NONLINEAR ANALYSIS OF A
SPACECHARGEWAVE
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
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TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . Ii
LIST DF ILLUSTRAT IONS . . . . . . .
CHAPTER
1. INTRODUCTION .
Amplificatlun by a Modulated Electron Beam . . I
SmaltSignal Analysis . . . . . . . . 3
SpaceChargeWave Amplifiers . . . . . . 3
Finite Beaml Considerations
Ex tensions to SrallSW 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 SMALLSIGNAL ANALYST IS OF A SPACECHARGE'AVJE
AMPLIFIER . . . 4
Synopsis . . . . . . ... . . . . 14
Small57cnal 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 SPACECHARGEWAVE EQUATIONS USING
THE METHOD OF SUCCESS IVE APFROX IRATIO:IS . . . 27
Synopsis . . 27
Assumptions . . . 27
Derivation of the SpaceChirqeWave_ Equations5 2 8
Method of Sujccessive kpprolxiritlurs I. I . 32
4. GENERAL SOLUTIC11S FOR THE SPACECHARGEWAVE
EQUATIONS . . 37
Synopsis . . . I I I I I I . . . 37
Firstorder Solutions . . . . . . . . 37
SecondOrder Solutions . . . . . . . 40
ThfrdGrd. .ou i n . . . . . 43
TABLE OF CONTENTSContinued
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
Firstorder Solutions . . . . . . . 52
SecondOrder Solutions . .. ... . .. . 53
ThirdOrder 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
FirstOrder Solutions . . ... . . . . 73
SecondOrder Solutions . ... .. . . . .... 75
ThirdOrder 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 Spacecharge reduction factors for cylindrical
beams in cylindrical drift tubes . . . . . . 16
2.2 Model of a spacechargewave amplifier using
three drifttube sections . . ... . . .. . 18
2.3 Spacechargewave amplification with the first
transition at a current maximum . . . . . 22
2.4 Spacechargewave 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 ILLUSTRATIONISCont;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 drifttube 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 spacecharge
waves was first investigated by Hahn [1] and Ramo [2]. Passing a beam of
electrons thr ugh a ap, across which an rf 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 spacecharge 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
spacecharge 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 rf 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 rf current is present in the bear, This current
will induce a similar component of rf current in the output cavit., pro
vided the cavity is tuned to the rf 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 echarge 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 travelingwave 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 spacecharge 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.
SmallSignal 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 smallsignal analysis. Essentially
the equations are derived by utilizing the equations for the spacecharge
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 ac terms.
The separation of ac and dc terms then yields equations which are
functions of position (z) alone, since the variation with time (cjut)
cancels in the ac equation.
SipaCChargeWave 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 dc
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 velocityjump amplifier, was the first tof se is of
devices classified as spacechargewave amplifiers. others in thi
class are the rippledwall 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 spaechargewavc amplifier which incorporates
al) thcse posiblve alterratlves, has been performed by Peter, Bloom and
Ruetz 171 using smninlthetorj, It was found that ma>,u arn pll
fication resulted f the: driFttube potential and the drifttubeda tr
were increased while the bcim cross section was decreased at the po Int
kwtere the ac velocity %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 spacccharge wave amplifiers
f5 a construction by Dloom and Pctrer [QUI nf 3 trans mIssionIlne analog
which has a standing,atje pattern that repre,ants the amplitude pattern
o~f the spacecharge wave. The approach is bAsled on the smallsignal
anlsi hlch predicts that the maxlinum current aind velocity amplitudes
Interchange every quarter plasma wavelengths. A general discussion of
the llnearlied 5fmallslgnal analysis as applied to the varlou m~rowavee
amplifiers Is presented in books by Beck 9)], Harman [10], and Hutter [ll].
The smallsignal 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
spacecharge 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 spacecharge 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 spacecharge
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 HahnRalb 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 standingwave
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 largesignal analysis
has been the applicability of the reduction factor to largesignal
situations. Mlhran [13] has demonstrated experimentally that replace
ment of wp by RFu in the extrapolated smallsignal theory leads to
errors in both magnitude and location of current maxima.
Extensions to SmallSignal Theory
The smallsignal 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 smallsignal approach.
siral 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 = dc 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 = dc beam voltage,
Be = &/o.
This solution is based on the assumption of no spacecharge interaction,
and hence is a ballistic analysis. The Webster theory has been modified
to account for spacecharge effects by comparing (1.1) with the solution
derived by the smallsignal spacecharge 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
onehalf of its argument for small values of thdi 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 spacecharge 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 spacecharge 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
ihrect[lon of the spacecharge force past crossover, and for a change In
lthe magnitude of the force. Moreover, the theory Is onedimensional,
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 onedimensional. 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 largesignal 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 travelingwave 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 travelingwave
tube using a spacecharge 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 "peeloff" angle anywhere over a wide iranle of Beb
values. The "peeloff" angle is that normalized distance of the d rft
tube for which the normal ied rf current is onehalf 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 jihran 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 ac terms or asuming e "t variation in
time as the smallsignal 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
Schargewave equation is then grouped into terms according to degree.
The firstorder solution, which is proportional to a, is obtained by
IrI tailiing only the firstdegree terms. This solution Is then used to
2
obtain the secondorder 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 thirdorder 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 phaseshift 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 velocitymodulated 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 "peeloff" 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 spacecharge waves
propagating in a spacechargewacv amplifier which combines the features
of the velocityjump and rippledwall 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 smallsignal locations. Investigation In the second region
Includes the effect of different ratios of drifttube potentials, the
effect of changing the drifttube 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 drifttube
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 velocityjump 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
.Tndc potentials across the gaps, or modifying the drifttube
rippledwall principles In combination. The proper lengths of the drift
tube sections 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 spacechargejwae equation
are evaluated in Appendix B.
CHAPTER 2
SMALLSIGNAL ANALYSIS OF A SPACECHARGEWAVE AMPLIFIER
Synopsis
The smallsignal solutions of the HahnRamo theory are applied to
a spacechargewave amplifier composed of three drifttube sections.
The first and the third sections have Identical properties, while the
middle section possesses both a different potential and a different
drifttube 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
SIe proportional to the ratio of the plasma reduction factors and the
threefourths power of the voltage ratio. Maximum gain results if the
leniths of the drift tubes are some multiple of onequarter of a re
ced plasma wavelength.
SmallSlgnal Equations
The smallsignal 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 = T1. 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 ac
terms are small compared with dc 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 spacechargewave equations. Equations (2.1)
and (2.2) are valid for any region In which the dc velocity (vo) is
constant, such as inside a section of drift tube.
Initial Conditions
The velocityjump a.l. ifier, developed by Tien ajni Fl iIId [3],
utilizes a change in dc potential between drifttube 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 Spacecharge reduction factors for cylindrical beams
in cyIhIdrical drift tubes.
necessary to determine what happens to the magnitude of the ac current
and velocity as the beam passes through the transition point. Hutter
develops a proof, using the Lorentz force equation, which demonstrates
that the ac current is continuous, whereas, the ac velocity undergoes
a transition proportional to the ratio of the dc 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 ac excitation. A signal voltage Vm cos(ut) velocity modulates the
beam at the input cavity. The initial values of ac current and
velocity are derived In Chapter 3 in connection with the nonlinear
analysis. The smallsignal 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 dc beam voltage (V, .
The equations for the first drifttube 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
H1(
^^ HC
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 llSignal Analysis
If the first drift tube is onefourth of a reduced plasma wavelength
long (BqaZa = 900), then the magnitude of the ac 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 drifttube region. Solutions for the second rIEgion 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 ac 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 dc velocity Is proportional to (Vo) and the plasma frequency is
Inversely proportional to the square root of the dc 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 drifttube region are increased. The value of Rb can be increased
by Increasing the drifttube 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 drifttube
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 dc
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 0I
\ 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 ac 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 onehalf 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 onefourth 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 drifttube 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 drifttube 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 onehalf of a plasma
liimenth ilong in order to obtain the desired current amplification.
25
CD
x >.
o
I1
0 
0
ti rI
SII II II
I.
0
26
Conclus ions Based on Sma) I S 9na Theory
The smailsignal 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 smnallsignal solution for
oc velocity ha5 a minimum value of zeor, no reduction in the ac
velocity occurs if the beam voltage is increased at the point of zero
ac velocity. if the transition is located at someo other point, then
the amplitude uf thet ac 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 dc 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 SPACECHARGEWAVE 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 rf 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 rf fields.
5. The dc charge density of the ions (Ip.l) cancels the dc rchJr
density of the electrons (p J).
6. The velocity, in keeping ,ith the pc'lar;zaticon cocrd;nate s,steni,
is a singlevalued 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 SpaceChargeWave Equations
Several relations between position, current, velocity, and charge
ity are needed in order to derive the nonlinear spacechargev'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 ac 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 spacecharge 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 zaxis 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 spacecharge 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,
secondorder partial differential equation of fourthdegree.
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 firstorder 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 HahnRamo smallsignal solution,
which is also a linear function of a. The HahnRamo solution Is obtained
from (3.18) provided,
Pozi + fl = RP12 oZ (3.19a)
This is another way of saying that the smallsignal 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 secondorder and thirdorder 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 If
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 secondorder solution, whereas
RO 0 is used with the dc portion of the secondorder solution. The
thirdorder 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 wavenumber (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 thirdorder 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 firstorder solution (zl). This solution is substituted into the
right hand side of (3.22b) in order to evaluate the forcing function. The
!secondorder polarization (z2) is obtained by solving the linear, nohoo
geno~u5 equation (3.22b). The thirdorder polarizatio>n is obtained by
sol)ving (3.22c), ater First substituting the solutions for z, and Z2
into the riqtht hand 5ide of (3.22c) to evaluate the forcing function. For
this investigation a thi rdorder 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 SPACECHARGEWAVE EQUATIONS
Synopsis
In this chapter the general solutions for the first, second, and
thirdorder 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 drifttube 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 next chapt.i...er. ..
FiurEt _Drder Solutions
Since the smuallIlfignal sutian to the ace*'"hartewave 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 firstorder polarization (zj) is obtained
y tlituting (4.4) into (4.1). The polarization is normalized by
Tplying (4.1) by Bqla. The normalized firstorder 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 doublesubscripted
terms.
The firstorder 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 fTrstorder 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
SecondOtrder Soluti~on
The substitution of (45) Into (3.22b) determines the forcing
function for the secondorder 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 aic 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 firstorder solutions, the secondorder 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 secondorder 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 secondorder 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)
ThirdOrder Solutions
The forcing function for the thirdorder 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 thirdorder polarization
is
23 = F3(z) cos(tt Bez) + G3 () sin(wt Bcz). (4.21)
Th 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 thirdorder 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 thirdorder 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 thirdorder 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 spacechargewave
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 firstorder polarization ;n Region A will be denoted as 21a,
,hlle in Region B, Lhc solution ; donrted 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 drifttube
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 twocavity 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 closelyspaced 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 ac 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 closelyspaced
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 secondorder
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 irbe
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
FirstOrder Soluirons
Two of the firstorder 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 smallsignal solutions obtained
by Hahn [)l and Ramo [2].
SecondOrder Solutions
Before obtaining the secondorder 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 secondorder 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 secondorder 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 secondorder
(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 secondorder
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
IOrder Solutions
thirdorder 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 thirdorder 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 thirdorder 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 thirdorder 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 thirdorder 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
,[A1(, sin(z) + A 17COW() + A18 cos(b)a r/lE) 4 l9
cos !baaz + A20 cos(3z)] slin(Lt .z'0 (54 )
The constints A, Lhroug)h A 2D are evaluatf 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
smallsignal 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 snailsignal solution, is shown in Fig. 5.2.
Saturation of fundamental current
Smallsignal 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 regii.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 thirdorder 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 Smallsignal solution
14 0.3
o.4'
Gpab = 1,0
\ I
0.15
/
online r solution /
0.13 Smallsignal 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 smallsignal 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 firstorder 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 outofphase 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 thirdorder solutions, with the thirdorder 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 fourthorder solution, and therefore falls to show saturation effects.
The second harmonic solution obtained by this thirdorder 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 twocavlt' klystron discussed in the
previous chapter. The model for the spacechargewave amplifier is
identical to a twocavity 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 velocityjump amplifier proposed by Tien and Field
[3), and the rippledwall 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.ciii
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
eond drift tube Is positioned. The drifttube 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 witholut 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 ac terms are all consi derably smallr than th.
iiiiiiiiiiiiiiiiii' ,,,,,,
dc 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 smallsignal assumption used in the analyst ; of
Ith velocityjump amplifier by Tien and Field; however, the solution in
Region B contains secondorder and thirdorder tcri.s In addition to the
ima3lsignal 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 ac velocity term is approximately 10 per
velocity.
Consequently,
inb (Z ) = na (a). (6.11
n = I, 2, 3
FirstOrder Solutions
The firstorder 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 firstorder 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 firstorder 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 firstorder velocit
in Region A is obtained from (5.18b).
v (7 (4t/2) CCS(Z ) Sin(Lot X Z ( )
la a
11he firtorder velocity in Raeq~n B is found from (47).
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 firstorder veiiccity (vlb) is norm.al;2cd with rcspcct to the dc
velocity V 00) in Region A, rather than with rcsptect tco the dc
voICfIty (vab) in Region 5. This ;s done in order that the t%,o normalized
firstorder velucity terms voa3 and vob, may be comtparcd di rectly, The
boundary co*nditiop 61)i applied to (6,15) and (1).Thri remainIng
two oqvaticans needed to evaluate the firstcrdcr 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 firstorder 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.
SecondOrder Solutions
The secondorder 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 secondorder 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 secondorder 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' :! ! !:'
:'*iiiiiiii'
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 secondorder 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 secondorder 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 secendorder constants. The solutions to this set of equations are
also discussed in Appendix D.
ThirdOrccr So lutions
The thlrdorder current in Reg ion B Is obtained f rr (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 ), (633
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) + K23b 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) + r1~ ,in(BQ2b b, ]*E/Q2
[12b cos(3 BQIb ;) + k16b sin(3 BQlb ;](8 EQlb)
(6.35)
The thirdorder 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 thirdorder 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 thirdorder 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 thethrde
constants are obtained by comxparing (640) 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
Irorder constants. The solutions for this set of equations are dis
IedJIn Appendix 0. The thirdorder solutions, given by (6.33) and
Illlllllllllre complete once these four constants have been evaluated.
Transition Near a Velocity Maximum
Ardiing to the smallsignal 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 drifttube 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 smallsignal analysis are that best
amplification is obtained if the voltage and radius are changed at the
eact 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 drifttube 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 twocavity 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 drifttube
voltage, rather than the change in drifttube radius.
The effect of varying the drifttube 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 drifttube 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 amplificatlnThe 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 drifttube 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,
Smallsignal 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 twentytwo 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 overall 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 I87
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 smallslgnal 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 ocovity 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 fttubf? 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' 'elcity minitkem, and the second near a
velocity maximum. This typce of veloci tyJurip ampl ifier was discussed on
a linear bASIs in Chapter 2, and tht !rdflsignal 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 anpl ification Is
posstblo, at least for small values of a
Transition Near a Velocity Minimum
The smallsignal 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 drifttube potential is reduced, and the reduction factor is decreas
ed by reducing the drifttube 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 drifttube 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 drifttube 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.iBB^
maximum velocity In Region B to maximum velocity in Region A iI oinly pro
portional to the onefourth power of the voltage ratio, Howevrll It also
appears at flrs t.l s tre that the velocity should liii.creasie wi inmcrmeasing
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 reductionfactor 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 drifttube radius In Region B Is demon
strated in Fig. 6.5. The drifttube 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 drifttube potential can be compensated at least
partially by Increasing the drifttube radius. Amplification is achieved
in the third section if the drifttube potential is decreased at the sec
ond transition. The amplification can be further enhanced also by restor
ing the drifttube 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 drifttube radius. The drifttube potential is rllllrned
to the original value (Va) at the second transition. The amp!,ficati
resulting from the decrease In drifttube 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 drifttube ractus
