transit time can be put in the

form of )2

sin( 1''

0' gtX

θωθθ −+=

o where '0

10 Tωθ = , round the trip de transit angle of the center of the bunch

electron is == '0

10 Tωθ

)(2

0

0

VVemwLV

r + and

o 'X is the Bunching parameter of the reflex klystron oscillator '0

0

1'

2θβ

VVX i=

o all the electrons that cross the grid gap during the gap voltage variation from

positive peak to negative peak spend different amounts of time in the repeller

region and comeback to the gap at one instant forming into a ‘Bunch’. The

electron bunch gives maximum energy to the gap field when it faces maximum

deceleration which is possible only when the transit time of electrons in the

repeller space is 41'

0 −=nT where n is a positive integer two onwards. So the

round trip transit angle referring to the center of bunch electron 10θ should be

_________________________________________________________________________


144

( )

−==−=

412'

012'0 nTtt πωωθ

= N 2π = 2πn - π/2 where N = mode number of the reflex klystron.

o The transit time depends on the repeller and anode voltages

• Power output of the RKO: By comparing with that of the two-cavity klystron the

beam current of a reflex klystron oscillator can be written

∑∞

=

−−−−=1

'02

1002 )](cos[)(2

ngtnnxJIIi θθω

The fundamental component of induced current is )(2 '102 XJII iβ=

The ac power delivered to the load 2

21IVPac = = )( '101 XJIV iβ

_________________________________________________________________________


145

but ⇒−

=)

22(

2 '

0

1

ππβ n

XVV

i )2

2(

)(2 '1

'00

ππ −=

n

XJXIVPac

• Efficiency of the RKO:

dc

ac

PP=η ; with 00IVPdc =

)2

2(

)(2 '1

'

ππη

−=

n

XJX

Maximum output occurs at 2=n . )( '1

' XJX reaches a

max value of 1.25 at 4.2' =X and 5.0)( '1 =XJ giving

ηmax = 22.7%

• Repeller voltage versus mode number:

200 2

1 vmeV = ⇒ 200 2

1 vemV =

mL

VVenvn

VVemLv r

r ω

ππππωθ2

)()2

2(

22

)(2 0

2

00

010

+−=⇒−=

+=

From the above two relations

222

02

0 4

)()2

2(

21

Lm

VVen

emV

r

ω

ππ +−=

⇒ me

L

n

VVeV

r22

2

0

0

8

)2

2(

)( ω

ππ−=

+ me

L 8)(

22

2'0

ωθ=

• Power output versus repeller voltage:

000

01

11

2)()( IV

mVe

LVVXJXP r

AC ω+=

• Frequency versus Repeller voltage

_________________________________________________________________________


146

men

LVVV

r

)2

2(8

1)( 222

0

0 ππω

−=+

from this relation differentiating rV with respect toω .

emV

n

LddVr 08

)2

2( ππω −=

This expression gives change of frequency with respect to the repeller voltage as

082

)2

2(

mVe

L

n

dVdf

r π

ππ−=

0

0

82'

mVe

Lπθ=

_________________________________________________________________________


147

• Electronic admittance Ye : It is the ratio of the induced current i2 at t2 in phasor form

to the voltage V2 across the gap at t2.

)

2(

'

'1

10

2

0

02/

1

'10

'0

10 )(2

2)(2 θπ

π

θ θββ −

−

−

==j

ij

ji

e eX

XJVI

eVeXJIY

• Equivalent circuit of the Reflex Klystron: it consists of a parallel combination of

one capacitor, one inductor both representing the energy storage elements of the cavity,

three inductances representing copper losses, beam loading and load conductance.

o C and L represent energy

storage Elements of the cavity

o Gc = Copper losses of cavity

o Gb = Beam loading conductance

o Gl = the load conductance

_________________________________________________________________________


148

• The necessary condition for oscillations is that the magnitude of the negative real part of

the electronic admittance should be greater than or equal to the total conductance of the

cavity circuit. i.e.

|- Ge | ≥ G where G = Gc + Gb + Ge and Ye = Ge + jBe

MAGNETRON TYPES:

1. Split-anode magnetron: it uses the static negative resistance between two anode segments. It operated at frequencies below µ wave region.

2. Cyclotron-frequency magnetron: It operates under the influence of synchronism between an alternating component of electric field and a periodic oscillations of the electrons in a direction parallel to the field. It operates at frequencies in the µ wave range, power output very low and very efficiency.

3. Travelling-wave magnetrons: they depend on the interaction of electrons with a travelling electromagnetic field of linear velocity.

TRAVELING WAVE MAGNETRON • Cavity Magnetron is a high power, high efficiency microwave oscillator which

depends upon the interaction of electrons with a traveling electromagnetic wave for

its operation..

• It is a diode with several connected re-entrant cavities in the anode structure. The

connected re-entrant cavities results in the existence of a rotating rf field in the

interaction region whose angular frequency is dtdφβω 0= where β0 = phase

constant = (2πn)/NL

• It is a crossed – field device as it employs axial dc magnetic field and radial dc

voltage. If the values these fields are adjusted property, the electrons follow cycloid

paths in the interaction space.

• In total there exists three fields in the interaction region of magnetron.

_________________________________________________________________________


149

π Dc electric field in radial

direction

π Static magnetic field in

axial direction

π Rotating rf field

• Hull Cut-off field is the magnetic

field/electric field required to return the electrons to the cathode after they have just

grazed the anode in the absence of RF field.

• Mode π oscillations: Self-consistent oscillations can exist only if the phase

difference between the adjoining anode poles is nπ / 4 where n =1,2,3 etc. For best

results n = 4 is used in practice which gives a phase difference of π between two

successive poles and the resulting mode is known as π - mode.

• Strapping: Magnetrons using identical cavities in the anode block employs

strapping to prevent mode jumping. For π mode of operation it consists of two rings

of heavy gauge wire connecting alternate anode poles.

π At very high frequencies, rising–sun anode structure is used to prevent mode

jumping.

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150

• Frequency pulling: The resonant frequency of the magnetron alters with the

changes in the load admittance. Such frequency variation is known as frequency -

pulling.

• Frequency Pushing: It is the variation of the frequency of the magnetron due to the

changes in anode voltage..

• Phase–focusing effect: In cavity magnetron the electrons are bunched by an effect

known as phase – focusing effect without it, the favoured electrons (i.e. which

contribute energy to the RF field) would fall behind the phase changes of the electric

field across the gaps, since such electrons are retarded at teach interaction with the RF

field. The wheel spoke bunches rotate counter clockwise with correct velocity allowing

continuous interchange of energy. In this interchange, the RF field receives much more

than it gives.

FUNCTIONING:

_________________________________________________________________________


151

• The dc field strengths are to be adjusted to cut-off values i.e. 0V should be set to

2

2

222

0 18

−=

babB

meVoc for a given 0B or 0B should be set to

−

=

2

2

2/1

0

1

8

bab

emV

Boc

for a given 0V .

• Under the influence of the dc fields and the rotating rf field the emitted

electrons from the cathode revolve around the cathode with a cyclotron angular

frequency m

eBC

0=ω

• When the cyclotron frequency of the electrons is equal to the angular frequency

of the field i.e. cω = dtdφβ0 where β0 = phase constant = (2πn)/NL the

interactions between the field and electron occurs and the energy is transferred .

ANALYSIS:

Equations of Electron motion:

• The Lorenz force acting on an electron because of the presence of both the electric

field E and the magnetic flux B is given by

F = - e ( BvE ×+ ) = m dtdv

• In Cylindrical-coordinate system the general expression for the acceleration is a =

dtdv = zp v

dtzdv

dtd

dtdv

dtd

dtd

2

222

2

2 1 +

+

− φ

φρρ

φρρ

π When the electron is in the interaction space 0=zv and only Bz ≠ 0 , ρE ≠

0 , with this information we can write the equations of motion for electrons

in φ direction as

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152

dtd

ρ1

dtdB

me

dtd

zρφρ =

2

dtd )(

21 2

2

ρωφρdtd

dtd

c=

where zc B

me=ω is known as the cyclotron angular

frequency of the electrons

tconsdtd

c tan21 2

2

+=

ρωφρ ; the integration constant can be found from the fact at

a=ρ , 0=dtdφ . Now the constant = cω

21 so

−= 2

2

121

ρωφ a

dtd

c (1)

Since the magnetic field does no work on the electrons, the kinetic energy of the electron

is 22222 221

φρ vvVmeveVmv p +==⇒=

222

+

=

dtd

dtdV

me φρρ (2)

When the electrons just graze the anode, 0, VVb ==ρ and OCBBanddtd ==0ρ

And (2) becomes )1(21

2

2

ba

dtd

c −= ωφ (3)

0

22 2 V

me

dtdb =

φ (4)

Substituting (3) and (4) 0

2

2

22 21

21 V

me

bab c =

−ω

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153

Hull cut-off magnetic field equation

−

=

2

2

2/1

0

1

8

bab

emV

Boc

If B0 > Boc for a given V0 the electron will not reach the anode.

Hull Cut-off voltage equation: 2

2

222

0 18

−=

babB

meVoc

If V0 < Voc for a given B0 the electrons will not reach the anode.

• Cyclotron Angular frequency Cω : Since the magnetic field is normal to the

motion of electrons, the travel in a cycloid path, the

out-ward centrifugal force is equal to the pulling force

meB

rvevB

Rmv

C ==⇒= ω2

• Uses: It is a high power microwave source used in

π Radar transmitters

π Linear particle accelerators.

• Performance chart: It is a chart containing the contours of constant power output,

constant efficiency, constant flux density and occasionally constant frequency.

Plotted on coordinates of direct anode voltage Vs direct anode current at a specified

load admittance.

• Load admittance diagrams: it is a chart containing contours of constant power or

efficiency and of constant frequency plotted in coordinates of load resistance and

load reactance. When polar coordinates are used these diagrams are called “Rieke

Diagrams”

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154

• Potential in the interaction region:: 0( ) ln ln bV Va aρρ =

;

• Velocity of the electron in the interaction region::2/1

)(2)(

= pV

mev ρ

• Powers - 10 kW to 5 Mw

• Efficiency - 50 % (40 to 70%)

TRAVELLING WAVE TUBE (TWT)

• TWT is a linear beam tube in which the interaction between the beam and the RF field is

continuous over a length.

• Complete bunching results due to continuous interaction between axial RF field and

beam ultimately giving high gain

• TWT uses non-resonant microwave circuit and is capable of enormous bandwidths.

• Slow-wave structures are special circuits used in microwave tubes to reduce the wave

velocity in a certain direction so that the electron beam and signal wave can interact;

Ex: Helix, Folded back line, Zigzag line, inter-digital line, corrugated wave-guide and

coupled cavities are slow-wave structures widely used.

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155

• HELIX: When the signal is travelling along the coil with velocity C, the phase velocity

of the wave in the axial direction is

d

PCdP

CPvp ππ≈

+=

22 )(= =ψsinC

βω

• The axial electric field E (x, y, z) is periodic along axial direction i.e z- direction with

periodicity equal to patch. This field can be expressed as linear combination of several

spatial harmonics using Fourier theorem.

∑∞

∞−=

−=n

jpn z

eyxzyxE ),(),,( ; L

nn

πββ 20 += [ ]PL =

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156

=nβ phase constant of the ihn mode

The phase velocity of various spatial components of the wave is

Lnv

npn πβ

ωβω

20 +

==

• PRINCIPLE OF WORKING:

o There exists a spatially periodic travelling wave co-axial to the helix when the

signal is travelling along the length of its conductor.

o The electron beam is accelerated to a velocity, which is slightly more than the

phase velocity of the axial wave.

o The axial wave accelerates the electrons during one half cycle and decelerates

during the second half cycle but at any point of time there exists more electrons

in the decelerating half cycle

resulting in net transfer of energy

from the electrons to the wave.

o The strengthened wave offers more

deceleration to the incoming

electrons, increasing electron

concentration in its region thereby

increasing energy transfer to the

beam manifold.

o It results in the exponential growth

of the signal along the length of the

helix.

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157

• Brillouin Diagram: It is ω - β diagram for a helical slow-wave structure with several

spatial harmonics. The second

quadrant of the ω -β diagram

indicates the negative phase

velocity that corresponds to the

negative n .The shaded areas are

the forbidden regions for

propagation. It is because if the axial phase velocity of any spatial harmonic exceeds the

velocity of light, the structure radiates energy.

• Attenuator: Oscillations are possible in this high gain device, especially if poor load

matching causes significant reflections along the slow wave structures very close

coupling of slow-wave circuits further aggravates the problem. Attenuator is used to

prevent oscillations in TWT and it has the subsidiary effect of reducing the gain. The

attenuator may be a lossy metallic coating such as Aquadag or KANTHAL on the

surface of the glass tube.

Both forward and reverse waves are attenuated, but the forward wave is able to

continue and grow past the attenuator because bunching is unaffected.

• Electronic Equation: It determines the convection current induced by the axial electric

field.

0

1212

0

0

2)()(2 RE

jjE

jVIji

e

e

e

e

γββ

γββ

−=

−= , where γ is the propagation constant of the axial

waves ee jβαγ += ,0ve

ωβ = , γγ =0 with 0=i

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158

• Circuit Equation: It determines how the spatial ac electron beam current affects the

axial electric field of the slow-wave helix.

20

200

2

1 γγγγ−

−= ZE where 0Z = characteristic impedance of helix

• TWT gain parameter 3/1

0

00

4

≡

VZIC

• Power gain AP = - 9.5 + 47.3 NC db

• Propagation constants: The four propagation constants represent four different modes

of wave propagation in the O-type helical TWT.

o )2

1(23

1CjC ee ++−= ββγ : It represents forward traveling wave. The beam

velocity is slightly more than the phase velocity of this wave. Energy transfer

takes place from the beam to the wave so its amplitude increases as it progresses.

o )2

1(23

2CjC ee ++= ββγ : It represents forward traveling wave. The beam

velocity is slightly more than the phase velocity of this wave Energy transfer

takes place from the wave to beam so its amplitude decreases as it progresses.

o )1(3 Cj e −= βγ It represents wave traveling in the forward direction. The beam

velocity is less than the phase velocity of this wave. There exists no transfer of

power between the beam and wave.

o )4

1(3

4Cj e −−= βγ It represents a wave traveling in the reverse direction with

no transfer of energy. The beam velocity is less than the phase velocity of the

wave

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159

GUNN DIODE

• Gunn oscillators and amplifiers are most important microwave devices that have been extensively used as local oscillators and power amplifier covering the frequency range 1 to 100 GHz in which Gunn diode is a critical part.

• Gunn diode is an n-type slab of GaAs, InP, InAs, InSb and CdTd. • Gunn diode exhibits dynamic negative resistances when it is biased to a

potential gradient more than a certain value known as threshold field Eth due to Gunn effect or Transferred Electron Effect (TEE).

• In any n-type semi-conductor the drift velocity, the following relations govern current, field and drift velocity.

µµµµ qndEdJEqnJand

dEdvEv d

d =→==→=

When the field is less than the Eth , increase in the field E causes the dv to increase resulting in the positive mobility µ . Hence an increase

in the E causes J to increase resulting in positive resistance. When the field is in between Eth and Ev increase in the field E causes

the dv to decrease due to the onset of TEE resulting the negative mobility µ . Hence an increase in the field E causes J to decrease resulting in the manifestation of differential negative resistance.

When the field is more than Ev increase in field E causes dv to increase resulting in the positive mobility µ due to the disappearance of the TEE. Hence an increase in the E causes J to increase resulting in positive resistance.

• The threshold field values are GaAs-3.3kv/cm, InP-10.53kv/cm, InAs-

1.63kv/cm, InSb-0.63kv/cm CdTd-13.03kv/cm • TEE is ‘a field induced transfer of conduction band electrons from a high

mobility lower energy satellite valley to low mobility higher energy satellite valley’

o It is bulk material property i.e. it takes place at each and every point in the body of the Gunn.

o Due to this effect the mobility of the electrons in the diode become negative.

• In the InP diode o There exists three satellite valleys in its conduction

band where as in others it two. o The peak to valley current ratio is larger because the

electron transfer proceeds faster with increasing field. o Thermal excitation of the electrons has less effect

leading to the lower degradation of the peak to valley

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160

current ratio because the larger energy separations between lower and its nearest valley.

• In the InAs and InSb diodes TEE can be observed only under hydrostatic pressures that reduce the energy difference between the satellite valleys. Their energy difference is more than that of the forbidden gap under normal pressures.

• The electrons drift through the diode with velocities depending upon the field intensity and its maximum when the diode is biased to threshold value.

• Peak velocities in various diodes are GaAs-2.2, InP-2.5, InAs-3.6, InSb-5.0 and CdTd-1.5 times 107 cm/sec.

• Noise is due to ‘AM noise’ normally small, due to amplitude variations plus ‘FM noise’ which is due to frequency deviations.

• The upper frequency of the TEDs is limited to 150GHz mainly due to the ‘finite response time’.

• The output power falls as 2

1f

• Gunn oscillators and amplifiers are being widely used as local oscillators and power amplifiers covering 1 to 100GHz range

GUNN DOMAINS

The transfer to lower mobility valley starts with the electrons located in a small region where the field intensity is more due to lower carrier concentration. These regions are called high field domains. The domains travel to anode shifting all the electrons in their path to lower mobility valley. The velocity of the domains is slightly more than the drift velocity of the electrons.

• Domains start to form whenever the electric field in a region of the sample increases above the threshold value and with the stream through the device.

• If additional voltage is applied to the diode with a domain then the domain will increase in size and absorb more voltage than was added and the current will decrease.

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161

• The domain disappears after reaching the anode or in the mid-way if the field drops to a value less than sustain field value Es.

• Decreasing the field slightly lower than the threshold value can prevent the formation of new domain.

• The domain modulates the current through the device as the domain passes the regions of different doping and cross-sectional areas.

• The domain length is inversely proportional to the doping concentration.

TWO-VALLEY MODEL THEORY:

• It has been proposed by Kroemer to explain the manifestation of negative resistance in certain type of bulk semiconductor materials.

• In the conduction band of n-type GaAs a high mobility lower valley is separated from a low mobility upper valley by an energy difference of 0.36ev.

• Under equilibrium conditions the electron densities in both the valleys remain same.

• When the applied field is lower than the field corresponding to the energies of the electrons in the lower valley then the no transfer of electrons takes place from one to other valley. The mobility of the carriers is positive.

• When the applied field is higher than the field corresponding to the energies of the electrons in the lower valley and lower than the field corresponding to the energies of the electrons in the upper valley, then transfer of electrons takes place from high mobility lower to low mobility upper valley. The mobility of the carriers becomes negative.

• When the applied field is higher than the field corresponding to the energies of the electrons in the higher valley, then no transfer of electrons takes place because by that time all the electrons of the lower valley must have been transferred to the upper valley. The mobility of the carriers is positive.

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162

• The nobilities of the electrons in the two valleys must satisfying the relation

P

l

ul

lul

ul EnnfwhereP

dEdn

nE

f==>

−

−

+− µ

µµµµ ,1

RWH THEORY

Ridley, Watkins and Hilsum proposed this theory to explain the phenomenon of Negative Differential Resistance (NDR) in bulk materials. Its salient features are

• Bulk NDR devices are classified into two groups. One voltage controlled NDRs and second current controlled NDRs.

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163

• The characteristic relation between Electric field E and the current density J of voltage controlled NDRs is ‘N’ shaped and that of the current controlled NDRs is ‘S’ shaped.

• The electric field is multi-valued in the case of voltage controlled NDRs and it is electric current that is multi-valued in case of current controlled NDRs.

• The differential resistivity increases with field in case of voltage controlled NDRs and decreases in case of current controlled NDRs.

• A semi-conductor exhibiting bulk NDR is inherently unstable because a momentary space charge, which might have been created due to random fluctuation in the carrier density, grows exponentially with time because the relaxation time is negative.

• Because of NDR, the initially homogeneous semi-conductor becomes heterogeneous to achieve stability. It results in ‘high field domains’ in voltage controlled NDRs and ‘high current filaments’ in current controlled NDRs.

• The high field domain starts forming at a region where the field intensity is higher extending further perpendicular to the direction of current flow separating two low

field regions. The width of the domain is

−−=

12

1

EEEELd A

• The high current filament starts forming at a region where the field intensity is higher extending further along the direction of the current flow separating two low

current regions. Its cross-sectional area is

−−=

12

1

JJJJAa A

According to RWH theory for the semiconductor to exhibit negative resistance,

• The separation energy between the lower valley and the upper valley must be several times larger than the thermal energy of the electrons at room temperatures i.e.

• The separation energy between the valleys must be smaller than forbidden energy gap between the conduction band and valence band.

• Electrons in the lower valley must have high mobility, small effective mass and low density of state whereas those in the upper valley must have low mobility, large effective mass and high density of state.

As Si and Ge don’t meet these criteria, they can not exhibit dynamic negative resistance.

GUNN MODES

• Major factors that determine the modes of operation are Concentration and uniformity of the doping Length of the active region Operating bias voltage Cathode contact property

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164

Type of the circuit used. • An important boundary separating the various modes of operation is

2120 10 −= cmLn

• The TEDs with Ln0 products smaller than 21210 −cm exhibit a stable field distribution.

Gunn oscillation mode:

>

≅−212

7

10sec/10

cmnLcmfL

• This mode is operated with the field more than the threshold value i.e. E > Eth • The high field domain drifts along the specimen until it reaches anode or low

field value drops to below the sustaining field value ie E < Es

• The frequency of oscillation is given by eff

dom

Lvf = where domv is the velocity

of the domain and effL is the effective length the domain travels before a new domain gets nucleated.

Transit time domain mode: [ ]sec/107 cmfL≈ • The high field domains are stable in the sense that they propagate with a

particular velocity but don’t change in any way with time. • When the high field domain reaches the anode the current in the external

circuit increases. • The frequency of the current oscillations depends on among other things,

the velocity of the domain across the sample. If the velocity increases the frequency increases and vice versa. It also depends upon the bias voltage.

• The shape of the domains in GaAs and InP TEDs is triangular. • In this mode the oscillation period is transit time. The efficiency is below

10%.

Delayed domain mode: sec]/10sec/10[ 76 cmfLcm <<

• In this mode the domain is collected by the anode when E < Eth and the new domain formation gets delayed until the rise of the field to above threshold.

• The oscillation period is greater than the transit time. • The oscillations occur at the frequency of the resonant circuit.

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165

• The efficiency of this mode is about 20% Quenched domain mode: sec]/10sec/10[ 76 cmfLcm <<

• While the domain is traveling, the bias field drops to a value less than ES during negative half cycle quenching the domain. A new cannot form until the field again rises above the Eth .

• Oscillations occur at the frequency of the resonant circuit rather than the transit time frequency.

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166

• The operating frequencies are higher than the transit time frequency. • Formation of multiple high field layers takes place. • The upper frequency limit for this mode is determined by the speed of

quenching. • In this mode the efficiency can be 13%.

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167

LSA mode:

><

>304

7

102102

sec/102

xf

nx

cmXfL

o This is the simplest mode of operation. o As the frequency is high the domains do not get sufficient time to form. o Most of the domains find them selves in the negative conduction state

during a large fraction of voltage cycle. o A large portion of the device exhibits a uniform field resulting in efficient

power generation at the circuit controlled frequency. o This mode is suitable to generate short pulses of high peak power o Its maximum operating frequency is much lower than the that of the TT

devices.

Stable amplification mode:

=

≅−21211

7

1010sec/10cmtonL

cmfL

o In this mode the devices exhibits stable amplification at the transit time frequency.

o Negative conductance is utilized to prevent the formation of the domains. o There exists three regions of amplification depending on the fL product

range from 87 105.010 xto Bias ciruit oscillation mode:

• This mode occurs when there is either GUNN or LSA oscillation and fL is small.

• When the diode is biased to the threshold GUNN oscillation begin leading to sudden decrease in the average current of the circuit driving it to oscillations.

• The frequency of the oscillations may be in the range from 1KHz to 100 MHz.

IMPATT

• The IMPATT diode is now one of the most powerful solid-state sources for the generation of microwaves. It can generate higher CW power outputs in millimeter-wave frequencies i.e. above 30 GHz of all solid-state devices. These are compact, inexpensive, moderately efficient and with improved device fabrication technology these diodes also have become reliable under high temperature operation

• IMPATT stands for ‘IMPact ionization Avalanche Transit Time’. • IMPATT diodes employ ‘impact ionization’ and ‘transit time’ properties of semi-

conductor structures to get negative resistance at microwave frequencies.

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168

• Impact ionization or avalanche multiplication: ’it is a process related to semi-conductors in which the generation and multiplication of hole-electron pair takes place due to knocking off the valence electrons into conduction band by the highly energetic carriers when the electric field is increased above certain value’.

The rate of pair production is a sensitive non-linear function of field. • The negative resistance occurs from the delay, which cause the current to lag behind

the voltage by half cycle time, have two components: One is Avalanche time delay caused by ‘finite buildup time of the

avalanche current.’ Other is transit time delay by the finite time for the carriers to cross

the drift region. • These diodes are made from GaAs, Ge, Si. • Extremely high voltage gradient 400kv/cm back biasing the diode is required for its

operation. • In all the structures there exists two regions

o Avalanche region: in this region avalanche multiplication takes, doping concentration and field intensity are high.

o Drift region: in this region avalanche multiplication does not take place, doping concentration and field levels are low.

o Depletion region is AR plus DR.

• Maximum negative resistance is occurs when the transit angle θ = π at which the

operating frequency becomes L

vf d

2= where vd is drift velocity of the carriers and L

length of the drift region. • IMPATT is the name of a diode family. It’s basic members are

o Read diode ++++ −−−−−− pipndualitsorninp o Single drift diode ++ −− pnp o Double drift diode or RIMPATT diode ++ −−− nnpp o Pin diode ++ −− nip

• The noise measure in GaAs is low when compared to Si and for Ge it is in between

GaAs and Si. The main reason for the low noise behavior of GaAs is that for a given field the electron and hole ionization rates are essentially same, where as in Si these are quite different.

• The highest powers, frequency and efficiency are obtained from double drift diodes that are also known as RIMPATTs. The power-frequency2 product is highest for these diodes. The improved performance results mainly from the fact that holes and electrons produced by the avalanche are allowed to give energy to RF signal while

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169

traversing the drift region. In the case of single drift diodes only one type of carriers is so utilized.

• Comparison: o When compared to GUNN diode these diodes have more efficiency around

30%, more powerful around 15w CW and their frequency can reach up to 200GHz where as in the case of GUNN it is only 100GHz.

o But when compared to GUNN diodes these are more noisy. o Below 40GHz GaAs IMPATTs have higher powers and efficiency than do

Si IMPATTs. o Between 40-60 GHz GaAs IMPATTs show higher power and efficiency

whereas Si IMPATTs give high reliability and yield. o Above 60GHz Si IMPATTs outperform the GaAs IMPATTs. o Around 10GHz efficiency is close to 40%.

• Power output:

At lower frequencies the power output is thermal-limited and varies as f –1 ;

At higher frequencies (>50 MHz) the power is electronic limited and varies as f –2

• Difficulties: The noise is high mainly because of the statistical nature of the

generation rates of electron-hole pairs in the avalanche region. Highly sensitive to operational conditions. Large electronic-reactance, which can cause detuning or even

burnout of the device unless proper care is taken. • Applications:

In microwave links In CW radars In electronic counter measures.

TRAPATT

• TRAPATT stands for ‘TRApped Plasma Avalanche Triggered Transit’ • TRAPATT diode is a high power, high efficiency device. • For a TRAPATT diode, the design and performance are more complicated because

of strong device-circuit interaction that dictates most of the device performance. • Silicon ++++ −−−− ppnornnp structures are used to get high powers. The

doping of the depletion region is generally such that the diodes are well punched through at break down i.e. depletion region extends from ++ −− nntonp junction.

_________________________________________________________________________


170

• OPERATION: o It is mounted at a distance of λg/4 from a short in a wave-guide or coaxial

line so that a high RF field exists across the diode o Initially the diode charges up like a linear capacitor, driving the magnitude of

the field above the breakdown voltage. o High field avalanche zone or shock front passes through the diode and fills

the depletion layer with a highly conductive dense plasma of electrons and holes whose space charge depresses the voltage to low values.

o The plasma generated takes time to get removed from the depletion region followed by the residual charge from the ends of the depletion layer, raising the voltage across the diode.

o The diode once again charges up like a fixed capacitor until current moves to zero. The same voltage is maintained across the diode until the current rises again.

o As the voltage across the diode is low during the drift of the pulse, drift velocity becomes less leading to more transit time, dissipation becomes less giving rise to higher efficiencies, operating frequencies lower and active regions become thinner.

o This diode requires a circuit that can support harmonics of fundamental frequency of high voltage amplitudes. The rich harmonic content is necessary to get the required phase delay in the current at such low frequencies.

• Difficulties:

o It has higher noise figure when compared to IMPATT diodes. o Its operation is quite complicated and requires good control over the device

and circuit. o The upper operating frequencies are practically limited to below millimeter-

wave range i.e. 10GHz. o It is highly sensitive even to small changes in circuit or operating conditions

or temperature. • Performance:

o The output power of a series connection of five diodes under pulse condition reaches 1.2kw with a efficiency of 25%.

o The upper frequency limit is close to 10GHz and highest obtained efficiency is 75%

o Its high pulse power output is much larger than most other microwave semi-conductor devices.


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171

1. As frequency increases the inter electrode capacitance [A ] a) increases b) decreases c) Remains constant d) zero

2. The degenerative feed back in vacuum tubes is caused by [ ]

a) Inductance of cathode lead b) Cpg c) Cgk d) CPK

3. energy of the electron moving opposite to the flux

lines in field the K.E of an electron [A ] a) Increases b) decreases c) do not change d) zero

4. The factor that determines the location of the catcher cavity is [ ]

a) position where bunching forms b) position where fundamental frequency component is maximum c) Where Velocity of electrons is minimum d) Where Velocity of electrons is maximum

5. The basic principle of operation of klystrons is [ ]

a) Resonance b) Degenerative feed back c) Velocity modulation d) Taking high velocity electrons.

6. A two cavity klystron without a feedback path works as [ ]

a) Coupler b) Wave guide c) oscillator d) Amplifier

7. The electrons of the bunch in a reflex klystron must remain in the repeller field for the

minimum number of cycles [ ] a) 3 ¾ b) 1 ¾ c) ¾ d) 4

8. If the electrons in a Reflex Klystron remain in the repeller field for 1 ¾ cycles, the

mode of operation is [ ] a) 3 b) 1 c) 2 d) 4

9. Due to debunching, the power output of the Reflex Klystron [ ]

a) Decreases b) increases c) No charge i d) none of the above

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172

10. ______ limits the tuning range around the center frequency of a Reflex Klystron in a particular mode of operation [ ]

a) bunching b) half power points of the mode c) Velocity of electrons d) input Voltage

: 11. The Velocity to which the electrons are accelerated [A ]

in Klystrons is

a) meV02 b)

meV2

0

c) e

mV2

0 d) e

mV02

12. Velocity modulation takes place in [A ]

a) Cavity grid gap b) Drift space c) Repeller region d) Cathode region

13. Density modulation takes place in [B ]

a) Grid gap b) Drift space c) Cathode region d) Depletion region

14. Max efficiency of Reflex Klystron is [A ]

a) 22.7% b) 27.2% c) 72.2% d) 77.2%

15. Max efficiency of two cavity Klystron is [ ]

a) 58% b) 85% c) 55% d) 88%

16. Magnetron is [A ]

a) High power oscillator b) Low efficiency oscillator c) Low Power oscillator d) High efficiency Amplifier

17. The paths of electrons in Magnetron are [A ]

a) Cycloid b) Circular c) Elliptical d) Parabolic

18. Reflex Klystron gives max power in [A ]

a) 1 ¾ mode b) ¾ mode c) 2 ¾ mode d) 3 ¾ mode

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173

19. Stagger tuning is done to improve [A ]

a) Band width b) Efficiency c) Gain d) Power output

20. The multi cavity klystron, the cavities are spaced [C ]

a) Equally b) Increasing distance c) Decreasing distance d) In any way

21. Beam coupling coefficient ρ is [B ]

a) ( )sin 2g

g

θθ

b) ( )sin 2

2g

g

θθ

c) 2/g

gSinθ

θ d)

g

gSinθ

θ

22. Equation of Velocity modulation is [A ]

a) )( 1tv =

−+

221 1

0

10

gi tSinVVv

θωβ

)( 1tv = V0

−+

221 1

0

1 gi tSinVV θ

ωβ

)( 1tv = V1

−+

221 0

10

0 θωβ tSinVVi

)( 1tv =

−+

221 0

10

1 θωβ tSinVVi

23. Bunching Parameter of two cavity Klystron is [ ]

a) 00

1

2 vL

VVi ωβ b)

0

02V

LiViv ω

β

c) L

vVVi

ωβ

0

1

2 d)

00

1 22 VV

Vi ωβ

24. Optimum length of drift space is Lopt [ ]

a) 1

006.3VVv

iβω b)

1

00

6.3 VVv

iβω

c) 00

16.3vVViβ

ω d)

0

006.3VivV

βω

25. Bunching limits in multi-cavity Klystrons are from [ ]

a) Positive Peak to Negative Peak b) Negative

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174

c) Positive peak to positive peak d) Negative Peak to negative peak 26. Reflex Klystron can give output [A ]

a) 10 mw b) 1 Mw c) 10 Mw d) 100 Mw

27. Magnetron has ________ electrodes [A ]

a) Two b) Three c) Four d) One

28. Magnetron is [D ]

a) Crossed Field Device b) Oscillator c) Electron beam rotates d) All

29. Change of frequency with change in load is [ ]

a) Frequency Pulling b) Frequency Pushing c) Mode jumping d) Skipping

30. The magnetic field at which electron from cathode, grazes over the anode and falls

back into the cathode in a Magnetron is [ ]

a) Threshold field b) Hull cut-off c) RWH cut-off d) None

31. The total phase of the n-cavity Magnetron, to generate sustained oscillations should be

equal to [ ]

a) n π b) 2πn

c) 2πn d) none

32. Change of frequency with Anode Voltage is [ ] a) Frequency Pulling b) Frequency Pushing c) Mode jumping d) Skipping

33. The magnet surrounding the body of a TWT serves ______ purpose [A ]

a) Focus electrons into a tight beam b) for higher Velocities c) For denser bunching d) for larger amplification

34. The following is wide – band width MW amplifier [D ]

a) 2- cavity Klystron b) Reflex Klystron c) Multi cavity Klystron d) TWT

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175

35. Back wave oscillations with in TWT can be reduced by [A ]

a) attenuators b) buncher cavity c) paddles d) slugs

36. The electronic admittance diagram of reflex klystron, in shape is [C ]

a) straight line b) circular c) spiral d) parabolic

37. ( )XJ X is maximum when X is equal to [ ] a) 0.582 b) 2.408 c) 1.25 d) 0.52

38. The transit angle 1

0θ in the repeller region of reflex klystron is [ ] a) 2π n b) 2π n - π

c) 2π n – n d) 2π n - π / 2 39. The noisiest diode out of the following is [ ]

a) TRAPATT b) IMPATT c) GUNN d) LSA

40. Diode which give no harmonics is [ ]

a) TRAPATT b) GUNN c) IMPATT d) BARITT

41. structure of the Read diode is [ ]

a) P+ - n – P+ b) n+ - P – P+ c) n+ - P – I – P+ d) P+ - I – n+

42. Diodes which does not work at high frequencies (> 40 GHz) is [ ]

a) GUNN b) IMPATT c) TRAPATT d) All the above

43. Gunn devices function based on [A ]

a) Transfer electron effect b) Avalanche Transit effect c) Trapped plasma d) Impact ionization

44. High power pulsed outputs can be obtained in [ ]

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176

a) GUNN b) TRAPATT c) IMPATT d) all the above

45. The growing RF wave travels in direction to that of electron beam [ ] a) same b) opposite c) alternating a&b d) none of the above

46. the back wave oscillator can be tuned by changing the [ ]

a) length of slow wave structure b) changing the load at the output c) changing the accelerating anode potential d) taking reflectors

47. Gunn diodes are [D ] a) Junction devices b) Avalanche devices c) Both d) None

48. Gun effect is [C ]

a) Transferred electron effect b) Bulk material effect c) Both d) None

49. Gunn Effect can be observed in [D ]

a) Ge b) Si c) both d) None 50. Gunn diodes exhibit [C ]

a) Positive resistance b) Negative resistance c) Both d) None

51. Gunn domains start near [A ]

a) Cathode region b) Anode region c) Middle of the diode d) None

52. Domains travel at a Velocity equal to [B ]

a) Drift Velocity of electron b) More than drift Velocity of electron c) Less than drift Velocity of electron d) None

53. For Gunn oscillation modes fL is nearly [ ]

a) 107 cm.- sec. b) 106 cm./ sec. c) 106 cm- sec d) 107 cm / sec.

54. For transit time domain mode nL is [ ]

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a) > 1012 cm-2 b) < 1012 cm-2

c) 106 cm / sec. d) 107 cm / sec. 55. In delayed domain mode the domain, [ ]

a) Reaches the anode b) Disappears on its way to anode c) Both d) None

56. The quenched domain mode the domain [ ]

a) Reaches the anode b) Disappears on its way to anode c) Both d) None

57. The mode which provides amplifications at transit time frequency is [ ]

a) LSA mode b) Transit time domain mode c) Both d) None

58. Gun effect occurs only when doped with [A ]

a) n-type b) p-type c) No doping d) All

59. For an ideal cavity ‘Q’ value is [B ]

a) Zero b) Infinity c) One d) None 60. The limits to the frequency of LSA mode are set by [ ]

a) 104 < fn < 105 b) 104 >

fn < 105

c) 106 < fn < 107 d) None

61. Which of the following does not exhibit Gunn effect [D ]

a) Ga As b) Ia P c) In As d) Ge As

62. Avalanche multiplication for amplification is used by [ ]

a) TED’s b) ATT’s c) Paramps d) TWT’s

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63. The process of formation of electron cloud in a Magnetron is called as a) Polarization effect b) Avalanche effect c) Phase dissemination effect d) Phase focusing effect [D ]

64. MASER utilizes the principles of [A ] a) Stimulated emission b) spontaneous emission c) Simultaneous emission d) all the above

BASICS OF MICRO-WAVE LABORATORY

Microwave bench in the lab is a rectangular wave-guide run over which various components like source, attenuator, frequency meter, tunable probe etc. are mounted. It provides an unexcelled tool for learning basic concepts of standing waves and mismatched transmission lines at microwave frequencies. Its length is proportional to wavelength, as a result at low frequencies it becomes unwieldy long and at high frequencies it becomes too small to work comfortably with it.

The mode of the wave that exists in the bench of the lab is TE10 i.e. dominant

mode. So the cut-off wavelength is 2C aλ = where ‘ a ’ is the inner distance between the sidewalls of the wave-guide. The equipment is designed to work in the X-band, which ranges in frequency from 8.2 to 12.4 GHz, wavelength from 2.5 cm to 3.75 cm. The guide wavelength ranges from 2.98cm to 6.47cm, the guide dimensions are 2.286 X 1.016 cm with cut-frequency 6.557 GHz and cut-off wave length is 4.56 cm.

If the source is Reflex Klystron, it is required to be operated in 4

31 made giving max possible power output. To achieve this condition set the beam voltage to around 300 V and increase repeller voltage until max deflection is observed in the VSWR meter. If the source is Gunn, it must be operated in the middle of its negative resistance region by varying its bias voltage until max deflection is observed in VSWR meter.

The micrometer head provided at the source end of the bench is to change to the

frequency of the microwave source. The power output and frequency of the source are dependent upon the output impedance and power reflected. The isolation must be sufficient to prevent reflected wave entering back into the source.

While measuring guide wavelength, the termination should be short, which can

give sharp and hence easily locatable minima leading to accurate measurements. Even though the distance between two consecutive maxima is 2gλ only minima should be used to measure gλ as they are more accurately locatable than the maxima.

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VSWR meter consists of an ac amplifier tuned to 1 KHz approximately. For

VSWR meter to be of any use its input and hence the output of the microwave source must be a modulated signal to this frequency. If the source is RK the modulation is done internally, in case of Gunn oscillator it is performed externally with PiN diode.

VSWR meter is designed to measure VSWR. Its scale is calibrated to read

VSWR directly. It can also be used as a reference to measure power levels. Most of the measurements require a power level in the bench at which it can give a deflection in VSWR meter when its gain is 30 db. For the double min method to be used, the VSWR on the line must be more than 3 db. Other wise 3db points do not exist over the standing wave pattern.

Slide-screw tuner is a wave guide equivalent of transmission line stub with two

degrees of freedom. It is designed to provide the necessary mismatch to establish high VSWR over the line. If the depth of insertion 4d λ< it provides capacitive susceptance and for 4d λ> it is inductive susceptance.

In the wave-guide detector or tunable probe, for better response the short must be

maintained at a distance of 4gλ from the diode and also in the tunable probe. The diodes of the wave-guide detector and movable probe give voltage or current proportional to the power incident over the surface of the diode. This fact can be used to measure microwave power ratio with ammeter or voltmeter. Relative power

in db is 1 2

2 1

I VP = 10log db or 10log dbI V

.

The length of the slotted section is such that to accommodate at least three minima (one guide wavelength) at the lowest frequency of operation. Low power levels in the bench necessitate too much insertion of the probe leading to distortion of the standing wave pattern giving rise to erroneous results. So attenuation of the wave should not be too high.

BLOCKS OF THE BENCH • The source used in the microwave bench is either Reflex Klystron or Gunn

oscillator. In either case the frequency of the wave can be varied using the micrometer head provided at one end of the bench.

• Isolator always follows the source. Its purpose is to prevent the reflected wave entering back into the source. In the lab the Isolator that is configured with three port circulator and matched termination is used.

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• The attenuator used in the bench set up is flap attenuator providing attenuation of 0 to 25 db. The amount of attenuation provided by the device can be read from the micrometer scale provided.

• Wave-meter: It is designed to measure the microwave frequencies in the X band directly. Outwardly it is cylindrical in shape with a rotary cap at the top, rotary scale which is a tuning dial directly calibrated in frequency and a vertical pointer over a transparent plastic enclosure attached to the fixed base which has a wave-guide through it. In the grooves over its surface two rings move upwards when the scale is rotated clock-wise and downwards when it is rotated anti-clockwise. At the top of the scale it is 12.4 and at the bottom of the scale it is 8.2 frequencies in GHz. Inside it is a circular cavity with a movable short attached to the cap to allow the mechanical tuning of the resonant frequency, and the cavity is loosely coupled to wave-guide with a small aperture. In operation, power will be absorbed by the cavity as it is tuned to the frequency of the wave travelling through the wave-guide. The absorption can be monitored by a ‘dip’ in the deflection of the VSWR or power meter connected to the system.

• The standing wave detector is designed to observe the standing wave pattern existing in the slotted section and consists of

• Slotted wave-guide: It is a piece of rectangular wave-guide with a non-radiating slot over its broad wall. Probe can be inserted through the slot into the guide to sense the field

• Tunable probe: It is movable with its probe into the slot along the slotted section. The output of the tunable probe is proportional to the power of the wave into which its probe is inserted and it is normally given to the VSWR meter. The cap of the tunable probe can be pulled out or pushed in to match the slotted section to the wave-guide.

• Vernier scale: This is provided along the length of the slotted section to locate the position of the tunable probe exactly thereby the nodes or antinodes of the sw pattern.

• Rack and pinion arrangement: it is to move the probe and place it at any desired location over the SW pattern.

• VSWR meter: It is basically a high gain voltmeter consisting of basic meter movement and a high (to be able to measure low quantities) variable (to have multi-range facility) gain (60db) ac (to avoid the drift problems associated with dc amplifiers) amplifier. To vary the gain three knobs are provided one in steps of 10db and the remaining two in continuous manner. Its scale has two parts one to measure absolute VSWR: top ‘1’ to ‘ ∞ ’and just below to it ‘3’ to ‘10’, another part below the ordinary scale to measure VSWR in db’s: from ‘0’ to ‘10’. In addition both have extended scales to measure ‘accurately’ the VSWR in between ‘1.3’to ‘2’.

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• Wave-guide detector: It consists of a diode across, with a movable short inside a piece of wave-guide. For maximum response the short must be maintained at a distance of λg/4. It is designed to detect the presence of wave. Its output is proportional to the power of the wave incident. So it is a square law device.

FREQUENCY • Dip method or wave meter method:

• The wave meter is connected in the bench with attenuator on one side and the waveguide detector on the other side. The output of the wave-guide detector is given to the VSWR meter and power flow in the bench is adjusted until proper deflection is observed in the meter.

• The wave meter is rotated until it is one end of the scale i.e. the indication of the meter is either 12 or 8 GHz.

• Rotate the wave meter in the opposite direction slowly but continuously by pressing the centre finger of the left hand over its cap while observing the deflection in the VSWR meter.

• At one point of time a ‘dip’ in the deflection of the meter can be observed. Stop the rotation of the wave meter and note down the indication of the meter in between rings against the pointer. This is the frequency of the wave running in the bench.

• Slotted line method: • This method uses cut-off wavelength and guide wavelength to calculate the free space

wave length. From the free space wave length frequency can be calculated.

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182

• ( )2

01g

cf and λλλ λ λ

= =−

• As the mode of the wave in the bench is dominant, the cut-off wavelength is twice the inner distance between the sidewalls of the wave guide. By measuring this distance using a scale and multiplying it with two cut-off wavelength can be obtained.

• To find the guide wavelength terminate the bench with a short resulting in the formation of the standing wave pattern in the slotted section. The output of the tunable probe is given to the VSWR meter and the distance between two consecutive minima is measured using the Vernier scale provided. Twice this amount gives the guide wavelength.

VSWR • Low VSWR:

• This method can be used to measure the VSWR when it is less than ten with reasonable accuracy.

• The bench is terminated with the DUT for which VSWR is to be determined resulting in the formation of the standing waves in the slotted section. In the laboratory, the DUT is usually a Horn antenna. The output of the tunable probe is connected to the VSWR meter.

• Place the probe over a maximum and using the gain varying knobs provided over the front panel of the VSWR meter move the pointer of the meter to ‘1’ over the scale.

• Then move the probe to minimum and note down the indication of the pointer over the scale which gives the VSWR of the wave over the bench.

• In the case of the pointer drops to no deflection position while moving to minimum, then increase the gain of the meter by 10db, move the probe to minimum, note down the indication of the pointer on the scale 3-10 which is the VSWR of the wave.

• Double minimum method: • Double minimum method can be used only if the SWR over the line is more than 3db

and it requires to be used only when SWR is more than 10. To be able to apply this method, a VSWR more than 10 has to be established first over the line. In the laboratory it is done using a match terminated slide screw tuner. With match terminated slide screw tuner connected to the slotted section, place the tunable probe

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183

over a maximum of the standing wave pattern and move the pointer of the VSWR meter over to ‘1’ by varying the gain. Now move the probe to a minimum and vary the position and depth of the probe of the slide screw tuner until the pointer in the VSWR meter is over ' ∞ ' of the top scale. Increase the gain of the meter by 10db and if the pointer is still over' ∞ ' (or 10 of the scale below) then the SWR over the line is 10 or more. If the pointer stays over in between two extreme positions of the scale even after increasing the gain, then the setting of the slide screw tuner should be changed in such an amount in such a direction so that the pointer is over ' ∞ '. Now the VSWR over the line is 10 or more and we can use the double minimum method to measure it accurately.

• The bench is to be terminated with the DUT, which can establish high VSWR i.e. more than 3db over the slotted section. The output of the tunable probe is given to the VSWR meter.

• Move the tunable probe over to a minimum and by varying the gain place the pointer

on ‘3’ in the db scale of the VSWR meter. • Move the tunable probe to either side until the pointer moves to ‘0’ in the db scale.

Note down the position of the tunable probe over the Vernier scale. Let it be d1. • Now move the probe in the opposite direction until the pointer again stands over the

‘0’ after passing over the ‘3’ in the db scale. Note down the position of the probe. Let it be d2.

• Now replace the termination of the bench with short and measure the distance between two consecutive minima. Twice this distance gives the guide wavelength gλ .

• The VSWR can be obtained using the formula ( )1 2

VSWR g

d dλ

π=

To establish to high VSWR in the lab: ATTENUATION:

Power ratio method: o The DUT for which attenuation is to be measured is placed before the slotted section of

the bench terminated with matched load. o The output of the tunable probe is given to a power meter.

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184

o Let us suppose the indications of the power meter are P1 and P2 with the DUT and with out DUT in the bench.

o Then the attenuation of DUT is 2

1

A in db 10log PP

=

o In case of the non-availability of the power meter, the power ratio can still be obtained by measuring the output current or voltage of the tunable probe using multi-meter or CRO. The ratio 1 2 1 2 1 2P P V V I I= = as

o This method uses two different points on the characteristic of curve of the diode detector at which the detector may not be obeying square law characteristic leading to erroneous readings.

RF substitution method: o Place the DUT before the slotted section and connect the tunable probe output to VSWR

meter. Termination of the bench must be matched. Note down the deflection of the pointer in the VSWR meter.

o Replace the DUT with standard variable precision attenuator and vary its attenuation until the deflection of the pointer is same as that in the previous step.

o At this position the attenuation of the standard attenuator which can be noted down gives the attenuation of the DUT

IMPEDANCE Slotted line method: o The bench is terminated with the DUT for which impedance is to be measured. And the

position of a minimum is located along with the measurement of SWR ρ .

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185

o Replace the termination with a short. Measure the guide wavelength and shift in minimum both in magnitude and direction.

o If the shift is towards left the load is inductive and if it is right the load is capacitive.

o Use the formula shown below to calculate the impedance of the DUT. θ

θ

j

jL

ee

ZZ

Γ−Γ+

=11

0

where the magnitude of the reflection coefficient 11

+−=Γ

ρρ and dβπθ 2±= where

d is shift in the minimum and gλπβ 2= phase shift constant, + in case of right shift and – in

case of left shift. 0Z is characteristic impedance of the slotted section. Magic Tee method: o The matched source and null detector are connected to the side arms of the magic tee. o The standard variable precision impedance and unknown impedance are connected to the

coplanar arms of the magic tee. o The standard variable precision impedance is varied until the null is observed in the

detector. o The indication over the standard variable precision impedance is the impedance to be

known. Power

Bridge methods:

o Bolometers are devices which change their resistance with temperature. When µ wave power

falls over its surface, it gets converted into heat rising its temperature. With change in temperature the resistance changes. The change in the resistance, which can be measured conveniently using bridge methods, is a measure of the µ wave power incident.

o Bolometers can be divided into two categories one Barretters whose resistance rises with temperature and thermistors whose resistance falls with temperatures.

o Barretters are thin short platinum wires used to measure low µ wave power levels. They change 5ohm per milli-watt of incident µ wave power. These are very delicate and sensitive devices useful to measure very low power levels less than few milli-watts. They have

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186

positive temperature coefficient of resistance. Thermistors are semi conductor devices with negative temperature coefficient of resistance. Used to measure low and medium µ wave power levels. The change in resistance is 60 ohm per milli-watt of incident µ wave power.

o Power meter: it is a balanced bridge circuit in which one of the arms is a bolometer. The µ wave power incident over this arm changes its resistance driving the bridge into unbalance. The amount of unbalance which is proportional to the incident µ wave power is amplified using the bridge amplifier and measured using a voltmeter. The voltmeter is calibrated to read the power directly.

o Single bridge circuits give erroneous readings due to mismatch at the µ wave input port and also due to sensitivity of thermistor to ambient temperature. These shortcomings can be overcome by adopting double identical bridge. Calorimetric methods: o This method is useful to measure high µ wave powers. It involves conversion of the

µ wave energy into heat, absorption of heat by some liquid or dielectric and then measurement of the rise of the temperature of the liquid/dielectric.

o Static calorimeter: it consists of a 50ohm coaxial cable filled with a dielectric load with a high hysterisis loss. The incident µ wave power is dissipated in the load. The average

input is t

TmCP p18.4

= watts where t is time in sec,T is the temperature in 0C and m

is the mass of the medium in gms. o Circulating calorimeter: in this method the power is made to incident on the water

flowing at a constant rate through a water load. The heat introduced into the fluid makes the exit temperature to be higher than the input temperature. The incident power is then measured using the relation TCdvP p18.4= watts where v is rate of the flow of the fluid in cc/sec, d is the specific gravity of the fluid in gm/sec, pC is the specific heat in cal/gm.

‘Q’ of cavity: By transmission:

• This method is used when the cavity for which ‘Q’ is being measured has two ports or openings. It is to be connected before the slotted section in the bench with tunable source. The termination of the bench must be matched and the output of the tunable probe is given to a power meter.

• The power that is transmitted by the cavity is measured using the power meter at different frequencies and a graph is drawn. It resembles inverted ‘U’.

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187

• From the graph find the resonant frequency, the frequency at which the transmitted power is maximum and also the half-power frequencies at which the transmitted power half of the maximum.

• Using the relation resonant frequency

bandwidthQ = find the Q of the cavity.

By measuring VSWR:

• This method is useful when the cavity has single opening or port. The bench with tunable source is terminated with the cavity and output of the tunable probe is connected to VSWR meter.

• The VSWR due to the cavity is measured at different frequencies and graph is drawn whose shape is similar to ‘U’.

• The resonant frequency rS , frequency at which the VSWR is lowest and the lowest VSWR both can be noted down from the graph

• From the lowest VSWR using one of the following relations which ever gives more

than one find the half-power VSWR. 21 22

1 12 4r r

r r

ρ ρ ρρ ρ

= + + + or

2

1 22

1 12 4

r r

r r

ρ ρρρ ρ

= + + +

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188

• Using the computed 1

2ρ from the already drawn graph find the half-power

frequencies and from them band-width.

• Now using the formula resonant frequencybandwidth

Q = find the Q of the cavity.

Parameters of DC

The important characteristics of the DC are coupling factor, directivity and isolation.

• Measure the output power of the source inP . In case of non-availability of the power meter, use the wave-guide detector –CRO combination to measure the voltage proportional to the power. Let it be inV

• Give input at the port 3. Measure the output power at port 1 with port 2 match terminated. Let it be cP . If the voltage proportional to power is measured using wave-guide detector–CRO combination, let it be cV

• Give input at the port 2. Measure the output power at port 1 with port 3 match terminated. Let it be dP . If the voltage proportional to power is measured using wave-guide detector–CRO combination, let it be dV

• Give input at the port 2. Measure the output power at port 3 with port 1 match terminated. Let it be TP . If the voltage proportional to power is measured using wave-guide detector–CRO combination, let it be TV

• Now coupling in db ( ) ( )10log 10login c in cC P P V V= = , directivity in db

( ) ( )10log 10logc d c dD P P V V= = and isolation in db ( )10log in dI P P=

( )10log in dV V= . If the measurements of correct I C D= + • Precaution: After the power output of the source inP in measured, the settings of the

source, attenuator or waveguide detector should not be changed.

s-parameters

• S-parameters are complex quantities and to measure them network analyzer is required. If the device is assumed an ideal, reciprocal with equal arm lengths, then the s-parameters become pure real quantities.

• Let us try for the s-matrix of the magic tee assuming it an ideal one. As power meters are not usually available, we can use wave-guide-CRO combination to measure the relative powers.

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189

• Step I: Measure the output of source. Let it be inV . The settings of the sources and attenuator should not be varied until the completion of the experiment.

• Step II: Give input to port1 and measure output at ports 2,3 and 4 while maintaining matched terminations at the other ports. Let them be 21 31 41, andV V V .

• Step III: Give input to port3 and measure output at ports1,2 and 4 while maintaining matched terminations at the other ports. Let them be 13 23 43, andV V V .

Now 31 31 ins V V= , 21 21 ins V V= , 41 41 ins V V= , 23 23 ins V V= , 43 43 ins V V= . As the device is reciprocal 31 13s s= , 21 12s s= etc. Then the diagonal elements of the s-matrix can be found using unity property. If the measurements are correct them the diagonal elements must be zeros.


10. The bench cannot be designed to work at lower frequencies [A ] a) Becomes too lengthy b) becomes too small

c) Standing waves do not occur at lower frequencies d) none

11. The purpose of the microwave bench is [C ] a) to test transmission line theory principles b) to observe standing wave pattern

c) both d)none

12. The microwave bench in lab designed to work in the band [A ] a) X b) L

c) S d) C 13. The minimum frequency that can exist in the Bench (approx.) is [A ]

a) 9 GHz b) 12 MHz c) 9 MHz d) None

14. The maximum frequency that can exist in the Bench (approx.) is [B ] a) 9 GHz b) 12 MHz c) 9 MHz d) None

15. The minimum free space wave length of the Bench in the lab is [ ]

a) 2.5 cm b) 3.3 cm c) 3.3 mm d) None

16. The max free space wave length on the Bench in the lab is [ ] a) 2.5 cm b) 3.75 Cm

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190

c) 3.3 mm d) None

17. The inner dimensions of the waveguide run of the bench are [ ] a) 2.5 cm b) 3.75 Cm c) 3.3 mm d) None

18. For a=2.3 cm, the maximum guide wave length on

the bench can be [ ] a) 2.0 cm b) 6.47 cm c) 2.98 cm d) None

10. For a = 2.3 cm, the minimum guide wave length on [ ]

the bench can be a) 2.98 cm b) 3.3 cm c) 6.47 cm d) None

47. The mode used in the laboratory bench is [A ]

a) Dominant mode b) degenerate mode c) any one of the above two d) none

48. The cut off wave length of the bench is [A ]

a) 2a b) 2b c) c f d) none

49. For radio frequencies above 1000 MHz the method

used for Impedance measurement is [A ] a) Slotted line b) Impedance Bridge c) Either of the method d) None

50. The isolator used in between source and slotted section is [B ] a) Avoid harmonics in the source b) to prevent the reflected wave entering back into the source c) A & B d) None

51. In VSWR meter the amplifier is of high gain because [A ]

a) Input to meter in low b) To provide multi range facility c) Both d) None

52. In VSWR meter, the amplifier’s gain is variable because [B ]

a) Input to meter in low b) To provide multi range facility c) Both d) None

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191

53. The amplifier in VSWR meter is tuned to a frequency [A ] a) 1 KHz b) 2 KHz c) 10 KHz d) 100 KHz

54. The lower part of the VSWR meter scale represents [A ]

a) VSWR b) Power c) Attenuation d) None

55. The PiN diode in the bench is used to modulate the output of [B ]

a) Reflex klystron b) Gunn oscillator c) Both c) None

18. In VSWR meter the amplifier used is ac type because [A ] a) to avoid drift problem of dc amplifiers b) Cheap c) Small size d) All

56. The diode detector was input/output relation that is [C ] a) Non linear b) Square c) Both d) None

57. The mode of the wave over the bench is [C ]

a) Dominant b) TE10 c) Both d) None

58. Wave meter is [ ]

a) Cavity b) Resonator c) Both d) None

59. In wave meter the pointer is [B ]

a) Oscillatory b) Rotary c) Stationary d) None

60. In the bench the source is always followed by [B ]

a) Attenuator b) Isolator c) Wave meter d) Detector

61. In the bench the source is modulated by a frequency [A ]

a) 1 KHz b) 10 KHz c) 100 KHz d) None

62. Tunable probe exists over / in [B ]

a) VSWR meter b) Slotted section

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192

c) Attenuator d) None 63. The method used to measure high VSWR is [B ]

a) Slotted line method b) Double minimum method c) Both d) None

64. Low VSWR method can be used to measure VSWR up to [A ]

a) ten b) five c) three d) None

65. In low VSWR method, in VSWR meter the pattern maximum [A ]

is set to

a) One b) zero db c) Both d) None

66. In high VSWR method, in VSWR meter the pattern minimum [C ] is set to a) One b) zero db c) Three db d) None

67. While measuring guide wave length, the termination of the bench [A ] must be a) Short b) Matched termination c) Open d) None

68. While measuring guide wave length, always minimum points are [C ] considered because

a) Minima are sharp b) minima are easily locatable c) Both d) None

69. Guide wave length does not depend upon [A ] a) Termination b) frequency

c) Mode of the wave d) none 70. While measuring attenuation, the termination of the bench must be [B ]

a) Short b) Matched termination c) Open d) None

71. Short is required at a distance from diode in wave guide detector [B ]

a) 4λ b)

4gλ

c) 2λ d) None

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193

72. Gyrator gives a phase shift to a wave travelling from output [B ] to input

a) 180o b) 0o c) 90o d) None

73. Calibration standards are [C ]

a) Rotary type phase shifter b) Rotary type attenuator c) Both d) None

74. When the depth of penetration is > λ /4, the susceptance

of the probe is [ ] a) Inductive b) Capacitive c) Nil d) None

75. The range of VSWR in db indicated VSWR meter is [B ] a) 0-2db b) 0-10db c) 0-5db d) 0-100db

38. For low power measurement the technique used is [A ]

a) Bolometer technique b) Calorimeter technique c) Reflect meter technique d) Calorimetric wave meter technique

39. The temperature coefficient of the Thermisters is [ ]

a) Positive b) negative c) Zero d) None

40. The ‘Q’ of Cavity with single opening can be measured [ ] a) Transmission method b) Double minimum method c) Both d) None

41. Barretters have [ ] a. Positive temp efficient of resistance b. Negative temp coefficient of resistance c. Both d) None

42. The magic T can be used for the measurement of [B ] a) Frequency b) Impedance c) Q factor d) None

56. Impedance of the line to the left of voltage minimum is [ ]

a) Inductive b) capacitive c) High resistance d) low resistance 57. Impedance of the line to the right of voltage minimum is [ ]

a) Inductive b) capacitive c) High resistance d) low resistance

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194

58. Impedance of the line at the voltage minimum is [ ]

a) Inductive b) capacitive c) High resistance d) low resistance 59. Impedance of the line at the voltage maximum is [ ]

a) Inductive b) capacitive c) High resistance d) low resistance 60. Thermo couple is a junction of [B ]

a) Two similar metals b) Two dissimilar metals c) Identical but different lengths d) None

61. The relation between VSWR (S) & reflection coefficient (P) is [ ]

a) S = PP

+−

11 b) S =

PP

−+

11

c) S = P

P+1

d) None

62. If rP is reflected power, iP is the incident power, [B ]

the reflection coefficient ‘P’ is a) r iP P b) r iP P

c) i rP P d) i rP P

63. Q – factor is measured using [ ]

a) Reflect meter method b) Transmission method c) Power ratio method d) None

64. The bolometer that is having a negative temperature coefficient [ ] of resistivity that is called a) Barrater b) Varistor c) Thermisters d) Calorimeter

65. In laboratory experiments the output from Reflex [ ]

Klystrons are modulated by square waves because a) It is easy generative a square wave b) It prevents frequency modulation c) Detector circuit is easy to design d) The termination is less complicated

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195

66. In microwave power measurement using bolometer the principle of working is the variation of

a) Inductance with absorption of power b) Resistance with absorption of power

c) Capacitance with absorption of power d) All 67. We use two 20db directional couplers along with two [ ]

detectors in which technique of impedance measurement a) Slotted line b) Reflecto-meter c) Heterodyne technique d) None

68. We use two matched detector in which

technique of q factor measurement [ ] a) Slotted line b) Reflectometers c) Heterodyne technique d) None

56. .The technique used to measure the dielectric constant is [ ] a) Slotted line b) Waveguide method

c) Reflect meter method d) Wave meter method 57. for impedance measurement the following oscillator is used [ ] a) Reflex klystron tube oscillator b) Gunn oscillator

c) a and b d) a or b

microwaves components

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