Title: Nonlinear analysis of a space-charge-wave amplifier
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Title: Nonlinear analysis of a space-charge-wave amplifier
Physical Description: vi, 127 leaves : ill. ; 28 cm.
Language: English
Creator: Houts, Ronald Carl, 1937-
Publication Date: 1963
Copyright Date: 1963
 Subjects
Subject: Amplifiers, Vacuum-tube   ( lcsh )
Microwaves   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis - University of Florida.
Bibliography: Bibliography: leaves 124-126.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00097955
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000559203
oclc - 13447092
notis - ACY4651

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NONLINEAR ANALYSIS OF A

SPACE-CHARGE-WAVE

AMPLIFIER








By
RONALD CARL HOUTS












A DISSERTATION PRESENTED TO THE CRADUATF COUNCIL OF
THE UNIVERSITY OF FLOP.IA
IN PARTIAL FULFILLMENT COIF THIE PiEQUIPIMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
August, 1963


aIii



illlri,













ACKNOWLE DCMENTS


The author wishes to express his sincere appreciation to Dr. A.D.

Frland for his counsel and guidance during the course of this research.

Iso would like to acknowledge the guidance given him throughout his

ilte .program by his supervisory committee and in particular to Dr. W.E.

IIIIIlwh was his chairman but is now on a year's leave of absence, and to

IIW. PItIrson who is his new chairman.

The *uthur also wishes to thank the Department of Health, Education,

ilare without whose support he would not have been able to study for

dree n Doctor of Philosophy. The use of computer time, provided by

J rjijty of Florida Computing Center, is gratefully acknowledged.

author's special thanks go to his wiFe, Marilyn, who has been In

'an important partner in this venture.






> ii
::::::::::::::::::V:::::::





:::::::::::::::::::::::::::::A:















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS . . . . Ii

LIST DF ILLUSTRAT IONS . . . . . . .

CHAPTER

1. INTRODUCTION .

Amplificatlun by a Modulated Electron Beam . . I
Smalt-Signal Analysis . . . . . . . . 3
Space-Charge-Wave Amplifiers . . . . . . 3
Finite Beaml Considerations
Ex tensions to Srall-SW final Theory. . . 6
Exatenslion of Webster's theory . . . . . 6
Electron disk approach . . . . . . 9
Paschk:'s successive approxinati )n . . 10
Scoape of the Present Investigation . . . . 12

2 SMALL-SIGNAL ANALYST IS OF A SPACE-CHARGE-'AVJE
AMPLIFIER . . . 4

Synopsis . . . . . . ... . . . . 14
Small-57cnal Equation . . . . 14
initial Conditlors . . 15
Sma I -S ic al Arta I S i s . . . . . . 19
Conclus ions Bas(ed on Srfad I -Signal Theory , . 26

3.DERIVATION OF SPACE-CHARGE-WAVE EQUATIONS USING
THE METHOD OF SUCCESS IVE APFROX IRATIO:IS . . . 27

Synopsis . . 27
Assumptions . . . 27
Derivation of the Space-Chirqe-Wave_ Equations5 2 8
Method of Sujccessive kpprolxiritlurs I. I . 32

4. GENERAL SOLUTIC11S FOR THE SPACE-CHARGE-WAVE
EQUATIONS . . 37

Synopsis . . . I I I I I I . . . 37
First-order- Solutions . . . . . . . . 37
Second-Order Solutions . . . . . . . 40
Thfrd-Grd. .ou i n . . . . . 43











TABLE OF CONTENTS--Continued


Page


CHAPTER


5. PARTICULAR SOLUTIONS FOR REGION OF FIRST
DRIFT TUBE . .. . . . . . . . .. . 48

Synopsis . . . ... . . . . . . . 48
Boundary Conditions . . .. .. . . . . 48
Continuity of current . ... . . . . 49
Conservation of energy .. .. . . ... 50
First-order Solutions . . . . . . . 52
Second-Order Solutions . .. ... . .. . 53
Third-Order Solutions .... . . ... .... 55
Results . . . . . . . . . . . 59
Saturation of fundamental current . . . .. 60
Locations of velocity and current maxima
and minima . . . . ... . . .. . . 60
Phase shift .. . ... . . . . . . 64
Second harmonic generation . .... ..... . 67

6I. PARTICULAR SOLUTIONS FOR REGION OF SECOND
DRIFT TUBE . . . .. . . . . . . ... 69

Synopsis . . . ... . . . . . . . 69
Boundary Conditions . ... . . . . . . . 70
First-Order Solutions . . ... . . . . 73
Second-Order Solutions . ... .. . . . .... 75
Third-Order Solutions . ... .. . . . . 80
Transition Near a Velocity Maximum . . . . .. 84
Transition Near a Velocity Minimum . ..... . 91

7I SII ARY AND CONCLUSIONS . . . . . . . .. 99


)F PRINCIPLE SYMBOLS . . . . . . . .

)F PARAMETERS USED IN EQUATIONS . . . . . .


ION OF ELECTRON OVERTAKING . .

FOR OBTAINING SOLUTIONS FOR THE C
RATION IN REGION B . . . .


ONSTANTS
. . . . . 118

.. 124














LIST OF ILLUSTRATIONS


Figure Page

2.1 Space-charge reduction factors for cylindrical
beams in cylindrical drift tubes . . . . . . 16

2.2 Model of a space-charge-wave amplifier using
three drift-tube sections . . ... . . .. . 18

2.3 Space-charge-wave amplification with the first
transition at a current maximum . . . . . 22

2.4 Space-charge-wave amplification with the first
transition at a velocity maximum . . . . . . 25

3.1 Current flow In an Infinitesimal beam element . .. 30

5.1 Fundamental frequency current vs. position . ... .. . 61

5.2 Fundamental frequency velocity vs. position . . .. 62

5.3 Maximum fundamental frequency current as a
function of a . . . . . . . . . . 63

5.4 Location of current and velocity maxima and
minima as a function of a . ... ....... . .65

5.5 Phase shift at the point of maximum current as
a function of a . . . . . . . . .. 66

5.6 Second harmonic current vs. position . . . ... 68

6.1 Current in Region B as a function of drift-
tube radius . . . . . . . .... ... .. .86

6.2 Current in Region B as a function of drift-
tube voltage ............... .... .

6.3 Maximum current in Region B as a function of the'
location of the first transition . . . . .... 8..

6.4 Velocity in Region B as a function of the drift-
tube voltage .... . 92



......... .m ;v


:: :











LIST OF ILLUSTRATIONIS--Cont;nued


Figure Page

6.5 Velocity in Region B as a function of the drift-
tube radius . . . . . . . . ... . . 94

6.6 Maximum velocity In Region B as a function of C . .. 96

6.7 Location of the first transition for maximum velocity
In Region B . . . . . . .. . . . .. 97

6.8 Location of the maximum velocity in Region B . . .. 97

6.9 Effect of drift-tube lengths upon the velocity
amplification . .. . . . . . . ... . 98














CHAPTER I


INTPODUCT ION


Amplification by a Modulated Electron Beam


The ability of a modulated electron beam to propagate space-charge

waves was first investigated by Hahn [1] and Ramo [2]. Passing a beam of

electrons thr- ugh a -ap, across w-hich an r-f electric field is applied,

causes the electrons to be decelerated or accelerated depending on the

phase of the electric field at thle time of passage. The accelerated

electrons tend to catch up with the electrons which left the gap at an

earlier time, but were decelerated by the applied electric field. This

grouping of electrons Is referred to as bunching. It is assumed that

the charge density of an unmodulated electron beam is zero, as there

are equal numbers of electrons and positive ions. The positive Tons

result from the ionization of the residual gases by the electron beam.

In the modulated electron beam, the charge density in the region where

the electrons are bunched is negative; whereas, in the region between

bunches, the charge density is positive. Because of this difference

in charge density, the space-charge forces between the electrons mDvI

electrons from the region of high density into the region of l!w density.

Since regions of lower electron density appear both ahead of and behind

a region of high density, some electrons are accelerated wiithi respect t

the average beam velocity while others are decelerated. T, .tatn.Io,











observer, there woIuld appear to be two wa'.es propagating with velocities

slightly higher and lower than the average bearn velocity. As the bunches

of electrons spread apart in the axial direction under the influence of

space-charge repulsion, new bunches of electrons form from the de.el-

erated electrons of one bunch, and the accelerated electrons of the

following bunch formed by the r-f field a short tine later. A second

gap is positioned at some point along the bear where a ne, bunch of

electrons has formed. This large grouping or bunching of electrons neans

a sizeable component of r-f current is present in the bear, This current

will induce a similar component of r-f current in the output cavit., pro-

vided the cavity is tuned to the r-f frequency at which the beam ir ex-

cited. The cavity may be tuned Instead to a harr'cnic of the fundamental

I:llllfreiquency, in which case the device is considered a frequency, multiplier

i rather than an amplifier. The resultant voltage produced in the cutput

ci. ty Is an amplified version of the input excitation. The analyses of

ubn and Ramo shows that the optimum position for the output gap is one-

foerth of a plasma wavelength, where the plasma frequency is the fre-

cy at which the bunched electrons oscillate under the effect of

p e-charge forces. The bunching process takes place without further

nce by external fields If the beam is enclosed in the region be-

n cavities by a conducting cylinder called a drift tube. The two

e with the drift tube separating them are the essential elements

he device called the klystron. Because of the use of

Srennt cavities, the klystron has an inherent bandwidth limlta-

nd other device called the traveling-wave tube has been











developed. Essentially the drift tube is replaced by a helix, which

supports an electromagnetic wave which propagates with an axial veluc-

Ity less than the speed of light. Thus the electromagnetic wave can

Interact with the space-charge wave on the beam in a continuous fashion.

This helix eliminates the need for the tuned output cavity, which has a

limited bandwidth, since the amplified signal is taken from the helix

directly.


Small-Signal Analysis


The action of the electron beam may be described by specifying

how the velocity v(z,t) and current i(z,t) modulations fluctuate. A

first approximation for the solutions of velocity and current may be

obtained by a process referred to as small-signal analysis. Essentially

the equations are derived by utilizing the equations for the space-charge

force on an electron; the continuity of current: the relations between

velocity, current, and charge density; and Maxwell's equations. In

this approach it is assumed that the time variation is sinusoidal, and

the equations are linearized by discarding the product of a-c terms.

The separation of a-c and d-c terms then yields equations which are

functions of position (z) alone, since the variation with time (cjut)

cancels in the a-c equation.



Sipa-CCharge-Wave Amplifiers


Tien and,,Fiiiiiiiiid iirexaTnai d the principles, of klystrf operation

and foundihHI'It FuIrther ampIfIcution culd be obtained if the drift tube











was divided into setonsh o f appropriate length, eac havingr a d-c

potential different from thle adjacnt sections. Additional energy is

available for increasing the amplitude of velocity modulation over that

which would ocur 'in a single section of drift tube. Thisdvie

called the velocity-jump amplifier, wa-s the first tof se is of

devices classified as space-charge-wave amplifiers. others in thi

class are the rippled-wall ampipfier, where the sections of the drift

tube take on different diameters without changing potentials, and the

ripplec'--trear, uniplifier, where the drift tube Is unchanged, but the

beaF1 Cil Jeter fIuJctuatOs The fonrure device w s investigated by

Dirdsall ind Whinnery [41], and thle latter by Birdsall15[1 and flihran [6].

A general aonelysls ot a spa-e-charge-wavc amplifier which incorporates

al) thcse po-siblve alterratlves, has been performed by Peter, Bloom and

Ruetz 171 using smn-inlthetorj, It was found that ma>,u arn pll-

fication resulted f the: driFt-tube potential and the drift-tubeda tr

were increased while the- bcim cross section was decreased at the po Int

kwtere the a-c velo-city %jo, miniliui_ At a velocity maxIimu all para-

neters were then retujrned tc their original values. One interesting

approach for investigating t.e operation of spacc-charge -wave amplifiers

f5 a construction by Dloo-m and Pctrer [QUI nf 3 trans mIssion-Ilne analog

which has a standing-,atje pattern that repre,ants the amplitude pattern

o~f the space-charge wave. The approach is bAsled on the small-signal

anlsi hlch predicts that the maxlinum current aind velocity amplitudes

Interchange every quarter plasma wavelengths. A general discussion of

the llnearlied 5fmall-slgnal analysis as applied to the varlou m~rowavee











amplifiers Is presented in books by Beck 9)], Harman [10], and Hutter [ll].

The small-signal theory is presented in Chapter 2. and the nonlinear analy-

sis Is the subject of the following chapters.



Finite Beam Considerations


Since beams are of finite diameter and located close to metallic

conductors, i.e., the walls of the drift tube, it is apparent that the

space-charge forces, which cause the bunching and debunching of the beam,

will be reduced because of the fringe effect of the electric field. This

fringe effect increases as the beam radius (b) approaches that of the

drift tube (a), and is more prominent in beams having a small normalized

beam radius (B b). The effect of reducing the space-charge force Is a

reduction in the plasma frequency (w ), i.e., the frequency at which the

beam bunching takes place. This modification is obtained by multiplying

the plasma frequency by a parameter called the plasma reduction factor

(R1). Hahn made the first analysis of finite beams in his original

report, and discovered that an infinite number of modes of space-charge-

wave propagation can exist. Ramo suggested that for a reasonable approxi-

mation only the fundamental mode need be considered. More rigorous sum-

mation techniques demonstrate that the effect of the higher order modes

Is most pronounced In beams which effectively fill the drift tube and

have high Beb values. Branch and Mihran [12] have solved the trans-

cendental equations which result from the Hahn-Ralb theory for the radial

propagation constant needed to determine the reduction factor. Curves











are presented .lth RI plotted against either Beb c.r b,'l, '.;th the

remaining variable as a parameter, for several beam geometries. The

plasma reduction factor affects the wa.elcngths of the standing-wave

envelopes of the velocity and current modulations, and the amplitude

of the current is also inversely proportional to the reduction factor.

The plasma reduction factor Is derived under the assumptions of small-

signal theory, and one of the problems invrol.ed in large-signal analysis

has been the applicability of the reduction factor to large-signal

situations. Mlhran [13] has demonstrated experimentally that replace-

ment of wp by RFu in the extrapolated small-signal theory leads to

errors in both magnitude and location of current maxima.



Extensions to Small-Signal Theory


The small-signal theory correctly predicts the fundamental c.pera-

tion of these microwave devices for small ratios of buncher voltage (Vm)

*to beam voltage (Vo). As the signal level is raised, experimentally

oblirved phenomena such as harmonic generation, saturation of the fund-

amental current, phase shift, and growth of the second harmonic component

re |not even predicted, much less analyzed, by the small-signal approach.

||s||i||ral approaches have been conceived to extend the l near analysis into

a nonlinear region of operation. These schemes are discussed in the

f9Igtnl sctions.


mn of Webster's theory

olne f tie earliest analyses of klystron operation was performed











by Webster [14]. The Webster solution for the fundamental component of

current is


If = 2loJl(Xo), (1.1)


where

Xo = ballistic bunching parameter p(a/2)Bez,

I = d-c beam current,

JI = Bessel function of 1st order and Ist kind,

= gap coupling coefficient,

a = voltage modulation index (Vm/Vo)

Vm = amplitude of signal voltage applied to buncher,

V0 = d-c beam voltage,

Be = &/o.


This solution is based on the assumption of no space-charge interaction,

and hence is a ballistic analysis. The Webster theory has been modified

to account for space-charge effects by comparing (1.1) with the solution

derived by the small-signal space-charge wave theory,



if = io(a/2)l.B0 sin(3Bp)/(BpZ), (1.2)



where Bp = wp/Vo. The solutions given by (1.1) and (1.2) yield identi(al

results for small values of a. The Bessel function may be replaced by

one-half of its argument for small values of thd|i| argument. Calling

the argument X, and comparing (1.1) and (1.2) undea the assumption

that X is small gives



...,,iii.... .iiiil.











X = 4(r.-'2)B3e sin(BpZ) '(3p ). (1.3)



This solution for the bunching parameter when space-charge effects are

accounted for is substituted into (1.1), and used as the solution for If

regardless of the value of X. There is no mathematical justification

for this substitution, and it is used simply because an extrapolated

solution is better than no solution at all. Comparison of the ballistic

bunching parameter (X,) defined in (1.1) with the solution obtained in

(1.3) shows that the bunching parameter including space-charge effects

is related to Xo by the relation



X = XK sin(Bpz)/(Bpz). (1.4)



The term sin(Bpz)/(Bpz) Is called the debunching factor. As a Increases

the modified Webster theory Indicates that the current does not increase

proportionally, i.e., the fundamental component of current saturates.

Although the modified solution has been extrapolated past the point of

IeIectron crossover, the predicted ballistic paths of the electrons are

Incorrect. The modified theory fails to account for a change in the

ih|rect[lon of the space-charge force past crossover, and for a change In

l|the| magnitude of the force. Moreover, the theory Is one-dimensional,

III I ia fringing effect of the finite beam Is neglected. Mlhran [13]

traes the- development of a further modification to Webster's theory

or predicting the electron paths past crossover. This theory, developed

by iv, ,c irrects the first two errors of the extrapolated theory; however,










the analysis is still one-dimensional. In order to approximate the effect

of the fringe field, the plasma reduction factor is introduced into the

equations by replacing Bp by RI p. Hihran demonstrates that klystron

design based on this approximation results in significantly less than

optimum efficiency. He cites a typical large-signal case here the

efficiency is 20 per cent less than an empirical optimum.


Electron disk approach

Tien, Walker, and Wolontis [15] devised the idea of replacing the

electron beam by a set of charged disks which have radii equal to the

beam radius, and are evenly distributed along the axis of the beam.

They applied this electron disk approach to the traveling-wave tube for

small values of the gain parameter (C), and later Tien [161 extended

the analysis to larger values of C. Webber (17, 18] applied the disk

model to the klystron, and Rowe [1g] applied it to the traveling-wave

tube using a space-charge field based on an electron distribution in

time rather than in space. Although the disk approach includes the

effect of space charge In an approximate manner, the idea of replacing

a plasma of electrons by 16 or 24 charged disks Is questionable.

Webberrs klystron computations do not fit Mlhran's [13] experilmntally

derived values of "peel-off" angle anywhere over a wide iranle of Beb

values. The "peel-off" angle is that normalized distance of the d rft

tube for which the normal ied r-f current is one-half db below 'Tits

ba 1 I s t c max I nm. This mode does, however, predic. t the growth

the second ha rmwnitlc which |ji|hran 120] verified experimentally.











Paschke's successi.r: approximation

An alternative approach has been applied by Paschke [21 who con-

sidered the electron beam as a fluid just as was done for the linear

case by Hahn and Ramo. Ma.well's equations arc applied to the model,

and the resulting nonlinear partial dfferent;al equation is solved

without discarding products of a-c terms or a-suming e "t variation in

time as the small-signal analysis does. The technique for solving the

nonlinear equation Is referred to as the method of successi.'e approxi-

mations, it is assumed that the dependent jarlable may be expressed

as a power series in the voltage modulation coefficient (a). The space-

Scharge-wave equation is then grouped into terms according to degree.

The first-order solution, which is proportional to a, is obtained by

IrI tailiing only the first-degree terms. This solution Is then used to
2
obtain the second-order solution which Is proportional to a and is

IfouII dy retaining terms in the general equation which are proportional

to III This process is repeated until it Is assumed that the additional

rims have a negligible contribution, i.e., the power series is con-

gent. In the approach used by Paschke, a third-order theory was

ildered to be sufficient to investigate the phenomena oF saturation,

a Ihift, and second harmonic generation. Thus the solution is of




i = i (1.5)
n=I


final to an. The approach has a fundamental assumption,











which h Is shared ,with the disk model, that the electric field does not

vary over the cross section of the beam. It has been shown by Diving [22]

and Mihran [13] that this assumption is valid for thin beams (Beb : I).

The main limitation to the theory Is the fact that It is only 'alid for

velocities which are sinqle-.alued functions of space, i.e., no electron

overtaking is permitted. In succeeding reports, Paschke considered finite

beams k.ith fringe fields in order to evaluate the second harmonic com-

ponent [231, saturation of the fundamental and third harmonic generation

[241, and phase shift of the fundamental [25]. The phase-shift problem

has also been investigated by Engler [26] without using some of the

approximations utilized by Paschke [25]. giving [22, 27] applied the

method of successive approximations to a velocity-modulated electron

beam and obtained an infinite series solution. The results included

the effect of higher order modes, whereas Paschke had assumed that only

the fundamental mode was generated. 0lving demonstrated that Paschke's

Intuitive idea of using plasma reduction factors in the nonlinear region

Is justified for thin beams where higher order modes of the harmonics

may be neglected. Since this restriction also applies to the approxi-

mation of constant electric field over the beam cross section, the method

of successive approximations introduces no new restrictions other than

the one of no electron overtaking. Although this method does not con-

sider overtaking, Paschke [24] showed that his method fits the experi-

mental values of "peel-off" angle defined by Mihran [1ii 3] very well in

the region where Beb 1 1, I.e., in the region wiAlere the model is valid

With the method of succesSibe approximations It is po:slihl to











investigate right up to the point of electron o.crtaking with some

assurance that the results can be verified by experiments.



Scope of the Present Investiqation


Paschke's method of analysis is applied to the space-charge waves

propagating in a space-charge-wacv amplifier which combines the features

of the velocity-jump and rippled-wall amplifiers. Beam diameter changes

are not considered as It is relatively difficult to produce this vari-

ation physically, whereas the other two changes can be done with com-

Iarative case, Some of the phenomena investigated In the first region

include the saturation of the fundamental current as the bunching volt-

ae is Increased, phase shift, second harmonic generation, and the shift

in locations of the velocity and current maxima and minima from the pre-

di d small-signal locations. Investigation In the second region

Includes the effect of different ratios of drift-tube potentials, the

effect of changing the drift-tube radius between sections and the location

f the transition between the two regions in order to cause maximum current

vlcity variations for fixed sets of beam potentials and drift-tube

tries. These changes will be Investigated with the thought in mind

St current excited in an output cavity can best be increased for

fxed value of a by properly utilizing the velocity-jump and


I c [28] applied this nonlinear analysis to a device with an
Mte lies of gaps and drift tubes; however, no provision was made
.Tnd-c potentials across the gaps, or modifying the drift-tube













rippled-wall principles In combination. The proper lengths of the drift

tube se-ctions in order to optimize the output current il11 also be ex-

amined, as it; I the saturation effect whlch limits [he output current

as a Increases.

All symbols utilized in this paper are defined in Appendix A, and

all constants defined In the deri ,ation of the space-charge-jwae equation

are evaluated in Appendix B.














CHAPTER 2


SMALL-SIGNAL ANALYSIS OF A SPACE-CHARGE-WAVE AMPLIFIER


Synopsis


The small-signal solutions of the Hahn-Ramo theory are applied to

a space-charge-wave amplifier composed of three drift-tube sections.

The first and the third sections have Identical properties, while the

middle section possesses both a different potential and a different

drift-tube radius. Investigation includes both Increasing and decreas-

ing the potential and radius of the second drift tube, relative to the

values of these parameters in the first region. The gain is found to

SI|e proportional to the ratio of the plasma reduction factors and the

three-fourths power of the voltage ratio. Maximum gain results if the

leniths of the drift tubes are some multiple of one-quarter of a re-

ced plasma wavelength.




Small-Slgnal Equations


The small-signal equations developed by Hahn and Ramno, as given by

(ii 013, Are


[I cos(B qz) J(W/Wq) ( /v 0) v sin(B qz)] ej(Wt ez)

(2. I)











v = [vI cos(Bqz) j(Wq) (voIo1) I] sln(Bqz)] ej(W Bez)

(2.2)


where the i subscript indicates an Initial value at the entrance to the

region described by the equations, and j = T-1. The other variables

were defined in connection with (1.1) and (1.2), and are tabulated for

easy reference in Appendix A. These equations are derived on the

assumptions that the variation with time is given by eJut, that a-c

terms are small compared with d-c quantities, and that the theory, which

Is derived for a beam of infinite cross section, can be applied to beams

of finite diameter by use of the plasma reduction factor as tabulated by

Branch and Mlhran [12]. A plasma reduction factor curve for a beam of ra-

dius (b) Inside a drift tube of radius (a) is shown In Fig. 2.1. A list of

the properties required of the plasma, In order to derive the small-

signal equations, is stated In Chapter 3 In connection with the deri-

vation of the nonlinear space-charge-wave equations. Equations (2.1)

and (2.2) are valid for any region In which the d-c velocity (vo) is

constant, such as inside a section of drift tube.




Initial Conditions


The velocity-jump a.l. ifier, developed by Tien ajni Fl iIId [3],

utilizes a change in d-c potential between drift-tube sections t.o

further amplify the current and velocity modulatior, In order to

use (2.1) and (2.2) to deicrpbe the aplificati n process. t is








2.0 2.4 2.8 3.2 3.6 4.0
1.0 I i-| Beb
R

upper scale
0.9



0.8
lower scale


0.7

b/a 1 /2

0.6 b/a = 2/3























0.4 0.8 1.2 1.6 2.0

NORMALIZED BEAM RADIUS (Beb)



F I Space-charge reduction factors for cylindrical beams
in cyIhIdrical drift tubes.











necessary to determine what happens to the magnitude of the a-c current

and velocity as the beam passes through the transition point. Hutter

develops a proof, using the Lorentz force equation, which demonstrates

that the a-c current is continuous, whereas, the a-c velocity undergoes

a transition proportional to the ratio of the d-c jelocitles in the two

regions.


'b(z ) = a( t) (2.3)


'b(zt) = (Va"'b) V(a (), (2.4)


where the subscripts a and b refer to the first and second regions

respectively, and z, is the position at which the transition is located.

A model of the amplifier is shown in Fig. 2.2.

Before the beam enters the input cavity, located at zo, there is

no a-c excitation. A signal voltage Vm cos(ut) velocity modulates the

beam at the input cavity. The initial values of a-c current and

velocity are derived In Chapter 3 in connection with the nonlinear

analysis. The small-signal solutions are



il(zo) = 0 (2.5a)


....o)' (c1/2) Voa' ................li H1 i
v, (z) = (a/2) Voa, (,, 5b)



where a is the ratio of signal voltage (Vm) to d-c beam voltage (V, .

The equations for the first drift-tube region are found by substI iting










1..



LJ






i i I
c.



C,








I '
Li U
I 5


SI L
s t' .











I I

Ii
SI
I
I I I





.L
I I


I U



3 sa _ I I I-





H--1---(







^-^ H-C
EEEEEIEE.
::1::::::::.
EEEEEEEEEEE:

EEEEEEEEEEEEE.T











(2.5) into (2.1) and (2.2).



i = j(a,'2) (wI.'qa) i0 sin(qaZ ) (2.6)


va = (a 2) v"a cos(BqaZ) (2.7)


The exponential term, which carries only phase shift information, has

been omitted from (2.6) and (2.7), and is also omitted in the rest of

the equations in this chapter. To derive any instantaneous solution,

it Is necessary to insert the exponential term, and take the real por-

tion of the product. Onl/ envelope magnitudes are considered, therefore

this exponential term may be omitted.



S'ra ll-Signal Analysis


If the first drift tube is one-fourth of a reduced plasma wavelength

long (BqaZa = 900), then the magnitude of the a-c variations at za are



a(za) = (a/2) (/w qa) i (2.8a)

Va(za) = 0. (2.8b)


Equations (2.8) are utilized to obtain entrance conditions for the

second drift-tube region. Solutions for the second rI|Egion are obtained

by substituting (2.8) into (2.1) and (2.2).










ib = (a2) (W,"rdqa) ir cos qbzl

vb = j (a/2) (Eqb/wqa cob sirn(qbZ)


(2.9a)

(2.9b)


If the length of the second drift tube is 3qbzb = 900, then


(2.10a)


ib(zb) = 0


vb(Zb) = ("/2) (Wqb-'jqa) "ob. (2.10b)


The entrance conditions for the third section are obtained by comparing

(2.10) and the requirement given by (2.4).


l,(zb) = 0


12. a)

(2.11 b)


v (2b) (Vb/Va)j b(zb)


itions for the third region are obtained by substituting (2.11)

I) and (2.2). Since the third region has the same properties

first region, the parameters use the a subscript.


ic, j (a/2) (/q) (Wqb/qa) (VbVa) sin(B z) (2.12a)


v, = (a/2) (cqb/ga) (Vb/lV, ob cos(1qaz)


(2. 12b)


maximum a-c velocity in the third region (vcm) to the

ty in the first region (vam) is


)











Vcm/ am (Vb'Va (Rb/Ra) (wpbpa) (Vob/oa. (.13)


The d-c velocity Is proportional to (Vo) and the plasma frequency is

Inversely proportional to the square root of the d-c velocity. Thus

(2.13) reduces to


vv am (Pb/Ra) (Vb/Va)34 (2.14



The ratio of the current may.ima is identical to (2.14).

Examination of (2.14) indicates that the best amplification is ob-

tained if the values of the potential and plasma reduction factor In the

second drift-tube region are increased. The value of Rb can be increased

by Increasing the drift-tube radius, as shown in Fig. 2.1. If the size

of the drift tube is not increased, the value of Rb is less than R

because the normalized beam radius (Bebb) decreases as the drift-tube

potential is increased. The variations in the envelopes of the current

and velocity modulations are shown In Fig. 2.3, for a choice of drift

tube and beam parameter values dealt with on a nonlinear basis in later ||

chapters.

The non I flier Investigation includes only the first t.w regions

The value of nafliim.l velocity in Region C, which Qccurs Just after the

second transition (zb), is obtained by multlply*rr the vaTl e of the

velocity just before the second transition by the ratio of the d-c

velocities (vob/vaI) This yie ds adequate Information for de.strat

Ing the velocity saturation effect, The modifIed Webter theory,













-


o0 0
u .
N-


Ci-'
U hi
'tic


\






\ o


X 1 \
o \





1- -
a u
L 0




I
u >



;/


/ -7 r-

C 0-I
\ o o- o



\ .0


c
C

a ___


olno
- ,


(N 0
n Ii
ii
'N


N
, a"











o











% ,
'3

o










0
N g
C


(N


0


L


N u
r


0

o Ci

.c




-.

















0
S -c
00 ^
LL


% U 4 N1-


h-










discussed in Chapter 1, can be utilized to predict the a-c current present

In the output cavity located a quarter plasma wavelength beyond the second

transition. The current prediction is obtained by replacing a in (1.3)

by a new a', where a' is equal to 2vcm /oa' and substituting (1.3) into

(1.1) In place of X,. In addition Bp Is replaced by Rl]p to account for

the effect of the fringe field.

An expression similar to (2.14) results If the first transition is

made at one-half of a plasma wavelength (6 z = 1800). The entrance
qa a
conditions for the second drift region are

II(za) = 0 (2.15a)


vi(a) = (a/2) (Va/Vb) voa. (2.15b)


The new solutions for the second drift space are found by applying (2.15)

to (2.1) and (2.2).


S= (a/2) (W/wqb) (Va/Vb) i1 sln(8qbz) (2.16a)

Vb = (a/2) (Va/Vb)i voa cos(B bz) (2..16b)


The second drift tube must be one-fourth of a reduced plasmI wavelength

long in order to allow the second transition to occur at a current

maximum. The entrance conditions for the third region are



l ij(za) (a/2) (W/wqb) (Ya/b 2174)










The solutions in the third region, assuming once again that thi: region

possesses the same properties as the first region, are obtained by sub-

stituting (2.17) into (2.1) and (2.2).



ic = (a/2) (w i qb) (Va'Vb) ;o cOS(Bqaz) (2.18a)

3/Li
v = j (a/2) ( aR b) (V V ) v sin(B z) (2.18b)
c a b a b oa a


The variations in the envelopes of the current and velocity/ modulations,

with the first transition occurring at the point of maximum velocity

variation, are shown In Fig. 2.4. The ratio of the maximum current or

velocity In the third region to the corresponding maximum value in the

first region is


Icm/iam Vcm /am = (RaRb) (Va'Vb)314. (2.19)


It is apparent from examining (2.19) that in order to obtain maximum

amplification it is necessary to decrease the drift-tube potential and

lradius in the second region. This is just the opposite approach from

that of the first analysis; however, comparison of (2.14) and (2.19)

dicates that the degree of amplification is the same. The only

d ffierese comes In the length of the drift-tube sections. With the

rist tirinsition occurring at a current maximum, all drift tubes are

fourth of a reduced plasma wavelength long; whereas, in the second

oach, the first and third sections must be one-half of a plasma

liimenth ilong in order to obtain the desired current amplification.






25













CD









x >.
o















I-1
0 -





0




ti rI





SII II II
I-.-


0





26





Conclus ions Based on Sma) I -S 9na Theory


The smail-signal theory predicts that the amplification is inde-

pendent of the degree of voltage modulation, I.e., it Is independent

of a. The experimental fact that amplifiers actually reach saturation

quite quickly as cc increases cannot be ascertained from this small-

signal theory. This, theory also predicts that the best amiplification

is obtained if the voltage is decreased at a velocity maximum, and in-

creased at a velocity minimum. Because the smnall-signal solution for

o-c velocity ha5 a minimum value of zeor, no reduction in the a-c

velocity occurs if the beam voltage is increased at the point of zero

a-c velocity. if the transition is located at someo other point, then

the amplitude uf thet a-c velocity is reduced, These simple results

are contrasted in Chaptcr 6 with the- results obtained by a nonlinear

teory, Among othcr things, It Is shown that the best amplification is

ntobtnined by placlvc the d-c velocity transitions at positions

cwespondl rg, to mu It!pl es of a quarter plasma wavelength, and that

saurtio~n effects quickly come into play as the signal amplitude is














CHAPTER 3


DERIVATION OF SPACE-CHARGE-WAVE EQUATIONS USING

THE METHOD OF SUCCESSIVE APPROXIMATIONS


Synopsis s


Following the approach used by Paschke, the electron beam is con-

sidered as a plasma, where the effect cf the individual electron is

ignored. The nonlinear partial differential equation which describes

the polarization of an electron is derived by examining the current

flow in an infinitesimal beam element. Linear equations are obtained

by the method of successive approximations.




Assumptions


in order to simplify the analysis the following assumptions are

made:

I. The electron velocity is small compared with the velocity of

light.

2. The r-f electric field component in the axial dTrectlani|||

is approximately constant over the beam cross section.

3. The effects of temperature, particle colli| ions, and potertle

depression iiaused by space charge are Jpniiiio':l'llllred.

4. Ion ras is m:liaslsumtied to be infinite, hence the ians are
.................... ............. ~.ilillilli......iiiiiiiiii i..... ... .,, , , , , , , , , ,
........ 'iiiiiiiii" ... iiiiiiiiiii :iiiiiiiiiiiiiiiiiiiiiii iiiii:::::::::::::::::::.iiiiiiiiiiiiiiiiiiiiiiiiiii


M m m m m mmiiiiliiiii m.ii l "ii liiiiiiiiiii iiiiliiiiii ll m m miiiii llillillilillilli l











affected by r-f fields.

5. The d-c charge density of the ions (Ip.l) cancels the d-c rch-Jr

density of the electrons (p J).

6. The velocity, in keeping ,ith the pc'lar;zaticon cocrd;nate s,steni,

is a single-valued function Cf spae, i e., electron overtaking

does not occur.

7. The effect of the transverse electric field associated L.;th the

plasma surrounded by a conducting cylinder is accounted trr b,

using the appropriate plasma reduction factor.

8. All variables are assumed to be functions only of the independent

variables time (t) and axial position (z).

9. An infinite magnetic field confines the motion or the electron

beam to the axial direction.




S Derivation of the Space-Charge-Wave Equations


Several relations between position, current, velocity, and charge

ity are needed in order to derive the nonlinear space-charge-v'ave

IIHns which describe the electron plasma. The charge density is

b by


i Pt P, + p(zt). (3.1)


current density is related to the charge density and











(3.2)


It = ;, + i = (PC. + )(vo + v).


where


I PCVo'


(3.3)


The a-c current density and charge density are related by the principle

of continuity of current,


(3.4)


Applying Newton's notation to partial differential equations gives


f,'iz f ,


(3.5a)


where f is any function of axial position (z) and time (t). Thus applying

(3.5) to (3.4) gives


I = P. (3.4)


The velocity at any point in the plasma is a func

the space-charge forces at that point. Using Ne|

relationship is given by


v) v = -


- a;.iaz = c,0at.


df/dt f, (3.5b)











There is an associated electric field transverse to the z-axis vwhen-

ever an electron beam of finite size is in the vicinity cf a metallic

drift tube. This transverse electric field, which is time var)iing,

indicates the presence of a displacement current (h) per unit length

of beam. This coupling current enters into the continuity equation for

the total current flowing thr..ugh an infinitesimal length of beam as

shown in Fig. 3.1.
M dz




i ( + oe i)A (I + EOE)A
( ?+ (1 4 EoE) A dz



M dz


ri Fg. 3.1 Current flow in an infinitesimal beam element.


Sui lamln of the currents shown in Fig. 3.1 gives


+ Co(E) + M/A = 0. (3.7)



a new variable (z ) is introduced. This variable repre-

a e plarization. or displacement, of an electron from its position

dled beam. This choice of variable is in keeping with the

n sstem of coordinates established by Bobroff [29]. The

n i created to the electric field due to space-charge by











- !po !p = E,, E + t ,


.here F is the coupI irii flu. ,h ;lic represents the presence .:f the trans-

verse electric field. The coupling flu.,. is related to the c,-upling

current (M) by the relation


(f) = M/A.


(3.9)


Substitution of (3.8) and (3.-) into (3.7) givvs



..(Z p)


Integration of (3.10) with respect to a yields



i = po zp.


Comparison of (3.10) and (3.4) in terms of i gives


(3.10)


(3H1)


p= p ( )'. (3.12


Integration of (3.12) with respect to time yields



P = P,0p (3.1j

The velocity is obtained by substituting relations for i and p into (3



V z p + p(314
z iiiii
== = = ...= = = = =


iiiii, ii


(3.3)










An equation for velocity is obtained by solving (3.8) for E and substi-

tuting the result into (3.6).



v + (v + v) v = l(Pp + f)/' (3.15)



Substituting (3.14) into (3.15) results in an equation in.olv.inq only

the polarization (zp) of the electron. The final form is a nonlinear,

second-order partial differential equation of fourth-degree.



SZp. + 2(Zp) .o + Zp.'/o" Tl(pOzp + f).o, 'E 2
'jp 'o '.u 00
+ 2[ i (i )' p'Pp '- p) pp P o 0

+[ p(Zp')2 + p" (ip)2) ,2 + 2 Z" [z ( Z) zzp'v],'
I ii(Poz, + f) [3Z, 3(p')2 + (p )3 ]'J,,vo 0 (3.16)



Theil velocity is related to the polarization (z ) through (3.14), and

the l current density is related to the polarization through (3.11).




Method of Successive Approx.inations



The solution to (3.16) is obtained by the technique known as the

hWd of successive approximations. In this approach it is assumed

he polarization may be expressed as

3
zp z, (3.17)
n=1


...........iini iii











where it is assumed that Zn is proportional to an. Three equations are

formulated by substituting (3.17) into (3.16) and separating terms by

powers of a. The variables ;, v, and f are also related to a by series

similar to (3.17).

Substitution of (3.17) into (3.16) leads to a first-order equation

of the form


(zI)" + 2(z ) '/v + zi., 2 (qTl )(PoZ| + fl)/V,2 = 0.(3.18)



Equation (3.18) has a solution (2l) which is directly proportional to a.

This solution should be identical to the Hahn-Ramo small-signal solution,

which is also a linear function of a. The Hahn-Ramo solution Is obtained

from (3.18) provided,



Pozi + fl = RP12 oZ (3.19a)



This is another way of saying that the small-signal plasma reduction

factors may be utilized to account for the effect of the fringe field.

Following the technique used to obtain (3.19a), Paschke has approximated

the second-order and third-order fringe effects by the relations



Poz2 + f o 02 pz (3" P21b

Poz3 +f3 123 f poz3 (3.19c)
Thenotiai; ^ -0 21,3 > Poz3 ilr n I-f


The notation < RgQ >f fllu thv aiple of ENlr, and indfcat that
::::::::::::::::::::::::::::::::










R2, the value of reduction factor associated with 2B b, Is utilized when

seeking the second harmonic portion of the second-order solution, whereas

RO 0 is used with the d-c portion of the second-order solution. The

third-order solution has a component of fundamental frequency and a

third harmonic portion. Hence, RI Is used with the fundamental frequency ,

portion, and R3 with the third harmonic. The value of R.3 is determined

by examining the reduction factor curve of fixed b/a at the point 3Beb.

Solving [22] has shown that the application of plasma reduction factors

is valid for thin beams (Beb < I).

Substituting (3.19a) into (3.18) gives



z + 2 (il)'/v + 'l/V 2 + (R1 )2 zi = 0, (3.20)

where the plasma wave-number (Bp) is defined as


p il /ol/e /v (3.21)


Three linear partial differential equations are obtained from (3.16) by

||tituting (3.17) and (3.19) into (3.16) and separating according to

a.wer of a.



z" + 2(il) '/v + ij/V 2 + (BpR1)2 z = 0 (3.22a)

1+ 2) /vo 2/v 2 + B 2 / R022 > z2 = 3(R13 )2 Z1z1
;;iiiiii,,, *


- ilz ]/v 2 2 [zI" il (zi) z1 ]/v


(3.22b)









. . ? 4 2 2
z + 2(z.) ,'v + 3 + pI 2 P z=

2 [ z 2 + z2 2z Zi(2)' -2( o2

z -- '2 ,, ] 2
+ [ 2(z ) Z1 221l z (Z 'i

+2 [( 2 + (2) 1 1i ~2 z2 ;1" ]Vo

+ B 2 [ 3z, 22 2 2 3 < RI- z(l') 2 > z
(3.22c)


The velocity ma, also be expressed as a series solution

3


v ; n.


(3.23)


where vn is proportional to a".
Expanding the denominator of (3.14) into a power series yields

S= (vo zp' + zp) [ + zpl + (')2 + (')3 + .-- ]. (3.24


Equations for the first-, second-, and third-order velocity terms are
obtained by substituting (3.17) into (3.24) and comparing coefficients

of powers of a.


VI = vo Z1' + z1 (3 25

V2 = v1 zl + z2 + 2' (325


v3 z31 + I z2 t 2 1 (3225


)





36





The solution for the electron polarization (z p) is obtained by first

solving the linear, homogeneous, partial differential equation (3.22a) fo

the first-order solution (zl). This solution is substituted into the

right hand side of (3.22b) in order to evaluate the forcing function. The

!second-order polarization (z2) is obtained by solving the linear, nohoo

geno~u5 equation (3.22b). The third-order polarizatio>n is obtained by

sol)ving (3.22c), a-ter First substituting the solutions for z, and Z2

into the riqtht hand 5ide of (3.22c) to evaluate the forcing function. For

this in-vestigation a thi rd-order approximation is considered to be suff i-

cient to) determinle the sntuiration effect. Consequently, the electron

polarization (z. ) j5 appro~xiciatcd by the sun of zl, Z2 and 23.













CHAPTER 4


GENERAL SOLUTIONS FOR THE SPACE-CHARGE-WAVE EQUATIONS


Synopsis


In this chapter the general solutions for the first-, second-, and

third-order components of the polarization, current, and velocity are

derived. Unlike the analyses of Paschke and Engler, boundary condi-

tions are not substituted upon completion of each order of the solu-

tion. Consequently, the general solutions are considerably more complex

than the solutions obtained in their reports. Both Engler and Paschke

restricted their analysis to the investigation of a single drift region,

corresponding to Region A of this report. Since the primary investigation

in this report is the effect of drift-tube potential and reduction factor

changes on the amplification process, at least two unique drift regions

are needed. The additional labor required to obtain a general solution

is necessary since the particular solution in Region B is obtained by

comparison of the general solution obtained in this chapter ""with the

particular sol| iol n for Region A obtained in the ne||||||xt chapt.i...|er. ..




FiurEt- _Drder Solutions


Since the smuallIl-fignal sutian to the ace*'"harte-wave equation


has a.










z1 = Fl(z) cos(wt Bez) + GC(z) sin(wt B z) (4.1)


is chosen. Equations for the functions Fl(z) and Gl(z) are obtained b,
substituting (4.1) into (3.22a).


[F1,(z) + (RIBp)2 F (z)] cos(tw Bez)

+ [G (z) + (R I )2 GI(z)] sin(wt B z) = 0 (4.2)


Since (4.2) must hold for all values of time,


Fl (z) + (RIBp)2 F1(z) = 0 (4.3a)

GI"(z) + (RIp))2 CG (z) = 0 (4.3b)


iThe product RIB is called the reduced plasma wave number and is defined
Illy the symbol (B ,). The solutions to (4.3) are


Fl(z) = A cos(B3qz) + B sin(qliz) (4.4a)

G (z) = C cos(Bqlz) + D sin(Bqlz). (4.4b)


e ral solution for the first-order polarization (zj) is obtained
y tlituting (4.4) into (4.1). The polarization is normalized by

Tplying (4.1) by Bqla. The normalized first-order polarization is










zi = qla zl = [A cos(BGqZ) + B sin(Bqlz)] cos(wt Bez)

+ [C cos(Gqlz) + D sin(Bqiz)J sin(wt Bez),.5)


here the bar (") indicates that the parameter has been multiplied by

8qla. The subscript a of 1qla indicates that this is the value of Bql
in Region A. The absence of a second subscript for the other terms in

(4.5) simply indicates that the terms are applicable for either region.

The particular solutions for Region A or Region B have double-subscripted

terms.

The first-order current (il) is found by substituting (4.5) into

(3.11).


ii/To = BE [C cos(Bqz) + D sin(Bqlz)] cos(wt Bez)


BE [A cos(Bqlz) + B sin(Bqlz)] sin(ut Bez),

(4.6)
where

BE = 8e/qla-


The fTrst-order velocity is found by substituting (4.5).into (3.2a 15



VI/vo = BQl IB Cos(Sq1z) A in(ql)] cos(t z)

+ 811 f cUS.!ffz) e s(iEJ;'3 sJn(wt I *, (4,7)


where










Second-Otrder Soluti~on

The substitution of (4-5) Into (3.22b) determines the forcing

function for the second-order polarization equation.



z2'' + 2 ( z2) 1v +i2' /V2 + 2 < RC) 2 Z2 f2, 48

where

ff2 = KI1 cos(2B qlZ + 2 sin(213 l)

K + K, cus(213qlz + K sin(26qlz)] cos 2(wt B,z)


[K 6+ K 7cos(213q z) + K. sin(213 qIz)] sin 2(wt B e Z.



The constants KI through K8 are defined In Appendix B. The general

solutlon for z2 i5 of the form



ZZ FO(z) + F2(z) cos 2(wt Bez) + cG2(z) sin 2(wt '3z). (4.10)


The reducti101 factor RO 1s assoc iated w! th the ai-c p -rt ion of (4.8) and

R2 with the second harmonic portion. 7hrec tequatfons are obtained by
substitutinq (4.10) Into (4.8) and compa3ringq coefficients of the same

functlon 4f time.


Fo"(2) + (ROS p) 2 EO(Z) = K cos(2Bqlz) + K2 s'n{28 tz) (4.11a)

F2A. +w ( Ro3p 2 F 2(z) K + K 4 rcs2tR .z + 1< ''n(23q1Z (4.11b)










G2"() + (R2Bp)2 G2(z) = K6 + K7 cos(2BqlZ) + K8 sin(ZBqlZ) (4.11c)


Since RO = 0, the solution to (4.11a) is found by integrating twice with

respect to z.


FO(Z) [Ki cos(2BqlZ) + K2 sin(2Bql)]/(2Bq1)2 + E z + H,

(4.12)

where E and H are constants of integration. The complementary function

for (4. lb) is


F2(z)c = I sln(Bq2Z) + J cos(Sq2Z),


(4.13)


where


Bq2 = R2B p


(4.14)


The form of the particular

portion of (4.lib).


Integral is determined by the


F2(z)p C1 + C2 cos(28qlz) + C3 s


The arbitrary i|

(4.15) Into (4



F2()p


C"l .liii,.C2, and C3 are e

I colllllpalring coefficie|









and s = R ql/q2.


The general solution to (L. 11b) is the sum of (4.13) and (4.16).
Comparison of (4.11b) and (4.11c) indicates that the form of the
solution to (4.11c) Is identical to (4.16) provided K3 Is replaced
by Kg,etc. The complementary function involves toic, new constants of
integration called L and H. The general solution for z2 is the sum
of the general solutions for FO(z). F2(z), and C2(z). As in the case
of the first-order solutions, the second-order solutions are normalized.


;2 = E + H Bqla [KI cos(2Bflz) + K2 sin(2t'qiz)]/ (21 )2

+ [T sin(Bq22) + j cos(qq2z) + 3qla LK3 + g K4 cos(2Ijqiz)

+ 9 Kg sin(2Bqlz)}/(rlq2)2] cos 2(w't GBz) + [[ sin(Bq2z)

+ H cos(Bq2z) + Gqla (K6 + g K7 cos(2G 1z) + g Kg sin (2Bqlz))

/(Bq2)21 sin 2(wt Bez) (4.17)


he second-order current is found by applying (4.17) to (3.11).


2 BE [ cos(q2z) + i sin(B qz) + Bq a (K + g K

hcos(2BqlZ) + g Kg sin(2Bqlz))/(Bq2) 2] cos 2(ot Bez)

S2 BE [ cos(8q2z) + sin(8q 2z) + Sqla (K3 + g K4

(28 z)4g 4 KS1C sin(2Bqlz))/(Bq2)2] sin 2(Qt Bez)
(4.18)









The second-order velocity is found by substituting (4.5), (4.7), and
(4.17) Into (3.25b).


v2"vo = E + BE BQ1 P10/2 + (BQ)2 P24 -[K2/(63ql)] cos(2BqlZ)

+ [K./(6i3q)] sln(Zqlz) + [BQ2 C cos(Bq22) BQ2 J

-sin(Bq2z) K6/B + (I + 2s2) g [K5 cos(28qZ) K4

*sin(2Bqlz))/(3Gql)] cos 2(wt BeZ) + [BQ2 L coS(Bq2Z)

BQ2 i sin(Bq2z) + K3./Be + (1 + 2s2) g (K8 cos(2Bqlz)

K7 sin (23qiz)}/(3Bq])] sin 2(wt SBe) (4.19)



Third-Order Solutions

The forcing function for the third-order polarization equation is
evaluated by substituting solutions for zI and z2 into (3.22c).


z3" + 2(i3) '/Vo + vo2 + < R3 p2 3 [ K cos(Bqlz)

+ K0I cos(blBq2z) + K,1 cos(b2Bq2Z) + K12 cos(3B[qz) + Kli |lin(Bql;

+ K. sin(b Bq2z) + K sin(b BqZ) + K16 sin131. I I(t

+1K17 cos (Bqlz) + K18 cos(blBq2z) + K19 coIs(b2i2z) +

*coS(3Brq;li) *+ KIii iInii 1l) +i K22 sn(b bai2ii) + '(21 iin(b26J22)










+ K24 sin(3BqiZ)] sin(wt Bez) (4.20)


Only that portion of the forcing function which is of fundamental

frequency has been retained. The other portion, which Is a third

harmonic function of time, and hence is associated i,;th the generation

of a third harmonic component of polarization, is omitted. The reason

for this omission is simply that the problem is concerned Ilth the

fundamental frequency, and harmonic components are carried only when

they are required to evaluate solutions which involve fundamental

frequency components. The notation < P 2 > Is replaced by P12 because

Ibf the omission of the third harmonic component.

The form of the general solution for the third-order polarization

is

23 = F3(z) cos(tt Bez) + G3 () sin(wt Bcz). (4.21)


|T||h equations needed to determine F3(z) and Gj(z) are obtained by sub-

stituting (4.21) into (4.20) and comparing coefficients of the same

function of time.


F3 (z) + (RI p)2 F3(z) = A, (4.22a)



LG3 (z) + (R1 I)2 G3(z) Bi (4.22b)


fre A presents the coefficient of cos(at 8,z) and B1 of sin(wt Bez)

gsven in .(4.20). The solutions of the complementary functions of (4.22)








are
F3(z)c P cc.s(qiZ) + Q sin(Bqil) (4.23a)


G3(z)c = R cos(Bqiz) + S sin(Bqlz). (4.23b)


The solutions for the particular integrals are determined by the forcing
function in (4.20) Since cos(Bqlz) is part of both the forcing function
and the complementary function, the general form of the particular Inte-
gral must be modified to Include the term cl z cos(3qlz). A similar
substitution is made for the sin( qlz) term. The various coefficients
are evaluated by substituting the general form into (4.20) and comparing
with the forcing function. The general solutions for F3(z) and G3(z) are


F3(z) = [P K13z/(2Bql)] cos(qlz) + [Q + Kgz/(2B1q)] sin(Bq1z)

+ [K I cos(blBq2z) + K14 sin(blBq2Z)]/(b3 q22)

[K11 cos(b2q2 q) + K 5 sin(b2Bq2z)]/(bBq22)

[K12 cos(3Bqlz) + Kl6 sin(3 qlz)]/(88Bq2) (4.24)


G (z) = [R K21 /(28 )] cos(qI z) + [S + K 17z/(20 )] sin(Bqlz)

+ [K8 Cos(blBq2z) + K22 sin(b Bq2z) ]/(b 3 Bq2RhIiII !ii .

[K19 cos(b2Bq2Z) + K23 sin(b2 aq2z)J]/(bO11q2

[K20C(38qlz) + K24 sin(38qlz) ]/(i ij (4 125)










The general solution for the normalized third-order polarization ;s



23 = 3(z) cos(Ct fe') + G3(z) sin(ut 0ez), (4.26)


where the bar (1) indicates that the term is multiplied b/ 8qla.

The third-order current is obtained by substituting (4.26) into

(3.11).

13/io BE 63(2) cos(wt lez) BE F3(z) s;n(ct Gez) (4.27)


The third-order velocity is determined by substituting zl, z2, 2z, .',

and v2 into (3.25c).


o /v = [F'(z) + Fa(z) + Fc(z)] cos(Wt 8e)


+ [G3 (z) + Fb(z) + Fd(z)] sin(wt B3z), (4.28)


here F3 (z) and 6 (z) are the derivatives with respect to z of F3(z)

and 03(z) respectively, and the functions F8(z), Fb(z), F((z), and

(i)are defined in Appendix B.
In this chapter the general solutions to the space-charge-wave

ans are determined. Boundary conditions are applied in the next

r at the input gap to determine the particular solutions in

ocn A. In the following chapter, boundary conditions are applied

a point where the drift tube properties change to determine the

ular solutions In Region B. The solutions and wariables which




47





apply in one particular regi.,n .ill be denoted b; a second cubscr;pt.

Thus the first-order polarization ;n Region A will be denoted as 21a,

,hlle in Region B, Lhc solution ; donr-ted as lb'














CHAPTER 5


PARTICULAR SOLUTIONS FCR REGION OF FIRST DRIFT TUBE


Synopsis


In this chapter particular solutions are obtained from the general

solutions of Chapter 4 by applying boundary conditions at the input

cavity located at Z = 0. The solutions for this initial drift-tube

region, known as Region A, are utilized as entrance conditions for

obtaining solutions for Region B. The information obtained from the

solutions in Region A pertains to a basic two-cavity klystron amplifier,

driven at low signal levels, with the nonlinear beam effects taken into

account. The phenomena investigated include current saturation, shift

in the locations of velocity and current maxima and minima as signal

voltage increases, phase shift, and second harmonic current generatir.n.

llconstants identified with Region A will bear an a subscript.

NprIcal data used to plot the various curves presented In this chapter

were obtained by use of a Fortran compiler applied to the IBM 709 computer.




Boundary Conditions


A inal voltage (Vs) is applied to a pair of closely-spaced grids

atz = 0. The signal voltage is given by










V,(t) = Vm sln( t), (5.1)



where w is the angular frequency at which the voltage varies. A previously

unmodulated beam of electrons passes through the pair of grids. The effect

of this a-c voltage on the electron beam is the establishment of velocity

modulation. The extent of the effect Is determined by applying the

principles of continuity of current and conservation of energy. The usual

klystron assumptions [30] discussed in the following sections are applied

at the Input gap.


Continuity of current

The principle of continuity of current Is used to determine the

Initial current at z = 0. Since the beam Is unmodulated before entering

the gridded gap, the current modulation upon exit is assumed to be neg-

ligible. Hence,



Ti(o) = 12(o) = i3(0) = 0. (5.2)



This conclusion Is based on the assumption that the two grids are so

closely spaced that there is no time for electron bunching to occur,

I.e.,the transit time Is zero. Using the assumption of closely-spaced

grids, it may be assumed that the beam coupling coefficient (4) is unity.

For gaps with finite grid spacing the coefficient is always less Ci h

one. Since it is assumed that the transit time II zerdi ii t follows, that

the Initial polarization (zp), or displacement from the undisturbed post

tion, is also zero. This condition Is used to consllete the second-order

polarization solution.










Conservation of energy

The velocity of the electron beam upon exit from the Input cavity is

obtained by applying the principle of conservation of energy at z = 0,

along with the assumption that yi = 1.



(m/2) (a2 oa2) = -e Vsn(wt) (5.3)


Transposing,

va2 oa2 + 2 Vm sin(it). (5.4)


Since

oa2 2 r Va, (5-.5



where Va is the potential of the drift tube in Region A. (5.4) may be

written as


Va2 2 11 [Va + V sin(ut)], (5.6)


ii velocity in Region A is obtained by factoring V., taking the square

|ot, and replacing Ji2 Va by voa


va = Voa 4 + (Vm/Va) sin(wt) (5.7)


hrati V A is by definition the voltage modulation index (a).


a aVm/Va.


(5.8)









Application of the binomial expansion to (5.7) yields



va Voa [C + (a/2) sln(Ft) (a/2)2 sin2(Jt)i2 + (a./2)3

Ssin ( t)/2 + ... ]. (5.9)


Completing the square and cube in (5.9) gives


va f Voa (i + (a/2) sln(ut) (a/4)2 [I cos(2(t))


+ (a/4)3 (3 sin(rt) sin(3wt)) + ... ]. (5.10)


By applying Paschke's method of successive approximations the beam
velocity may be written as


va = Voa + Vla + VZa + V3a, (5.11)


where vna is proportional to an. The various velocity ,itermmay ir||be

evaluated at z = 0 by comparing (5.10) and (5.11) by powers of a.



v1() (W2) sIn(wt) (5,12A

V -to) /4 P cos(240il (5,12b


where










First-Order Soluirons


Two of the first-order constants of integration are evaluated by

applying (5.2) to (4.6).



?1(0) = xj [Ea cosiLt) sin(wt)] = 0, (5.13)

where

Sa = "ea' qla. (5.14)


sincee (5.13) must hold for all values of time,



A8 = F = 0. (5.15)


The *thrr two constants are determined by applying (5.12a) and (5.15)

to (4,7)


V10) Ba Cos(wt) + D. sin(wt) = (a/2) sin(wt) (5.16)



Ba = 0 (5.17a)


Da = a/2. (5.17b)


-order viriatIons are obtained by substituting (5.15) and (5.17)

itlT (47).


2) sin(z) cqas(!t. xaz)


MI


(5.18a)









vla = (a'2) cos(z) sin(ut "a;). (5.18b)

where
S q= la (5.19)


These solutions are identical wlth the small-signal solutions obtained

by Hahn [)l and Ramo [2].



Second-Order Solutions


Before obtaining the second-order solutions, it Is necessary to

evaluate KR through K8 as given in Appendix B. The values in Region

A are obtained by substituting (5.15) and (5.17) Into the general

expressions.

K1 = K3a = Va = Ka 0 (5.20a)


K2a = 3(a/4)2 (5.20b)

K = 3 (i/4)2 (5.20c)
5S

K6a = xa(a/4)2 (5.20d)


K7a = 3x,(a/4)2 (5.206)


Two of the second-order constants of integration are obtainhed by substi-||.i

touting (5.2) and (5.20) Into (4.18).










2a(0) 2 x [Ha + Sata (c/412 (I + 3 ga)] cos(2wt)

2 xJJa sin(2ct) = 0. (5.


ce,

J~ =0 (5.



a = ta g ( (/2) (5.


remaining second-order constants are determined by substituting

12b) and (5.20) into (4.19) and comparing functions of time.



S a(0) = Ea + (a/4)2 (1/2) (/4 + ( as a) sin(2wt)

+ /s 2 2 ) g (a/)2
+ [Tals (e/4) (I + 2 sa2 9a )


Scos(2wt) = (a/4)2 [1 cos(2wt)].




Ea =-(3/8) (a/2)2


a 3 (a/4)2 Saga (1 2 sa2)


21)



22a)


22b)


(5.23)




(5.24a)

(5.24b)


La = 0. (5.24c)


on of boundary conditions, it was mentioned that the

l..i(zp) is 5 ro provided the grids are closely spaced.

IIEILL!) is determined by applying this boundary

22),and1 (5.24) to (4.17).


Hen


..........=i i ....










Ha = 0


The second-order

(4.18).


(5.24d)


current Is found by substituting (5.22) and (5.24) Into


2 2 2
12a = (1.2) (a'2) ta[l + 3 ga cos(2z) 4 ga (1 Sa

cos(z,'s )] cos 2(rt xaz) + (3/2) (a/2)2


Staga [a sin(2z) (I 2 sa2) sln(2,'s)

sin 2(wt xaz) (5.25)


The second-order

into (4.19).


velocity is found by substituting (5.22) and (5.24)


V2a = (a/4)2 [cos2 ( +) + (1 3 ga (1 2 s2) cos(z/sa)

+ ga (1 + 2 sa2) cos(2))} cos 2(wt Xa8) + xag

*((1 + 2 Sa2) sln(2i) 4 sa(1 sa2) sin(i/sa))


-sin 2(wt xaz)] (5.


In order tV

through K24, as

Equations (S, I15

t hemse plrameteri


I-Order Solutions


third-order soluI









Ka (9/4) (c/2)3 taa (5.2a )

K = (3/4) (a/2)3 a [2 s (b sa ( 6 s + 1) '(4 s )] (5.27b)
10a aa a 2a a a a

l = (3/4) (a/2)3 x ag [2 s (b s 2) + (6 s 1), ( s )]
1a a a a a (5.2c)

KR2a (15/4) (a/2)3 s taga (5.27d)

K13a K4a = KI5a = K16a = 0 (5.27e)


k17a = 18a ig9a = K20a = 0 (5.270)

I i (a/4)3 9g [54 s 2 + 2 2 + 14 t 16 s 2 t 9].2 (5.279)

21a ( 3 ba g ) [9 (1 2 s (b 2s2)
K22 = 2 (a/4)3 bla (g /sa) [9 (1 2 sa2)/4 + xb (b 2s2)
..... l a.a Xatb2a b4a a


S- 2 (a/4)3 b2a (a/s ) [9 (1 -


= (a/4)3 g [90 s 9 + 14 t 2
a a a


Two of the third-order constants

ying (5.2) to (4.27).


(5.27h)
2 sa2)/ + at bla(b3a + 2 sa)

(5.27i)
- 2 x 2],2 (5.27i)
a


of integration are evaluated by


73(0) = xa C3(0) cos(at) xa F3(0) sin(wt) = 0


Condition must hold for any value of time,


G3(0) = 0


(5.28)


(5.29a)









F3(0) = 0. (5.29b)

Appl,inq (5.29) ro (4.24) anJ (4.25) gives


2. -b S 2. b3
Pa + V10 s ba II Sa b4a a 2S = 0 (5.30


R + K18 a /b3 K 19 5 b K208 = 0. (5.30b)


The third-order constantly Pa and R are fuund by transposing (5.30) and
applying (5.27f).


a 12.'8 + sa2 llb4a 1- Kob3a (5.31)


P.a = 0 (5.32)


The remaining third-order constants are determined by applying (5.12c)
to (4.28).


;3a(0) = [F3a (0) + Faa(0) + Fr (0)] cos(wt)

+ ['(0) + Fb() + F) Fda(0)] sin(wt)

= 3 (a/4)3 sin(wt),

where F3'(0) Implies differentiation of F3(z) with respect to z ind
evaluation at z = 0. Applying (5.27) to the derivatives of (Ii24)
and (4.25) gives

..::EEEE6=









3a '() = Q (5.34a)


G3a'(O) a + (sabla'b a) K22a 21a'2 (Sab2alb4a)23a

(3/8) K24a. (5.34b)


The values of Fa(z) through Fd(z) for Region A, and evaluated at z = 0, are


Faa(O) = 0 (5.35a)

Fba(0) = 9 (a,4)3 (5.35b)

Fa(0) = 0 (5.35c)

Fda(0) = 3 (a/4)3. (5.35d)


SiIbituting (5.34) and (5.35) into (5.33) gives


Scos(wt) + [S K2/2 + (sab a/b3) K22 (sab2/b4a)

K,23 -(3/8) 24a 12 (a/4)31 sin(t) = 3 (a/.4)3 sin(ct).
(5.36)
rdorder constants are evaluated by comparing functions of time.


a= 0 (5.37)


21/ (s bl/b 3) 22 + (3/8) 24a + (sab2a/b4a) K23a

+5(V 1 e(5.38)









Since the third-order constants have been established, the thlrd-

order solutions for the current and velocity can be evaluated,



13a aa3 [A 6sin~z) + A 7 in(b ]azis ) + A8 sln(b2a/5sa

+ A 9sinf3 ) + AO 10zCos(Z-)] cos(wt xoz a x a 0


CAI cos(2) + A2 cos(ble /sa) + A 3cos(b2az/s a) + A 4


ccs(3z) + A9 5 z in(;)] sin(wt xa z)



v'3, = G1 [All -z cos() + A12 sin(z) + A 1 n3 lzie

+ Ag si n(b,,; 5,) + Al 155 r(31) cos(cot x,-) + t3

,[A-1(, sin(z) + A 17COW() + A18 cos(b)a r/lE) 4 l9


cos !baaz + A20 co-s(3z)] slin(Lt .z'0 (54 )


The constints A, Lhroug)h A 2D are- evaluat-f n Ajppcndix B.




Results

The, fundavintal cclponent of current is foumd by addinrg (5)amnd




ifa + ia*3a 5.1











in Fig. 5.1 the amplitude of the fundamental frequency current is plotted

against distance along the beam axis (z) for three values of a. The

small-signal solution (ila) is also shown for the purpose of demon-

strating saturation effects. Several interesting points are noted

from examination of this figure. These are separately discussed below.

The amplitude of the fundamental frequency velocity, along with the

corresponding snail-signal solution, is shown in Fig. 5.2.


Saturation of fundamental current

Small-signal theory predicts an increase In the amplitude of the

fundamental current, which is linearly proportional to a; ho%,ever, it

is apparent from Fig. 5.1 that the amount of increase becomes pro-

portionally smaller as a increases. This saturation effect is a well-

known experimental fact. There is some reason to doubt that the

apparent growth in current with distance for a = 0.3, as ascertained

from this theory, is physically real, Inasmuch as this data comes from
2
Ia reg|ii.on where overtaking has occurred. Fig. 5.3 shows the effect of

aturtion as maximum current is plotted against a. It should be

ed that the apparent Increase in the slope past a = 0.3 is well

iide! the range where this third-order approximation Is valid.


ins vf velocity and current maxima and minima

i ii also apparent from Fig. 5.1 and Fig. 5.2 that the locations

nairum and minimum current and velocity shift as a is increased.

inIl is of the maximum and minimum values of current and velocity


of iihetaklng is discussed in Appendix C.




61

f Nonl inear soultion

1.6 Small-signal solution




14 0.3


























o.4'






Gpab = 1,0


\ I









0.15
/

online r solution /
0.13 Small-signal solution



.1 a = 0.3


.\ I
0.09 ,. /I /,
a=0.2 I/
I .. I /







I/ 'IN/ I /






b,'a = 2/3
Geab 1 .0




WO= 80 120" 160" 2000

Fi 5.2 Fundamntal frequency velocity vs. position.




63




f



Solut ion







1 20





tlieory


0'a


/Nun I nea r
solutAt iun
















00 3 oA0





Fig, 5,3 f ladimi fuimimntol Trequcnvy curroot as a fura:tton of a











as a function of a are shown in Fig. 5.4. These shifts from the locations

predicted by small-signal theory are appreciable even for a moderate value

of a such as 0,2. This shift in position has been experimentally observed

by Mihran [13].


Phase shift

Since this analysis is a nonlinear approach, phase shift should be

expected. The fundamental current is given by the sum of (5.18) and (5.39).

and may be expressed in the form



fa fa cos(Wt xa z f). (5.42)



Examlnation of (5.39) shows the presence of both a cos(wt) and a sin(jt)

component. The first-order term, as given by (5.18), has only a cos(wt)

cllmponent. Hence, the magnitude of the fundamental current Is given by



( j 2 + ( )2 (5.43)
fa la 3ac 3as


e the subscripts c and s indicate cos(kt) and sin(ai) coefficients,

spectively. The phase shift is given by



6fa = ten- +I3as (5.44)
[a +3ac


Te shift increases markedly with increasing a, as indicated in

S his phase shift Is evaluated at the point of maximum current

















160'




















140 0.2
K = 1.0

120 G ab = 1.0





100





80 "





60






0 0. I 0.2

Fig. 5.4 -. ..Loaton of current aIlllve:
mll in) as a function llllfa.


v
ma3


/n


r:i i l

/


vin









eFO


28"





24o





200





16"





120





8


K 1.0
Sb/a = 2/3
40 G e b = 1 .0







I 0.1 0.2 0.3 0.4 a

F;ig. S Phase shift at the point of maximum current as a
function of a.











for each value of a These results indicate that the technique of ignor-

inq the out-of-phase ccnponent, as employed by Paschke [23, 24], leads

to considerable error, particularly/ for larger values nf a.


Second harmonic generation

The magnitude of second harmonic current is shown in Fig. 5.6.

Examination of (5.25) indicates that the magnitude of second harmonic

current increases as the square of the value of a. Comparison of Fig. 5.1

and Fig. 5.6 would tend to Indicate that better performance could be

obtained by tuning the klystron output cavity to the second harmonic

rather than the fundamental. This conclusion is unwarranted. however,

since the fundamental current solution is based on terms from the first

order and third-order solutions, with the third-order term contributing

to saturation. On the other hand, the second harmonic solution shown

neglects an additional second harmonic term which would be contributed

by a fourth-order solution, and therefore falls to show saturation effects.

The second harmonic solution obtained by this third-order approximation

Is analogous to the fundamental frequency solution obtained by small-

signal analysis. i


.li

















; = 1.0
ba=23
b b 1.0
ea


Fig. 5.6 Second harmonic current vs. position.


a = 0.3













CHAPTER 6


PAPTICLLAP SULLUTIOIlS FOR PEGION OF SECOND DRIFT TUBE


Synops is


In this chapter methods are investigated for amplifying the current

beyond that obtained in the simple two-cavlt' klystron discussed in the

previous chapter. The model for the space-charge-wave amplifier is

identical to a two-cavity klystron, except the drift tube is divided

into sections, each of which na/ have a different potential and a

different diameter. The additional amplification results from changes

in the drift tube potential and radius. Both of these changes effect

the value of the plasma reduction factor (RI). This amplifier represents

a combination of the velocity-jump amplifier proposed by Tien and Field

[3), and the rippled-wall amplifier investigated by Birdsall and Whinnery

[4]. A third method for obtaining amplification is to vary the beam

diameter; however, this technique is not considered because of the addi-

tional complexity introduced both in the theoretical analysis and in iii

construction of a working unit. The analysis includes the effect of

placing the transition in either the vicinity of maximum velocity m.ci-ii

nation, or in the vicinity of minimum velocity modulation. Th InvEt-

nation also Includes a search for the optimum location of the transition,

in so far as concerns the achievement of maximum current modulating at

some point further down the beam. The solutionsll foir current ard vel

69










are obtained from the general solutions in Chapter 4, by applyin3 boundar;

conditions at the transition. All data for the figures in tnis chapter

are obtained from computations performed by the IBM 709 computer. The

data for the input to the transition are obtained from the sclutIrns for

Region A, using the beam parameters specified in Chapter 5.3




Boundary Conditions


The modulated electron beam drifts under the influence of space-

charge forces until it reaches the position z = Za, at which point the

e|ond drift tube Is positioned. The drift-tube potential changes from

Va to Vb at the junction of the two tubes. From the conservation of




(m/2) (vb2 va2) = lel (Vb V a),




vb2= 2 (Vb V a + a2. (6.1)


g he relation

vo2 = 2 -. V, (6.2)



help I||| r: that the computer program for Region B was correct,
o were performed withol|u||t making any change in either drift-
bl oIr plasma reduction factor. The results of these compu-
mparadu wit' the solutions for Region A extended through
overvd by Region B. The results were identical to three
r2ures. The solutions obtained for Region A were compared
lutions obtained by Englel.lur HI26].


70










(6.1) may be expressed as


,% = 2 2 + v 2 (6.3)
b ob oa a '
or

C. 2 v (6.4)
b ob a Oa oVb (6.4


Expanding (6.4) in a binomial series gives



"b t vob [I + y'/2 'y2/g y3/'16l, (6.5a)

where

y = (Ca2 Voa2)Vob2. (6.5b)


The velocity (va) may be written as a power series in a according to

the method of successive approximations.



va = ya + Via + V2a + V3a' (6.6)


where the various order terms (v ) are proportional to ~n, Substitution

of (6.6) into (6.5b) yields



y [2 voa (va + V + 2 V+ v2a2 + U3 2


+ 2 (va V2a + Vla V3a + V2a V3a) b2 (6


It is assumed that the a-c terms are all consi derably smallr than th.

iiiiiiiiiiiiiiiiii' ,,,,,,










d-c terms. Consequently, (6.7) may be approximated by



y = 2 voa (Vla+ '2 + 3a).'b2 (6 8)


Equation (6.8) is substituted into (6.5a), and the approx.imation that

Vna/voa < 1 is applied. The resulting velocity in Pegion B i



vb Vb [I + voa I"la + +2a + va)" (b)2]. 6.9)


The method of successive approximations is applied to vb. Comparison

of terms In (6.9) yields



vnb(Za) Va/Vb na(a) , (6.10)
n 1, 2. 3


IEsenclally this is the small-signal assumption used in the analyst ; of

Ith velocity-jump amplifier by Tien and Field; however, the solution in

Region B contains second-order and third-order tcri.s In addition to the

ima3l-signal solution.

he second boundary condition is the continuity of current. Hutter

S .sh.ws that the current is continuous across a velocity jump by

s nmatic considerations, and alternately as a limiting case of

ll eterson equations.


.2 the largest a-c velocity term is approximately 10 per
velocity.










Consequently,


inb (Z ) = na (a). (6.11
n = I, 2, 3




First-Order Solutions


The first-order current in P.cgion B Is found from (4.6).



ilb = BEb [Cb cos(BQlb z) + 5b sin(BQlb z)] cos(wt BEb z)


BEb [Ab cs(BQlb ) + b sin(BQlb z)] sin(6t BEb z)
(6.12)

The first-order current in Region A is obtained from (5.18a), and is

evaluated at the boundary z = Za2



ila(Z) = (a/2) xa sin(Za) cos(wt xa Za) (6.13)


Two equations for the first-order constants are determined by applylngi

the continuity of current condition to (6.12) and (6.13). The two

equations are obtained by comparing coefficients of ~"n liiiii) *d 1 nift)



(a/2) x8 sin(Za )co(allS a


BEb cb cosIQtI 1 + Db Sln(lQlb ) cosZE] Za
+==========================================





74





(a/2) X. sin(2a) sin(xa Za =


SEb ICb Cos(BQlb Za +b sin(BQlb Z] sin(BEb Z.


BE b [Ab cos(SQ] b Za )+Eb -in(BQ1 b a)] cos(BE b Z



The other two, equations needed to evaluate the four co nstants ar

derived frcci the veoucity boundary condition, The first-order velocit

in Region A is obtained from (5.18b).



v (7 (4t/2) CCS(Z ) Sin(Lot X- Z ( )
la a


11he fir-t-order velocity in Raeq~n B is found from (4-7).



V ib(Z b/VaBQ b b c.oS(EQb b sin(BQI b ]

Scos('t BEb 7) + I(Vb/V -Qlb Ib cos BQlb Z)






The first-order veiiccity (vlb) is norm.al;2cd with rcspcct to the d-c

velocity V 00) in Region A, rather than with rcsptect tco the d-c

voICfIty (vab) in Region 5. This ;s done in order that the t%,o normalized

first-order velucity terms voa3 and vob, may be comtparcd di re-ctly, The

boundary co*nditiop 61)i applied to (6,15) and (1).Thri remainIng

two oqvaticans needed to evaluate the first-crdcr constant!, are obtained










by conparing cocfficients of cos(wt) and sin(lt).


(a 2 (Va Vb) cos(Z.) sin(x Z) =


BQIb [b cOs(BQIb Z) Ab sin(EQIb Za)] ccs(BEb Za)

BQlb [Ob cos(EQIb Z) Cb sin(EQlb Za)] sln(BEb Za)
(6.173)


(a,'2) (Va/Vb) cossZ') cos(xa3 Zl


BQlb [Bb coslEQlb a) Ab sin(BQIb 2 )] sin(EEb Za)

+ Qb [Db cos(BQIb Z Cb sln(PQlb Za)] cos(BEb Z)
Sb b a' b b a b z
(6.17b)

The first-order solutions are established once the constants Ab, Bb' Cb'
and Db are evaluated and substituted Into (6.10) and (6.16). Cramer's
Rule5 is used to evaluate the constants.


Second-Order Solutions

The second-order current in Region B is obtained from (4.18).


2b(Z) = 2 BEb [Ylb(z) + Mb cos(BQ2b z) + Lb si n(lQ2 iZ)

cos 2(at BEb ;)- 2 BEb[Y2b(z) b Dos(IiI2 Z)


5See Appendix D for a discussion of the solutions for these ,ll










+ Ib sin(BQ2b z)] sin 2(wt BEb Z),


where

Ybb(z) = Sb [ 6b + b K7b cos(2 BQIb ;) + b KSb



Y2b( ) = b [K3b + gb kb (2 b ) b Sb




The second-order current In Region A is obtained from

ated at the transition (Za).


sin(2BQI b z)] 'BQ2b

(6.19)

sin(2 BQIb -)]/BQ2b

(6.20)


(5.25), and c.alu-


12a(Za) a Yla(Za) cos 2(ut x Z) + Y2a(a)


Ssin 2(wt x, Z ),


(6.21)


Yla (Z) = (1/2) (a/2)2 t 2 [I + 3 g cas(2 Z )
1a a a a a


- 4 g9 (I a2) cos(Zaisa)]


(6.22)


2a(Za) H (3/2) (a/2)2 ta ga [sa sin(2 2)


- (1 2 s 2) sin(Z/s )].


(6.23)


s equated to (6.21), using the boundary condition for

rent given by (6.11), and coefficients of cos(wt) and


(6.18)









Yla Za) cs5(2 Za) 2a(Za) s;n(2 xa a =

2BE [ l(Z ) M cos(PQ2 Z ) + L sin(l Q2 Z )]
b Ib a b b a b b a

Scos(2 BEb a) + 2 EEb [Y2b(Za) + b cos(BQ2b Za)

+ I sin BQ2 Z )] s.n(2BE Z 1 (6.24a)
b b a b a


'la(Za) sin(2 xa Za) + y2a(Za cos(2 xa Za) =

2 BEb Ib(Z) b cos(BQ2b Za) + Lb sin(BQ2b Za)]

sin(2 BEb 2a) 2 EEb ['2b(Za) + 3b cos(BQ2b Za)

+ lb sin(SQ2b Za)] cos(2 CEb Za) (6.24b)


The second-order velocity in Region B ;s obtained from (4.19), and
normalized with respect to voa.


v2b(Z) =-Vba [Eb + BEb BQIb PlObi2 + (BQJb) 2b/4

(K2b/6) cos(2 BQlb z) + (Klb/6) sin(2 BQlb z)1

+ Vb/V [BQ2b (Tb cos(BQ2b ) Jb sin(BQ2b Z))

+ Y3b(z)] cos 2(wt BEb z) + Vb'Va [BQ2b [b

cos (BQ2 z) Mb sn(BQ2b z)) + Yb(z)]

sin 2(wt BE zI,2
iii 0ll l 'll+0l' :! ! !:'
:'*iiiii-iii'








where

Y ( ) = ,, + (9 /3) (CI
3b 6b b b

.4b sin(2 BQIb i)]


S2 cos(2 eQ )
b Sb b


Y4b(Z) = 3b"b + (9b/3) (I + 2 b2) [b cos(2 BQIb 2)

R7b sln(2 BQ lb )] .



he second-order velocity in Region A Is found from (5.26),

ii the point of discontinuity (Z ).


2a(Z) = (/4)2 [cos2 (Z 1 + Ya(Z ) ccs 2(wt x Za)


+ Y4 (Z ) sin 2(wt xa Z )a,




Y (Z) + a (I + 2 sa2) cos(2 Za) 3 9 (1 2 sa2)


V 4(Z) = x (1 + 2 s 2) sin(2 2 ) 4 g a (I s 2)
=aa aaaa aa a


(6.26J




(6.27)


and evaluated


(6.28)




cos(Z a, )

(6 29a)

sin(Z /s ).
a a

(6.29b)


.25) and (6.28) are compared at z = Za by using the boundary

.I6.10). Three equations for the second order constants are

-uparing the coefficients of cos((-t), and sin(wt), and the

lint terms.

(V ab) cos2 (Zg) = Eb + (1/2) BEb BQlb PlOb + (1/4)









*(BQb)2 P2b (1.6) 2b cos(2 BQlb a + 16) lb

-sin(2 BQIb Za) (6.30)


-(a/4)2 (a/Vb,) [Y3a(Z.l cos(2 xa Za) Ya(Za) sln(2 xa Z )

= [BQ2b Ib cos(BQ2b Za) ab sln(BQ2b Zol b + Y3b(Z)]

Scos(2 BEb Za) EBQ2b Lb cos(BQ2b Za) b sin(BQ2b Za))

+ Y4b(Za) ] sn(2 BEb Za) (6.31a)


-(a/4)2 (Va/Vb) [Y3aZa) sin(2 a. Za) + Y4a(Za) cos(2 xa Za))

= [BQ2b (b costBQ2b Za) ib sin(BQ2b Za) + Y3b(Za)]

sin(2 BEb Za) + [BQ2b [Ib cos(BQ2b Z) Mb sin(BQ2b Za))

+ Y4b(Za)] cos(2 BEb Za) (6.31b)

The second-order constant Eb may be obtained from (6.30).


Eb (/4)2 (V /Vb) cos2 (Za) (1/2) BEb BQ b PlOb (1/4

(B 2 2b (1/6) K2b cos(2 BQlb Z,) )

sln(2 6Qlb Za) (6,32)


Equations (6.24) and (6.


resent a set of four




80




four secend-order constants. The solutions to this set of equations are

also discussed in Appendix D.



Third-Orccr So lutions

The thlrd-order current in Reg ion B Is obtained f r-r (4.27).


i 3b) BE b R bcos(EQ1 bZ) + S b in(BQ1 b ) + y 5b W]

Sccs(wt BE b z) BE b[F b cos(BQI b Z + Q

JInfQl b ) + y 6b (Z s in(wt BE b ), (6-33




y5b(2 I 2 ) b l -b sn(EQI 2 6 F21b COS(SQ]b Z) sb/b 3b

[K l8b cos,(FQ2 b blb Z 22b sin(BQ2 b b zb )] B2 bL


S(sb/bhb) 119b rco(BQ2b b2b Z) + K-23b in(BQ2b bb z)]

/BQ2b K20b cos(3 Fjtb z) + K 24, sin(3 '3Q'b Z)/( SQb)




Y 6 (Z) (212) IK9b ir(BQI z) 13b c" ( BQ1 ) sb /b 3b


)0b cs(BQ7 b b lb z )+ b, sin(BQ2, b,, -,)]/BQ2b

s b ibcos(BQ2b b2b z) + r-1~ ,in(BQ2b b, ]*E/Q2









-[12b cos(3 BQIb ;) + k16b sin(3 BQlb ;](8 EQlb)
(6.35)

The third-order current In Region A is obtained from (5.39) and evaluated
at the transition (Za).

I3a(Z ) (Sa ) cost -x Z ) ) sin(t x ),
3a a 5a a a a(a) Sa a Za)
(6.36)
ijhere

Y(Za) = xa 3 [6 sin(Z) A7 sin(bla Za/sa) + A8 sin(b2a Za.'a)

+ Ag sln(3 Za) + A10 Za cos(Z )] (6.37)


6a(Za) = Xa a [A1 Cos(2a) + A2 cos(bla Za/sa) A3 cos(b2a /a)

+ A4 cos(3 Za) + A Za sin(Z )]. (6.38)


Equations (6.33) and (6,36)are compared at z = Za by using the continuity
of current boundary condition. Equating coefficients of cos(wt) and sin(wt)
yields two of the equations needed to evaluate the third-order constants.


Ya (Z ) cos(x Z ) + Y (Z ) sin(x Z ) a

BEb [Rb cos(BQlb Za) + Sb sin(BQl1b Za) YSb(Za)]

cos(BEb Za) + BE b [. cos(BQlb aZ) + b1

sin(BQlb Z) + Y6b(Za)Z sin(BEb )(










Ya(Za) sin(xa Za) 6a(Za) cos("a Za) =

BEb [.b cos(BQlb a ) + Sb sin(BQlb Za) + Yb(Za)]


sin(BEb Z ) BEb [Pb cos(BQIb Z ) + Qb sin(Qlb Z )

+ Y6b(a)] cos(BEb Za) (6.39b)


The third-order velocity in Reqion A is obtained from (5.40) and
valuated at z = Za.


3a(Z) Y 7(Z) cOS(ut x Z + Y8(Z) sin(.t x Z ),(6.40)
3a a 7aa a 8a QZa)

here

I Y 7a (Z) a3 [A 1 Za cos(Za) + A2 sinl(Z ) + A 3 sin(b Za ".)


+ A14 sln(b2a Z/S) + A5 sin(3 Za )
14 2a 88 15I a


(6.41a)


(Za) =3 A 16 Za sin(Za) + A17 cos(Za) + A18 cos(ba Za/sa)


+ A19 cos(b2a Za/a) + A20 cos(3 Z )].


(6.41b)


rder velocity in Region B is obtained
!fty is normalized with respect to voa
shed by multiplying (4.28) by VbJ/Va,


from (4.28). The third-
rather than vab This


; [BQl & b cos(BQlb Z) BQlb b sin(BQlb z)




83




+ y (2 05<< (wt BE 7,) + Fv -1 [BQI C 01 :Q
7b b b a b b b

BQ b~ R b =irfl(Q b ; + Y 8 Z-) in( t BE bZ), (6.42)



y 2 7b 3QI b 2) [R 9b CO(BQ bz) + R 13 in(BQI b )j
"7bb /b b si(B2 b b (Q ~
S(2b b b4b 'Ib n b2 b2b 15b co' (Qb 2b

(blb 'b/b3b) f b '0 (B02, 'b ) FICb -Jn(EQ2t b )b

(3/6) I1 sn( )-A~ o3 BQ, ) b + Obwz

+ Feb bb ) K b/2) cenn(Plt z)z)3/1

(6,43a)

YP( ) Qi (2 z 2) r, n(POI, z) + K coI(DQi z)]

+ (b b ) T ca (C 2 b S( hin(EQ2 b z
lb b, 3b 22b h lb 7 18b b lb )1

+ (b2b b/bDIh 19b r F(,2 b b-')b 2) 23b cc)' (F2b t2tz


+ 38 F20b n( ~ b 24b Cs3E )+(bz

+ Fq ft + (1" 17/2) sin(,60f z) r- (21b 12) cot(DQl] 7-)




T remaining pair of tequatilns requ rod to uvluatc thethr-de
constants are obtained by comxparing (6-40) and 6,42) et Zausg
the boundary condition given by (.O










(Va/Vb) [Y7a (Za) cos(x Z,) Yg (Z,) s;n(x Za)] =

GEQIb [b cos(BQIb Z) Pb sin(BQIb Za)] cos(BEb E )

BQlb S cos(BQl Z ) R sin(BQI Z ) sin(BE Z )
b b b a b b a b a

V7b (Za) cos(BEb Za b (Za) sin(BEb Za) (6.44)


(Va/Vb) r7aa (Za) sin(x' Za) + Ya (Za) cS(Xa za)l

BQlb [Q cos(BQIl Za) b sin(BQb Za)] sin(BEb Za)

+ BQl, bSb cos(BQTb Za Pb sin(BQlb Za)] cos(BEb Za)


SY7b (Za) sin(BEb Za) + Y8b (Za) cos(BEb Za) (6.44b)


nations (6.39) and (6.44) comprise a set of four equations in the four

Ir-order constants. The solutions for this set of equations are dis-

IedJIn Appendix 0. The third-order solutions, given by (6.33) and

Illlllllllllre complete once these four constants have been evaluated.



Transition Near a Velocity Maximum

Ardiing to the small-signal results of (2.16), the best amplifi-
ar a transition near a velocity maximum Is achieved if the drift-

g (Vb) Is reduced, and the plasma reduction factor (Rlb) is

a ad by decreasing the drift-tube radius (a) In Region B. These


i.......................










parameters are changed in relation to the values they possessed in Region

A. The conclusions drawn from the small-signal analysis are that best

amplification is obtained if the voltage and radius are changed at the

e-act location of maximum velocity variation, and that the amplification

obtained Is independent of a, i.e., the amplifier does not saturate.

The effect of varying the drift-tube radius is demonstrated in

Fig. 6.1 for a fixed voltage transition, Vb Va/2. It is seen from

Fig. 6.1, that a 10 per cent increase in amplification is achieved if

the b/a ratio is changed from 0.667 to 0.91 in Region B; whereas, the

gain drops about 5 per cent if the ratio is changed to 0.5 in Region B.

Comparison with the solution for Region A, which corresponds to the

conventional two-cavity klystron, shows that a 25 per cent increase in

amplification is achieved by changing the b/a value to 0.91 and reducing

the potential by a factor of two in Region B. The major contribution to

the increased amplification, however, comes from the change in drift-tube

voltage, rather than the change in drift-tube radius.

The effect of varying the drift-tube voltage is demonstrated In

Fig. 6.2 for a b/a transition from 0.667 to 0.91. The ampli"'fication in-

creases as the drift-tube potential in Region B decreasesi,,, The cLurvu| fur|

Vb = Va/10 also demonstrates another Important consi(eratian, namely the

location of the output cavity. If the position of thi seond vty

shifted by five degrees in relation to its optimum alue, the r

cation is reduced to sullch l an ll tent that a much sller Vl e tr

would yield better amplificatln-The actual shape the c

to one change In tentia me t n ubt,






Conven tiocna I
klystron with
K = 1.414
a' = 0.2


b/a = 1I 1.1
b/a 23 -
b/a = 1/2 -


.r

iF


I -


,/


0.7




0.6




0.5




0.4




0.3


-


I n'imui -urrent
in Pe.qzi,'n A


= 167'
= 0,


/



/

I








K= 1.0

(b/a) = 2/3
8 b= 1.0
ea


Vb/Va = 1/2


I lIl I I
190 210 230 250"


Current In Region B as a function of the drift-tube radius.








ffb

1.4





1.2





1.0





0.8





0.6





0.4





0.2 -





0


Vb/Va = 1/L4


Vb/Va = 1/2


Vb/Va "


K = 1.0

(b/a) = 2/3 (b/a)

Beab = 1.0 illl
.momm


190 21 0'


Fig. 6.2


Vb Va I 10










a considerable portion of the c>cle. It is also evident from exaninatic.n

of FTg. 6.2 that the optimum length of the second drift tubt is a definite

function of the voltage ratio; whcreas, the curves in Fig. t show that

the change in optimum length with plasma reduction factor is negligible,

Small-signal theory predicts that the location of the veloc t/-jump

must be at the point of maxinum velocity variation. In order to investi-

gate how the predicted amplification is affected by the nonlinear theory,

the location of the transition is varied. The maimun current obtained

|at some point in Region B for a particular transition locati:.n is plotted

Ilin Fig. 6.3 against the location of the tran5;tic n. The cptlinum location

for a = 0.1 is actually about twenty-two degrees closer to the input

llvity than the point where ma..imum velocity variation occuis. The cp-

tiinun amplification is about 5 per cent above that obtained b, locating

Ilthe transition at the point of maximum velocity variation. It should be

noteIId that the position of the cavity is nut too critical, in that a

five degree shift in location from the optinum position results in less

han I.. per cent loss in amplification for this example. Thus the

centributions to the over-all amplification due to changes in the drift-

be radius or location of the transition are negligible, at least for

a iues of a.

The reason for this increased current is the fact that while the

fluctuations are decreasing as the transition is moved toward

e the current fluctuations are increasing. Since the current

i t s at the transition, the current excitation in Region B

mithe iiicurrent amplitude at the transition. Thus a compromise













0.69




0.68 -





0.67





0.66








0.6


0.64


1 4 1I I I I-8-7|
144" 152* 160 168* 176.
11111 11111111,=. jj


Fig. 6.3 Maximum current in Region B
of the first transition.


as a function of t


Vb/Va = 1/2
(b/a) 1/1.1

a = 0.1


K = 1.0

(b/a)A = 2,3

Beab = 1.0












s n,de between max num velocity excitation and virtually no initial

current, and the case where little velocity excitation is achieved, but

a large current is present at the transition. in the small-slgnal theory

the location of maximum velocity and minimum current co incide, but thi

does no~t hold for the nonlinear analysis, as w.as previously show.n in

Fly. ".A.

The. fact that the velocity at the transition is less than the velo-

city at the entrance in Region A indicates that this amplifier design

mnight not be correct. The curve labeleda' 0.2 In Fig. 6.1 supports

th15 coniclusion, ThIS curve corresponds to the current obtainable fro

in conventional t o-covity klystron with the same signal voltage as the

space -cha rge -wavc apriptIficr, but em~ploying the beam characteristicso

Region E, namely th, iower d-:- potential which yields a beam of higher

perveance. To exceed the current output of this klystron, it is necss ary

to use an amplifier which ha the saen initial beam characteristics, then

cffplpy -an increase in dri ft-tubf? potential at the first transition, and

finally return to the? original potential at the second transition. The

flrs~t trarsitio'n is made near a' 'el-city minitkem, and the second near a

velocity maximum. This typce of veloci ty-Jurip ampl ifier was discussed on

a linear bASIs in Chapter 2, and tht !rdfl-signal results are shown in

Ftq. 23, lp the next section this amnpiificatlon process, Is investi-

gattd on a ronl irear bass k,here it is shown that a-npl ification Is

posstblo, at least for small values of a











Transition Near a Velocity Minimum


The small-signal solutions for a transition at a velocity minimum

are given In (2.9). It would appear that the variation in reduction

factor is just as important a consideration as changing the potential

because of the four to one ratio in the powers to which these parameters

are varied. Since the current amplification is relatively insensitive

to a voltage transition at a velocity minimum, a second transition Is

needed in the vicinity of a velocity maximum. As shown In the previous

section, the best amplification is achieved at the velocity maximum if

the drift-tube potential is reduced, and the reduction factor is decreas-

ed by reducing the drift-tube radius. Hence, the voltage and radius of

the drift tube must be Increased at the first transition, then returned

to their original values at the second transition.

The effect of varying the drift-tube voltage is shown in Fig. 6.4.

It is apparent that the velocity variations are relatively insensitive

to the change in potential; however, the velocity definitely decreases

as the drift-tube voltage in Region B increases. It is also apparent

that the optimum location of the second transition is a definite function

of the voltage ratio. The fact that the variations are relatively iin-

sensitive to changes in potential follows from (2.9b). The lrat io of.i|BB^

maximum velocity In Region B to maximum velocity in Region A iI oinly pro-

portional to the one-fourth power of the voltage ratio, Howevrll It also

appears at flrs t.l s tre that the velocity should liii.creasie wi|| in|||mcrmeas|ing

values of Vb, iillllt than decria:se. The explanation I ) foulllnd by examr





























1N
0 C0

II II II


r N
-n
<^ P r


II II II


C. Va
.a~


::..


Cj
INr






aI i
o





.0






r-








a 0
0 TP











o'







C
*u












'3

o











DI
*O



















o C*















VA
1 0
















--
.0
IL














0
CT


























N
5a










CQ
03 C










0






cu
c








o'

(U



N






a
0

CO











the plasma reduction-factor curves shown in Fig. 2.1. As the voltage

increases the normalized beam radius (B b) decreases, thus decreasing

the value of the reduction factor. Since the ratio of reduction factors

is raised to the first power, a small gain due to the increase in Vb is

more than compensated by the decrease in Rib.

The effect of varying the drift-tube radius In Region B Is demon-

strated in Fig. 6.5. The drift-tube radius In Region B should be increased

because the best amplification is achieved for smaller values of b/a. It

is apparent from Fig. 6.5 that the decrease in reduction factor resulting

from the increase in drift-tube potential can be compensated at least

partially by Increasing the drift-tube radius. Amplification is achieved

in the third section if the drift-tube potential is decreased at the sec-

ond transition. The amplification can be further enhanced also by restor-

ing the drift-tube radius to the value of Region A.

The principle purpose of the first transition is to change the drift-

tube potential from a value Va to a higher value Vb, without decreasing

the velocity amplification, or perhaps to even increase it somewhat by

increasing the drift-tube radius. The drift-tube potential is rllllrned

to the original value (Va) at the second transition. The amp!,ficati

resulting from the decrease In drift-tube potential war the IsuIjct of a

preceding section.

The locations of the two tlransitions are quite ImpOrtitF Small

signal theory predicts that each drift tube should be a quarter plaa

wavelength long. Their elocity at the second trlanstlon i5 sh wn


















Vb






.05


aru velocity; n Regio A


.04



bl a 0 91










K = 1.0 VbV = .


(f8b/3 = 2/3 Z = 82'








1,00, 140 180" 0 26o' 300'


flg t.5 Velocicty In Region B d- a function of the drift-tube ractus




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