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Sunday, 30 June 2019

Functional Block Diagram of 8085 Microprocessor

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The internal architecture (Functional Block Diagram of 8085 Microprocessor) is shown in figure.


The following are the functional blocks in the 8085 Microprocessor.

1. Accumulator
2. Temporary register
3. Arithmetic and Logic Unit (ALU)
4. Flag register
5. Instruction Register
6. Instruction Decoder and Machine cycle encoder
7. General purpose registers
8. Stack Pointer
9. Program Counter
10. Incrementer / Decrementer
11. Timing and Control unit
12. Interrupt control
13. Serial I/O control
14. Address buffer and Address / Data buffer

1. Accumulator (A-register)

It is an 8-bit register. It is associated with ALU. The accumulator is also called A-register. During the arithmetic / logic operations, one of the operand is available in Accumulator. The result of the arithmetic / logic operations is also stored in the Accumulator.

2. Temporary (TEMP) register

It is an 8-bit register. It is also associated with ALU. This register is used to hold one of the data (from memory or general purpose registers) during an arithmetic / logic operation.

3. Arithmetic and Logic Unit (ALU)

The Arithmetic and Logic Unit includes Accumulator, Temporary register, arithmetic and logic circuits and flag register. The ALU can perform arithmetic (such as addition and subtraction) and logic operations (such as AND, OR and EX-OR) on 8-bit data. It receives the data from accumulator and or TEMP register. The result is stored in the accumulator. The conditions of the result (such as carry, zero) are indicated in the flags.

4. Flag register

It is an 8-bit register. But only five bits are used. The flag positions in the flag register are shown below.

Flag register of 8085: The flags are affected by the arithmetic and logic operations in the ALU. The flag register is also known as Status register or Condition code register. There are five flags namely Sign (S) flag, Zero (Z) flag, Auxiliary Carry (AC) flag, Parity (P) flag and Carry (CY) flag.

 Sign (S) flag: Sign flag is set (1) if the bit D7 of the result in the accumulator is 1, otherwise it is reset (0). This flag is set when the result is negative. This flag is used only for signed numbers.

 Zero (Z) flag: Zero flag is set (1) if the result in the accumulator is zero, otherwise it is reset (0).

 Auxiliary Carry (AC): Auxiliary Carry flag is set (1) if there is a carry from bit position D3of result in the accumulator, otherwise it is reset (0). This flag is used for BCD operations.

 Parity (P) flag: Parity flag is set (1) if the result in the accumulator has even number of 1s, otherwise it is reset (0).

• Carry (CY) flag: Carry flag is set (1) if the result of an arithmetic operation results in a carry from bit position D7, otherwise it is reset (0). This flag is also used to indicate a borrow condition during subtraction operations.


5. Instruction register

When an instruction is fetched from memory, it is stored in the Instruction register. It is an 8-bit register. This resister cannot be used in the programs.

6. Instruction Decoder and Machine cycle encoding

This unit decodes the instruction stored in the Instruction register. It determines the nature of the instruction and establishes the sequence of events to be followed by the Timing and control unit.

7. General purpose registers

There are six 8-bit general purpose registers namely B, C, D, E, H and L registers. B and C registers are combined together as BC register pair for 16-bit operations. Similarly D and E registers can be used as DE resister pair and H and L as HL register pair. The HL register pair is also used as memory pointer (M-register) for storing 16-bit address in some instructions.There are two more 8-bit temporary registers W and Z. These registers are used to hold data during the execution of some instructions. W and Z registers cannot be used in programs.

8. Stack Pointer (SP)

Stack is a portion of memory (RAM) used as FILO (First In Last Out) buffer. This is mainly used during subroutine operations. Stack Pointer is a 16-bit register used as a memory pointer (16-bit address) for denoting the stack position in memory. The Stack pointer is decremented each time when data is loaded into the stack and incremented when data is retrieved from the stack. Stack pointer always points to the top of the stack memory.

9. Program Counter (PC)

The Program Counter (PC) is a 16-bit register. It is used to point the address of the next instruction to be fetched from the memory. When one instruction is fetched from memory, PC is automatically incremented to point out the next instruction.

10. Incrementer / Decrementer

This unit is used to increment or decrement the contents of the 16-bit registers.

11. Timing and Control unit

The internal clock generator is available in this unit.This unit has the micro programs for all the instructions to carry out the micro steps required in completing the instructions. This unit receives signals from the Instruction decoder and Machine cycle encoding unit and generates control signals according to the micro-program for the instruction.

12. Interrupt control

There are five hardware interrupts available in 8085 Microprocessor namely TRAP, RST 7.5, RST 6.5, RST 5.5 and INTR for interfacing the peripherals with the microprocessor. These interrupts are handled by the Interrupt control unit. INT A signal is generated by the Interrupt control unit as an acknowledgement for an interrupting device. If two or more interrupts occur at the same time, service is given according to the priority basis.

13. Serial I/O control

Serial data is transmitted to the peripherals through SOD pin and received through the SID pin. The SOD and SID pins are handled by the Serial I/O control unit using the SIM and RIM instructions.

14. Address buffer and Address / Data buffer

The Address buffer is an 8-bit unidirectional buffer from which the higher order address bits A8 – A15 leaves the microprocessor to the memory and peripherals. The Address / Data buffer is an 8-bit bidirectional buffer used for sending the lower order address bits A0 – A7 and sending and receiving the data bits D0 – D7 to the memory and peripherals.

Construction of DC Machines

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Introduction

D.C. generator is a electrical machine which converts mechanical energy (or power) into electrical energy (or power). The energy conversion is based on the principle of the production of dynamically or motionally. Induced e.m.f .

Whenever a conductor cuts magnetic flux and e.m.f is produced in it according to Faraday’s laws of electromagnetic induction. This e.m.f causes a current to flow if the conductor circuit is closed. Mechanical energy is given on the generator by means of rotation of armature it is known as prime mover.

e.g. prime mover-diesel engine, turbine wised mill.

Construction of dc machines

The construction diagram of DC machines is shown in figure. It consists of following main parts are

1.         Yoke (or) magnetic frame
2.         Magnetic poles
3.         Field coils
4.         Inter poles (or) Commutation poles
5.         Commutator
6.         Brushes
7.         Bearing and end cover

Yoke (or) magnetic frame

1. It provides mechanical support for the machine and act as the magnetic flux produced the pole.
2. It forms the portion of magnetic circuit. it carries the magnetic flux produced by the poles.
3. It is made up of cast iron for smaller machine for larger machines it is made up of case steel.



The field magnet consists of pole cores and pole shoes.

   (1) The spread out the flux in the air gap
   (2) They support the field coils.

Pole cores and pole shoes are built with these laminations of steel. These laminations are held together using rivets. The cores are laminated to reduce the eddy current loss. The magnetic poles are fitted inside the yoke by means of screws.

Field coils

Field coils are usually wound with enameled copper wire. The magnetic field strength depends upon the current flowing through the coil. The north and South Pole depend upon the direction of the current flow through the field coil.

Inter poles (or) Commutation poles

1. The function of inter pole is to improve the commutation and to reduce the armature reaction.
2 . The exciting coils on the interlopes are connected in series with the armature.

Armature

The armature core is keyed to the machine shaft and it rotates between the field poles. It consists of slotted steel laminations. These laminations are stocked to form a cylindrical core.

The laminations are insulated from each other by thin coating of varnish. The purpose of the lamination core is reducing the eddy current loss. Armature winding divided into two,

(1) Lab winding: for low voltage, high current machine
(2) Wave winding: for high voltage, low current machine

Commutator

The commutator is made up of copper segments insulated from one another by mica sheets. The number of segments is equal to the number of armature coils. The segment is connected to armature conductor. Armature conductors are soldered to the commutator segment in a suitable manner to give rise to the armature winding.

Functions of commutator

1. Collection of current
2. Current from the armature conductor.
3. It converts alternating current induced in the armature conductors are to unidirectional current.

Brushes

Brushes are made up of carbon and rest on the commutator. The function of the brushes is to collect current from the commutator to the external stationary load. The brushes are put inside the brush holders. The brush holders are kept pressed against the commutator by a spring as shown in figure.

Bearing and end cover

Bearing

Ball bearings (or) rollers bearings are fitted inside the end cover. Armature shaft is mounted over these brings.

End over

End over are made up of cast iron fabricated steel. They are fitted to both ends of yoke.

Armature winding

Type of winding

(1) Lab winding
(2) Wave winding

Lab winding

1. Lab winding is used in large output current and low voltage dock machine.
2. In lab winding, the number of parallel paths (A) is equal to the number of poles (P).  i.e., A = P
3. The number of brushes is made equal to the number of poles.

Wave winding

1. Wave winding is used in low output current and high voltage d.c machine.
2. In wave winding, the number of parallel paths (A) =2
3. Number of brushes = 2

Saturday, 29 June 2019

Self Excited and Separately Excited DC Generator

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Types of DC Generator:
What is excitation?

The current flows through the field winding for production of magnetic flux are called excitation.
Depending upon the method of excitation D.C generators are classified as,

(1).Separately excited D.C Generator
(2).Self excited D.C Generator

(1).Separately excited D.C Generator

In separately excited D.C generator, the exciting field current supplied by a separate source as shown in figure. When current flows through the field winding, the main pole becomes magnet and the flux lines are produced.

Applications:

If it has constant d.c output voltage, these generators are used for

1. Battery charging
2. Supply source to d.c motor



(2). Self Excited D.C Generator

In self excited d.c generator, the emf induced in the armature supplies the exciting field current. Therefore, need not separate d.c supply for exciting field poles.

Depending upon the way in which the field winding is connected with armature , self excited d.c generators are classified as,

1. D.C shunt generator
2. D.C series generator
3. D.C compound generator

1. D.C Shunt Generator

In D.C shunt generator, field winding is connected in parallel with the armature winding as shown in figure. Its field winding has many turns of thin wire having high resistance. Then armature rotates emf is induced due to the residual magnetism. Due to induced e.m.f, field current (Il) flows and hence flux and voltage increases.

D.C shunt generator, Ia = IL + Ish
Where, Ia = armature current
IL = load current
Ish= shunt field current

Applications

1. It is used in battery charging
2. It is used in electroplating.
3. It is used as exciter for alternator.

2. D.C Series Generator

In d.c series generator, the field winding is connected in series with the armature winding as shown in figure. It carries a large current. Its field winding has few turns of thick wire having low resistance.

Ia = IL = Ise

Where,
Ia = armature current
IL = load current
Ise = shunt field current

For series generator no load characteristics is absent because the circuit is closed only when load is connected. The load current flows through the field winding and produces flux. In series generator, as the terminal voltage increases with the load current . Booster to compensate for load voltage drops.

Applications

1. It is used booster.
2. It is used for supply to arc lamps.

3. D.C Compound Generator

Compound generator is a machine which consists of series field winding and shunt field winding. In d.c compound generator both series and shunt field winding are connected with the armature winding as shown in figure (a) & (b). The flux of the pole depends on the total flux of both field winding. 

Depends upon the connection field winding d.c compound are classified into two types.

1. Long shunt d.c compound generator
2. Short shunt d.c compound generator

1. Long Shunt Compound Generator

In long shunt compound generator, the series field winding is connected in series with the armature winding and shunt field winding is connected in parallel with this arrangement as shown in figure.
In long shunt compound generator, Ia = Ise
Ia = Ish + IL

2. Short Shunt Compound Generator

In short shunt compound generator, the series field winding is connected in series with the armature winding and shunt field winding is connected is parallel with this arrangement as shown in figure.
Ise = Ic
Ia = Ish + IL

Depends upon the connection of field winding the d.c compound generator is further classified into two types.

1. Cumulative compound generator:
2. Differentially compound generator

1. Cumulative compound generator

If the series field flux adds with shunt field flux it is called as cumulative compound generator.
In cumulative compound generator, the flux produced by the shunt field winding and flux produced by the series field winding acts in the same direction. Hence the total field flux acting in this type of generator is sum of these two fluxes.

Total flux, Φ = Φsh+ Φse

2. Differentially Compound Generator:

If the series field flux opposes with the shunt field flux, it is called as differential compound d.c generator.

In differentially compound generator , the flux produced by shunt field winding and the flux produced by the series field winding acts in the opposite direction. Hence the total field flux acting in this type of generator is difference of these two fluxes.

Applications:

1. Differentially compound generators are used for D.C welding purpose.
2. It is used in power offices and hotels.
3. It is used where power is to be transmitted to a long distance.

DC Generator Working Principle

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Introduction:

An electrical generator is a machine which converts mechanical energy in to electrical energy.

DC Generator Working Principle:

Electrical generator is based on the principle that whenever a conductor cuts magnetic flux an e.m.f is induced in the conductor. This e.m.f causes a current to flow if the circuit is closed. The direction of induced e.m.f is given by “Fleming’s right hand rule.”

The important components of d.c generator are,

1. A magnetic field
2. Conductor (or) group of conductors
3. Motion of conductor with respect to magnetic field.


In D.C generators , a stationary magnetic field is produced by field magnets. The armature consisting of conductor is rotated inside this magnetic field by a prime mover. The prime mover may be a turbine or diesel engine petrol engine. The nature of e.m.f induce in the armature conductors is alternating (A.C). The A.C e.m.f converted into unidirectional e.m.f (D.C) by means commutator. The commutator rotates along with the armature.

The method of producing emf in a single loop generator is explained with the help of figure.
Figure (a) & (b) shows a single loop of copper coil ABCD moving in a magnetic field. The two ends of coil are joined to two slip rings ‘a’ and ‘b’. These slip rings are insulated from each other. Two collecting brushes press against the slip rings a, b as shown in the figure. The brushes collect the current induced in the coil and supply it to the external load.



When the coil rotates inside the magnetic field the flux linked with the coil changes and hence e.m.f is induced in the coil, which is proportional to the rate of change of flux linkages. Imagine the coil to be rotating is the clock- wise direction.

When the plane of the coil is right angles to flux lines i.e., in position -1(where θ = 0) the flux linked with the coil is maximum, but rate of change flux linkages is minimum. Because in this position the coil side AB and CD do not cut or share the flux. Hence no e.m.f is induced in the coil at position-1.
Now the coil moves to at position-3 from position-1. When the coil reached the position-3 (where θ = 90˚), the coil plane horizontal to the flux lines the flux linked with coil is minimum, but the rate of change of flux linkages is maximum. Therefore maximum e.m.f is reduced in the coil position 3.
In the next quarter revolution from position 3 position 5 (90˚ to 180˚) the flux linked with coil is gradually increases but the rate of change of flux linkages is decreases. Therefore maximum e.m.f is reduced is zero position 5.

Now the coil moves from position 5 to position 7 (180˚ to 270˚) the e.m.f induced in the coil is in the reverse direction. Therefore at position 7, the e.m.f induced is negative maximum. Then the coil moves from position 7 to position 1, the flux the linked with the coil gradually increases, but the rate of change of flux linkages decreases the e.m.f induced is zero at position 1. Thus the emf induced in the coil is an alternating e.m.f as shown in figure.

If the slip rings are replaced by split rings, the alternating e.m.f will become unidirectional current (D.C). The split rings are made out of a conducting cylinder which is cut into two segments insulated from each other by thin sheet of mica. The coil ends are joined to this segment. Carbon brushes rest on the segments.

Figure (a) shows the connection of coil ends with split rings ‘a’ and b. in the first half revolution current flows along ABLMCD in the brush No: 1 is contact with segment ‘a’ acts the positive end of e.m.f and ’b’ act as the negative end.

In the next half revolution the direction of current in the coil has reversed as shown in figure (b). But at the same time the positions of segment a and b have also been reversed. The segment ‘a’ is coming in contact with brush no: 2 and becomes negative end of induced e.m.f. Again the current in the load resistance flows in the same direction is from L to M. The current is unidirectional current due to rectifying action of split rings (also called as commutator).
The unidirectional current is shown in the fig (c). To minimize the ripple in D.C current , the number of coil in the armature should be increased.

Fleming’s Right Hand rule

Keep the thump, fore finger and middle finger of the right hand mutually perpendicular to each other. If the fore finger points the direction of magnetic flux lines and the thump points the direction of motion of conductor, then the middle finger points the direction of induced e.m.f (or) current.

Friday, 28 June 2019

Hay Bridge Derivation

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Hay Bridge Derivation

The schematic diagram of the Hay bridge is shown in Figure below. It is a modified form of Maxwell bridge. The difference between Hay bridge and Maxwell bridge is that the resistor R1 is in series with the capacitor in Hay bridge. The unknown inductance Lx and its resistance part are represented by Lx and Rx respectively as shown in the schematic diagram. One of the ratio arm used capacitor C1 and a variable resistance R1. The other two arms have R2 and R3.

The balance condition is obtained as follows:



From the schematic diagram we find that:

Z1 = R1 – (j/ωC1)
Z2 = R2
Z3 = R3
and Z4 = Rx + jωLx

Therefore

(R1 - (j/ωC1))( Rx + jωLx) = R2 R3

R2 R3 = R1 Rx + jωLxR1 + (Lx/C1) – (jRx/ ωC1)

Taking real terms:

R2 R3 = R1 Rx + (Lx/C1) ------------------------ 1

Taking imaginary terms we have:

Rx/ ωC1 = ωLx R1 ------------------- 2

Both the equations 1 and 2 contain Lx and Rx therefore we must solve these equations simultaneously.

Taking equations 1 and 2 and multiplying equation 1 with ωR1 we have:

ωR1R2R3 = ωR12Rx + Lx ω R1/C1 ------------------ 3

Dividing the equation 2 with C1

Rx/ ωC12 = ωLx R1/ C1--------------------- 4

Subtracting 4 from 3 we have,

ωR1R2R3 - Rx/ ωC12 = ωR12Rx
ωR1R2R3 = ωR12Rx + Rx/ ωC12
ωR1R2R3 = Rx ((1 + ω2 C12R12 )/ ωC12)
Rx = ω2 C12R1R2R3/(1 + ω2 C12R12)

From equation 2
Lx = Rx/ ω2 C1R1
Lx = ω2 C12R1R2R3/(1 + ω2 C12R12) ω2 C1R1
Lx = R2R3 C1/(1 + ω2 C12R12)

The solution yields
Rx = ω2 C12R1R2R3/(1 + ω2 C12R12) ------------------- 5
Lx = R2R3 C1/(1 + ω2 C12R12) ------------------- 6

From the equations 5 and 6, it is seen that they contain the angular velocity (ω). Therefore the frequency of the voltage source must be accurately known as it appears. However this is not true. As we known that Q = 1/ ω C1R1 substituting the value of Q in the equation 6 he expression for Lx becomes:

Lx = R2R3 C1/(1 + (1/Q)2) ---------------- 7

For values of Q greater than ten the term (1/Q)2 will be smaller than 1/100 hence can be neglected.

Therefore:  Lx = R2R3C1------------------- 8

Hay's Bridge Advantages:

1. This bridge gives very easy derivation for unknown inductance for high Q coils and is appropriate for coils having Q greater than 10.

2. The bridge also gives simple derivation for Q.

3. From the expression for Q we find the resistance R1 to be appearing in the denominator. Hence for high Q coils value of R1 must be small. It shows that the bridge requires only a low value of resistor R1, unlike Maxwell bridge which required impractically large value of resistance.

Disadvantages of Hay's Bridge:

This bridge is not suitable for measuring inductance values with Q values smaller than 10. The reason is that the factor (1/Q)2 cannot be neglected in expression (7), in such cases.

Monday, 24 June 2019

Scattering Matrix in Microwave Engineering

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Microwave Hybrid Circuits:


A microwave network or microwave hybrid circuit consist of several microwave devices such as sources, attenuators, filters, amplifiers etc coupled together by transmission lines for the transmission of microwave signal. The point of interconnection of two or more devices is known as a junction. The measurement of z,y,h and ABCD parameter is difficult at microwave frequencies due to following reasons.

1. Non availability of voltage and current measuring equipments.
2. Short circuit not easily achieved for wide range of frequencies.
3. Presence of active devices make the circuit unstable so microwave circuits are analyzed using scattering parameters or ‘S’ matrix. S matrix relates the amplitude of reflected waves with incident waves.



Scattering Matrix in Microwave Engineering :


It is a square matrix which gives all the combinations of power relationship between input and output ports of a microwave junction. The elements of ‘S’ matrix are known as scattering parameters or scattering coefficients.
Microwave 2 port network
Consider the microwave 2 port network.

a1 – amplitude of incident wave at port 1
a2 – amplitude of incident wave at port 2
b1 – amplitude of reflected wave at port 1
b2 – amplitude of reflected wave at port 2

The incident and reflected waves can be related using ‘S‘ matrix as
[b] = [s] [a]







b1 = s11a1 + s12a2
b2 = s21a1 + s22a2

s11 is the reflection coefficient at port 1 when, a2 = 0
s11 = b1/a1 where a2 = 0

s22 is the reflection coefficient at port 2 when, a1 = 0
s22 = b2/a2 where a1 = 0

s12 is the attenuation of wave travelling from port 2 to port 1 when, a1 = 0
s12 = b1/a2 where a1 = 0

s21 is the attenuation of wave travelling from port 1 to port 2 when, a2 = 0
s21 = b2/a1 where a2 = 0

In a microwave network if the incident power is Pi, reflected power is Pr, output power is Pa then, the losses defined are

1. Insertion loss = 10 log (Pi/Po)
2. Transmission loss or attenuation = 10 log((Pi - Pr)/Po)
3. Reflection loss = 10 log (Pi/(Pi - Pr))
4. Return loss = 10 log (Pi/Pr)

Properties of S matrix:


1. S matrix is always a square matrix of order n x n.

2. Under perfect match condition the diagonal elements of S matrix are zero.

3. S matrix is always symmetric. ie, Sij = Sji

4. S matrix is an unitary matrix. Ie, [S][S]*= I. where, I is an identity matrix.

5. The sum of product of each term of any row or column multiplied by complex conjugate of corresponding term of another row or column is zero.





6. In a two port network, if the reference plane are shifted from one and two to 1’ and 2’ the new S matrix is given by




This property is known as phase shift property.

7. Since S matrix is symmetric [S]T = [S]

Friday, 21 June 2019

Z Transform of U(n)

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Find the Z Transform and ROC of U(n) ?

The Z transform of a discrete time signal x(n) is given by,






Here given x(n) = u(n)

Therefore,






We know that U(n) = 1; n ≥ 0
                             = 0; n < 0
Therefore,






X(z) = Z0 + Z-1 +Z-2 + Z-3 + Z-4 +………….
        = 1 + Z-1 +Z-2 + Z-3 + Z-4 +………….

It is clear that the infinite series is a Geometric Progression (GP)

The sum of the GP is given by

Sum = First Term / (1 – Common Ratio)

The common ratio (r) is given by

r = second term/first term
  = third term/second term

So, r = Z-1 /1 = Z-2/ Z-1 = Z-1

Hence, the sum of the series is given by





ROC of U(n)


The ROC of U(n) is given by

|r| < 1

|Z-1| < 1

|1/Z| < 1

|Z| > |1|
ROC of U(n)



Thursday, 20 June 2019

Velocity Modulation in Klystron Amplifier

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Klystron Amplifiers:

The 2 cavity klystron amplifier is widely used microwave amplifier, operated by the principles of voltage and current modulators.

Basic Operations:

A high velocity electron beam produced by the accelerating anode is passed through a buncher cavity, drift space, catcher cavity and finally collected by the collector terminals. The electrons injected from the cathode is accelerated by applying a DC voltage ‘Vg’. They arrive at the first cavity that is the buncher cavity or input cavity with uniform velocity. At the buncher cavity these electrons encounter signal voltage or gap voltage. The electrons that pass through the zeros of the gap voltage pass with unchanged velocity. The electrons that pass through positive half cycles of the gap voltage undergo acceleration in velocity. The electrons that pass through negative half cycles of the gap voltage undergo retardation in velocity. (As a result of these the electrons get bunched together as they travelled through the drift space). The variation in electron velocity in drift space is called velocity modulation. (The buncher cavity velocity modulates the electron beam). This electron beam induces a RF current in this field is opposite to the input cavity. Thus the kinetic energy is transferred from the electrons to the field capture cavity. The second cavity is called capture cavity since it captures energy from the bunch electron beam. The electrons emerging from the capture cavity are collected by the collector terminal.




Velocity Modulation in Klystron Amplifier:

The velocity of electrons before entering the buncher cavity is given by,
Vo = 2ev0/m

Where m is the mass of the electron
e is the charge of electron
v0 is the cathode potential

On substituting the values of e and m, the equation reduces to

Vo = 0.596 x 106 (v0)   m/sec ------------- 1

When the microwave signal is applied to the input terminal, the gap voltage is given by

Vs = V1sinωt ----------------- 2

V1 is the amplitude of the signal.
The average transit time through the gap at a distance ‘d’.

Τ = d/Vo = t1 – t0 -------------- 3

Where t0 is the line at which beam reaches the buncher cavity. t1 is the time at which the beam leaves the buncher cavity.

The average transit angle, θg = ω(d/Vo) = ω(t–  t0) ------------ 4

The average microwave voltage in the buncher cavity is






= V1/T ω (cos ωt0 – cos ωt1) ----------------- 5

From eq (4)
ωd/V0 = ω(t1 – t0)

ωt1 = ω(d/Vo + t0) ------------- 6

Subsituting eq (6) in eq (5)

<Vs> = V1/Tω [cosωt0 – cos(ωd/Vo + ωt0)] ------------- 7

Let ωt0 + ωd/2Vo = ωt0 + θg/2 = A
ωd/2Vo = θg/2 = B
A + B = ωt0 + θg/2 + θg/2 = ωt0 + θg 
A – B = ωt0 + θg/2 - θg/2 = ωt0

A + B = ωt0 + θg  ,  A – B = ωt0   ----------------- 8

Substitute eq (8) in eq (7)

<Vs> = V1/Tω [cos(A - B) – cos(A + B)]
         = 2V1/Tω.  sin A sin B
         =  2V1/Tω.  sin (ωt0 + θg  ) sin (θg/2)

Let ωT = θg

Therefore, <Vs> = 2V1/ θg.  sin (ωt0 + θg  ) sin (θg/2)
                    <Vs> =  V1 βi sin (ωt0 + θg/2)

βi = sin (θg/2)/g/2)