Microwave Engineering-Chapter 11-Sample

11

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Microwave Linear BeamTubes

11.1 Introduction

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In the last three chapters, we discussed the different types of microwavenetworks. In this and later chapters, we will discuss the different types ofmicrowave sources and amplifiers. Like low-frequency oscillators and amplifiers,microwave oscillators and amplifiers are also of two types: (a) vacuum tubes and(b) semiconductor devices. In the current and next chapter we will discuss thedifferent types of microwave tubes, after which we will discuss semiconductordevices.As conventional tubes cannot work at frequencies greater than 1GHz,because of lead inductance, inter-electrode capacitance, transit angle effects andgainbandwidth product limitations, special types of tubes are required for highfrequency operations. High-frequency tubes are generally categorized into twoclasses: (a) linear beam tubes (or O-type) and (b) crossed field tubes (or M-type).In this chapter we will discuss linear beam tubes and in the next chapter, crossedfield tubes.In practice, there are different types of linear beam tubes, such as two-cavityklystron, reflex klystron, travelling-wave tubes (TWTs), forward wave amplifiers(FWAs), backward wave amplifiers and oscillators (BWAs and BWOs), andtwystron. Out of these, klystron and reflex klystrons use resonant cavities,and hence are resonant structures. On the other hand, TWTs, FWAs, BWAs,and BWOs are non-resonant structures. A twystron is a hybrid structure anduses combinations of klystron and TWT components. Based on their operatingprinciple and structure, microwave linear beam tubes can be classified as shownin Fig. 11.1.In a linear beam tube, electrons emitted from the electron gun receivepotential energy from the DC beam voltage and accelerate towards the anode.As a result, the potential energy is converted into kinetic energy before theyarrive at the interaction region. In the interaction region, these electrons facethe microwave field, and either accelerate or decelerate depending on the phaseof that field. This acceleration or deceleration results in bunching of electrons,

11_Micro_Engineering_Chapter_11.indd 497

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498 M I C R O W A V E E N G I N E E R I N GLinear beam tubes (O-types)

Cavity

Slow wave structure

Resonant

Klystron

Twystron

Forward wave

Backward wave

Helix travelling-wave tube

Backward wave amplifierBackward wave oscillator

Coupled cavity travelling-wave tube

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Reflex klystron

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Fig. 11.1 Classification of linear beam tubes

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which then drift down the tube and arrive at the output structure. At the outputstructure, these bunched electrons induce a current and give up their kineticenergy to the output microwave field. Finally, they are collected by the collector.Throughout the journey, electrons of the electron beam remain together withthe help of a focusing magnetic field whose axis coincides with the axis of theelectron beam.

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Note The name O-type linear beam tubes was derived either from their French nameTPO (tubes propagation des ondes) or from the word original (meaning original typesof tubes).

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11.2 High-frequency Limitation of Conventional Tubes

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Conventional tubes, such as triodes, tetrodes, and pentodes, cannot work atfrequencies greater than 1GHz because of the following reasons:

11.2.1 Lead Inductance and Inter-electrode Capacitance EffectLeads that are connected to electrodes in conventional tubes have an inductiveeffect, which, along with the inter-electrode capacitances between grid to plateand grid to cathode, sets a higher cut-off for the operating frequency. To elaborate,at microwave frequencies, the parasitic reactance of lead inductances and interelectrode capacitances becomes very large compared to that of microwave resonantcircuits, and hence such ordinary tubes cannot operate at microwave frequencies.This problem can be overcome by reducing the length and area of the leads.However, such attempts, in turn, minimize the power-handling capability of thetubes. Furthermore, at microwave frequencies, the input conductance of the tubesoverloads the input circuit and thereby reduces their efficiency. To illustrate thesefacts, let us consider the triode circuits shown in Fig. 11.2.


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M I C R O W A V E L I N E A R B E A M T U B E S 499Cgp P

g

P+

+

g

+

Cgk

+ +

Vg KVin

R

L

C V0

Vg+

Vin

Vk

Lk

+

gmVg

rpR

K

L

C V0

Lk

(a)

(b)

Fig. 11.2 Triode circuit (a) Without inter-electrode parasitics(b) With inter-electrode parasitics

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An equivalent circuit of the triode has been constructed assuming that interelectrode capacitances and cathode inductances are the only parasitic parameterspresent. Since C gp C gk and Lk 1 (C gk ), the input voltage and current canbe written as follows:Vin = Vg + Vk = Vg (1 + j Lk g m ) (11.1)

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I in = jC gk Vg

(11.2)

ig

2 C gk Lk g m1 + 2 L2k g m2

+j

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Yin =

C gk

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or

jC gkVgjC gk (1 j Lk g m )I in==Vin Vg (1 + j Lk g m )1 + 2 L2k g m2

1 + 2 L2k g m2

(11.3)

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Yin =

hted

Therefore, the input admittance can be expressed as follows:

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Since 1 2 L2k g m2 , Eq. (11.3) modifies to the following form:Yin = 2 C gk Lk g m + jC gk

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(11.4)

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The input impedance can be written as follows:1

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Z in =

2

Lk C gk g m

j

1

3 L2k C gk g m2

(11.5)

Equation (11.5) reveals that input resistance is inversely proportional to the squareof the frequency, whereas input reactance is inversely proportional to the cube offrequency. Therefore, at high frequency the input impedance tends to zero or shortand the output power decreases rapidly.The input admittance and input impedance of a pentode can similarly bewritten in the following way:Yin = 2 C gk Lk g m + j (C gk + C gs )and

Z in =

12

Lk C gk g m

j

C gk + C gs2 2 3 L2k C gkgm

(11.6)(11.7)

where C gs is the capacitance between the gate and the screen.


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500 M I C R O W A V E E N G I N E E R I N G

11.2.2 GainBandwidth Product LimitationIn ordinary vacuum tubes, the maximum gain is generally achieved by resonatingthe output circuits, as shown in Fig. 11.3. If we assume that rp Lk , then theload voltage can be expressed as follows:+gmVg

rp

R

g mVg

G + j C 1 ( L)1 1where G = + rp R

C V1

L

Vl =

Fig. 11.3 Output tunedcircuit of pentode

(11.8)(11.9)

rp is the plate resistance, R is the load resistance, andL and C are the tuning elements.

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The corresponding resonance frequency can be expressed as follows:1fr =2 LCThe maximum gain at resonance can be expressed as follows:Am = g m G1Furthermore, G = C Lor 2 LC GL 1 = 0

(11.10)(11.11)

(11.12)

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Since the bandwidth is between the half power points, the half power points canbe expressed as follows:

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GG21GL G 2 L2 + 4 LC=+2LC2 LC2C4C

1 =

G2C

and

2 =

G+2C

2 G + 1 2C LC

(11.13)

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or

-C

1, 2 =

G 2 + 1 2C LC

(11.14)

The bandwidth can be expressed as follows:BW = 2 1 =or

G+2C

G 2 + 1 G + 2C LC 2C

G 2 + 1 2C LC

G 21BW = 2 + 2C LC

(11.15)

G 21Now since , the bandwidth becomes of the following form: 2C LC G 2GGBW 2 = 2= 2C 2C C

(11.16)

Hence, the gainbandwidth product can be written as follows:Am ( BW ) = g m C


(11.17)

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M I C R O W A V E L I N E A R B E A M T U B E S 501

Equation (11.17) reveals that the gainbandwidth product of a vacuum tube isindependent of its frequency and is a constant. Therefore, an ordinary resonantcircuit cannot be used with a microwave tube. In practice, microwave devices useeither re-entrant cavities (in case of klystrons) or slow-wave structures (in case ofTWTs) to obtain a high gain over a broad bandwidth.

11.2.3 RF LossRF loss at high frequencies can be of two types: (a) skin effect loss and (b)dielectric loss.

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Skin effect loss At a high frequency, current has a tendency to concentratearound the surface rather than being distributed throughout the cross section.This is known as skin effect. It reduces the effective surface area, which in turnincreases the resistance and hence the loss of the device. Resistance loss is alsoproportional to the square of the frequency.Losses due to skin effect can be reduced by increasing the current-carryingarea, which, in turn, increases the inter-electrode capacitance and thus limits highfrequency operations.

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Dielectric loss Dielectric loss in a material is proportional to frequency, andhence plays an important role in the operations of high-frequency tubes. This losscan be avoided by eliminating the tube base and reducing the surface area of thedielectric materials, and can be reduced by placing insulating materials at the pointof minimum electric field.

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Radiation loss At higher frequencies, the length of the leads approaches theoperating wavelength, and as a result these start radiating. Radiation loss increaseswith the increase in frequency and hence is very severe at microwave frequencies.Proper shielding is required to avoid this loss. Radiation loss can be minimized byenclosing the tubes or using a concentric line construction.

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11.2.3.1 Transit Angle EffectTransit angles pose another limitation to the use of conventional tubes at highfrequencies. The transit time of a tube is the time taken by an electron to travelthe inter-electrode distance. Mathematically, an electron transit angle is defined asfollows: d g = d v0(11.18)where g : transit time across the gapd: distance between cathode and gridv0 = 0.593 V0 106 : velocity of electron(11.19)V0: DC voltageAt low frequencies, an electron leaves the cathode and travels to the anodewithin a small fraction of the positive half-cycle of grid voltage, and hence thetransit time effect is negligible. However, at a high frequency, transit time islarge compared to the time period of the signal and so cannot be neglected. Atthese frequencies, even if the electron leaves the cathode during positive gridpotential, the grid potential may become negative or even go several cycles before


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11.3 Klystron Amplifiers

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the electron passes through it. In practice, the potential between the cathode andgrid may alternate 10100 times during this time. In the positive half-cycle, gridpotential attracts the electron beam and supplies energy to it, whereas in thenegative half-cycle, it repels the electron beam and extracts energy from it. Asa result, the electron beam oscillates back and forth in the region between thecathode and thegrid, and may even return to the cathode. The overall result is areduction of the operating frequency of the vacuum tube. Furthermore, since thetransit time is not negligible, the transconductance of the device at microwavefrequencies become complex with a relatively small magnitude. This, in turn,indicates a decrease in the output.This analysis indicates that to use a vacuum device at microwave frequencies,transit angle must be reduced, either by increasing the anode voltage or bydecreasing the inter-electrode spacing. However, the increase in anode voltagewill increase the power dissipation, whereas the decrease in inter-electrode spacingwill increase the inter-electrode capacitance. The increase in inter-electrodecapacitance can be reduced by reducing the area of the electrodes, but this willreduce anode dissipation and hence the output power.In microwave tubes, the transit angle can be reduced by first accelerating anelectron beam with a very high DC voltage and then velocity modulating it. Infact, this is the basic principle of operation of a klystron tube, as discussed in thefollowing section.

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A klystron is a widely used microwave amplifier that works on the principleof velocity and current modulation. It generally consists of an electron gunassembly, a buncher cavity, a catcher cavity, and a collector, as shown inFig.11.4. The microwave field to be amplified is fed into the buncher cavity andthe amplified output is obtained from the catcher cavity. If electrons from theelectron gun assembly pass the first cavity (or buncher cavity) at a zero cavitygap voltage (or microwave signal voltage), they remain unaffected and leave thecavity without any change in velocity. However, if they pass the cavity duringRF output

RF inputAnode

Collector

Cathode

V(t1)

V0

+

Vg

+0

d

t0

t1

Drift space i(t1)

BunchedelectronbeamDistance scaleTime scale

L+dt2

L+2dt3

Fig. 11.4 Schematic diagram of two-cavity klystron amplifier


7/17/2014 11:27:30 AM


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the positive half-cycle, they face an accelerating field and leave the cavity witha higher velocity. In contrast, if they pass the cavity during the negative halfcycle, they face a decelerating field and leave the cavity with a lower velocity.This process is known as velocity modulation, and results in the bunching ofelectrons. Bunched electrons then move towards the second cavity or the catchercavity. Owing to bunching, the density of electrons in the catcher cavity variescyclically with time.Since the electron beam has been modulated by an RF field, it contains an ACcomponent and is current modulated. If maximum bunching occurs approximatelymidway between the second cavity grids during its retarding phase, then thekinetic energy of the electron beam is transferred to the field of the second cavity,resulting in signal amplification. After giving up their energy in the catcher cavity,the electron beam emerges from it with a reduced velocity and is finally collectedat the collector.A reflex klystron can provide up to 500kW CW power and 30MW pulsedpower at 10GHz, with a power gain of about 30dB and efficiency of about 40%.In the following section, we will perform a quantitative analysis of a two-cavityklystron under the following assumptions:

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1. The electron beam has a uniform cross-sectional density.2. Space charge and debunching effect are negligible.3. Magnitude of the input microwave signal is much smaller than the DCaccelerating voltage.4. Electrons leave the cathode with a zero initial velocity.5. The transit time of electrons across the cavity gap is very small in comparisonwith the time period of the input RF signal.6. The cathode, anode, cavity grids, and collector of the tube are all parallel.7. The cavity grids of the tube do not intercept any passing electron.8. RF fields are totally confined within the cavities and are zero in the drift space.

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The velocity of an electron, entering the buncher cavity under the influence ofa high DC voltage, V0, can be expressed as follows:v0 =

2eV0= 0.593 V0 106m

(11.20)

When the buncher cavity has been excited by an input microwave signal, the gapvoltage can be written as follows: Vs = V1 sin (t )(11.21)where V1 (V1 V0 ) is the amplitude of the input signal. Since V1 V0 , the averagetransit time through the buncher gap distance d is given by the followingequation: d v0 = t1 t0 (11.22)Therefore, the average gap transit angle is as follows: g = d v0 = (t1 t0 )

(11.23)

Now, the average microwave voltage in the buncher gap can be expressed asfollows:


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504 M I C R O W A V E E N G I N E E R I N Gt

1V1Vs = V1 sin (t ) dt = 1 cos (t1 ) cos (t0 ) t0

cos (t ) cos t + d 00v0 d sin sin ( g 2) 2v0 g d = V1sin t0 + Vs = V1sin t0 +2 2v0 ( g 2) d 2v 0Vs =

or

or

V1

(11.24)

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gap and can be denoted by the following equation:

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During the derivation of Eq. (11.24), we have used the trigonometric relation A + B A B dcos ( A) cos ( B ) = 2 sin sin where, A = t0 and B = t0 + . 2 2 v0sin ( g 2)The termis known as the beam coupling coefficient of the input cavity( g 2)

hted

i = sin ( g 2) ( g 2)

(11.25)

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g g 2eV0 iV12e sin t0 + 1 +V0 + iV1 sin t0 + =2 V0m 2 m

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v (t1 ) =

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A simple program can be written to plot the beam coupling coefficient as a functionof gap transit angle (Fig. 11.5).At the buncher cavity, the input signal is superimposed on the DC voltage V0,and hence the velocity of the electron, exiting from the buncher cavity, can bewritten as follows:

(11.26)

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The term iV1 V0 is called the depth of velocity modulation. Since i tb > ta,L can be estimated such that it satisfies the following condition:L = v0 (td tb ) = vmin (td ta ) = vmax (td tc )

(11.30)

Under such circumstances, all the electrons leaving the cavity between ta and tcwill arrive at a distance L from the buncher cavity at the same time td and forma bunch. Equation (11.30) can also be written as follows:


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506 M I C R O W A V E E N G I N E E R I N Gz

Distance

L

BunchingcentreSlower

Vs = V1 sin (t)

FasterSameta

tb

2

tc

t

td

2

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Fig. 11.6 Bunching of electrons

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0Bunchergrid

(11.31)

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L = v0 (td tb ) = vmin td tb + = vmax td tb 2 2

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Note Figure 11.6 is basically the graph of the electron paths in a two-cavity klystron tube,and it describes the process of electron bunching in a velocity-modulated tube. Such adiagram is also known as an Applegate diagram.

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Now from Eq. (11.28) or (11.29), we can write the following expressions:(11.32)

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V vmin = v0 1 i 1 2V0

(11.33)

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and

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V vmax = v0 1 + i 1 2V0

Substituting Eqs (11.32) and (11.33) in Eq. (11.31), we can write the followingrelations:

or

V L = v0 1 + i 1 td tb 2V0 2 V V L = v0 (td tb ) + v0+ v0 i 1 (td tb ) v0 i 122V02V0 2

(11.34)

V L = v0 1 i 1 td tb +2V0 2 or

V V L = v0 (td tb ) + v0 v0 i 1 (td tb ) v0 i 1 22V02V0 2


(11.35)

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Comparing Eq. (11.31) with Eq. (11.34) or (11.35), we get the following expression:v0

VV v0 i 1 (td tb ) v0 i 1=022V02V0 2

or

V iV1=0(td tb ) i 12 2V02V0 2

or

iV1V (td tb ) = i 12V02 2V0 2

or

td tb =

2V0V0V0=as 2iV1 2 iV1 2 iV1

(11.36)

Substituting Eq. (11.36) back in Eq. (11.31), the following relation is obtained:

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V0iV1

(11.37)

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L = v0 (td tb ) = v0

hted

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Note It may be noted that the distance given in Eq. (11.37) is not unique for maximumbunching.

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Within the drift region, electrons move with a velocity v (t1 ). Now, if the catchercavity is placed at a distance L from the buncher cavity, then the transit time oftheelectron can be expressed as follows:

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1 g LL iV1T = t2 t1 == 1 +sin t1 2V02 v (t1 ) v0

w

(11.38)(11.39)

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where T0 = L v0

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or

1 V g i 1T = T0 1 +sin t1 2V02

is the DC transit time. Since iV1 V0, performing binomial expansion of thesecond bracketed term, we get the following equation: V g T = T0 1 i 1 sin t1 (11.40)2V02 Therefore,g g VT = t2 t1 = T0 T0 i 1 sin t1 = 0 X sin t1 (11.41)2V02 2 Lwhere 0 = T0 == 2N (11.42)v0is the DC transit angle between the cavities, N is the number of electron transitV(11.43)cycles in drift space, and X i 1 02V0is the bunching parameter of a klystron.


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The instant at which an electron arrives at the catcher cavity is as follows:

or

V g t2 = t1 + T0 1 i 1 sin t1 2V02 V g t2 = t0 + + T0 1 i 1 sin t0 + 2V02

or

V g t2 = t0 + + T0 1 i 1 sin t0 + g 2V02

or

V g t2 = t0 + + T0 1 i 1 sin t0 + 2V02

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(11.44)

Alternatively, from Eq. (11.41), we can write the following expression:

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g t2 t1 = t2 t0 = 0 X sin t1 2 g t2 0 = t0 X sin t0 + 2

hted

or

(11.45)

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Substituting Eq. (11.23) into Eq. (11.45), we get the following relation:

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g g g t2 0 + = t0 + X sin t0 + 2 2 2

(11.46)

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g t2 0 g = t0 X sin t0 + g 2

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g where the term t2 0 + represents a catcher cavity arrival angle and the2 gterm t0 + a buncher cavity departure angle. A simple program can be written2 to plot the catcher cavity arrival angle as a function of the buncher cavity departureangle (Fig. 11.7).If we assume that dQ0 amount of charge passes through the buncher cavity gapat a time interval dt0, then we can write the following equation:dQ0 = I 0 dt0

(11.47)

where I 0 is the DC current. According to the law of conservation of charges, thesame amount of charge will also pass the catcher cavity at a later time interval dt2.Hence, we can write the following equation: I 0 dt0 = i2 dt2 (11.48)The absolute value of time ratio is necessary in Eq. (11.48), because a negativevalue would indicate a negative resistance.


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MATLAB Program

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% Plot of catcher cavity arrival angle as a function of buncher cavitydeparture angleclear all;clf;for X = 0:0.5:1.5n = 0;for th = -pi:pi/180:pin = n + 1;th_dep(n) = th;th_arr(n) = th - X * sin(th);endaxis([-pi, pi, -pi, pi]);xlabel('Buncher cavity departure angle');ylabel('Catcher cavity arrival angle');plot(th_dep, th_arr, 'k');hold on;end

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X = 0.0X = 0.5

hted

2

ig

1

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123

2

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3

X = 1.0

X = 1.5

-C

Catcher cavity arrival angle

3

101Buncher cavity departure angle

2

3

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Fig. 11.7 Plots of catcher cavity arrival angle vsbuncher cavity departure angle

Differentiating Eq. (11.46) with respect to t0, we get the following relation:

or

g dt2= X cos t0 + 2 dt0

g dt2 = dt0 1 X cos t0 + 2

(11.49)

Substituting Eq. (11.49) into Eq. (11.48), we get the following expression: g I 0 dt0 = i2 (t0 ) dt0 1 X cos t0 + 2


7/17/2014 11:28:06 AM


or

g i2 (t0 ) = I 0 1 X cos t0 + 2

(11.50)

In terms of t2, we get the following equation:I0I0=1 X cos t + g 1 X cos t T + g 0 0 22 2 I0i2 (t2 ) = (11.51) g 1 X cos t2 0 2 i2 (t2 ) =

or

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where we have used Eqs (11.23) and (11.42).A simple program can be written to plot the normalized beam current as afunction of catcher cavity arrival angle (see Fig. 11.8).Since bunches are formed at periodic intervals, the beam current in the catchercavity is also a periodic waveform of period 2 . Therefore, the current i2 can beexpanded in Fourier series as follows:

hted

i2 = a0 + an cos (nt2 ) + bn sin (nt2 ) 1i2 d (t2 )2

(11.52)(11.53)

1i2 cos (nt2 ) d (t2 )

(11.54)

bn =

1i2 sin (nt2 ) d (t2 )

(11.55)

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an =

w

and

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where a0 =

ig

n=1

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Substituting Eqs (11.48) and (11.46) into Eqs (11.53)(11.55), we get thefollowing relations:II11a0 =idt=I 0 d (t0 ) = 0 [t0 ] = 0 ( + ) = I 0()222 2 22(11.56)

an =

1i2 cos (nt2 ) d (t2 )

or

an =

g 1I 0 cos n 0 + g + t0 X sin t0 + d (t0 ) 2

or

an =

1 nt + n + n nX sin t + g d (t )Icos()00g000 2

(11.57)

and

bn =

1 nt + n + n nX sin t + g d (t )Isin()00g000 2

(11.58)


7/17/2014 11:28:13 AM


MATLAB Program

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%Plot of normalized beam current of klystron as a function of catchercavity arrival angleclear all;clf;for X = 0:0.5:1.5n = 0;for th = -pi:pi/180:pin = n + 1;theta(n) = th;i2bI0(n) = 1/(1 - (X * cos(th)));endaxis([-pi, pi, 0, 20])xlabel('Cather cavity arrival angle');ylabel('Normalized beam current')plot(theta, abs(i2bI0), 'k');hold onend

hted

20

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16

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1084

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2

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Normalized beam current

18

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0 3

2

X = 1.5

X = 1.0X = 0.5

101Catcher cavity arrival angle

X = 0.02

3

Fig. 11.8 Plot of beam current as function of catcher cavity arrival angle

To solve the integration given in Eqs (11.57) and (11.58), we need to split thecosines and sines of a sine function in the form of cos ( A B ) and sin ( A B ). Thiswill result in the following equations: g 1an = I 0 cos (nt0 + n g + n 0 ) cos nX sin t0 + 2 g + sin (nt0 + n g + n 0 ) sin nX sin t0 + d (t0 )2


(11.59)

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and

bn =

1t + g I0 sin ( n t0 + n g + n 0 ) cos nX sin 0 2 g cos (nt0 + n g + n 0 ) sin nX sin t0 + d (t0 )2

(11.60)

Cosines and sines of a sine function of Eqs (11.59) and (11.60) can be expressedas follows:

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g g sin nX sin t0 + = 2 J1 (nX ) cos t0 + 2 2 g g +2 J 3 (nX ) cos 3t0 + + 2 J 5 (nX ) cos 5 t0 + + 2 2 (11.62)

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g g cos nX sin t0 + = 2 J 0 (nX ) + 2 J 2 (nX ) cos 2 t0 + 2 2 g g +2 J 4 (nX ) cos 4 t0 + + 2 J 6 (nX ) cos 6 t0 + + 2 2 (11.61)

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Substituting Eqs (11.61) and (11.62) in Eqs (11.59) and (11.60), the integrationcan be carried out. The final expression of the Fourier coefficients can be writtenas follows:bn = 2 I 0 J n (nX ) sin (n 0 + n g )

(11.64)

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(11.63)

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and

an = 2 I 0 J n (nX ) cos (n 0 + n g )

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where J n (nX ) is the nth-order Bessel function of the first kind.Substituting Eqs (11.56), (11.63), and (11.64) in Eq. (11.52), we get the followingrelation:

{

i2 = I 0 + 2 I 0 J n (nX ) cos (n 0 + n g ) cos (nt2 )n=1

}

+ sin (n 0 + n g ) sin (nt2 ) or

i2 = I 0 + 2 I 0 J n (nX ) cos (n 0 + n g nt2 )

(11.65)

n=1

Substituting Eq. (11.23) in Eq. (11.65), we get the following expression:

i2 = I 0 + 2 I 0 J n (nX ) cos (nT0 + n nt2 )or

n=1

i2 = I 0 + 2 I 0 J n (nX ) cos {n (t2 T0 )}

(11.66)

n=1


7/17/2014 11:28:20 AM


Equation (11.66) reveals that the magnitude of the fundamental component ofbeam current at the catcher cavity can be expressed as I f = 2 I 0 J1 ( X )(11.67)This fundamental component has its maximum amplitude at X = 1.841 (11.68)VSubstituting Eq. (11.43) in Eq. (11.68), we get i 1 0 = 1.841(11.69)2V0Further substituting Eq. (11.42) in Eq. (11.69), we get the following relation:iV1 Loptimum= 1.8412V0v0or

Loptimum = 1.841

2V0 v0 3.682v0V0=iV1iV1

(11.70)

Comparing Eqs (11.37) and (11.70), we can find that the following relation holds:Loptimum

v0V0 iV1= 0.85iV1 3.682v0V0 3.682

l

=

eria

L

(11.71)

op

yr

ig

hted

M

at

That is, L is approximately 15% less than Loptimum. This is partly due to theapproximations we have made for deriving Eq. (11.37) and partly due to the factthat the maximum fundamental component of current will not coincide with themaximum electron density along the beam because harmonic components alsoexist in the beam.The current induced at the walls of the catcher cavity by the passing electronbunch is proportional to the input microwave voltage V1. The fundamentalcomponent of induced microwave current in the catcher cavity is given by thefollowing relation:i2,ind = o i2 = 2 0 I 0 J1 ( X ) cos { (t2 T0 )}

-C

(11.72)

ev

ie

w

where o is the beam coupling coefficient of the catcher gap. If the buncher andcatcher cavities are identical, then o = i , and the magnitude of the fundamentalcomponent of current induced in the catcher cavity can be expressed as follows:I 2,ind = o I 2 = 2 o I 0 J1 ( X )

Pr

(11.73)

The output equivalent circuit of a klystron is shown in Fig. 11.9, where Rsho isthe wall resistance of the catcher cavity, RB is the beam loading resistance, RL is theexternal load, and Rsh is the effective shunt resistance. Therefore, the output powerdelivered to the load and the catcher cavity is given by the following equation:2

Pout = ( o I 2 ) Rsh 2 = o I 2V2 2

(11.74)

0I2

0I2

Rsho

RB

RL

Rsh

V2

Fig. 11.9 Output equivalent circuit of klystron


7/17/2014 11:28:28 AM


where V2 is the fundamental component of catcher gap voltage.Therefore, the efficiency of the klystron amplifier can be expressed as follows: IVP = out = o 2 2 (11.75)Pin2 I 0V0If coupling is perfect, then 0 = 1, V2 = V0, and the maximum beam currentapproaches the following value:I 2,max = 2 I 0 J1 (1.841) = 2 0.582 I 0 = 1.164 I 0(11.76)

ig

hted

M

at

eria

l

Thus, the maximum electronic efficiency is expressed as follows: IV1.164 I 0V0max = o 2 2 = 0.58(11.77)2 I 0V02 I 0V0Therefore, the maximum electronic efficiency of a klystron is about 58%. Inpractice, the efficiency of a klystron is about 1530%.The equivalent mutual conductance of a klystron amplifier is as follows:i2,ind 2 o I 0 J1 ( X )Gm =(11.78)V1V1From Eq. (11.43), we get the following expression:2V XV1 = 0 (11.79)i 0Substituting Eq. (11.79) in Eq. (11.78), we get the following relation:

where G0 = I 0 V0

op

yr

2 o i J1 ( X ) 0 I 0 J ( X ) 0= o i 1G02XV0X

-C

Gm =

(11.80)(11.81)

= o2 0

ev

Gm

ie

w

is the DC beam conductance.If we assume that o = i , then Eq. (11.80) modifies as follows:G0

J1 ( X )X

(11.82)

Pr

A simple program can be written to plot the normalized transconductance as afunction of bunching parameter (Fig. 11.10).For the maximum output X = 1.841 and Eq. (11.82) can be written as follows:Gm G0 = o2 J1 (1.841) 0 1.841 = 0.582 o2 0 1.841 = 0.316 o2 0 (11.83)

The voltage gain of a klystron is as follows:Av = V2 V1 = o I 2 Rsh V1

(11.84)

Substituting Eqs (11.73) and (11.79) in Eq. (11.84) and assuming that 0 = i ,weget the following relation:

or

Av =

o I 2 Rsh 2 o I 0 J1 ( X ) Rsh o 0=V12V0 X

Av =

o2 J1 ( X ) Rsh 0 I 0J (X )= o2 0 G0 Rsh 1XV0X


(11.85)

7/17/2014 11:28:35 AM


MATLAB Program

M

020 = 30

hted

020 = 25

ig

020 = 20

10

0

0.5

1

ev

0

w

-C

020 = 5

op

020 = 10

5

yr

020 = 15

ie

Normalized transconductance

15

at

eria

l

%Program to plot the normalized transconductance of klystron as afunction of bunching parameterclear all;clf;for beta0sqth0 = 5:5:30n = 0;for x = 0:0.01:4n = n + 1;GnbG0(n) = beta0sqth0 * besselj(1,x)/x;X(n) = x;endaxis ([0, 4, 0, 15]);xlabel('Bunching Parameter');ylabel('Normalized Transconductance');plot(X, GnbG0, 'k')hold onend

1.5

2

2.5

Bunching parameter

3

3.5

4

Pr

Fig. 11.10 Plot of normalized transconductance vsbunching parameter

Substituting Eq. (11.82) in Eq. (11.85), we get the following expression:Av =

GmG0 Rsh = Gm Rsh G0

(11.86)

g P V 2P0V12 1 2 1i i cos = 0 12 F ( g ) 2 2 22V0 22V0

(11.87)

Power required to produce bunching can be expressed as follows:PB =

where P0 = V02 G0

(11.88)

is the DC power required to produce the electron beam andF ( g ) =

g 1 2 1i i cos 2 22


(11.89)

7/17/2014 11:28:38 AM


The power required to produce bunching can also be expressed as follows:PB = V12 GB 2

(11.90)where GB is the bunching conductance.Substituting Eqs (11.88) and (11.90) in Eq. (11.87), the following relation is obtained:

or

V12V2GB = V02 G0 F ( g ) 1 222V0GB G0 = F ( g )

(11.91)

eria

l

A simple MATLAB program can be written to plot GB G0 or F ( g ) as a functionof g (Fig. 11.11). The figure shows that when g = 3.5, the equivalent bunchingconductance is about one-fifth of the electron beam conductance, or alternativelybunching resistance is about five times the electron beam resistance.

at

MATLAB Program

Normalized electronic conductance

Pr

ev

ie

w

-C

op

yr

ig

hted

M

%Program for calculation of GB/G0clear all;clf;n = 0;for th = 0:pi/360:4n = n + 1;Theta(n) = th;if th == 0;beta = 1;elsebeta = sin(th/2)/(th/2);endF(n) = (beta^2 - (beta * cos(th/2)))/2;endplot (Theta, F)xlabel ('Theta (G) in Radians');ylabel('Normalized Electronic Conductance');0.250.20.150.10.0500

0.5

1

1.522.5g in radians

3

3.5

4

Fig. 11.11 Plot of normalized electronic conductance asfunction of g


7/17/2014 11:28:42 AM


Power delivered by the electron beam to the catcher cavity can be written asfollows:

or

V22V2V2V2= 2 + 2 + 2 2 Rsh2 Rsho 2 RB 2 RL

(11.92)

1111=++ RshRsho RB RL

(11.93)

The loaded Q-factor of the catcher cavity circuit, at the resonance frequency, canbe expressed as follows:1111=++QL Q0 QB Qext

(11.94)

M

at

eria

l

where QL is the loaded quality factor of the whole catcher cavity, Q0 is the qualityfactor of the catcher cavity walls, QB is the quality factor of the beam loading, andQext the quality factor of the external load.

hted

Note A klystron may have the following types of cavities: coaxial, radial, tunable, toroidal,and butterfly cavities.

Pr

ev

ie

w

-C

op

yr

ig

An extended interaction in a klystron can be obtained by coupling two or moreadjacent klystron cavities. A five-section extended interaction cavity is shown inFig. 11.12.The average energy of electrons leaving the buncher cavity over a completecycle can be almost equal to that of the electrons entering the cavity during thatperiod, provided that the buncher cavity gap is negligible. However, when thisgap is not negligible, the average energy of the electrons leaving the bunchercavity over a cycle is larger than that of the electrons entering the cavity duringthat period. The difference in the average energy between the input and outputelectrons is a result of an interaction between the electrons and the RF field in thebuncher cavity. This effect is known as beam loading.Since two-cavity klystrons are associated with a considerable amount ofnoise, they are not generally used in receiver circuits. However, they find wideapplications in troposcatter transmitters, UHF TV transmitter power amplifiers,and ground stations for satellite communication. Two-cavity klystrons can work inthe frequency range starting from C-band up to 60GHz and can provide an outputpower of 100250kW, with a possible power gain of 3060dB over a bandwidthof 1060MHz. The efficiency of such tubes is about 3040%.

Fig. 11.12 Coupling of cavities in klystron for extended interaction


7/17/2014 11:28:46 AM


EXAMPLE 11.1 A typical two-cavity klystron amplifier has the following parameters:V0 = 1 kV, R0 = 50 k, I 0 = 20 mA, f = 4 GHz, gap spacing = 0.75mm, spacing betweenthe two cavities = 5cm, and Rsh = 25 k. Find the (a) input gap voltage to give the maximum voltage V2; (b) voltage gain, neglecting beam loading in the output cavity; (c)efficiency of the amplifier, neglecting beam loading; and (d) beam loading conductance. Itisprovided that at X =1.841, J1 ( X ) = 0.582.Solution Given: V 0 = 1 kV, R0 = 50 k, I 0 = 20 mA, f = 4 GHz, Rsh = 25 k,d = 0.75 mm, and L = 5 cm

(a)For maximum V2 , J1 ( X ) must be maximum. Therefore, J1 ( X ) = 0.582 atX =1.841. The velocity of the electron just leaving the cathode is as follows:v0 = 0.593106 V0 = 0.593106 1000 = 1.8752 107 m/s

d 2 4 109 0.75103== 1.0052 radv01.8752 107

eria

g =

l

The gap transit angle is as follows:

sin ( g 2)

( g 2)

=

sin (1.0052 2)= 0.9584(1.0052 2)

hted

i = 0 =

M

at

The beam coupling coefficient is given by the following relation:

yr

L 2 4 109 5102== 67.0135 radv01.8752 107

op

0 = T0 =

ig

The DC transit angle between the cavities is expressed as follows:

w

(b) The voltage gain is expressed as follows:

ev

2V0 X2 1000 1.841== 57.329 Vi 00.9584 67.0135

ie

V1,max =

-C

The maximum input voltage is then given by the following equation:

2

(0.9584) 67.0135 0.582 25103 02 0 J1 ( X )Rsh == 9.7296R0X50 103 1.841

Pr

Av =

(c) Now,I 2 = 2 I 0 J1 ( X ) = 2 20 103 0.582 = 23.28103 AV2 = 0 I 2 Rsh = 0.9584 23.28103 25103 = 557.7888 V

and =(d) GB =

or

GB =

0 I 2V2 0.9584 23.28103 557.7888== 0.3111 = 31.11%2 I 0V02 20 103 1000

g G0 2 0 0 cos 2 2 1.0052 127(0.99584) 0.9584 cos = 7.865310 3 2 2 50 10


7/17/2014 11:29:00 AM


The beam loading resistance is expressed as follows:RB =

11== 1.2714 106 GB 7.8653107

Practice Problem11.1 A typical two-cavity klystron amplifier has the following parameters: V0 = 1.2 kV,R0 = 48 k, I 0 = 25 mA, f = 3 GHz, gap spacing = 1mm, spacing between the twocavities = 4.5 cm, and Rsh = 20 k. Find the (a) input gap voltage to give maximumvoltage V2; (b) voltage gain, neglecting beam loading in the output cavity; (c) efficiencyof the amplifier, neglecting beam loading; and (d) beam loading conductance. It isgiven that at X =1.841, J1 ( X ) = 0.582. 110.8489 V, 5.0682, 26.30%, 7.3701107

eria

l

11.4 Multi-cavity Klystrons

ev

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-C

op

yr

ig

hted

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A multi-cavity klystron, shown in Fig. 11.13, is designed by cascading a numberof cavities to achieve an enhanced gain. These intermediate cavities act asbunchers and are placed at a distance from the previous cavity for which thebunching parameter X equals to 1.841. Since velocity modulation increases as thebeam progresses through various cavities, the requirement of X = 1.841 results ina decrease in the spacing between consecutive cavities. To keep the inter-cavitydistance constant, and maintain X = 1.841, beam voltage (V0 ) in consecutivecavities must be increased.During the analysis of a two-cavity klystron, the space charge effect wasassumed to be negligible because, in low-power operations, electron density ofthe beam is small. However, during high-power operations, electron density ofthe beam is high and therefore repulsion between electrons cannot be neglected.The space charge force within an electron bunch depends on the size and shapeof the electron beam. For example, for an infinite electron beam, the electric field

Pr

RFoutput

RFinputIntermediatecavities

Collector

Heater Electronbeam

InputCathodeAnode cavity

Drifttube

Outputcavity

Cooler

Magnet coils

Fig. 11.13 Schematic diagram of multi-cavity klystron amplifier


7/17/2014 11:29:07 AM


acts only in the axial direction, whereas for a finite beam, the electric field is radialas well as axial. Thus, the axial component for a finite beam is less than that foran infinite beam. Due to a reduced axial space charge force, plasma frequency isreduced and plasma wavelength is increased.If we assume that both charge density and velocity perturbation variations aresimple sinusoidal in space and time, then we can write the following equations:and

= B cos ( e z t ) cos (q t + )

(11.95)

v = C sin ( e z t ) sin (q t + )

(11.96)

B: constant of charge density perturbationC: constant of velocity perturbation e = v0 : DC phase constant of electron beamq = R p: perturbation frequency or reduced plasma frequency

(11.97)(11.98)

eria

l

where

hted

M

at

R: space charge reduction factor (varies from 0 to 1)e 0p =: plasma frequency and a function of electron beam densitym 0(11.99) : phase angle of oscillation

and

op

vtot = v0 + v

-C

tot = 0 +

yr

ig

The total charge density, electron velocity, and beam current density can thereforebe expressed as follows:

J tot = J 0 + J

(11.100)(11.101)(11.102)

Pr

ev

ie

w

where 0 is the DC electron charge density, v0 is the DC electron velocity, J 0 is theDC beam current density, and J is the instantaneous RF beam current perturbation.At any point in the beam, the instantaneous convection current density can beexpressed as follows:J tot = tot vtot = (0 + )(v0 + v )

or

J tot = 0 v0 0 v + v0 + v = J 0 + J

where J = v0 0 v and

J 0 = 0 v0

(11.103)(11.104)(11.105)

Here, v is very small and can be neglected.Differentiating Eq. (11.104) with respect to z, we get the following relation:J(11.106)= ( v0 0 v )z zSubstituting Eqs (11.95) and (11.96) in Eq. (11.106), we get the following relation:J=v0 B cos ( e z t ) cos (q t + ) + 0 C sin ( e z t ) sin (q t + )z z

{


}

7/17/2014 11:29:15 AM


or

J= Bv0 e sin ( e z t ) cos (q t + ) + C 0 e cos ( e z t ) sin (q t + )z(11.107)

Further substituting Eq. (11.97) in Eq. (11.107), we get the following expression:J= B sin ( e z t ) cos (q t + ) + C 0 e cos ( e z t ) sin (q t + )z(11.108)

Differentiating Eq. (11.95) with respect to t, the following equation is obtained:

= B sin ( e z t ) cos (q t + ) + Bq cos ( e z t ) sin (q t + )t(11.109)

hted

M

at

eria

l

Now, from continuity condition:

(11.110) J = tTherefore, equating the RHS of Eqs (11.108) and (11.109), we get the followingrelation:C 0 e = Bq

(11.111)

yr

ig

Substituting Eqs (11.95) and (11.96) in Eq. (11.104), we get the followingexpression:

-C

op

J = v0 B cos ( e z t ) cos (q t + ) + 0 C sin ( e z t ) sin (q t + ) (11.112)

Pr

ev

ie

w

Further, consecutive substitution of Eq. (11.111) and Eq. (11.97) in Eq. (11.112)gives the following expression:BqJ = v0 B cos ( e z t ) cos (q t + ) +sin ( e z t ) sin (q t + )eqJ = v0 B cos ( e z t ) cos (q t + ) +v0 B sin ( e z t ) sin (q t + )or(11.113)Since q 1, Eq. (11.113) can be approximated as follows:J = v0 B cos ( e z t ) cos (q t + )

It can be shown that Eq. (11.114) can be written as follows:1 J 0J =iV1 sin ( q z ) cos ( e z t )2 v0qwhere q = q v0 is the plasma phase constant.The modulated velocity can be expressed as follows:Vv = v0 i 1 cos ( q z ) sin ( e z t )2V0


(11.114)

(11.115)(11.116)

(11.117)

7/17/2014 11:29:22 AM


Electrons leaving the input gap of a klystron amplifier have a velocity given bythe following relation: Vv (t1 ) = v0 1 + i 1 sin ( )2V0

(11.118)

where V1: magnitude of input signal = d v0 = t1 t0

(11.119)

D: dap distance

Since electrons under the influence of space charge forces exhibit a simpleharmonic motion, velocity at a later time t can be expressed as follows:(11.120)

eria

l

Vvtot = v0 1 + i 1 sin ( ) cos { p (t )} 2V0

ig

hted

M

at

If the two cavities of a two-cavity klystron are identical and the second cavityis placed at a point where RF current modulation is maximum, then the magnitudeof the RF convection current at the output cavity of that klystron can be writtenas follows:1 I 0i2 =(11.121)i V1 2 V0q

1 I 01 I 0 2 o i V1 = o V1 2 V0q2 V0q

-C

I 2 = o i2 =

op

yr

Therefore, magnitudes of induced current and voltage in the output cavity can beexpressed as follows:

1 I 0 2 o V1 Rshl 2 V0q

w

ie

V2 = I 2 Rshl =

(11.123)

ev

and

(11.122)

Pr

Thus, the output power delivered to the load in a two-cavity klystron amplifier canbe expressed as follows:21 I 22Pout = I 2 Rshl = 0 o4 V1 Rshl 4 V0q

(11.124)

Using Eq. (11.124), the power gain of the two-cavity klystron can be expressedas follows:222PoutPout1 I 0 4 V1 Rsh Rshl 1 I 0 4== = o o Rsh Rshl (11.125)2Pin4 V0q 4 V0q V12 RshV1

The efficiency of a two-cavity klystron can be expressed as follows:2224PoutPout1 I 0 o V11 I 0 V1 4Rshl = === o Rshl PinI 0V0 4 V0q I 0V04 V0 V0q


(11.126)

7/17/2014 11:29:27 AM


This analysis assumes two cavities. Now we will perform a simple analysis ofa four-cavity klystron under the following assumptions:1. All the cavities are identical and have the same unloaded Q and couplingcoefficient.2. Two immediate cavities are not externally loaded.3. Input and output cavities are matched.

at

eria

l

If V1 is the magnitude of the input cavity voltage, then the magnitude of theRF convection current density injected into the first immediate cavity gap canbe expressed by Eq. (11.115), and the induced current and voltage in the firstintermediate cavity by Eqs (11.122) and (11.123), respectively. The gap voltageof the first intermediate cavity produces a velocity modulation on the beamin the second intermediate cavity. The RF convection current in this secondintermediate cavity can be expressed as follows:1 I 01 I 0i3 =(11.127)i V2 = o V2 2 V0q2 V0q

hted

M

Substituting Eq. (11.123) in Eq. (11.127) after replacing Rshl by Rsh, we get thefollowing relation:(11.128)

ig

21 I i3 = 0 o3 V1 Rsh 4 V0q

yr

The voltage of the second intermediate cavity will be as follows:

-C

op

21 I 0 42V3 = o i3 Rsh = o V1 Rsh 4 V0q

(11.129)

Pr

ev

ie

w

This voltage produces further velocity modulation and is converted to an RFconvection current at the final output of the four-cavity klystron. The outputconvection current density can be expressed as follows:1 I 01 I 0i4 =(11.130)i V3 = o V3 2 V0q2 V0qSubstituting Eq. (11.129) into Eq. (11.130), the following equation is obtained:3

1 I 2i4 = 0 o5 V1 Rsh8 V0q

(11.131)

Therefore,31 I 0 62I 4 = o i4 = o V1 Rsh8 V0q

(11.132)

The output voltage can therefore be expressed as follows:V4 = I 4 Rshl


31 I 0 62= o V1 Rsh Rshl8 V0q

(11.133)

7/17/2014 11:29:33 AM


The output power is, therefore,61 I 2 42Pout = I 4 Rshl = 0 o12 V1 RshRshl 64 V0q

(11.134)

Multi-cavity klystrons are used as medium- or high-power amplifiers for bothCW and pulsed applications. In general, such tubes can operate in the range from250MHz to 60GHz, with a typical CW output power of 100kW in the VHF rangeand 250kW in the X-band. The pulsed output power may be as high as 25MW.In addition, multi-cavity klystrons can provide a power gain of 30dB at UHF upto 60dB at X-band, in the bandwidth range of 860MHz. The mechanical tuningrange is about 300600MHz.Multi-cavity klystrons are widely used in UHF television transmitters, troposphere scatter transmitters, and ground stations for satellite communication.

hted

M

at

eria

l

EXAMPLE 11.2 A typical four-cavity klystron has the following parameters: beamvoltage = 15kV, beam current = 1.5A, operating frequency = 10GHz, signal voltage =10 V (rms), gap distance = 1cm, input and output beam coupling coefficient = 1, DCelectron charge density = 106 C /m 2 , RF charge density = 108 C /m 2, and velocityperturbation = 105 m /s. Calculate (a) DC electron velocity, (b) DC phase constant,(c) plasma frequency, (d) reduced plasma frequency for R = 0.5, (e) DC beam currentdensity, (f) instantaneous beam current density, (g) transit time across the input gap, and(h) electron velocity leaving the input gap.

ig

Solution Given:

-C

op

yr

V0 = 15 kV, I 0 =1.5 A, f =10 GHz, V1 =10 V (rms), d =1 cm, 0 = 106 C /m 2, = 108 C /m 2, v =105 m /s, and R = 0.5(a) DC electron velocity is expressed as follows:v0 = 0.593106 15103 = 0.7263108 m/s(b) DC phase constant is given as follows: 2 10 109e = == 8.651102 rad/mv0 0.7263108

(c) Plasma frequency is given by the following relation:

Pr

ev

ie

w

p =

e 01.759 1011 106== 1.4095108 rad/sm 08.8542 1012

(d) Reduced plasma frequency at R = 0.5 is as follows:

q = R p = 0.51.4095108 = 0.7047 108 rad/s

(e) DC beam current density is expressed as follows:J 0 = 0v0 = 106 0.7268108 = 72.68 A/m 2

(f) Instantaneous beam current density is as follows:J = v0 0v = 108 0.7268108 106 105 = 0.6268 A/m 2

(g) Transit time across the input gap is expressed as follows:

=

d102== 0.1376 109 s = 0.1376 nsv0 0.7268108


7/17/2014 11:29:42 AM


(h) Electron velocity leaving the input gap is given by the following relation:

oror

Vv (t1 ) = v0 1 + i 1 sin ( )2V010sin (2 10 109 0.1376 109 )v (t1 ) = 0.7263108 1 +32 1510

v (t1 ) = 0.727 108 m/s

Practice Problem

eria

l

11.2 A typical four-cavity klystron has the following parameters: beam voltage= 12kV,beam current = 1A, operating frequency = 8 GHz, signal voltage = 9V (rms), gapdistance = 0.9 cm, input and output beam coupling coefficient = 1, DC electron charge82density = 106 C /m 2, RF charge density = 10 C /m , and velocity perturbation= 105 m /s.Calculate (a) DC electron velocity, (b) DC phase constant, (c) plasma frequency, (d)reduced plasma frequency for R = 0.4, (e) DC beam current density, (f) instantaneousbeam current density, (g) transit time across the input gap, and (h)electron velocity 0.6496108 m/s, 7.7379102 rad/m,leaving the input gap.

hted

M

at

1.4095108 rad/s, 0.5638 108 rad/s, 64.96 A/m 2 , 0.5496 A/m 2 , 0.1385 ns, 0.6498108 m /s

ig

11.5 Two-cavity Klystron Oscillators

Pr

ev

ie

w

-C

op

yr

In the last two sections, we discussed two- and multi-cavity klystron amplifiers. Ifa fraction of the output of such an amplifier is fed back into the input with a unityfeedback loop gain and a phase shift of an integral multiple of 2 (i.e., in positivefeedback), then it will produce an oscillation, resulting in a klystron oscillator.Typical two-cavity klystron oscillators (Fig. 11.14) can produce an output powerin the range 210W, in the frequency band 550GHz. Two-cavity oscillatorswith an output power of 200W are also available, which find applications in CWDoppler radars, frequency modulators, and high-power microwave links, and aspump sources in parametric amplifiers.A two-cavity klystron has the advantage of producing a relatively high CWpower as compared to their size. However, it also suffers from some majordisadvantages like frequency tuning. Cavities used in a two-cavity klystron havehigh Q values with narrow bandwidths, and thus, individual tuning is awkward.RF outputCathode

Anode

Bunchercavity

Collector

CatchercavityFeedback path

Fig. 11.14 Two-cavity klystron oscillator


7/17/2014 11:29:47 AM


In addition, maintaining the positive feedback is also difficult. Therefore, twocavity klystrons are generally used for fixed-frequency applications.

11.6 Reflex Klystrons

Pr

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-C

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ig

hted

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eria

l

The problem of frequency tuning of a two-cavity klystron can be solved using aspecial klystron structure, called reflex klystron, shown in Fig. 11.15. Fig. 11.16shows a commercial packaged reflex klystron. A reflex klystron is a single-cavitystructure and therefore its tuning is very easy.In a reflex klystron, the DC voltage generates a wide band of RF noise inthe cavity. However, the RF noise frequency that corresponds to the resonancefrequency of a particular mode of the cavity, sustains and triggers the initial RFoscillation in the device. When the electron beam from the cathode enters the cavity,its velocity is modulated by this RF voltage or, more precisely, by the cavity gapvoltage. Electrons entering the cavity gap at the positive half-cycle are acceleratedand move with faster velocities, those entering the cavity gap at the negative halfcycle are decelerated and move with slower velocities, and those entering thecavity gap at the zero gap voltage move with an unchanged velocity. These resultin velocity modulation. Velocity-modulated electrons then proceed to the repellerterminal and experience a repelling force. As a result, their velocities start decreasingand finally become zero before reaching the repeller terminal. The zero-velocityelectrons still experience the repelling force and hence start moving, but towards thecavity. Therefore, after a certain time,RF outputelectrons return to the cavity and areAnodefinally collected by the cavity walls orCollectorCathodeother grounded metal parts of the tube.In practice, the total times takenby the individual electrons to getVElectron 0+velocity-modulated and return backbeamto the cavity are not same. This isVs =V1 sin (t)+ Vr because the electrons moving with at0 t1,t2Time scalefaster velocity, and hence with a highert0 d Distance scale L + dkinetic energy, penetrate more distancetowards the repeller than those movingFig. 11.15 Schematic diagram of reflexwith a slower velocity, as shown inklystron

Fig. 11.16 Commercial packaged reflex klystron


7/17/2014 11:29:48 AM


Fig. 11.17. Thus, electrons that arevelocity modulated at an instant awill take more time to return to thecavity than those velocity modulatedat instants b and c. In a reflex klystron,the repeller voltage is adjusted so thattVa b cdall the electrons, velocity modulatedt0Tin between a and c, come back to thecavity at the same instantd.tPractically, the best possible time33mode1 mode44for electrons to return to the cavity gapis when the voltage existing across thegap will apply maximum retardationFig. 11.17 Applegate diagram of reflexto them, that is, when the gap voltageklystronis positive maximum. This causes theelectrons to fall through the maximum negative voltage between the gap grids, thustransferring themaximum amount of energy to the gap. If the power delivered bythe bunched electrons to the cavity is greater than the power loss in the cavity, thenthe RF field inside the cavity will increase in amplitude and the oscillation will besustained. Figure 11.17 reveals that the electrons returning after 34 and 1 34 cyclesfrom b satisfy the condition for delivering maximum power to the RF field andhence for sustaining the oscillation. In practice, each of these numbers represents amode of klystron. Theoretically, an infinite number of modes can exist. The transittime of the electrons corresponding to these modes can be written as follows:

Distance from cavity gap

l

op

yr

ig

hted

M

at

eria

l

Cavity grid voltage

s

-C

1t0 = n T = NT where n is an integer4

(11.135)

Pr

ev

ie

w

The lowest-order mode (3/4) occurs for the maximum value of the repeller voltageand hence for the minimum transit time of electrons in the repeller space. On theother hand, higher-order modes occur at lower repeller voltages and hence forhigher transit times of electrons in the repeller space. Since for the lowest-ordermode, the repeller voltage and hence the acceleration of the bunched electrons ontheir return are maximum, the corresponding power output is also maximum.The variation of the output power for different modes can be explained asfollows. As the mode number increases, electron bunches are formed more slowly.As a result, electrons get more time for mutual repulsion and hence spread more.In addition, higher-order modes are associated with a long drift time, providingmore time for the mutual repulsion and causing further spreading of electrons.Spreading of electrons from the bunch, also known as debunching, thus becomesmore prominent as the mode number increases. Due to the debunching effect, thereturning electron bunch is less populated, as compared with lower-order modes,and thus less power is delivered to the output cavity.Plots of power output as a function of repeller voltage and operating frequencyas a function of repeller voltage, for different modes of reflex klystron, are shownin Fig. 11.18. The figure reveals that lower-order modes are better for working ona fixed frequency, whereas higher-order modes are better when frequency tuning isrequired. In practice, two types of tuning mechanism can be used: (a)mechanical


7/17/2014 11:29:50 AM

3N =343N =14

l

3N =24Repeller voltage

eria

Power output

Frequency


at

Fig. 11.18 Power output and frequency characteristics of reflex klystron

yr

f 2 f1MHz/V V2 V1

(11.136)

op

ETS =

ig

hted

M

tuning and (b) electronic tuning. In mechanical tuning, dimensions of the cavityare varied either by flexing a portion of the cavity wall or by changing the spacebetween the cavity grids. In electrical tuning, the repeller voltage is varied.Electronic tuning sensitivity (ETS) can be defined as follows:

Pr

ev

ie

w

-C

where f1 and f 2 are the frequencies in MHz at which mode power reduces to halfof its value at the top. Figure 11.18 reveals that ETS is higher for higher-ordermodes, though the output is small.The repeller can be overheated by the impact of high-velocity electrons anddamaged very quickly. Thus, when operating a reflex klystron, electrons should beprevented from reaching the repeller terminal. This can be done by connecting aresistor to the cathode of the klystron so that the repeller does not get more positivethan it. Alternatively, a protector diode can be used, with its anode connectedto the repeller and cathode connected to the cathode of the klystron. With thisarrangement, repeller voltage cannot become positive.Note When a reflex klystron is switched on, a high negative voltage is first applied tothe repeller and then a positive anode voltage is applied. This precaution prevents highenergy electrons from reaching the repeller terminal.

The analysis of a reflex klystron is similar to that of a two-cavity klystron tosome extent and is subjected to the following approximations:1. Cavity grids and repeller are plane and parallel, and also very large in extent.2. RF field is absent in the repeller space.3. Electrons are not intercepted by the cavity anode grid.


7/17/2014 11:29:52 AM


4. No debunching of electrons takes place in the repeller space.5. RF gap voltage is small compared to the beam voltage.Electrons entering the cavity gap from the cathode at time t0 can be assumed tohave a uniform velocity:v0 = 0.593 V0 106

(11.137)

When the electron leaves the cavity at z = d at time t1, it will have the followingvelocity: V g v (t1 ) = v0 1 + i 1 sin t1 (11.138)2V02 Vr + V0 + V1 sin (t )L

eria

E=

l

These velocity-modulated electrons will experience a net retarding electric field:(11.139)

t

yr

ig

hted

M

at

and return to the cavity at time t2. The associated force equation can be written asfollows:V + V0d2 z(11.140)m 2 = eE = e rLdtwhere we have assumed that the electric field is only along the z-direction and(Vr + V0 ) V1 sin (t ) .Integrating Eq. (11.140), we get the following equation:

-C

(11.141)

w

dz= v (t1 ). Therefore,dte (Vr + V0 )v (t1 ) =(t1 t1 ) + C1 = C1mL

ev

ie

At t = t1,

op

e (Vr + V0 )dz e (Vr + V0 )=dt =(t t1 ) + C1dtmLmLt1

(11.142)

Pr

Substituting Eq. (11.142) into Eq. (11.141), we get the following relation:dz e (Vr + V0 )=(t t1 ) + v (t1 )dtmL

(11.143)

Further integrating Eq. (11.143), we get the following expression:z=

e (Vr + V0 )mL

t

t

t1

t1

(t t1 ) dt + v (t1 ) dtt

or

z=

e (Vr + V0 ) t 2t tt1 + v (t1 ) t t + C21 2mLt1

or

z=

e (Vr + V0 )2mL


(t 2 2tt1 + t12 ) + v (t1 )(t t1 ) + C2

7/17/2014 11:29:59 AM


or

e (Vr + V0 )

z=

2mL

2(t t1 ) + v (t1 )(t t1 ) + C2

(11.144)

Now at t = t1, z = d. Therefore,d=

e (Vr + V0 )2mL

2(t1 t1 ) + v (t1 )(t1 t1 ) + C2 = C2

(11.145)

Substituting Eq. (11.145) back into Eq. (11.144), we get the following relation:e (Vr + V0 )

z=

2mL

2(t t1 ) + v (t1 )(t t1 ) + d

(11.146)

or

(t2 t1 ) + v (t1 ) = 0

T = (t2 t1 ) =

at

2mL

eria

2mLe (Vr + V0 )

2(t2 t1 ) + v (t1 )(t2 t1 ) + d

2mLv (t1 ) e (Vr + V0 )

M

or

e (Vr + V0 )

hted

d=

l

Since electrons return to the cavity gap at time t2, at t = t2, z = d. Thus, we canwrite that

(11.147)

op

w

V g T = T0 1 + i 1 sin t1 2V02 2 mLv0e (Vr + V0 )

(11.148)(11.149)

Pr

ev

where T0 =

ie

or

g 2mLv0 iV1sin t1 1 +2V02 e (Vr + V0 )

-C

T=

yr

ig

where T is the round trip transit time. Substituting Eq. (11.138) in Eq. (11.147),we get the following expression:

is the round trip DC transit time of the centre of the bunch electron.Multiplying Eq. (11.148) by , we get the following equation:

or

V g T = (t2 t1 ) = T0 1 + i 1 sin t1 2V02 g T = 0 + X sin t1 2

where 0 = T0

(11.150)(11.151)

is the round trip DC transit angle of the centre of the bunch electron andV(11.152)X = i 1 0 2V0is the bunching parameter of the reflex klystron oscillator.


7/17/2014 11:30:07 AM


{

n=1

}

eria

i2t = I 0 2 I 0 J n (nX ) cos n (t2 0 g )

l

To transfer maximum energy to the oscillator, the returning electron must crossthe cavity gap when the gap field is maximum retarding. Therefore, the round triptransit angle of the centre of the bunch is given as follows:1T0 = n 2 = 2N (11.153)41where N = n (11.154)4is the number of modes.Current modulation of the electron beam when it returns to the cavity from therepeller region can be determined using the same procedure used in the analysis oftwo-cavity klystrons. However, since the beam current injected into the cavity isnow in the negative z-direction, we can write the following relation:(11.155)

M

i2 = i I 2 = 2 I 0 i J1 ( X ) cos (t2 0 )

at

The fundamental component of the current induced in the cavity by the modulatedelectron beam is given by the following equation:(11.156)

hted

where we have neglected g as 0 g .The magnitude of the fundamental component can be written as follows:

ig

I 2 = 2 I 0 i J1 ( X )

-C

op

yr

The DC power supplied by the beam voltage is given by Pdc = V0 I 0 VIwhereas the ac power delivered to the load is expressed as Pac = 1 2 2Substituting Eq. (11.157) in Eq. (11.159), we get Pac = V1 I 0 i J1 ( X )

(11.157)(11.158)(11.159)(11.160)

iV1VV V 1 0 = i 1 T0 = i 1 n 2 = i 1 2n 2V02V02V0 42V0 2

Pr

X=

ev

ie

w

Now first substituting Eq. (11.151) and then Eq. (11.153) in Eq. (11.152), thefollowing relation is obtained:

or

V1 =

2V0 X i 2n 2

(11.161)

Substituting Eq. (11.161) in Eq. (11.160), we get the following relation:Pac = I 0 i J1 ( X )

2V0 I 0 X J1 ( X )2V0 X =2n i 2n 22

(11.162)

Therefore,=

2V0 I 0 X J1 ( X )Pac== PdcV0 I 0 2n 2


2 X J1 ( X )2n 2

(11.163)

7/17/2014 11:30:14 AM


The factor X J1 ( X ) reaches a maximum value of 1.25 at X = 2.408 andJ1 ( X ) = J1 (2.408) = 0.52. In practice, the mode corresponding to n = 2 or N = 1 34has the most output power. Thus, the maximum efficiency of a reflex klystron iscalculated as follows:=

2 2.408 0.52= 0.227 4 2

(11.164)

Therefore, the maximum efficiency of reflex klystron is 22.7%.Now, first multiplying Eq. (11.149) by and then substituting Eqs (11.153)and (11.137) in it, we get the following equation:

V0

2

(Vr + V0 )

2m L 0.593106

=

eriaatM

=

hted

(Vr + V0 )

e 2n 2

2e 2 2n 2

ig

or

V0

yr

or

61 2m L 0.593 V0 102 n =4e (Vr + V0 )

(

4m 2 2 L2 0.593106

op

or

2m Lv0e (Vr + V0 )

l

T0 =

2

)

22n 2 e=8 2 L2 m

(11.165)

Pr

ev

ie

w

-C

Equation (11.165) reveals that for a given beam voltage V0 and cycle numbern or mode number N, the centre repeller voltage can be determined in terms offrequency.Differentiating Eq. (11.165) with frequency, we get the following expression:d2(Vr + V0 )d

{

or

}

d V0 8 2 L2 m =d 2 e 2n 2

8V0 m L2dVr1=2d (Vr + V0 )e 2n 2

(11.166)

Substituting Eq. (11.165) in Eq. (11.166), we get the following relation:8V0 m L2dVr=2de 2n 2


2e 2n 22 2

V0 8m L

=

8mV0Le2n 2

7/17/2014 11:30:21 AM


8mV0dVr2L=dfe2n 2

or

(11.167)

Equation (11.167) establishes the relation between the repeller voltage and thefrequency of operation of a reflex klystron.The output power can be expressed in terms of repeller voltage Vr as follows:Pac =

V0 I 0 X J1 ( X )(Vr + V0 )

L

e2mV0

(11.168)

(11.169)

hted

M

I 2 2 J1 ( X ) j 2 0 Ye = Ge + jBe = 0 i 0eV0 2X

at

eria

l

Equation (11.168) can be used to calculate the power output at the centre frequency.If an electron returns to the cavity a little before the time (n 14 )T , the currentlags behind the fields and an inductive reactance is presented to the circuit. On theother hand, if the electron returns to the cavity a little after the time (n 14 )T , thecurrent leads the fields and a capacitive is presented to the circuit. The electronicadmittance can be written as follows:

-C

op

yr

ig

Equation (11.169) reveals that the phasor admittance is a function of DC beamadmittance, DC transit angle, and the second transit of the electron beam throughthe cavity gap, and is non-linear. The plot of electronic admittance is shown inFig.11.19. Any value of 0 for which the spiral lies in the area at the left of the lineG jB will yield an oscillation, that is,1 0 = n 2 = 2N4

N= 2

34

34

ev

jBe

j80

Pr

N= 1

N =03N=4

ie

N =1

w

N =2

j60j40

N=

N=

14N= 1

j20

Ge 80 60 40 20121N= 12N= 212G jB

N =3

14

N= 2

20 40 60 80 Gej20j40

0

j60G

j80jBe

Oscillatingregion

Non-oscillatingregion

Fig. 11.19 Electronic admittancespiral of reflex klystron


14

which is the same as Eq. (11.153).The equivalent circuit of a reflexklystron is shown in Fig. 11.20; Land C are the energy storage elementsof the cavity, Gc is the copper lossesin the cavity, Gb is the beam loadingconductance, and Gl is the loadconductance. The necessary conditionfor oscillation is as follows:

Ge G

(11.170)

where Ge is the negative real part ofthe electronic admittance, given byEq.(11.169), and

G = Gc + Gb + Gl = 1 Rsh (11.171)

where Rsh is the effective shuntresistance.

7/17/2014 11:30:28 AM


Pr

ev

ie

w

-C

op

yr

ig

hted

M

at

eria

l

Reflex klystrons are widely used for generation of low power in laboratoriesto find line and load characteristics. The most common measurable parametersare VSWR and the position of VSWR minimum. In practice, to avoid the use ofcostly microwave receivers and power meters, microwave signals are modulatedwith a low-frequency signal. This modulated signal is probed and detected with acrystal detector. The detected microwave signal carries the original amplitude andphase information of the microwave signal, and is measured using low-frequencyreceivers.In practice, two basic modulation techniques are used for this purpose: (a) amplitudemodulation and (b) frequency modulation. In amplitude modulation, the DC repellervoltage is adjusted at the left edge of the mode power curve, as shown in Fig. 11.21,and a low frequency (usually 1kHz)square wave voltage issuperimposed oniI2it. The amplitude of this square wave isso adjusted that it attains the maximumCGlGcGbV2 power point, as shown in Fig. 11.21.LThe resultant repeller voltage is also asquare wave with a frequency equalto the frequency of the modulatingFig. 11.20 Equivalent circuit of reflexsignal, and the output power of theklystronreflex klystron can be either 0 or Pmax,depending on the instantaneous repellervoltage. In amplitude modulation, caref0VRshould be taken that the negative halfPmaxcycle of the modulating amplitude doesnot enter a higher mode or a part of theVR0 t1 t2 t3 t4 tsame mode, to avoid oscillation at two0Vdifferent frequencies.t1To frequency modulate the outputt2signal of a reflex klystron, a saw tootht3voltage is superimposed on the repelt4ler voltage (Fig. 11.22), resulting intthe repeller voltage also becoming sawFig. 11.21 Amplitude modulationtooth in nature. The output frequencyfmaxof the reflex klystron attains any valueRFf0f2between f min and f max, depending onf1tf0VRthe instantaneous repeller voltage.Therefore, the output of the reflexfminklystron becomes frequency moduPmaxlated. In practice, amplitudes of theDC repeller voltage and the saw-toothV0Vt1wave are so adjusted that frequencyt2t3sweeping can take place only over thet4t5linear region of the frequencyvoltagetcurve, ensuring almost constant poweroutput and linear frequency sweeping.Fig. 11.22 Frequency modulationR


7/17/2014 11:30:31 AM


In addition to their laboratory use, reflex klystrons are used in local oscillatorsemployed in commercial, military, and airborne Doppler radars as well as inmissiles. Such klystrons can produce an output power of 10500mW at a frequencyrange 125GHz, with an h of about 2030%.Note Maximum power output of a reflex klystron is less than that of a two-cavity klystron.

EXAMPLE 11.3 A reflex klystron operates under the following conditions: V0 = 575 V,L = 1mm, Rsh =16 k, and f =10 GHz. If the device is operating at the peak of 1 34 mode,calculate (a) repeller voltage, (b) direct current required to produce a gap voltage of 250V,and (c) efficiency. Assume that 0 =1, given that at X = 1.841, J1 ( X ) = 0.582.

eria

l

Solution Given: V0 = 575 V, L =1 mm, Rsh =16 k, f =10 GHz, n = 2, 0 =1, andV2 = 250 V

(a) Now,

yr

575= 924.07650.6734 103

op

or Vr + 575 =

ig

hted

M

at

2211 2n 4 .175910 e V02 2 == = 0.6734 103222293mL8(Vr + V0 )8(2 10 10 10 )575= 0.6734 103or2(Vr + 575)

-C

or Vr = 924.0765 575 = 349.0765 V250V2== 0.0134 A = 13.4 mA2 J1 ( X ) Rsh 2 0.582 16 103

(b) I 0 =

(c) The efficiency is expressed as follows:

ie

w

ev

2 X J1 ( X ) 2 1.841 0.582== 0.1949 = 19.49%4 2n 22

Pr

=

Practice Problem

11.3 A reflex klystron operates under the following conditions: V0 = 625 V, L = 1 mm,3Rsh =14 k, and f = 8 GHz. If the device is operating at the peak of 1 4 mode, calculate(a) repeller voltage, (b) direct current required to produce a gap voltage of 175 V, and(c) efficiency. Assume that 0 =1, provided that at X = 1.841, J1 ( X ) = 0.582.

[145.7098 V, 1.07 mA, 19.49%]

11.7 Helix Travelling-wave TubesKlystrons, as described in the last few sections, are resonant structures and hencenarrow-band devices. In comparison, TWTs are non-resonant structures and hencewideband devices. A TWT incorporates a slow-wave structure within it, through


7/17/2014 11:30:43 AM


w

-C

op

yr

ig

hted

M

at

eria

l

which a wave propagates with a velocity almost equal to that of the electrons inthe beam. As a result, the interaction time between the travelling RF field and theelectrons in TWTs is much larger than that in a klystron and lasts over the entirelength of the circuit. Due to the interaction between the RF field and electrons, asmall amount of velocity modulation is introduced in the electron beam, whichlater transforms into current modulation. This current modulation, in turn, inducesan RF current in the circuit, resulting in amplification. In general, two types ofTWTs are available: (a) helix TWT and (b) coupled-cavity TWT. Helix TWTs arewidely used in broadband applications, whereas coupled-cavity TWTs are widelyused for high-power applications like in radar transmitters. It should be notedthat the wave in a TWT is a travelling wave, which it is not true for klystrons. Inaddition, a coupling effect exists between the cavities of coupled-cavity TWTs,which is absent in case of klystrons.A basic helix TWT, shown in Fig. 11.23, consists of an electron beam,focused by a constant magnetic field along the electron beam, and a slow-wavestructure. A solenoid or permanent magnet is used for focusing the electronbeam. The disadvantages of a solenoid are that it is relatively bulky and alsoconsumes power. Therefore, this arrangement is suitable for high-power tubeswhere power output is more than a few kilowatts. For satellite communication andlow-power applications, where the weight as well as power consumption shouldbe minimized, permanent magnets are used. In satellite application, to reduce thebulk, the electron beam is focused using a periodic permanent magnet (PPM).In a PPM, a series of small magnets are located right along the tube, with gapsbetween successive magnets. The beam is slightly defocused in these gaps, butagain refocused by the next magnet, as shown in Fig. 11.24. In a PPM, individualmagnets are interconnected.

RFoutput

ev

ie

RFinput

Effect of attenuator

Pr

Electron beamfocusing magnet

Controlanode

Attenuator

Tube body

BunchingCollector

Cathode

Heater

ElectronbeamRFoutput

Helix

RFinput

Gain or modulationcontrol voltage

+

Beam supplyvoltage

+

Control supplyvoltage

+

Fig. 11.23 Schematic diagram of helix TWT


7/17/2014 11:30:44 AM


Fig. 11.24 Focusing of electron beam using PPM

Pr

ev

ie

w

-C

op

yr

ig

hted

M

at

eria

l

The slow-wave structure used in a TWT is either a helix (more commonlyused) or a folded back line. The helical slow-wave structure has both advantagesand disadvantages. The main advantage is that it is inherently non-resonant andhence a large bandwidth can be obtained, whereas the main disadvantage isthat the helical turns in a helix TWT are in close proximity and hence there is apotential chance of oscillation to set up due to feedback at high frequency. Thehelical structure also limits the use of the tube at high frequencies because thediameter of the helix must be small to allow a high RF field at the centre, which,in turn, presents focusing difficulties, especially under operating conditions wherevibration is possible.In a helix TWT, the applied RF signal propagates around the turn of the helixand results in an electric field at the centre of the helix along the helix axis. Theaxial electric field propagates with a velocity close to the product of the ratio ofthe helix pitch to helix circumference and the velocity of the light. In practice,the ratio is so adjusted that the velocity of the axial electric field becomes almostequal to that of electrons in free space. More precisely, DC velocity of electronsis maintained at a slightly higher value than the phase velocity of the travellingwave. When electrons enter the helix tube, an interaction takes place betweenthem and the moving axial electric field. As a result, electrons transfer a net energyto the wave on the helix and a signal amplification takes place.To understand the energy transfer process, let us assume that three electronsare entering the helix at three different instants. The first electron enters the helixwhen the RF field is retarding, and hence will move with a slower velocity; thesecond electron enters the helix when the RF field is zero, and hence will movewith an unchanged velocity; and the third electron enters the helix when the RFfield is accelerating, and hence will move with a faster velocity. Thus, the firstelectron will take more time than the second and third electrons and the thirdelectron will take lesser time than the first and second electrons, to reach thecollector. Therefore, if the first electron enters the helix, at a time before the others,and the third electron at a time later than the others, the length of the helix can beadjusted so that all the three electrons can reach the collector at the same time, thusforming a bunch at the collector end. Bunching shifts the phase by 2 . As a result,electrons in the bunch encounter a strong retarding field and energy is deliveredto the RF field.In TWTs, a mismatch exists between the input and output couplers over a widefrequency range, which causes a wave to be reflected from the output coupler andreturned to the input. At the input, a part of the reflected signal is re-reflected, whichnow travels towards the load and is amplified by the tube. The total procedureresults in an unwanted oscillation in the circuit, which can be avoided by placingan attenuator near the centre of the helix. Bunched electrons emerging from the


7/17/2014 11:30:45 AM


attenuator induce a new electric field with the same frequency, which, in turn,results in amplified microwave signals.The motion of electrons in a helix TWT can be analysed in terms of the axialelectric field. If the travelling wave is propagating in the z-direction then thezcomponent of the electric field can be expressed as follows:Ez = E1 sin (t p z )

(11.172)

eria

l

where E1 is the magnitude of the z-component of electric field, and p is the axialphase constant and is given by the following relation: p = v p(11.173)Now, the force on the electrons exerted by the axial electric field can beexpressed as follows:dvm = eE1 sin (t p z )(11.174)dtVelocity of the velocity-modulated electrons can be assumed to bev = v0 + ve cos (e t + e )

(11.175)

hted

M

at

where v0 is the DC electron velocity, ve is the magnitude of velocity fluctuationin the velocity-modulated electron beam, e is the angular frequency of velocityfluctuation, and e is the phase angle of fluctuation.Substituting Eq. (11.175) in Eq. (11.174), we get the following relation:me ve sin (e t + e ) = eE1 sin (t p z )

ig

(11.176)

ev

ie

w

-C

op

yr

For interaction between electrons and the electric field, velocity of the velocitymodulated electron beam must be approximately equal to the DC electron beamvelocity; therefore,v v0 (11.177)and we can writez = v0 (t t0 ) (11.178)Substituting Eq. (11.178) in Eq. (11.176), we get the following equation:

or

Pr

me ve sin (e t + e ) = eE1 sin {t p v0 (t t0 )}

{

}

me ve sin (e t + e ) = eE1 sin ( p v0 ) t + p v0 t0

(11.179)

Substituting Eq. (11.173) in Eq. (11.179), the following expression is obtained:

{}me ve sin (e t + e ) = eE1 sin { p (v p v0 ) t + p v0 t0 } me ve sin (e t + e ) = eE1 sin ( p v p p v0 ) t + p v0 t0

or

(11.180)

Comparing both sides of Eq. (11.180), we get the following relations:me ve = eE1or

ve =

eE1me

e = p (v p v0 )


(11.181)(11.182)

7/17/2014 11:30:52 AM


and

e = p v0 t0

(11.183)

Equation (11.181) reveals that the magnitude of velocity fluctuation of the electronbeam is directly proportional to the magnitude of the axial electric field.This analysis neglects the space charge effect. If the space charge effect isconsidered, then the electron velocity, charge density, current density, and axialelectric field can be written as follows:v = v0 + v1e jt z

= 0 + 1e

J = J 0 + J1eand

(11.184)

jt z

(11.185)

jt z

(11.186)

Ez = E1e jt z

(11.187)

)(

)

hted

or

(

J = v = 0 + 1e jt z v0 + v1e jt z

M

at

eria

l

where = e + j e is the propagation constant of the axial waves. In Eq. (11.186),the negative sign added before J 0 ensures that J 0 is positive in the negativezdirection.For a small signal,

2 jt z )J = 0 v0 + ( 0 v1 + 1v0 ) e jt z + 1v1e (

J 0 v0 + ( 0 v1 + 1v0 ) e jt z

(11.188)

yr

ig

or

2( jt z )

J 0 = 0 v0 J1 = 0 v1 + 1v0

(11.189)(11.190)

w

and

-C

op

where we have neglected the term 1v1eas 1v1 0 .Comparing Eqs (11.186) and (11.188), we get the following expressions:

Pr

ev

ie

The force equation can be written as follows:dvm = eE1e jt zdt dz orm +v + v1e jt z ) = eE1e jt z t dt z ( 0or

m ( j v0 ) v1e jt z = eE1e jt z

or

v1 =

e mE1j v0

where we have substituted

(11.191)dz= v0.dt

From the law of conservation of charges, we can write the following equation:

or

J +=0tJ 0 + J1e jt z +0 + 1e jt z = 0zt

(


)

(

)

7/17/2014 11:31:02 AM


or

J1e jt z + j1e jt z = 0

or

1 =

J1J= j 1j

(11.192)

Substituting Eqs (11.191) and (11.192) in Eq. (11.190), we get the followingrelation: e m J J1 = 0 v1 + 1v0 = 0 E1 + j 1 v0 j v0 1 + jv0 J1 = e 0 E1m ( j v0 )

or

+ j v0e 0 E1J1 = m ( j v0 )

or

J1 =

or

J1 = j

l

or

m ( j v0 )

=j

e0 v0 E1

M

2

2

mv0 ( j v0 )

(11.193)

hted

e0 E1

at

eria

e0 E1e0 E1=2 m ( j v0 )( + j v0 )m ( j )( j v0 )

J0 eE1 v0 m ( j v0 )2

yr

J1 = j

ig

Substitution of Eq. (11.189) in Eq. (11.193) results in the following expression:(11.194)

Pr

ev

ie

w

-C

op

If the magnitude of the axial electric field is uniform over the cross-sectional areaof the electron beam then the spatial electric current i will be proportional to theDC current I 0 with the same proportionality constant for J1 and J 0, and hence canbe written as follows:I0I0 e ei= jE1E = j2 1v0 m ( j v0 )v0 m 22v0 j v0e I0ei= j 2E1or(11.195)mv0 ( j e )2where e = v0

(11.196)

is the phase constant of the velocity-modulated electron beam.Now from Eq. (11.20), we can write the following relation:mv02= 2V0 e

(11.197)

Substituting Eq. (11.197) in Eq. (11.195), we get the following expression:e I0i= jE(11.198)2 12V0 ( j e )Equation (11.198) is known as an electronic equation.


7/17/2014 11:31:09 AM


Convection current in the electron beam, given in Eq. (11.198), induces anelectric field in the slow-wave circuit, which is added to the field already presentin the circuit. As a result, the circuit power increases with distance.To study the coupling, let us assume the slow-wave helix as a distributed losslesstransmission line, as shown in Fig. 11.25. Therefore, using the transmission lineequations of Chapter 2, we can write the following equations:

and

I = jCV + i

(11.199)

V = j LI

(11.200)

Substituting Eq. (11.199) in Eq. (11.200), we get the following relation:

or

eria

l

jCV + i 2 LCV + j Li = V = j LI = j L

2V = 2 LCV j Li

(11.201)

hted

M

at

In the absence of the convection current i, the propagati

Microwave Engineering-Chapter 11-Sample

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