IV/IV B.Tech Regular Degree Examination, November 2017

Scheme of Evaluation

Subject code:- 14EC703

Subject Name:- Microwave Theory and Techniques

Faculty: Head of Department

D. Gopi chand

Assistant Professor

9494357618

Answer all questions:

1.a)Mention the advantage of microwaves?

a) High Bandwidth (1% BW at 600MHz is 6MHz at 60GHz is 600MHz).

b) High Antenna gain and Directivity (𝐠𝐚𝐢𝐧 ∝𝟏

𝛌𝟐 ∝

𝟏

𝛉𝐇𝐏𝐁𝐖𝟐 ).

c) Line of sight Transmission and reception.

d) High penetration capability helps in large distance

communications.(Ionosperic propagation and satellite communication)

f) Less power requirements of transmitter at microwave frequencies.

Note: Any 2 points from above points.

1.b)What is circulator?

A circulator is a multiport device in which the wave incident at port 1 is

coupled to port 2 only, a wave incident in port 2 is coupled to port 3 only, and

so on.

Note: Even circulator diagram with power flow indications can be

awarded marks.

1.c)What is Joint in microwave components?

As a waveguide system cannot built in a single piece always, sometimes it is

necessary to join different waveguides. This joining must be carefully done

to prevent problems such as Reflection effects, creation of standing waves,

and increasing the attenuation etc., The waveguide joints besides avoiding

irregularities, should also take care of E and H field patterns by not affecting

them. There are many types of waveguide joints such as bolted flange,

flange joint, choke joint etc.

1.d) Why slotted sections are used in waveguide systems?

The purpose of using Slotted sections in waveguide systems is to find the

position of voltage minima and maxima. The measurement of impedance,

VSWR, guide wavelength can be done with the help of these slotted

sections.

1.e) Give the applications of H plane Tee?

It can be used as power divider (3 dB splitter) and power combiner.

1.f) Define velocity modulation in Klystrons?

The variation of electron velocity in drift space is known as velocity

modulation. In klystrons drift space is the space between the two cavity

resonators.

1.g) Compare the efficiencies of two cavity klystron and Reflex

Klystron?

The maximum electronic efficiency of two cavity klystron is 58% .The

maximum electronic efficiency of reflexive klystron is 22.7%.

In general efficiency of two cavity klystron is about 40%. In general

efficiency of reflex klystron is about 30%.

Note: general efficiency is also can be considered and awarded marks.

1.h) Write Hull cut off voltage expression?

The expression for Hull cut off voltage is given by

𝑽𝒐𝒄 = 𝟏

𝟖

𝒆

𝒎𝑩𝟎𝟐𝒃𝟐 𝟏 −

𝒂𝟐

𝒃𝟐 𝟐

1.i) Give the significance of operating a magnetron in a π mode?

For sustaining the oscillation in a magnetron, the phase difference between

two adjacent anode plates should equal (2πn/N) where N is the number of

cavities and n is any integer, this gives rise to π mode operation when n =

N/2. Also magnetron operating in the π mode has greater power output and

hence most commonly used.

1.j) Expand TRAPATT and IMPATT?

TRAPATT: Trapped Plasma Avalanche Triggered Transit

IMPATT: Impact ionization Avalanche Transit Time

1.k) What is Schottky diode?

The Schottky diode is a semi conductor diode formed by junction of semi

conductor with a metal. It has low forward voltage drop and a very fast

switching action.

1.l) Define attenuation with respect to microwave bench?

Attenuation is defined as the ratio of input power to the output power in dB.

𝑨𝒕𝒕𝒆𝒏𝒖𝒂𝒕𝒊𝒐𝒏 𝒊𝒏 𝒅𝑩 = 𝟏𝟎𝐥𝐨𝐠(𝑷𝟏

𝑷𝟐)

Where P1 = power detected by the load without attenuator in the line.

P2 = power detected by the load with the attenuator in the line.

Unit-1

2. a) Explain with suitable examples, about the advantages of Microwaves?

Microwave Properties and Advantages:-

High Bandwidth: At high frequencies more bandwidth (information carrying

capacity) can be realized. A 1% percent bandwidth at 600MHz is 6MHz(the

bandwidth of one TV channel), and at a 60GHz a 1% bandwidth is 600MHz(

bandwidth of 100 TV channels).

Antenna Gain and Directivity: The gain of antenna is directly proportional to

electrical size. The beamwidth of an antenna is inversely proportional to the

electrical size of its maximum dimension. Thus shorter wavelengths at microwave

frequencies allow for smaller antennas.

𝐠𝐚𝐢𝐧 ∝𝟏

𝛌𝟐 ∝

𝟏

𝛉𝐇𝐏𝐁𝐖𝟐

Effect of Ionosphere: When low frequency waves are directed upward into the

atmosphere, they experience significant reflection due to ionosphere. The high

frequency waves which pass through ionosphere with little effect and are therefore

utilized in satellite communications and space transmissions.

Line of sight Transmission/Reception: the microwave receiving antenna must be

with the line of sight of the transmission antenna. Long distance communication on

earth requires that microwave relay stations be used.

Electromagnetic Noise Characteristics: The EM noise level in nature over 1-

10GHz frequency is small. This allows for the detection of low signal levels using

sensitive receivers.

Target reflection of Electromagnetic waves (Radar cross section): In general,

electrically large conducting radar targets reflect more energy. Thus the higher

frequencies of microwaves are preferred for radar systems.

Power Requirements: At microwave frequencies, the power requirements of the

transmitter become very small as compared to that at MF/HF, due to high gain of

the antennas at microwaves.

Note: Any four advantages with explanation carries 6M first two points are

mandatory.

2.b) Derive the expression for cut off frequency of circular cavity resonators?

Circular-cavity resonator:

A circular-cavity resonator is a circular waveguide with two ends closed by

a metal wall. The wave function in the circular resonator should satisfy Maxwell's

equations, subject to the same boundary conditions described for a rectangular-

cavity resonator. It is merely necessary to choose the harmonic functions in z to

satisfy the boundary conditions at the remaining two end walls. These can be

achieved if

where n = 0, 1, 2, 3, ... is the number of the periodicity in the φ direction

p = 1, 2, 3, 4, ... is the number of zeros of the field in the radial direction

q = 1, 2, 3, 4, . . . is the number of half-waves in the axial direction

Jn = Bessell's function of the first kind

Hoz = amplitude of the magnetic field

Where n = 0, l, 2, 3, .. .

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

q = 0, l, 2, 3, .. .

Eoz = amplitude of the electric field

The separation equations for TE and TM modes are given by

Substitute k2=ω

2εμ

TM110 mode is dominant where 2a>d and the TE111 is dominant when d>=2a.

Note: Circular cavity resonator equations 4M and diagram and explanation

2M.

3.a) Explain construction and working of directional coupler?

A directional coupler is a four-port waveguide junction. It consists of a primary

waveguide 1-2 and a secondary waveguide 3-4.

When all ports are terminated in their characteristic impedances, there is free

transmission of power, without reflection, between port 1 and port 2, and there is

no transmission of power between port 1 and port 3 or between port 2 and port 4

because no coupling exists between these two pairs of ports. The degree of

coupling between port 1 and port 4 and between port 2 and port 3 depends on the

structure of the coupler.

The characteristics of a directional coupler can be expressed in terms of its

coupling factor and its directivity. Assuming that the wave is propagating from

port l to port 2 in the primary line, the

Coupling factor and the directivity are defined as

Coupling factor is defined as the ratio of the incident power pi to the forward

coupled power Pf measured in dB. Coupling factor is a measure of how much of

the incident power is being sampled.

𝐶𝑜𝑢𝑝𝑙𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟 𝑑𝐵 = 10 log 𝑃𝑖

𝑃𝑓 = 10log(

𝑃1

𝑃4)

Directivity is defined as the ratio of forward coupled power Pf to the back power Pb

expressed in dB.

𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑑𝐵 = 10 log 𝑃𝑓

𝑃𝑏 = 10 log(

𝑃4

𝑃3)

Isolation: it is defined as the ratio of the incident power Pi to the power Pb.

𝐼𝑠𝑜𝑙𝑎𝑡𝑖𝑜𝑛 𝑑𝐵 = 10 log 𝑃𝑖𝑃𝑓 = 10log(

𝑃1

𝑃3)

where P1 = power input to port 1,P2 = power input to port2,P3 = power output from

port 3 ,P4 = power output from port 4.

Note: Construction 3M working 3M

3.b) Explain the Faraday rotation principle using an isolator?

Faraday rotation: When a DC or static magnetic field B0 is applied along the z-

direction. A plane TEM wave that is already linearly polarized along x-axis at t=0

is made to propagate through the ferrite placed in the z-direction. The plane of

polarization of this wave will rotate with distance. This phenomenon is called

Faraday rotation.

Any linearly polarized wave can be regarded as the vector sum of two

oppositely rotating circularly polarized waves (E0/2 vectors in above figure).

The ferrite material offers different characteristics to these waves with the result

that the phase change in one wave is larger than the other wave resulting in

rotation „θ‟ of linearly polarized wave at z=l.

The angle of rotation „θ‟ is given by

θ=𝑙

2(𝛽+ − 𝛽−)

l= length of the ferrite rod, β+= phase shift of the right polarized wave, β-=

phase shift of left polarized wave.

The Faraday rotation principle is shown by the ferrite materials those exhibits

the property if magnetic anisotropy, non reciprocal property, insulator nature.

These ferrite devices are used in designing the non reciprocal devices like

Gyrator, Isolator, and circulator.

Isolator:

An isolator is a two-port device which provides negligible attenuation for

transmission of signal from port-1 to port-2 but provides high attenuation for

transmission from port-2 to port-1.

Construction:

The Cut away view of Faraday rotation isolator is shown above. It consists of

circular waveguide carrying dominant mode i.e. TE11 mode, with transitions to

a standard rectangular waveguide with dominant mode TE10 at both ends but the

output end transition is twisted by 450. A thin circular ferrite is placed inside the

circular guide, supposed by poly foam, and the waveguide is surrounded by a

permanent magnet which generates DC magnetic field in the axial direction of

ferrite rod. The input resistive attenuator/ card placed along larger dimension of

rectangular guide and the output resistive attenuator/ card are displaced by 450.

The function of resistive attenuator is to absorb any wave whose plane of

polarization is parallel to the plane of resistive attenuator.

Principle of Operation:

An input TE10 dominant mode is incident on port 1 of the isolator. As the wave

is perpendicular to the resistive card, it passes through the ferrite rod without

any attenuation and the wave is rotated 450 clockwise due to Faraday rotation.

As a result of rotation, the wave arrives at output port i.e. port 2 without

attenuation.

If a wave tries to propagate from port 2 to port 1, it passes through resistive card

placed at port 2 as electric field is perpendicular to the plane of card and the

wave is rotated 450 clockwise direction(looking from port 1) due to Faraday

rotation by the ferrite rod. Since the wave coming towards port 1 is parallel to

the resistive card, the wave is absorbed by it.

Thus any reflected wave from output port will not reach input port.

The typical performance of these isolators is about 1-dB insertion loss in

forward transmission and about 20-30 dB isolation in reverse direction.

Note: Faraday rotation principle 2M isolator diagram 2M working 2M

Unit-II

4.a) Derive the S-matrix of a magic Tee?

The simplest type of power divider is a T-junction. A magic tee is a

combination of E-plane Tee and H-plane Tee. The magic tee has following

characteristics.

1. If two waves of equal magnitude and the same phase are fed into port 1 and

port 2, the output will be zero at port 3 and additive at port 4.

2. If a wave is fed into port 4 (H arm), it will be divided equally between port 1

and port 2 of the collinear arms and will not appear at port 3 (E arm).

3. If a wave is fed into port 3 (E arm), it will produce an output of equal

magnitude and opposite phase at port 1 and port 2. Output at port 4 is zero

i.e S43 = S34 = 0.

4. If a wave is fed into one of the collinear arms at port 1 or port 2, it will not

appear in the other collinear arm at port 2 or port 1 because the E arm causes

a phase delay while the H arm causes the phase advance. i.e S12 = S21 = 0.

For the purpose of double the power at one port and by taking the input power

from the two transmitters situated at two other ports, the property 1 is used.

The S-matrix for magic Tee is given by

[S] =

𝑆11 𝑆12

𝑆21 𝑆22

𝑆13 𝑆14

𝑆23 𝑆24

𝑆31 𝑆32

𝑆41 𝑆42

𝑆33 𝑆34

𝑆43 𝑆44

Due to E-plane Tee:

S13 = -S23

Due to H-plane Tee:

S14 = S24

Due to geometry, port 3 and port 4 are isolated

S34 = S43 = 0

From the Magic Property:

S12 = S21 = 0

If the port 3 and port 4 are matched to the junction

S33 = S44 = 0

From the Symmetry Property, Sij = Sji

S13 = S31; S14 = S41;

S23 = S32; S24 = S42;

The S-matrix by substituting above results are:

[S] =

𝑆11 00 𝑆22

−𝑆23 𝑆14

𝑆23 𝑆14

−𝑆23 𝑆32

𝑆14 𝑆14

0 00 0

From the unitary Property, [S] [S]* = [U]

𝑆11 00 𝑆22

−𝑆23 𝑆14

𝑆23 𝑆14

−𝑆23 𝑆23

𝑆14 𝑆14

0 00 0

*

𝑆11∗ 00 𝑆22

∗−𝑆23

∗ 𝑆14∗

𝑆23∗ 𝑆14

∗

−𝑆23∗ 𝑆23

∗

𝑆14∗ 𝑆14

∗0 00 0

=

1 00 1

0 00 0

0 00 0

1 00 1

R1C1: |S11|2 + |S23|

2 + |S14|

2 = 1 eqn (1)

R2C2: |S22|2 + |S23|

2 + |S14|

2 = 1 eqn(2)

R3C3: |S23|2 + |S23|

2 = 1 eqn (3)

R4C4: |S14|2 + |S14|

2 = 1 eqn (4)

From the above equations, we get

S23 = 1

2 and S14 =

1

2

S13 = - S23 = −1

2

Then S14 = S24 = 1

2

Substituting S23, S14 in above equations 1&2 gives S11 = S22 = 0

The final S-matirx will be shown below by substiting above values

[S] =

0 00 0

𝑆13 𝑆14

𝑆23 𝑆24

𝑆31 𝑆32

𝑆41 𝑆42

0 00 0

[S]=

0 00 0

−1

2

1

21

2

1

2−1

2

1

21

2

−1

2

0 00 0

Note: Diagram 2M,initial conditions 2M, correct S-matrix 2M. initial

conditions are specified using properties of magic Tee

4.b)What are the evaluation parameters of a S matrix?

The behavior of the network in terms of reflected and injected power waves can be

described by a set of linear equations. For the 2-port case, the outputs can be

replaced to the inputs by

𝑏1 = 𝑆11𝑎1 + 𝑆12𝑎2

𝑏2 = 𝑆21𝑎1 + 𝑆22𝑎2

𝑆11 = 𝑏1

𝑎1 𝑎2=0

=𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑝𝑜𝑤𝑒𝑟 𝑤𝑎𝑣𝑒 𝑎𝑡 𝑝𝑜𝑟𝑡 1

𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝𝑜𝑤𝑒𝑟 𝑎𝑡 𝑝𝑜𝑟𝑡1

𝑆21 = 𝑏2

𝑎1 𝑎2=0

=𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑒𝑑 𝑝𝑜𝑤𝑒𝑟 𝑤𝑎𝑣𝑒 𝑎𝑡 𝑝𝑜𝑟𝑡 2

𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝𝑜𝑤𝑒𝑟 𝑎𝑡 𝑝𝑜𝑟𝑡1

𝑆22 = 𝑏2

𝑎2 𝑎1=0

=𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑝𝑜𝑤𝑒𝑟 𝑤𝑎𝑣𝑒 𝑎𝑡 𝑝𝑜𝑟𝑡 2

𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝𝑜𝑤𝑒𝑟 𝑎𝑡 𝑝𝑜𝑟𝑡2

𝑆12 = 𝑏1

𝑎2 𝑎1=0

=𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑒𝑑 𝑝𝑜𝑤𝑒𝑟 𝑤𝑎𝑣𝑒 𝑎𝑡 𝑝𝑜𝑟𝑡 1

𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝𝑜𝑤𝑒𝑟 𝑎𝑡 𝑝𝑜𝑟𝑡2

S parameters can only determined under conditions of perfect matching at the input

or output port.

Determination of S11 and S21 parameters: S11, S21 can be computed if port 2 is

terminated with matched load(Z0).

Load impedance (ZL) = characteristic impedance(Z0)

By defining Zin as input impedance given ZL = Z0

𝑆11 = 𝑍𝑖𝑛 𝑍𝐿=𝑍0

− 𝑍0

𝑍𝑖𝑛 𝑍𝐿=𝑍0+ 𝑍0

From the above equation, it is known that reflection coefficient(S11 is the method

of specifying the input impedancezin).

S11 = b1/a1 is the input reflection coefficient (when a2 = 0)

S21 = b2/a1 is the forward transmission gain (when a2 = 0)

Determination of S22 and S12 parameters: S22, S12 can be computed if port 1 is

terminated with matched load(Z0).

input impedance (Zin) = characteristic impedance(Z0)

𝑆22 = 𝑍𝑜𝑢𝑡 𝑍𝑖𝑛 =𝑍0

− 𝑍0

𝑍𝑜𝑢𝑡 𝑍𝑖𝑛 =𝑍0+ 𝑍0

From the above equation, it is known that reflection coefficient(S22 is the method

of specifying the output impedance zL).

S22 = b2/a2 is the output reflection coefficient (when a1 = 0)

S12 = b1/a2 is the reverse transmission gain (when a1 = 0)

Note: formulas of S-parameters 4M, diagram explanation 2M

5.a) Derive the S-matrix of a H plane Tee?

H-plane tee (shunt tee): An H-plane tee is a waveguide tee in which the axis of its

side arm is "shunting" the E field or parallel to the H field of the main guide as

shown below

Characteristics of H-plane Tee:

If the H plane junction is completely symmetrical and waves enter through side

arm, the waves that leave through collinear ports are equal in magnitude and phase

Therefore 𝑆13 = 𝑆23

S matrix is of order 3X3

𝑠 =

𝑆11 𝑆12 𝑆13

𝑆21 𝑆22 𝑆23

𝑆31 𝑆32 𝑆33

Because of plane of symmetry of the junction 𝑆13 = 𝑆23

If port 3 is perfectly matched, S33 = 0

From symmetry property, Sij = Sji

S12 = S21 ,S13 = S31 , S23 = S32 = S13

With all above properties [S] becomes,

𝑠 =

𝑆11 𝑆12 𝑆13

𝑆12 𝑆22 𝑆13

𝑆13 𝑆13 0

From unitary property 𝑠 [𝑠]∗ = [𝑈]

𝑆11 𝑆12 𝑆13

𝑆12 𝑆22 𝑆13

𝑆13 𝑆13 0∗

𝑆11∗ 𝑆12

∗ 𝑆13∗

𝑆12∗ 𝑆22

∗ 𝑆13∗

𝑆13∗ 𝑆13

∗ 0=

1 0 00 1 00 0 1

R1C1: |S11|2 + |S12|

2 + |S13|

2 = 1

R2C2: |S12|2 + |S22|

2 + |S13|

2 = 1

R3C3: |S13|2 + |S13|

2 = 1

R3C1: 𝑆13𝑆11∗ + 𝑆13𝑆12

∗ = 0

S11 = S22; 𝑆13 = 1

2 ;

𝑆13 𝑆11∗ + 𝑆12

∗ = 0 ; 𝑆13 ≠ 0 ;

𝑆11∗ = −𝑆12

∗

𝑆12 = −𝑆11

𝑆11 2 + 𝑆11

2 + 1

2= 1;𝑆11 =

1

2 𝑆12 =

−1

2 𝑆22 =

1

2

𝑆 =

1

2

−1

2

1

2−1

2

1

2

1

21

2

1

20

Note: Diagram 2M, initial conditions 2M, correct S-matrix 2M. initial

conditions are specified using properties of H plane Tee

5.b) Compare H plane Tee with E plane Tee?

H-plane Tee E-plane Tee

In an H-plane Tee, the axis of its side

arm is parallel to the magnetic field or

shunting the electric field of the main

waveguide.

In an E-plane Tee, the axis of its side

arm is parallel to the electric field of the

main waveguide.

An H plane Tee is also called a parallel

or shunt Tee.

The E-plane Tee is also called series

Tee.

When the power is fed at port-3, that is

at the side arm, the resulting power is

equally divided between port 1 and port

2 with in phase.

When the power is fed at port3, that is at

the side arm, the resulting power is

equally divided between port1 and 2,

but phase shift of 1800 is introduced

between the two outputs.

When the equal input power is fed to

both ports 1 and 2, the maximum power

is obtained at port 3.

When the equal input power is fed to

both ports 1 and 2, the maximum power

is obtained at port3.

When the input is applied at any one of

the collinear ports i.e., port1 and port2,

the resulting power is obtained at port3.

when the input is applied at any one of

the collinear ports i.e., port1 or port2,

the resulting power is obtained at port3.

Note: diagrams can also be drawn as a difference. Any 4 valid points with neat

sketches can be awarded 6M.

Unit III

6.a) Derive the equation of motion of an electron in the repeller region and

hence find the round trip transit time and transit angle of the centre of the

bunch of electrons in the reflex klystron?

Let us assume that space charge effects of electron motion are negligible. The

electron velocity 𝑣0 before entering cavity gap at z=0 and time t0 is given by

𝑣0 = 2𝑒𝑉0

𝑚= 0.593𝑋 106 𝑉0𝑚𝑡𝑠/𝑠𝑒𝑐

V0 = DC beam voltage

The same electron leaves the cavity gap at z=d at the time t1 with velocity

𝑣 𝑡1 = 𝑣0[1 + 𝛽1𝑉1

2𝑉0sin(𝜔𝑡1 −

𝜃𝑔2

)

The same electron is forced back to cavity by the repeller at z=d and time t2 by the

retarding electric field E, given by

𝐸 =𝑇𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑡𝑎𝑔𝑒

𝐿𝑒𝑛𝑔𝑡𝑕=𝑉𝑟 + 𝑉0 + 𝑉1 sin(𝜔𝑡)

𝐿

E exists in the z direction and is constant in z direction.

The force equation for the electron in the repeller region is

𝑚𝑑2𝑍

𝑑𝑡 2= −𝑒𝐸 = −𝑒

𝑉𝑟+𝑉0

𝐿 where |𝑉1sin(𝜔𝑡) ≪ (𝑉𝑟 + 𝑉0) is assumed.

Integrating above equation twice

𝑑𝑧

𝑑𝑡=

−𝑒𝑉𝑟 + 𝑉0

𝑚𝐿 𝑑𝑡𝑡

𝑡1

= −𝑒 𝑉𝑟 + 𝑉0

𝑚𝐿 𝑡 − 𝑡1 + 𝐶1

At 𝑡 = 𝑡1 ,𝑑𝑧

𝑑𝑡= 𝑣 𝑡1 = 𝐶1; 𝑡𝑕𝑒𝑛

𝑧 =−𝑒(𝑉𝑟 + 𝑉0)

𝑚𝐿 𝑡 − 𝑡1 𝑑𝑡𝑡

𝑡1

+ 𝑣(𝑡1) 𝑑𝑡𝑡

𝑡1

𝑧 =−𝑒(𝑉𝑟 + 𝑉0)

2𝑚𝐿(𝑡 − 𝑡1)2 + 𝑣 𝑡1 𝑡 − 𝑡1 + 𝐶2

At 𝑡 = 𝑡1 , 𝑧 = 𝑑 = 𝐶2; then

𝑧 =−𝑒(𝑉𝑟 + 𝑉0)

2𝑚𝐿(𝑡 − 𝑡1)2 + 𝑣 𝑡1 𝑡 − 𝑡1 + 𝑑

Now, on the assumption that the electron leaves the cavity gap at z=d and time t1,

with a velocity of v(t1) and returns to the gap at z=d and time t2, then at t= t2, z=d

𝑑 =−𝑒(𝑉𝑟 + 𝑉0)

2𝑚𝐿(𝑡2 − 𝑡1)2 + 𝑣 𝑡1 𝑡2 − 𝑡1 + 𝑑

𝑣 𝑡1 =𝑒 𝑉𝑟 + 𝑉0

2𝑚𝐿(𝑡2 − 𝑡1)

The round trip transit time in the repeller space is given by

𝑇′ = 𝑡2 − 𝑡1 =2𝑚𝐿

𝑒 𝑉𝑟 + 𝑉0 𝑣 𝑡1 = 𝑇0

′ [1 + 𝛽1𝑉1

2𝑉0sin(𝜔𝑡1 −

𝜃𝑔2

)

Where 𝑇0′ =

2𝑚𝐿

𝑒 𝑉𝑟+𝑉0 is the round trip DC transit time

Multiplication of above equation by radian frequency results in

𝜔𝑇1 = 𝜔 𝑡2 − 𝑡1 = 𝜃0′ + 𝑋′ sin(𝜔𝑡1 −

𝜃𝑔2

)

𝜃0′ = 𝜔𝑇0

′

is the round trip DC transit angle and

𝑋′ =𝛽1𝑉1

2𝑉0𝜃0′

is the Bunching Parameter of the reflex klystron oscillator.

Note: Valid expressions and step by step derivation with boundary conditions

included can be awarded 6M. Boundary conditions substitution 2M bunching

parameter 2M, round trip transit time 2M.

6.b) A Reflex Klystron operates at the peak of the n=1 or ¾ mode. The dc

power input is 40mW, and the ratio of V1 to V0 is 0.278

(i) Determine the efficiency of the Reflex klystron

(ii)Find the total output in mW

(iii) if 20% of the power delivered by the electron beam is displayed in the

cavity walls, find the power delivered to the load?

Since, Pdc = 40mW n=1 mode 𝑉1

𝑉0= 0.278 =

2𝑋′

𝛽𝑖(2𝑛𝜋 − 𝜋2

)

Assume βi = 1, and substitute n=1 we get X‟ =0.6555

J1(0.6555) = 0.31045.

Since 20% of dc power is dissipated in wall net DC power available for conversion

is Pdc = 40mW – 20% of 40mW = 32mW.

The output AC power is 𝑃𝑎𝑐 = 𝑃𝑑𝑐2𝑋 ′ 𝐽1(𝑋 ′ )

2𝑛𝜋−𝜋

2

= 2.740𝑚𝑊~2.75𝑚𝑊

Note: correct formula 2M, correct bunching parameter and first order Bessel

function value 2M, correct answers 2M.

7.a) write short notes on the following:

(i) Beam coupling coefficient

(ii) Beam loading

(iii)Different types of electric currents

(iv) Assumptions made to explain the principles of operation of klystrons.

Beam Coupling Coefficient:

It is defined as beam coupling coefficient at the input cavity gap βi.

By increasing the transit angle θg decreases the coupling between the

electron beam and the buncher cavity i.e., the velocity modulation of given

microwave signal decreased.

Beam Loading:

The maximum bunching should occur approximately midway between the

catcher grids.

The phase of the catcher gap voltage must be maintained in such a way that

the bunched electrons, as they pass through the grids, encounter a retarding

phase.

When the bunched electron beam passes through the retarding phase, its

kinetic energy is transferred to the field of the catcher cavity.

When the electrons emerge from the grids, they have reduced velocity and

are finally collected by the collector.

Figure shows the output equivalent circuit

Rsho - the wall resistance of the catcher cavity.

RB - the beam loading resistance

RL - the external load resistance

Rsh - the effective shunt resistance

Different types of current:

The current induced by the electron beam in the walls of the catcher cavity is

directly proportional to the amplitude of the microwave input voltage V1 ,

the fundamental component of the induced microwave current in the catcher

is given by

Where β0 is the beam coupling coefficient of the catcher gap. If the buncher

and catcher cavities are identical, then βi = β0 . The fundamental

component of the current induced in the catcher cavity then has the

magnitude.

Assumptions made to explain the principles of operation of klystrons:

1. The electron beam is assumed to have a uniform density in the cross section of

the beam.

2. Space charge effects are negligible.

3. The magnitude of the microwave signal input is assumed to be much smaller

than the dc accelerating voltage.

Note: each bit carries 2M

7.b)Derive the Hull cut off voltage expression of a magnetron?

The Hull cut off condition is obtained under the condition that there is no RF field,

which is in turn defines anode voltage is a function of magnetic field. The

magnetic field tends to prevent the flow of electrons to the anode. On the other

hand, under right circumstances, the electrons leave the hub after getting interacted

with RF wave that is rotating about the cathode , and flow to the anode. It happens

electrons speed is reduced to RF rotation rate. In this process the electrons amplify

the wave and losses the energy.

Force acting on the electron 𝐹 = 𝐵𝑒𝑣

In the direction of φ, the force component is given by 𝐹𝜑 = 𝑒𝐵𝑣𝜌

Where 𝑣𝜌 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑛 𝑡𝑕𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡𝑕𝑒 𝑟𝑎𝑑𝑖𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝜌, from the

center of the cathode cylinder.

Torque in the direction of ϕ can be given as

𝑇𝜙 = 𝜌𝐹𝜙 = 𝑒𝜌𝑣𝜌𝐵

Angular momentum = angular velocity X moment of inertia = 𝑑𝜙

𝑑𝑡= 𝑚𝜌2

Time rate of angular momentum = 𝑑

𝑑𝑡(𝑑𝜙

𝑑𝑡𝑥𝑚𝜌2)

𝑑

𝑑𝑡 𝑑𝜙

𝑑𝑡𝑥𝑚𝜌2 = 𝑒𝜌𝑣𝜌𝐵

2𝑚𝜌𝑑𝜙

𝑑𝑡+ 𝑚𝜌2

𝑑2𝜙

𝑑𝑡2= 𝑒𝐵𝜌

𝑑𝜌

𝑑𝑡

Integrate above equation with respect to “t”

2𝑚𝜌𝜙 + 𝑚𝜌2𝑑𝜙

𝑑𝑡= 𝑒𝐵

𝜌2

2

For a particular direction, mρϕ can be considered a constant

𝑚𝜌2𝑑𝜙

𝑑𝑡+ 𝐶 = 𝑒𝐵

𝜌2

2

The value of C can be determined by applying boundary conditions(i.e., at surface

of the cathode ρ=a and 𝑑𝜙

𝑑𝑡= 0

𝐶 = 𝑒𝐵𝑎2

2

Substitute above C value we get

𝑚𝜌2𝑑𝜙

𝑑𝑡=𝑒𝐵

2(𝜌2 − 𝑎2)

𝑑𝜙

𝑑𝑡=𝑒𝐵

2𝑚(1 −

𝑎2

𝜌2)

When ρ=a(i.e. at cathode), 𝑑𝜙

𝑑𝑡 𝑎𝑝𝑝𝑟𝑜𝑎𝑐𝑕𝑒𝑠 𝑡𝑜 0

When 𝜌 ≫ 𝑎,𝑑𝜙

𝑑𝑡 𝑎𝑝𝑝𝑟𝑜𝑎𝑐𝑕𝑒𝑠 (𝜔)𝑚𝑎𝑥

(𝑑𝜙

𝑑𝑡)𝑚𝑎𝑥 = (𝜔)𝑚𝑎𝑥 =

𝑒𝐵

2𝑚=𝑒𝐵𝑐2𝑚

Where B= Bc is the cut-off magnetic flux density

We know potential energy of electron = kinetic energy of electron

𝑒𝑉0 =1

2𝑚𝑣2 =

1

2𝑚(𝑣𝜌

2 + 𝑣𝜙2 )

𝑣𝜌 =𝑑𝜌

𝑑𝑡 𝑎𝑛𝑑 𝑣𝜙 =

𝜌𝑑𝜙

𝑑𝑡

𝑒𝑉0 =1

2𝑚[

𝑑𝜌

𝑑𝑡

2+ 𝜌2(

𝑑𝜙

𝑑𝑡)2]

𝑑𝜙

𝑑𝑡= (𝜔)𝑚𝑎𝑥 (1 −

𝑎2

𝜌2)

𝑒𝑉0 =1

2𝑚[

𝑑𝜌

𝑑𝑡

2

+ 𝜌2 𝜔 𝑚𝑎𝑥2 1 −

𝑎2

𝜌2

2

]

At anode ρ=b, 𝑑𝜌

𝑑𝑡= 0, substituting these boundary conditions in above equations,

𝑚

2 𝑏2 𝜔 𝑚𝑎𝑥

2 1 −

𝑎2

𝜌2

2

= 𝑒𝑉0

Substituting above equations we get

𝐵𝐶 = 8𝑉0𝑚/𝑒 1/2

𝑏(1 −𝑎2

𝑏2)

The Hull cut-off voltage is given by 𝑉𝑜𝑐 = 1

8

𝑒

𝑚𝐵0

2𝑏2 1 −𝑎2

𝑏2 2

Note: Correct derivation and expressions with boundary conditions carries

4M

Unit IV

8.a) Explain the basic modes of operation of a Gunn diode?

Gunn modes of operation:

Depending on the device characteristics and external circuitry, the Gunn diode can

be made to oscillate in any one of these four modes.

Gunn oscillation mode

Limited Space Charge accumulation (LSA mode)

Stable Amplification mode

Bias circuit oscillation mode

The modes of operation are classified based on the condition

𝑛0𝑙 >𝜀𝑠𝑣𝑑𝑒|𝜇𝑛 |

Gunn Oscillation modes

This mode has following features:

The electric field is greater than the threshold.

The product of frequency and length is about 107 cm/s

The product of doping and length is more than 1012

/cm2.

Due to the cyclic arrangement of either accumulation layer or the high

voltage domain the Gunn diode is unstable.

There are three types of Gunn oscillation modes. They are

Transit time domain mode

Delayed domain mode

Quenched domain mode

Transit time domain mode: Transit time domain oscillation is the basic mode and

is not depend on the external circuit. When a domain is quenched at the anode the

current peaks are obtained. Then, another is nucleated near the cathode. At any

time, the entire electric field across the device is above the threshold electric field.

The frequency is given by

f=1/τ=𝑣𝑑/𝑙 𝑣𝑑 𝑖𝑠 𝑡𝑕𝑒 𝑑𝑜𝑚𝑖𝑛 𝑜𝑟 𝑑𝑟𝑖𝑓𝑡 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 l is the effective length and fl is the product of

frequency and length.

The oscillation time τ0 is equal to the transit time(τ). It is a low power(<2W), low

efficiency mode, and the operating frequency is between 1 GHz and 18GHz.

Delayed domain mode: This is also called inhibited mode. In this mode, the

domain is collected when the dc bias is less than the threshold electric field(Eth).

The next domain can only be formed when the field again reaches threshold. The

oscillation period is greater than transit time of the critical electric field(Eth), that is

τ0>τ. The frequency of oscillation is determined by the resonant circuit. The

efficiency of this mode is limited to 20%. The drift velocity (𝑣𝑑) or f l lies between

106cm/s and 10

7cm/s.

Quenched domain mode: In this mode, the domain collapses before it reaches the

anode; that is it quenched before it is collected, hence the name quenched mode.

The bias electric field reduces below the sustaining electric field(Es) in the negative

half cycle. When the bias electric field swings back more than the threshold

electric field, another domain is nucleated and the procedure repeats. The operating

frequencies are higher than the transit time frequency. The maximum efficiency of

this mode is 13%.

Limited Space Charge Accumulation mode:

This mode operates by using high-Q resonant cavity with current pulses from the

Gunn diode. The Gunn diode is placed in a resonant cavity, which is tuned to a

frequency of the LSA mode(f0), so that the circuit operates like a negative

resistance device and the domains do not have enough time to form. In this mode,

the device can be biased to several times higher than Eth. When the input field is

increase more than the threshold electric field, the device remains in the negative

resistance region. Oscillation time is obtained by the external circuit, which is

given by

𝜏0 = 2𝜋 𝐿𝐶 + 𝐿

𝑅(𝑉𝑏𝑉𝑡𝑕

)

Stable Amplification mode: This mode is defined in the region where there

product of frequency times length is about 107cm/s and product of doping times

length is in between 1011

and 1012

/cm2.

Bias circuit oscillation mode: This mode occurs only when there is either Gunn or

LSA oscillation, and it is usually at the region where the product of frequency

times length is too small to appear. The drop in current at threshold can lead to

oscillations in the bias circuit typically 1KHz to 100MHz.

Note: 4 modes and explanation 4M and diagrams 2M

8.b) Using a typical microwave bench, explain the measurement of

attenuation?

Attenuators are used to adjust the power level of the microwave signal. If the

network is perfectly matched, the reflected power is zero and the insertion

loss is similar to the attenuation provided by the microwave device or

component.

Attenuation is defined as the ratio of input power to the output power in dB.

𝑨𝒕𝒕𝒆𝒏𝒖𝒂𝒕𝒊𝒐𝒏 𝒊𝒏 𝒅𝑩 = 𝟏𝟎𝐥𝐨𝐠(𝑷𝟏

𝑷𝟐)

Where P1 = power detected by the load without attenuator in the line.

P2 = power detected by the load with the attenuator in the line. Power Ratio method:- Power ratio is a process of measuring the input and output power with the

device(setup 1) as shown in figure below and without device(or attenuator)

in the circuit shown below.

The P1 and P2 are the powers measured the setup 1 and 2.The attenuation is

the ratio of power (P1 / P2 ) which is expressed in decibels.

Disadvantages of the power ratio method:- The attenuation calculated will not be accurate. Because the powers

measured (P1 & P2 ) is non linear.

It is also true for the networks with low input power.

With this method, we can measure the attenuation up to 20dB

RF substitution method:- The attenuation through the device under test is compared with a standard

microwave attenuator at the same frequency in this method.

Thus the output power “P” is measured by this method.

The drawback of power ratio method can be overcome by this method as the

attenuation is measured at single power position.

In setup2 the attenuator is adjusted to get the same output power “P” as in

setup1. the attenuation value is measured in the precision attenuator.

Note: 2methods each method carries 3M

9.a) Explain the method of measuring VSWR using microwave bench setup?

VSWR stands for voltage standing wave ratio. In a perfectly matched

system, there is no variation in the field strength along the waveguide.

A mismatch leads to reflected waves, there by leading to standing waves

along the length of the guide,

Standing guide are the indication of the quality of the transmission. VSWR

= 1 for perfectly matched system. The ratio of maximum to the minimum

voltage gives the VSWR.

𝑉𝑆𝑊𝑅 =𝑉𝑚𝑎𝑥𝑉𝑚𝑖𝑛

= 1 + 𝛤

1 − 𝛤

Where Γ is the reflection coefficient.

LOW VSWR MEASUREMENT:

VSWR values below 10 are very easily measurement by this method.

The VSWR meter direct displays these values. The attenuator is adjusted to

give a maximum reading on the meter. Then, the minimum reading on the

meter is obtained by adjusting the probe on the slotted line. Thus VSWR is

defined as the ratio of first reading to the second reading.

The meter can be calibrated in terms of VSWR. Here attenuator is adjusted

so that travelling probe gives maximum deflection on the VSWR meter.

The VSWR of 1 corresponds to full scale deflection(i.e. 10mV in the meter).

By adjusting the travelling probe the minimum reading can be obtained on

the meter. The variation in deflections and the corresponding VSWR values

are tabulated.

Deflection

in the

meter

VSWR

5mV 2

3.33mV 3

2.5mV 4

2mV 5

1.69mV 6

HIGH VSWR MEASUREMENT:

VSWR greater than 10 can be measured by double minimum method. In this

method, the probe is inserted to a depth where the minimum value can be read

easily.

Then the probe should be moved to a point where the power is twice the

minimum ( Pmin = 2V2

min /RL i.e. Pmin= 2P).

Let d1 be the position. Then again, the probe is moved to twice the power

point on the other side of the minimum(say d2) as shown in fig below

For the dominant mode TE10 mode rectangular waveguide, λ0, λg , and λc are

related as below. 1

𝜆02 =

1

𝜆𝑔2

+1

𝜆𝑐2

λ0 is free space wavelength

λg is guide wavelength

λc is cutt off wavelength

For the TE10 mode λc = 2a where “a” is the broad dimension of the

waveguide

𝜆0 = 𝐶𝑓

𝑉𝑆𝑊𝑅 = 𝜆𝑔

𝜋(𝑑2 − 𝑑1)

Note: 2 methods and each method carries 3M

9.b) Explain Varactor and Crystal diode?

Crystal detectors

Point-contact diodes, commonly called CRYSTALS, are the

oldest microwave semiconductor devices.

Unlike the PN-junction diode, the point-contact diode depends on the pressure of

contact between a point and a semiconductor crystal for its operation.. One section

of the diode consists of a small rectangular crystal of n-type silicon. A fine

berylium-copper, bronze-phosphor, or tungsten wire called the CATWHISKER

presses against the crystal and forms the other part of the diode. During the

manufacture of the point contact diode, a relatively large current is passed from the

catwhisker to the silicon crystal. The result of this large current is the formation of

a small region of p-type material around the crystal in the vicinity of the point

contact. Thus, a PN-junction is formed which behaves in the same way as a normal

PN-junction.

Varactor diode:

A varactor diode is a variable capacitor junction diode. These two terminal devices

also called as varicaps. This is a special type of PN junction that is designed to

operate in a microwave range. It works on the principle of voltage variable nature

of the depletion capacitance.

Working: In a PN junction, due to density gradient, holes diffuse N side and

Electrons diffuse P side. This causes a few ions on either side of the junction to be

depleted of mobile charges. This region is known as depletion region or space

charge region.

If forward bias is applied, the carriers move toward the junction which reduces the

depletion width.

If reverse bias is applied, the carriers move away from the junction: as a result the

depletion width increases.

The variation of width with voltage may be considered a capacitive effect with the

depletion region as the dielectric, and P and N region as parallel plates. This

capacitance is known as transition capacitance or junction capacitance.

𝐶𝑗 =𝜀𝑠𝐴

𝑊

Note: both diagrams and symbol 3M working explanation 3M

Prepared by:

Gopi chand. Dasari

Asst. Prof

Dept of ECE

BEC, Bapatla.