ON SPACECHARGE WAVE PROPAGATION
IN CROSSED ELECTRIC, MAGNETIC, AND
CENTRIFUGAL FORCE FIELDS
By
WILLIAM EDWARD LEAR
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1953
ACKNOWLEDGMENT
The author wishes to express his sincere gratitude
to Dr. W. W. Harman, former chairman of the supervisory
committee, for his continued guidance and encouragement,
and to the present chairman and members of the committee
for many valuable discussions and suggestions. He is
also indebted to Mr. R. P. Derrough, whose cooperation
made possible the experimental part of this work.
LIST OF TABLES
Observed Osoillations in Small
Cathode Magnetron . .
Observed Oscillations in Magnetron
With Intermediate Cathode Radius .
Observed Oscillations in Outside
Cathode Magnetron .
iii
Table
5.1
5.2
5.3
Page
95
101
109
LIST OF ILLUSTRATIONS
Figure
1.1
2.1
2.2
2.3
3.1
3.2
4.1
4.2
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Page
Unstable SpaceCharge Condition
in a Plasma . . ..
Klystron Amplifier . . .
TravelingWave Tube . .
Electric Field of TravelingWave in
Moving Coordinate System . .
OutsideCathode Magnetron . .
Comparison of Theoretical Cutoff
Curves . . .
SpiralBeam TravelingWave Magnetron
Linear TravelingWave Magnetron .
Block Diagram of Circuit Used to
Determine Oscillation Frequencies .
Observed Oscillations, Tube No. 1 .
HighFrequency Oscillations,
Tube No. 1 . . .
Static Characteristics, Tube No. 1 .
Observed Oscillations, Tube No. 2
Frequency vs. r/rc, Tube No. 2 .
Static Characteristics, Tube No. 2 .
Observed Oscillations, Tube No. 3 .
Static Characteristics, Tube No. 3 .
3
12
28
52
59
69
80
83
93
97
98
99
105
S106
. 107
S 115
S 116
LIST OF ILLUSTRATIONSContinued
Figure Page
II.1 Angular Velocity in Outside
Cathode Magnetron .. . 156
11.2 Electron Trajectories in Outside
Cathode Magnetron ..... . 137
TABLE OF CONTENTS
Page
LIST OF TABLES
S & & .* & *
LIST OF ILLUSTRATIONS . . .
Chapter
I. SPACECHARGE WAVES INTRODUCTION
II.
ANALYSES OF SPACECHARGE WAVE TUBES
* *
The Klystron
The Electron Wave Tube
The TravelingWave Tube
The TravelingWave Magnetron
The Magnetron Oscillator
:I. THE CYLINDRICAL DIODE MAGNETRON WITH
OUTSIDE CATHODE . .
V. ANALYSIS OF THE SPIRAL BEAM
TRAVELINGWAVE MAGNETRON . .
V. EXPERIMENTAL RESULTS . . .
InsideCathode Tube With Small
Cathode
InsideCathode.Magnetron With
Intermediate Cathode Radius
The OutsideCathode Magnetron
Discussion of Experimental Results
CONCLUSIONS . . . .
APPENDIX I.
DYNAMICS OF THE ELECTRON BEAM 
THE VELOCITY POTENTIAL . .
APPENDIX II. D.C. CONDITIONS IN THE OUTSIDE
CATHODE MAGNETRON . .
* 58
* 79
. 92
* 0
* 0
* S
122
124
132
iii
* iv
* *
. 11
II
I
VI.
APPENDIX III.
TABLE OF CONTENTSContinued
SOLUTION OF THE DIFFERENTIAL
EQUATION FOR F . ...
BIBLIOGRAPHY . . . . .*.
BIOGRAPHY . . . . .
Page
142
145
147
vii
CHAPTER I
SPACECHARGE WAVES INTRODUCTION
Spacecharge waves are variations in charge density
in a cloud of electric charge, usually an electron beam,
which are propagated throughout the cloud. To understand
more fully the nature of these waves it is necessary that
we examine the charge conditions existing in an electron
beam in a vacuum tube.
An ordinary vacuum tube which may operate at a pres
6i
sure of 106 microns, for example, still contains roughly
101 gas molecules per cubic centimeter. Therefore, it is
not an accurate picture to conceive of an electron beam
moving along its way unhindered in such a tube. As a mat
ter of fact, a one milliampere beam may produce perhaps
1012 gas ions per centimeter of travel.l(a)
The positive gas ions thus produced are relatively
immobile compared to the electrons, so that a given rate
of ion formation will serve to neutralize the spacecharge
effects of a much higher rate of electron flow.
(a) rscrpt number refer to Bibliography.
Superscript numerals refer to Bibliography.
Now let us assume that such a condition of neutraliza
tion exists in an electron beam. If a portion of the beam
should be slowed down, say by a retarding electric field,
a dense cloud of electrons would result, and the region
ahead of this cloud would be positively charged due to the
presence of the relatively stationary positive.ions. This
condition is shown in Figure 1.1. However, such a configura
tion of charge is obviously unstable. Electrons from both
ahead of and behind the positively charged region will now
flow into this region with the result that it soon becomes
negatively charged, leaving two new positive regions ahead
and behind. These new regions are then filled with elec
trons from both sides, and thus the process continues with
two waves of varying charge density being propagated along
the beam, one forward and one backward. These waves travel
with a velocity which is characteristic of the beam and
which is superimposed on the average velocity of the beam.
An alternative way of looking at the phenomenon is in
terms of fields rather than charges. An axial electric
field exists between the regions of high and low electron
density, and since this field is timevarying it produces
an accompanying magnetic field. Thus two electromagnetic
waves are propagated along the beam, one forward and one
backward.
Tonks and Langmuir2 in their work on discharge tube
..+ ; +. .. .... ...
. r . .... . .... i .. ... .......... ........ .
pI ,+ s4 4 
 + d + p+
+ . , 
+ ... "`ii I + + 
i
I  
... + + 1 +
+ + '.
+ + + 1. +  ., i p + + { ,

+t.
S. ^ 
 ..... + .. .
i. i .
++ 1Z  .
.. .. 1 .. ... ... ... ... ,+ : 
. ,, . .. .. . ... .. ; .. .. .. ..
+ + j + r ... .. + + + + .. .. .. .... ..
S: + .. .. ... +:I .... + ; .. + _

+
i .... .... 
I +
j   '  t ;
+  U 4 I 
It I
I" ' i .
:!~~~~~~~~ ~~~~ ~ , ~  i    .' *''.< '*J'  :  .    r.^ ,
+ ..
II .
.. ... ++m * :=
SI Io o
 + +i P ^ *! ^ 0
*+ :" :... . ... .! I o,0 .i... .
. : .. .
.
'E 1 % ~ ~ ^ I" r I 0' i
*+ : + = + + i : : ,
1 + i 
i .+ .. {. + .. .. + 
. ... +, I
*I' 1 : ._ : 
phenomena have shown that the natural angular frequency of
oscillation of the electrons due to a disturbance in the
plasma (the name given by them to the region in which the
electronic charge is neutralized by the presence of posi
tive gas ions) is given by
(1.1) Cp= I vpc
where ( is the electronic charge density (a negative num
ber), C is the charge on the electron (a positive number),
yv is the mass of the electron, and 6 is the permit
tivity of the medium. MKS rationalized units will be used
throughout this work.'
Bohm and Gross3 have developed a similar theory for
oscillations in electron streams by assuming a solution of
the form exp j ( 0() obtaining the eigenequation
relating W and P and solving the equation for w with
fixed r The presence of complex values for oW is in
terpreted to mean instability in the beam and to give the
frequency of oscillation. However, Twiss4 casts doubt on
the validity of this approach and shows that in the case of
the twovelocity electron stream where such an analysis
would indicate instability, amplification actually occurs
rather than oscillation. The conclusion drawn is that there
must be two beams traveling in opposite directions to
produce the feedback necessary for spacecharge wave oscil
lations to occur. Another logical possibility for oscilla
tion, not discussed by Twiss, is the case where a single
beam is traveling in a circular path, as is the case in the
cylindrical magnetrons to be discussed later.
In addition to these naturalperiod waves which may be
excited by some transient disturbance of the plasma, there
is, of course, the possibility of the forced excitation of
space charge waves of any angular frequency u by the ap
plication of a timevarying electric field of that frequency
to a portion of the plasma. The velocity of propagation of
such forced vibrations will be of interest and will now be
determined.
Haeff7 has analyzed the more general case of an electron
beam with a continuous velocity distribution, and we shall
consider his results when we discuss the electronwave tube
in Chapter II. For the present, however, let us consider a
beam in which all electrons travel along the z axis with the
same average velocity, vo, and in which the average charge
density, Pa is constant. We now apply a longitudinal
timevarying electric field of angular frequency, uJ to
the beam and determine the velocity of propagation of the
spacecharge waves which are excited. The analysis is es
sentially the same as that given by Harman in Reference 1.
we make the assumptions that solutions will be of the
form exp ( W  P ) and that the alternating components
of charge density, P, and of velocity, It are small
compared to the average values. The charge density and
velocity in the beam may be written
(1.2) P = o op
(1.23)
(1.5) r o l + Ir(
Three fundamental laws will be employed to produce a
relationship which can be solved for P which in turn
gives the phase velocity of the waves. These laws are (1)
Poisson's equation, (2) the continuity equation, and (3)
the force equation. First, applying Poisson's equation
VV = V 
or
and, remembering our assumption that the variation of all
alternating quantities is as exp ( we have
(1.4) =
Here only the a.c. component of charge density is
used since it is assumed that the average charge density,
eo is zero due to the presence of positive ions. Or
stated otherwise, the potential in the beam is assumed to
have only an a.c. component, so when the a.c. and d.c.
terms of Poisson's equation are equated respectively, the
result is equation (1.4) plus a second equation, Po = o .
The current density, L in the beam is the product
of charge density and velocity. This gives, from (1.2) and
(1.3),
(1.5) L = P o + .o + o l = o +* ,
Here, applying the smallsignal approximation, we have
neglected the secondorder product P, T, Since the cur
rent is due to electron motion alone, the positive ions of
the plasma being considered stationary, the value of Po is
the electron charge density and is not zero as it was above.
Making use of the continuity equation in one dimension, we
have
or
(1.6) W o = r r(,o +o,;)
8
The force equation in the onedimensional case is
d r aV
(1.7) yv d e .
The derivative may be expanded and (1.7) rewritten in the
form
v(1.) e V
(1.8) Tt d )j
from which we get
(1.9) u.vr rv
If (1.6) is now solved for v1 we obtain
(1.10)
SPo r
Substituting (1.10) into (1.9) yields
(1.11)
* n II
o, ~ h
Or (1.11) may be solved for to give
e pr P V
^ n.f _
^+Iro1
(1.12) r r'
(\4AJrv'
We.next substitute this value for the a.c. charge density
into (1.4) to obtain an equation which can be solved for
r it is
e P
(1.13) 1 
(w +.d rro)
We have seen from (1.1) that the quantity in the numerator
is the square of the plasma frequency Op Making this
substitution and solving for r we obtain the desired re
sult. It is
t* WA)p
(1.14) C = j
Since the propagation constant, r of an unattenu
ated wave of frequency uw traveling with phase velocity vp
is C = J vp we see that equation (1.14) corresponds
to two waveshaving phase velocities of and L
Wp _ro respectively.. That is, when u >7A p
the two spacecharge waves are propagated along the beam
with respective velocities that are slightly less than and
slightly greater than the average velocity of the beam.
A number of practical electron tubes make use of the
fact that space charge waves may be propagated along an
electron beam. Among these are the klystron, magnetron,
10
travelingwave tube, electronwave tube, and travelingwave
magnetron. It will be of interest to compare the methods
of analysis which have been used on these various' tubes and
to consider them as devices belonging to the single class
of spacecharge wave tubes rather than as a group of unre
lated devices. This will be done in Chapter II.
CHAPTER II
ANALYSES OF SPACECHARGE WAVE TUBES
As stated in Chapter I, our purpose in comparing the
various methods of analysis of several tubes will be to
point up similarities, not differences. Obviously a de
tailed account of each method of analysis, some of which
are the subject of lengthy papers or of books, would be out
of place here. What we shall attempt to do, though, is to
see the physical picture underlying the operation of each
tube, the fundamental laws used in the analysis, the sim
plifying assumptions which have been made, and the end re
sults. The analyses, with the exception of the treatment
of initial conditions in the wave approach to the klystron,
are essentially the same as those given by the authors
cited.
The Klystron
A twocavity klystron is shown in Figure 2.1. An elec
tron beam is formed in the electron gun and is accelerated
by the d.c. anode potential Vo. In passing through the gap
of the resonant input cavity, the beam is further acceler
ated or decelerated by the alternating gap voltage V1sinWltl,
CAITYR
OWTUT
CAVITY
K\Ha.BNH BEAM
FIGURE 2ol
.ELBG Su


SCOLLS~rCTOB
where tI is the time at which an electron passes through
the input gap. Electron "bunches" appear in the drift space
as electrons which have been accelerated catch up with elec
trons which have been decelerated. The bunched electron
beam passing through the gap of the output resonator induces
a gap voltage which has a large component at the input angu
lar frequency W1. If the output cavity is tuned to this
frequency, the result is an output voltage which is an
amplified version of the input voltage. Obviously, if a
portion of the output is fed back into the input terminals
in the proper phase relation, the device will also serve as
an oscillator.
The most common method of analysis of the klystron9'101
is to consider the .particle mechanics problem of an electron
acted on by steady and alternating forces and from the re
sult to obtain the alternating current produced in the beam.
The fundamental laws used in such an analysis are conserva
tion of energy, conservation of electric charge, and the
equations of classical mechanics. Assumptions which are made
in the most elementary theory are (1) small signal (i.e.,
the a.c. voltage V1 is small compared to the d.c. beam
potential Vo), (2) spacecharge forces are negligible, and
(3) the transit time of electrons across the resonator gaps
is negligible. A more elaborate approach in which these as
sumptions are not made gives correction factors which must
14
be applied to the results of the simpler analysis.
The energy equation is
(2.1) "T ''= V
and since V in this case is the sum of the d.c. and a.c.
potentials seen by an electron, there results
(2.2) t = ;; (Vo+ V, sartW t.)
The time that it takes an electron to traverse the
S
distance S, between resonator gaps is T = 2, which, making
O v
use of (2.2), results in an approximate form for the transit
time
(2.3) T= To (I si n )
where To is the transit time of an electron at the d.c.
beam potential, Vo.
A conservation of charge equation is now written. It
is
(2.4) Io di, = T, t >
which says that if the quantity of charge Io dt1 passes
through the input gap in time dtl, this same quantity of
charge will pass through the output gap in time dt2 and
will have the new rate of flow, 12.
The time, t2, at which an electron arrives at the out
put gap is
(2.5) t, = ,T = t, + To To o51t .
Equations (2.5) and (2.4) may be combined to give an
expression for the current 12 at the output gap. It is
Io
(2.6) 
where x is a constant called the bunching parameter and is
given by
(2.7) r = TT N V,
N being the number of oscillation cycles corresponding to
the average transit time To.
Equation (2.6) is the desired current expression except
that it is in terms of the departure time t1 instead of ar
rival time t2, and since the relation between t1 and t2,
equation (2.5), is a transcendental equation, 12 must be
presented graphically as a function of t2. A Fourier series
analysis of this curve gives the expression
(2.8) I+ = Io[ I +2J.1 () s;J, t T7N)
+ 2 J,(X)i S ;(uijATTTN)
where Jn is the Bessel function of the first kind of
order n. The second term is the one of interest in an ampli
fier and gives for the fundamental component of output cur
rent
(2.9) Iz, = Io J.() s (.tL r)
A transconductance, gm, is defined for the klystron as
the ratio of the peak value of fundamental output current
to the peak value of input gap voltage. The voltage gain
expression may then be written by considering the equiva
lent circuit of the output cavity. This gain equation is
(2.10) Voltage gain = = 9m RR cos .
V, R5+RL
Here R is the shunt resistance of the output cavity, which
includes cavity losses and beam loading effects, RL is the
load resistance, and 42 is given by
(2.11) tn= s' l)x
(R5+ RL))XS
where Xs is the shunt reactance of the cavity.
As was mentioned previously, the effect of the
simplifying assumptions made in this analysis have been the
subject of many investigations, but they will not be dis
cussed here, since our object is not a thorough treatment
of klystrons but rather an understanding of the general
method. Also, we shall not consider the subject of reflex
klystrons, although it should be mentioned that the same
approach may be used to find the beam admittance, 12/V1'
When the conductance component of this admittance is nega
tive and greater in absolute value than the cavity shunt
conductance, oscillation will occur.
An alternative analysis of klystron operation, and one
which is of more interest here since our subject is space
charge waves, is one which considers the effect of the space
charge waves which are produced by the upsetting of the
equilibrium conditions in the plasma when the beam is acted
on by the external field in the input gap.
As we saw in Chapter I, two unattenuated spacecharge
waves propagate along the beam, one faster than the average
beam velocity and one slower, we shall consider only the
case where w > >up This condition is not necessary, but
it allows us to see the physical picture without a great
deal of mathematical embellishment.
Using the condition W >j'p in equation (1.14) and
combining it with equation (1.10) gives
(2.12)
Since the positive sign corresponds to the wave which
is being propagated with a phase velocity greater than Vo,
we see that the density and velocity variations of the fast
wave are approximately in phase while those of the slow wave
are approximately 180 degrees out of phase.
The total velocity and charge density at any distance
z from the input gap may then be written
(2.13) '
where the subscripts s and f represent slow and fast waves,
respectively.
The velocity of an electron passing through the input
gap is determined by the sum of the d.c. and a.c. acceler
ating potentials and is given by
(2.14) o) =
where V1 is, of course, assumed to vary as the real part of
exp j W t. But if V1 < Vo, (2.14) may be written in the
approximate form
(2.15)
where
S)
we
Comparison of (2.15) and (2.13) for z = 0 shows that
the a.c. component of velocity is
Vf
(2.16) VV = v, = +f *
Then from (2.12)
(2.17) p = PO = P4/ 9
and from (2.12), equation (2.17) may be written
(2.18) P'= If ^ ( v)
The a.c. component of current density, il, is the
a.c. part of the product of p and V and, neglecting second
order effects, was found in (1.5) to be
, = P, 0r + "; PO
_
(2.19)
Substitution of (2.16) and (2.18) in (2.19) gives
(2.20) = P. z . r'
at z = 0. We now impose the condition 1, = 0 at z = 0 and
obtain for the two components of a.c. velocity
V, i
"If 8Vo'o VC/
(2.21) ,
and for the components of a.c. charge density
J V,
But we are considering only the case for which W >>p ,
so the ratio is small compared to 1 and may be neglected.
The velocity and density waves may then be expressed approx
imately as
*.), a)
(2.23) o^ b W ^ ^
P P +(~e)EdV
The a.c. current density now becomes
(2.24) La 4V0 'V p puJ'
Again making use of the condition that w >pp we see that
the current becomes largest for the values of z which make
sin = 1. In other words, the output gap should be
placed at a position J x s away from the input
gap if it is to intercept maximum a.c. current.
Equation (2.24) then tells us that the magnitude of
the a.c. current density, 12, at the position of an output
gap located 5 I meters from the input gap will be
(2.25) L I a
We shall carry this analysis no further since it now
proceeds 'in the same fashion as the mechanics problem ap
proach. We are now in position, however, to compare the
currents obtained by the two methods. To compare equations
(2.9) and (2.25) we remember that we have assumed in the
wave analysis that V1 o Vo. If we use this same condition
in equation (2.9) we may use the smallargument approxima
tion
(2.26)
22
The value of Is in (2.9) then becomes
(2.27) 1 Z = o V .
But N is the number of oscillation cycles in the average
transit time, which, for the gap spacing I L becomes
so
(2.28) N /= 4 J 
So (2.27) now may be written
(2.29) TI = Tr W
4 Vo &p
Thus there is a factor of TT difference between the
currents predicted by the two analyses. Equation (2.29)
was derived with fewer approximations and we should expect
it to be more accurate than (2.25). Experimental evidence
shows that this is true. However (2.29) cannot be expected
to give precise results either since we assumed that a.c.
variations of charge density and velocity were small com
pared with the corresponding d.c. quantities, which is def
initely not true in the usual klystron.
The thing which we have not yet investigated, and per
haps the most important feature from our spacecharge wave
point of view, is the physical picture of the bunching
process in the wave analysis. We found that two waves of
charge density are propagated down the beam. The propa
gation constant of each was purely imaginary, so no ampli
fication takes place, but there is an interference pattern
produced. At some distance down the tube the two waves
will be in phase and will produce maximum charge density.
We found this distance to be M P meters.
The Electron Wave Tube
This tube follows naturally our spacecharge wave
approach to the klystron, since the electron wave tube
problem is solved in exactly the same npanner as was the
klystron problem. The electron wave tube may be con
structed in the same fashion as a twocavity klystron; i.e.,
it may have an electron beam passing through the gaps of
input and output resonant cavities. While the presence of
resonant circuits eliminates one of the advantages of this
tube, namely, broad bandwidth, nevertheless such an arrange
ment is perfectly feasible and might perhaps aid in the
transition of our thinking from the klystron to the electron
wave tube.
The fundamental difference between the tubes is the
fact that the beam of the electron wave tube is not a single
velocity beam. While the general case would be one of a
continuous distribution of velocities in the electron beam,
let us, for mathematical simplicity, first consider the case
of a beam having electrons of two average velocities, Vo,
and voL It will be unnecessary for us to consider the
details of this analysis, since it is precisely the same,
step by step, as the analysis presented in Chapter I for the
propagation constant, F of the spacecharge waves propa
gated in the singlevelocity beam. The only difference is
that equations involving a.c. quantities now have two parts
instead of one due to the presence of the second stream of
electrons. The resulting equation for r analogous to
equation (1.13), is
(2.50) +
dUo, w'. r)
Since the solution for r in (1.13) differed only slightly
from ~ we postulate a similar result here and write
(2.31) r =o
where vm is the arithmetic mean of to, and Uo ; that is
(2.32) Vr. = Vo, = o, + .
We now substitute (2.32) into (2.51) and (2.31) into (2.30)
and seek to solve the resulting equation for oC Equation
(2.31) tells us that a real part for a means either a
gaining or an attenuated wave, since the wave varies as
exp ( r ).
If d is assumed to be very small compared to either
Vro, or urL (i.e., the beam velocities are close together)
and if the two plasma frequencies Jp, and Wp, are assumed
equal (average charge densities of the two beams the same),
an approximate solution results which is
(2.33) WP =V. tri 4
The equation for the normalized propagation constant, ,
has real roots for values of y rJ, which is a measure of
the velocity separation of the two beams, between zero and
F2. One of these roots is negative, which means one of the
four waves indicated by equation (2.33) will be amplified.
Of the other three waves, two are unattenuated and the other
is attenuated.
When the velocity separation factor ,p is greater
than F2, four unattenuated waves are propagated, and while
interference of the type found in the klystron may exist,
there is no amplification. Within the range where  is
real it has a maximum value of 0.5 when /p = /
The maximum electronic voltage gain of the tube is then
(2.34) 9b); (d = zo og,9 exp(o.s )P. = 4. 54 *
(234 a' I~. r" r r
This exponential increase in the wave will not continue
indefinitely, however, since saturation will limit the a.c.
beam current to a maximum value approximately equal to the
d.c. beam current.
If the input and output cavities are replaced by non
resonant circuits, say by helices, then amplification should
occur over a very wide frequency range. And such is indeed
the case, wide bandwidth being one of the outstanding fea
tures of the electron wave tube.
We shall not extend the analysis to the case where the
two streams of electrons have different average densities,
since, as stated previously, our object here is the under
standing of general principles rather than a thorough analy
sis. The case of a continuous distribution of velocities
in a beam, rather than only two velocities, will, however,
be considered when the plane magnetron is discussed. Here
again we shall see electron wave type amplification taking
place.
Before leaving the subject of the electron wave tube,
we must not forget our primary purpose in comparing the
analyses of various tubes; i.e., we should get the physical
picture of the manner in which amplification occurs from
the spacecharge wave point of view. Suppose that the, two
spacecharge waves with which we are now familiar have been
set up in each of the two streams of the twovelocity beam.
The waves of the faster beam will be moving through those
of the slower beam. Let us consider a region of high elec
tron density in the faster beam as it overtakes a similar
region in the slower beam. Coulomb forces acting between
the two regions will cause a decrease in the velocity of
the fast bunch and an increase in the velocity of the slow
bunch. The result is an intermingling of the bunches which
gives an increased charge density. This in turn produces
an increased axial electric field with a corresponding in
crease in energy stored in the field. Thus we see that the
energy for the amplification process comes from a decrease
in the kinetic energy of the electrons of the faster beam.
The TravelingWave Tube
This tube is similar to the electron wave tube in that
interaction between waves takes place along the length of
the tube and results in amplification. Here, however, the
interaction is not between spacecharge waves but between
a spacecharge wave and a guided electromagnetic wave trav
eling near the beam. The structure used to guide the elec
tromagnetic wave may be one of many types, but the helix
shown in Figure 2.2 is a commonly used form. Its purpose
is to guide the wave along near the beam with a phase veloc
ity which is near the average beam velocity. For this rea
son it is called a "slowwave structure".
Analyses of the travelingwave tube have been of two
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types; in one13 a circuit approach is used and in the
other1415 a wave approach. The object in either method
is one with which we are now familiar; i.e., a solution
is obtained for the propagation constant r to determine
whether amplification is possible.
The problem is attacked from the circuit point of view
by using a lumped constant equivalent circuit for the slow
wave structure as in ordinary transmission line theory.
Transmission line equations are used to determine the volt
age which would be excited in this circuit by the beam cur
rent. Then the force equation and the continuity equation
are employed to determine the convection current due to a
voltage being propagated along the circuit as exp (jC tr ).
The two resulting equations must be consistent and can be
solved for r The assumption is made that a.c. varia
tions of electron velocity are small compared with the aver
age velocity. It is also assumed that the beam and slow
wave structure are in such close proximity that all the dis
placement current due to the beam flows into the equivalent
circuit as an impressed current.
The solution of the transmission line equations results
in the equation
(2.35) V = r
30
for the voltage at any position along the line. Here Fo
is the propagation constant of the line in the absence of
the beam, and Zo is the characteristic impedance of the
line. These two line constants are given by the familiar
expressions in terms of B, the shunt susceptance per unit
length of line, and X, the series reactance per unit length
of line. They are
ro = j ^
(2.36) X
o= T /Z1
I1 in equation (2.35) is the a.c. component of beam current.
The second expression relating V and Il is obtained by
an analysis similar to that used in obtaining the klystron
a.c. beam current by the wave approach. The force equation
is
(2.37) e e
at + V
where vo and vI are again the average and a.c. parts of
velocity, respectively. The derivative is expanded as in
(1.8) and when the magnitude of v, is neglected with respect
to vo the resulting equation is
_ '
(2.38) r)
The continuity equation is now employed to obtain a
relation between a.c. charge density p, and a.c. current
density il. With the assumed exp (tJtPj) variation
with t and z, this becomes
(2.39) p.
The total convection current density is
(2.40) L LO. +. =CL(v. )(po+P.)
which, neglecting the product v. gives
(2.41) L, = foT + p o
Substituting (2.38) and (2.39) into (2.41) yields
r4)
(2.42) L, = vo(r L
or, writing (2.42) in terms of current I rather than cur
rent density i, we have
Io p. r v
(2.43) I. = J v r
Here To is the d.c. beam current, Vo the d.c. beam potential,
and .o = *
Now equations (2.43) and (2.35) may be combined to
obtain the desired expression for r It is
.i. A. r, r
(2.44) V(r re(.r)I = 1 .
To solve this equation for f we use the same technique
that was used in the case of the electron wave tube; i.e.,
we look for waves which are traveling with velocities near
the average beam velocity vo and write
(2.45) r= + + = o +
where oL is assumed to be small compared with o Since
we are looking for a wave which travels at a speed near that
of the electrons, we will consider the case where the d.c.
beam velocity is equal to the velocity of the wave in the
absence of the beam. If (2.45) is then substituted in (2.44)
an equation in oc results which when solved shows that
three waves are propagated along the tube. One is unatten
uated, one is attenuated, and the other, the one of interest,
is amplified. A fourth root for the equation in r which
was lost in the mathematical approximations above, shows that
there is an unattenuated wave propagated in the s direction.
The negative real part of the root of oC which shows
amplification is T P ) 4 The voltage
gain in db is then
9,.t (ab) = A + zo l0,5 exp v P4 X ,v J .
The factor A is an attenuation term which is due to the
fact that the input voltage is divided equally among the
three waves which travel in the +z direction. Its value
in this case can be shown to be 9.54 db, if N is large
enough so that only the gaining wave is significant at the
output. In terms of the number of wavelengths N in the
interaction space the gain is
(2.46) ja.n (db)= 9. 4 + 47,3 CN,
where C = ( ) 13 This parameter may be put into a
more general form which does not include the characteristic
3 EL
impedance of the line. It can be shown that C gp sVo
where E is the peak magnitude of the electric field acting
on the electrons, and P is the power flow.
The fieldtheory approach to the travelingwave tube
has been carried out by Chu and Jacksonl4 for the particu
lar case of a cylindrical helix with a cylindrical electron
beam along the axis of the helix. The general method will
be outlined before any of the details are presented.
The tube is divided into three regions. These are (1)
the region inside the beam, (2) the region between the beam
and the helix, and (3) the region outside the helix. The
wave equations for the axial components of the electric and
magnetic fields are solved in the three regions. This is
easily done for the two charge free regions and for Hz (the
axis of the helix is the z axis in cylindrical coordinates)
inside the beam, since the equations are homogeneous. The
equation for Ez inside the beam is inhomogeneous, however,
since it includes the axial current density iz. Here, as
in preceding analyses, we once again call upon the continu
ity equation and the force equation with the smallsignal
assumption to give us an equation relating is and Es. The
wave equation is then solved for Ez. Maxwell's curl equa
tions are now employed to give the other field components
in the three regions.
The helix is idealized by assuming it to be a lossless,
infinitely thin cylindrical sheet, but conducting only in a
direction which makes the angle f with a normal to the
axis. This imposes boundary conditions at the radius of
the helix which allow evaluation of the constants in the
wave solutions for the two chargefree regions. To match
the solutions at the radius of the beam a method employed
by Stratton16 is used in which a radial wave admittance
looking toward the axis is defined for the two regions, and
the values of admittance at the beam radius are equated.
This results in an equation for the propagation constant
which can be solved and the nature of the roots examined
to determine whether a gaining wave is possible.
For convenience in handling the mathematics the wave
solutions are divided into TE and TM waves. It is shown
that the TM wave is the one primarily responsible for inter
action between the waves and the electron beam, since only
the TM wave has the axial component of electric field neces
sary to produce spacecharge waves of the type we have dis
cussed.
We shall omit the steps involved in determining all the
field components, since these steps are quite straightfor
ward, and start here with the radial admittance functions
for the TM wave inside and outside the beam. The normalized
radial admittance for the TM wave inside the beam is defined
by
(2.47) = *
Substitution of the values found for Hp and Ez gives for
Yri
SK I, (n o)
(2.48) YrZ = 7 (n r)1
where K p (r L+ K1) I 0 and Il are the
wChere 0
Bessel functions of the first kind of order zero and one,
respectively, and n is given by
I I
(2.49) nL Pil + 7TrTb6 Lar.3 LJr 
Here b is the radius of the electron beam and Io and vo are
the d.c. beam current and velocity, respectively.
A corresponding, but considerably more complicated,
expression for the radial admittance outside the beam is
k" je pcLL,(p aL) Ta(p aj k atpG.)
., (pr k, (p) L PPL)
(2.50) Yo [K, C)ot4, ,rJK rp )Ka) *(P
1.(prp + K.(p j (
Equations (2.50) and (2.48) may now be equated for
r a b, the beam radius. The resulting equation is quite
formidable, however, since the desired unknown r is con
tained in p and n, which in turn are in the arguments of
the Bessel functions. This difficulty is circumvented by
writing a simplified approximate expression for both (2.48)
and (2.50). Equation (2.48) is simplified by assuming that
nb << 1. This allows the replacement of the Bessel functions
by their smallargument approximations and results in an
expression for Yri which is
K n1 b
(2.51) /' = J p *
The assumption nb << 1 means essentially that the beam is
thin and that all electrons are acted on by the same axial
electric field. This same assumption was made in the cir
cuit approach when the gain parameter C in equation (2.46)
was written
1I ) '
( p. \P 8 V. *
The radial admittance function Yro, when plotted for
real values of p, shows a pole at p = p2 and zeros at
p : 0 and p = p1. This curve of Yro vs. p may be approxi
mated in the region of pl and p2, which turns out to be the
region of interest, by the expression
a Yr.] PP. E PPI
(2.52) Yr= (P P P) P. (P'P PFP'
Remembering now that p2 (CL+ K) and considering
that the phase velocity of the waves is small compared to
c, the velocity of light, we may approximately replace jp
by F This results in the expression for the radial
admittance,
(2.53) = c
where r. = f and r, = PL
Equations (2.51) and (2.53) may now be equated to give
the desired expression for r It is
(2.54) X, \
This is a cubic equation in r A study of the roots of
the equation for a particular set of tube dimensions and
d.c. beam current reveals that there is a range of d.c.
beam velocity for which one of the three waves is amplified.
But for velocities above or below this range, all three of
the roots of r are imaginary; i.e., the waves are neither
amplified nor attenuated. The velocity at which maximum
gain occurs is the synchronous velocity, which means that
the velocity of the electrons is the same as the wave ve
locity on the helix in the absence of the beam.
The theoretical gain obtained from the wave approach
agrees fairly well with that obtained by the simpler cir
cuit approach. As in previous analyses, we shall not go
further into the wave approach, although there are still
many points of interest which we have not investigated.
Instead we shall look for.the physical picture of the
amplification process in the traveling wave tube.
The theory shows that for an electron beam traveling
with a d.c. velocity equal to the phase velocity of the
free wave, the phase velocity of the forced wave will be
lower than the electron velocity. This proves to be true
even for a beam velocity somewhat lower than the synchro
nous velocity. So if an observer werq to station himself
in a coordinate system which is moving with the velocity of
the gaining wave, he would see the electric field of the
wave as a static field and the beam drifting slowly by in
the positive z direction. Electrons would see alternately
a retarding field, then an accelerating field, as they
drifted through the wave. They would move faster in the
accelerating field regions and slower in the retarding
field regions. The net effect would be a bunching of the
beam by the wave. But while the wave is bunching the beam
it is also extracting energy from the beam as it slows down
the bunches in the retarding field regions. If the process
were allowed to continue, complete bunching would result
and electrons would then be slowed down below the wave ve
locity by the strong retarding field. This would result in
a loss of synchronism and amplification would cease. The
normal travelingwave tube operates at a much lower level of
bunching than this, however.
40
It is apparent that the travelingwave tube has rather
low efficiency, since the only energy that can be trans
ferred from beam to wave is the kinetic energy correspond
ing to the difference between beam velocity and wave
velocity. The outstanding feature of the tube, as in the
case of the electron wave tube, is the broad bandwidth pos
sible if the slowwave structure is one (such as the helix)
for which the phase velocity of a wave is relatively inde
pendent of frequency.
The TravelingWave Magnetron
This tube is similar to the travelingwave tube in that
there is a wave guided along a slowwave structure and being
amplified by extracting energy from an electron beam. There
is a fundamental difference in the energy transfer in the
two tubes, however. In the travelingwave tube we have seen
that the beam, in being slowed down by the wave, gives up
kinetic energy to the wave. In the magnetron amplifier,
electrons in a retarding field region move closer to the
anode, giving up potential energy to the wave, with essen
tially no change taking place in their kinetic energy. This
change in potential energy can be quite large compared to
the change in kinetic energy which takes place in the
travelingwave tube. The travelingwave magnetron, there
fore, has considerably higher efficiency than the traveling
wave tube.
Analyses of this tube have again been of two forms, the
circuit approach and the wave approach. In the former
method, which we shall consider first, Pierce1 has extended
his circuit analysis of the travelingwave tube to include
transverse motion of the electrons and the presence of the
static electric and magnetic fields.
The voltage equation for the transmission line in the
travelingwave tube case was
rP1. ar,
(2.35) V= r
where the assumed variation exp (dt P ) is understood but
omitted for brevity. In the same analysis the linear charge
density f was found to be
r uJ
(2.39) 7d
Combination of these two equations gives
_, ,.,, r o fO
(2.55) V = 7 
We might intuitively expect an equation of the same form in
the present case with the replacement of the linear
charge density, by some term which takes account of the fact
that there are variations in charge density in the transverse
plane and that the displacement current into the line is
now a function of the distance of the charges from the
line. Such an equation does result, and we shall use it
as a starting point here without going through the prelim
inary steps in its derivation. It is
(2.56) V = & 
Here 4 is a function of x and y (beam and wave are travel
ing in the z direction) which, when multiplied by the line
potential V, gives the potential in the vicinity of the
line, and '= The function 4 is assumed to be
given by the equation
(2.57) C4 = r, t 
By making use of this assumed variation of we may
rewrite the potential expression of (2.56). It becomes
IT
(2.58) V =_____
(7L41L r
where
As in the case of the travelingwave tube analysis,
the force equation and the continuity equation are used to
obtain another expression relating V and F which can be
combined with (2.58) to eliminate V. The method of pro
cedure is the same, the only difference being that in the
present case there are the added complications of motion in
two dimensions and of the presence of the static magnetic
field B. The equation which results is
^ V r fo^V (dQ f)td.&"J
roo
Here M L where Uo) is the cyclotron radian frequency,
If (2.59) and an equation for r derived from the con
tinuity equation are now substituted in (2.58) the desired
equation for P results. It is
(2.6o) ^r r
where H = L.) O
It will be remembered that Po is the propagation constant
of the wave on the slowwave structure in the absence of
electrons. Again we look for solutions where the propaga
tion constant of the waves in the presence of electrons is
not greatly different from Po and let
(2.61) r= r(( ot).
With the assumption p4< equation (2.60) may be reduced
to a fourth degree equation in p We obtain four wave
solutions, one of which increases in amplitude with z. This
amplification occurs for two ranges of electron velocity,
but the one of interest, as in the travelingwave tube, is
that for which the electron velocity is near the circuit
phase velocity. One of the values of p shows the possi
bility of a wave which increases in the z direction, but
this is not generally utilized.
The gain in this tube varies with the quantity H much
the same as the travelingwave tube gain was dependent on
the factor C. But we see that H is proportional to (
whereas C was proportional to Since is
a factor less than unity, the gain of the travelingwave
magnetron will be somewhat less than that of a traveling
wave tube having the same circuit, beam current, and beam
potential. However, as was mentioned earlier, the nature
of the energy conversion makes the efficiency of this tube
much higher than that of the travelingwave tube.
The wave solution for the travelingwave magnetron has
been carried out by Brossart and Doehler22, for the plane
magnetron, which is a limiting case of the cylindrical mag
netron with large anode and cathode radii and small anode
cathode spacing.
Two types of tube construction are considered. In one
the electrons are emitted from a cathode which is external
to the interaction space and enter this space parallel to
the electrodes and with a velocity determined by the balance
of static magnetic and electric field forces. Spacecharge
forces are neglected in this case. In the other system the
inner electrode is the cathode. Here the electron trajec
tories are epicycloids, and spacecharge is neglected only
for the case of small currents. The analysis shows that the
gain expression is of the same form in the two cases, the
only difference being in the magnitude of one of the param
eters in the final equation.
The slowwave structure is a flat helix with its axis
along the ycoordinate axis. Wave solutions are then as
sumed to be of the form exp (?)J Pi) The system is
assumed to be infinite in the z direction.
The first step in the analysis is the solution of the
wave equation for the transverse and longitudinal compo
nents of electric field. These components are
(.2o6r) _' r
(2x2 ,,i 1 h 'b)% rd)
(2.62) E .
iisink (4 Pd)
Here AV defines the amplitude of the wave and depends on
the initial conditions, and t is the transit time given
by 2= i to where to is the time at which an elec
tron enters the interaction space; d is the spacing between
electrodes.
Next, the electron trajectories under the influence of
the timevarying fields are assumed to be small perturba
tions, dw and Jdt on the d.c. trajectories. The force
equation is then used in combination with equation (2.62)
to obtain expressions for dS and For the linear
trajectory case these are
dxr
(2.63) P; ra v W f.
J) ef S sC ( I d)
where Lo is the cyclotron frequency, o is the equilib
rium position of the electron, and f = "'j'r7 o
being the d.c. beam velocity. In arriving at (2.65) the
approximation )J1< 4 o has been made.
For the case of cycloidal trajectories the expressions
are more involved. They are
(2.64)
dwi. J '
,.  I
Equations (2.64) are deceivingly simple, because
d. v r I AV., r
and
e T
8, s (' Pd) A
with Q and T given by
0.Q J. (I[ t Ita ^ '
( Xr J1 ( x j,)
and
Tr J. (^ [ tr.r + X + rx,))s
where Xr is the amplitude of the cycloid.
The next step in the analysis is to make use of the
continuity equation in conjunction with the equations for
efX and cf to obtain an expression for the longitudinal
component of a.c. current, iy. The smallsignal
approximation is made. The result is
(2.65) Z : r I rd J co(r x) 6 r xi.r
Vo if .Ki4P rd)
for the linear trajectory case. Here Vo and I are the
d.c. potential between electrodes and the d.c. beam cur
rent, respectively. For the cycloidal trajectories
X. X.
(2.66) L Jd d r .4 dX if, t?
Xr Vaf S Aig rPd)
X:o
The final step of the solution is to write a conserva
tion of energy equation
(2.67) dP t dP, + d P = o .
Here dP is the apparent power given up by the electronic
current along a small path dy, dP, is the apparent power
taken by the helix along the path dy, and dcP& is the in
crease along dy of the apparent power which is propagated
in the direction of the wave. These three quantities are
found respectively from
dld
(2.68) dP = P + .)= do.
~dPL
H e r e. h p r te i e ph e e
curet log sal pthdy f s heaparntpoe
where the asterisks indicate conjugate quantities. Y and
0 are the attenuation and phase constants of the wave,
with the zero subscripts being used to denote these quanti
ties in the absence of the electron stream. The coupling
resistance R must be determined from the wave equation
solutions on the particular slowwave structure and is given
in this case by
(2.69) R; @(d V 6
where o and 4 are factors which depend on the dimensions
of the helix.
If now the proper substitutions are made in equation
(2.67) and the real and imaginary parts are equated, the
result is
(2.70) ( > ) + D ,
The factor D is a constant for a particular tube type and
marks the only difference in the result for the two cases
considered here. The quantity v in (2.70) is the phase
velocity of the wave in the presence of electrons.
A study of equation (2.70) shows that there are two
waves propagated in the forward direction, one of which is
strongly attenuated and the other strongly amplified. Each
is propagated with a phase velocity v equal to the arith
metic mean j (Vr + I) of the electron velocity and the
free wave velocity. The value of Y the real part of the
propagation constant, is given by
k, V JaX I s ') y wd6P 6'd (i +
(2.71) + 4 X s4d) ca zoM 4 ;
where q9 is the pitch angle of the helix, and
LeooXr) z [cos.jr 1 J ) 1t .Oxr)
+jx 3.c I )[ .+ X r) + ( X.
+ l2 0B ,) "
+ (,. x
In particular, for the case of electron velocity and
free wave velocity the same, and for a lossless helix, the
gain is shown to be
(2.72) gao.t = + 8.7 r. db.
with r given by (2.71).
For a physical picture of the amplification process let
us examine the case of electron velocity equal to wave
velocity (we remember that this case produced no amplifica
tion in the travelingwave tube). The interactions which
take place are more readily observed if we place ourselves
in a frame of reference which is moving along in the y
direction with the velocity of the wave and electrons. Then
the travelingwave appears as a static electric field as
shown in Figure 2.3. Electron motion is now influenced only
by the r.f. field, which appears to be stationary, and the
static magnetic field, since there is no static electric
field in our moving frame of reference.
Electrons are initially stationary with respect to the
fields, and, therefore, there is no magnetic field force,
S= e(ix ) However, as soon as an electron begins
to move under the influence of the electric field, there
will also be a magnetic field force. For example,.an elec
tron at A in Figure 2.3 will be forced to the right by the
electric field (arrows on the field lines represent lines of
force on an electron and are opposite to the electric field
direction). As it moves toward the right, there is a mag
netic field force which is directed downward, so the net ef
fect is a drift to the right and downward, away from the
anode.
An electron at B will move up and to the right, one at
C will move down and to the left, etc. The net effect of
the action of the fields will be a bunching of the electrons
. 4 
I
I. Iw
Ki r .
so t I
I  4
~ m
d I
+A U4
4~ K 4I
Th~wmvimIL7'7IIlir
in the region of the retarding field between B and C. And
in this region the magnetic field force carries the elec
trons closer to the anode causing them to lose potential
energy which they give up to the electric field.
In spacecharge wave terminology, and returning to a
stationary frame of reference, we have a traveling wave and
a spacecharge wave moving along together and mutually aug
menting each other as they travel down the tube.
In addition to the high efficiency (around 40% in ex
perimental tubes) already mentioned as a desirable feature
of the traveling wave magnetron, the tube also has the broad
bandwidth feature of the travelingwave tube and of the
electronwave tube.
The Magnetron Oscillator
The most familiar form of the magnetron oscillator,
and the one least susceptible to a complete analytical
solution, is the cavity magnetron. In basic principle it
is the same as the travelingwave magnetron amplifier just
discussed with the structure closed on itself to provide
the feedback necessary for sustained oscillations. And
since the device is to function as an oscillator, the slow
wave structure is a resonant one rather than broadband.
Here again we have waves and electrons traveling along
in synchronism, this time in a closed cylindrical path. And
as before, electrons in the retarding phase of the traveling
wave move toward the anode, giving up potential energy as
they go, while electrons emitted in the opposite phase are
sent back to the cathode. The result is the almost complete
bunching and very high efficiency characteristic of a cavity
magnetron oscillator.
The situation in the magnetron oscillator is obviously
one of largesignal, rather than smallsignal, behavior, and
it is also apparent that spacecharge effects can no longer
be neglected. This produces a problem of such complexity
that a completely analytical solution is too difficult and
other methods must be used. One approach23'24 is the method
of selfconsistent fields in which a potential distribution
is assumed, the motions of electrons in this potential field
are calculated by numerical methods, a charge density dis
tribution is determined from a sufficient number of trajec
tory calculations, and the potential due to this charge dis
tribution is then calculated. If the resulting potential
agrees with the assumed potential, the problem is solved; if
not, a new assumption must be made and the process repeated.
This approach has provided considerable insight into magne
tron oscillator operation under particular conditions and
has confirmed the fact that the revolving spacecharge is
in the form of spokes which extend to the anode in the regions
of retarding electric field.
An analysis for the frequencies of oscillation of a
cylindrical magnetron with a smooth anode has been carried
out by Harris.5 Full account has been taken of the space
charge, but the smallsignal approximation has been used.
The dynamics of the electron beam is handled by means of a
velocity potential. Since this method will be used in the I
analysis of the outsidecathode magnetron to be presented
in Chapter III, it is carried out in some detail in Ap
pendix I. The analysis by Harris shows that oscillations
should be possible in the smooth anode tube at integral
multiples of the Larmor frequency, .
The mechanism for amplification (and oscillation) in
the smoothanode tube is not so apparent since there is no
slowwave structure. However, since there is a continuous
distribution of velocities in the electrons rotating around
the cathode, we see that we have the conditions necessary
for electronwave amplification, and if the structure is
closed on itself, for oscillation.
The problem of amplification in a plane magnetron with
smooth anode has been solved using the velocity potential
approach by Macfarlane and Hay25 and by Bohm.26 The func
tion of the crossed static electric and magnetic fields in
this case is to provide a continuous distribution of veloci
ties in the electron stream, so it might perhaps have been
more appropriate to discuss this work in the section on the
electron wave tube. However, a tube of this sort qualifies
I
as a magnetron due to the presence of the crossed fields,
and if feedback of the proper phase were arranged, could
presumably operate as an oscillator.
The analysis by Macfarlane and Hay is similar except
for geometry to that carried out in Chapter III for the
outsidecathode magnetron, so only the results will be pre
sented here. It is found that amplifying waves can travel
along the beam having continuous velocity distribution,
called a "slipping stream" by the authors, for all fre
quencies. If the electron velocity varies linearly across
the beam from V_. to V1, a fractional velocity aL is
defined by
(2.73) =
It is found that for o4< 0.42 the tube behaves pri
marily like a twobeam tube and has a maximum gain of
2.1 ~ decibels per unit length, where Pp is the plasma
frequency and VTo is the average beam velocity. This is
compared with a gain of 4.35 decibels per unit length
for the two beam tube. This twobeam tube behavior contin
ues up to a frequency Ir Above this frequency
the tube has a low gain of a travelingwave tube nature.
For oC 0.42 the tube behaves primarily like a
travelingwave tube having a gain of about 0.53 decibels
'
per unit length for all frequencies above the plasma fre
quency. This is considerably less than the gain of
6 ( p) j decibels per unit length exhibited by the
traveling wave tube, but the slipping stream tube achieves
its amplification without the use of a slowwave structure.
This travelingwave tube action, as explained by Macfarlane
and Hay, is due to.the presence in the stream of resonance
layers which act as highly reactive impedance sheets and
can guide waves of slow phase velocity in the same manner
as a helix or other slowwave structure.
The case analyzed by Bohm is a limiting case of the
slippingstream tube where IV =0 and it is shown for
this case that no amplification will result.
In addition to the travelingwave type of amplifica
tion and oscillation in a magnetron, there are other types
of oscillation possible in which the magnetron is able to
sustain oscillations in an external resonant circuit. Two
types27 of oscillators, other than the travelingwave type,
are the negative resistance oscillator and the cyclotron
frequency oscillator. Since our primary concern is with
traveling spacecharge waves, we shall not discuss these
oscillators here. However, an analysis for the negative
resistance characteristics of the outsidecathode magnetron
will be carried out in Chapter III.
J *.
CHAPTER III
THE CYLINDRICAL DIODE MAGNETRON
WITH OUTSIDE CATHODE
The cylindrical smoothanode magnetron with the con
ventional arrangement of the inner electrode as the cathode
has been analyzed by Harris as discussed in the preceding
chapter. The opposite physical construction, a tube with
the outer electrode as the cathode as shown in Figure 5.1,
will now be considered. As our analysis will show, such a
configuration produces some unique results.
The velocity potential approach to the dynamics of the
electron beam will be used. The differential equation for
F1, the a.c. part of the velocity potential, has been de
rived by Harris and applies equally well whether the cathode
is the inner or outer electrode. This derivation is shown
in Appendix I, and the result, equation (1.26), will be used
as a starting point in our analysis. Thus we have
F,7( + ,L =
' 1. V7    . .T
j :_  .. .[ .... l Z ~~ : ,~' ~
.......h .. .... .....h..... . ... ,
1 .~.1
n~n ^^rn^^__^^___ i:_ ^^_4^
i ...7 4 Ir tt'
S I ' '
 _ ___ ._4K .4_ __ .._ __ __
_  +  ...
, .... I ^ ^ x !  . m^"
I I'~
  *  . 
lI ' i . .I
........... ...
.;  ........ .... i 
' ...., r 'E
S: '' r  .
t I .: ~ "

.. .. . .. .. _.. ...1.. ~. ... .. .. ... .
___ iiI
S .1 _
. _ I I 
, , "t i" 
I'~ I I 4
~1 I : I : 
__~~ I. Ir Il_ _
Si. j II i I I. ':I
Here = I k
L m r '
where vo is the d.c. beam velocity and h is the wave number
in the assumed variation of F1 as exp j(wk4e ).
We must have the values of vo and of po the steady
state space charge density, in order to evaluate Wp and ,
which appear in (3.1). An analysis of the static character
istics of the outside cathode magnetron is given in Appen
dix II. The results of this analysis are, from equations
(11.9) and (II21),
(3.2) Vr. = = 1 ( ./)
and
(3.3) z eo e k r4
e.
where Uo is the cyclotron frequency.
Substituting (3.2) into the expression for v yields
(3.4)
and substituting (3.3) into (1.1) we have
LWP r+
61
Performing the indicated operations in (3.1) and simplify
ing the equation gives
0 CL LC L
(3.5) + *) a Cj = 0 7
where
W L 9~L
bo
and
AuJ. 4 A)
It is apparent that some approximation must be made
which will make equation (3.6) more tractable. Let us as
sume that r>?>r This means physically that we shall
restrict our solution for F1 to the region near the inner
radius of the space charge when the tube is operating in
a cutoff condition. With this assumption only the higher
order terms in the coefficients are retained and (3.5) be
comes
62
wd) e , F', e,
(3.6) rduFr dr s r W
which reduces to
(3.7)
where
dC F 3 d F, L
dr F , F o
d r r dr r'
L L
VL L40
The solution to (3.7) is
c, r
or, letting Z + 4*L = f=4+
(3.8)
F C. r + c rt .
Applying the boundary conditions FI = 0 when r : rc gives
(3.9)
F, = C, r r(r
The inward radial admittance, that is the admittance
at the inner radius of the beam looking in the direction
of the charge free region, will be the same except for sign
as that found by Harris5 for the conventional smoothanode
z ++fT
magnetron. The admittance expression is
dF,
0dr
(3.10) y, 
From (3.9),
dr
and
Sdf,
F; di
Atvf dr
4 d dr
re
4*;
dr
may be found.
1 r X
rz
which may be written
(3.11) F: dr F r
where
and
, the positive sign being taken.
Substitution of (3.2), (3.3), and (3.11) into (3.10) yields
It is
I
rrc
d Lawt',o ^<~ <44 Wo J
(3.12) y, =  *
,/ r _ 
We now direct our attention to the chargefree region
in order that we might determine a radial admittance for
that region at the edge of the spacecharge cloud. The two
values of admittance will then be equated.
A proper solution to the wave equation in cylindrical
geometry gives the following expression for the axial com
ponent of magnetic field:
(5.13) H= J( + Nr)
where Jh and Nh are Bessel functions of the first and
second kind, respectively, of order h, and K = .
C
The solution of (3.13) cannot become infinite on the
axis in the case of a hollow cylinder, and we reason that
the same sort of solution will hold here even though there
is a small anode cylinder on the axis. We then retain only
the first part of the solution, setting AL = o.
We next make use of the Maxwell equation
vC) 
remembering that all field quantities are assumed to vary
as exp (wt e) and obtain
")A& J .llr),
where Au is the permeability of the medium, and J (kr)
is the derivative with respect to r of JC(Kr).
Then the radial admittance may be expressed as
,__ JL Cr)
(3.14) Y= Ee A Cr)
We will now make the assumption that rr <'.1 This
means physically that the phase velocity of the wave is
small compared with the speed of light. This assumption
seems justified since the wave, if any, will be traveling
in synchronism with a rotating wave of space charge, and we
expect the spacecharge wave velocity to be not too differ
ent from the d.c. beam velocity.
With this assumption we may now use the small argument
approximations for Jt and JA and (3.14) becomes
Kr
(3.15) Y1 = d ;L
We may now equate the two values of radial admittance,
equations (3.12) and (3.15), and obtain
(3.16)  =
A, Er ^ ~
Recalling that KL=  u= LA we may simplify
(3.16) to give the desired equation in w This is
(3.17) Lao ( ) 4+ ^ = "
Since we have assumed solutions of the form exp j(wdt e ),
it is apparent that W must be complex with a negative
imaginary part if oscillation is to occur. Application of
the quadratic formula to (3.17) shows that will be
complex if the following inequality is satisfied:
(3.18) 4( ) '
For the assumed case of rl>>~ we see that
and the inequality becomes
(3.19) 8 4 )(4 ^) "
Inspection of (3.19) shows that for mr the first
term in parentheses is negative and the second term is
positive. Therefore, the product is negative and can never
be greater than h2. And since h must be a positive integer,
an investigation of the relationship between h and m shows
that the only allowed value of i< 2 is for h = 2, in which
ease 0 = 0. Substitution of 0 = 0 and h = 2 in (3.19) re
veals that the inequality is again not satisfied.
So the conclusion may be drawn that, within the limits
of the assumptions made, the cylindrical smoothanode mag
netron with outsidecathode will not oscillate. This is a
fascinating possibility because, if, as has so often been
postulated, the marked deviation of the cutoff curve of a
magnetron from the theoretically predicted curve is due to
oscillations, then it should be possible to check experi
mentally the cutoff voltage expression derived in Appen
dix II, equation (11.13), for the outsidecathode tube.
In any event, equation (11.13) deserves some special
comment. It is repeated here for convenience.
e f v f 4 1
(3.20) VC 8.
An appreciation of the magnitudes involved here may be
obtained by considering a particular tube for which
r : 0.655 cm. and ra = 0.0191 cm. For an anode voltage
of 1000 volts, equation (3.20) predicts that a magnetic
field of only 103 webers/meter2 (10 gauss) is required
for cutoff. Contrast this with the value of 35 x 10"
webers/meter2 required for cutoff with the same anode po
tential on an insidecathode tube of the same dimensions.
Theoretical cutoff curves for the two types of construc
tion with the same dimensions are shown for comparison in
Figure 3.2.
Equation (11.21), the expression for the static space
charge density in the outsidecathode tube, is also quite
interesting when compared with the corresponding expression
for the conventional tube. The two expressions are exactly
alike, term for term, each being given by
WO^m e. /, ( 'V
(3.21) ao = e +
But in the outsidecathode tube I < rc so the charge
density is greatest at the edge of the space charge cloud,
farthest from the cathode. This should be important from
the standpoint of efficiency in the inverted multicavity
magnetron, since it puts the greatest number of electrons
in the region where interaction with fields takes place.
In Chapter II we saw that there are other types of
oscillation possible in magnetrons beside the traveling
wave type; this type, we have shown, cannot exist in the
..... I_: .. .. .... ..' .J = = := ... :f +' . .
____ :L 2L  4 ....I 4
~ 
S  " L T"" vi
__" i. ; :; 1' *" _  __'_  ,**** * _
S ..... .... + L .....
I'ti. _^__^_^ ____^_^
__ I. ,. i S : .
~~~~ H + j
II
t .. .1 t sg ._ _. ... i *1 ii
7tt j jT T 7i777 t TI
_, i i i +
___i .4 I3 : . . ; .. i
; ~ J
I i ',' : :11 "
I I
: t I
I
11f' ,
IA
 4 I $  I  4 I '+  f+ ,
i I i Til
I
q 
1 1 :
i
r ...
i 
1+'
.... 1 i
i
i i i
1 I i
outsidecathode tube with smooth anode. We should, there
fore, investigate the possibility that this tube can present
a negative resistance to an external tuned circuit.
We shall follow here the approach used by Brillouin21
for the insidecathode tube. We assume that the potential,
the radial current, and the radial position of an electron
have d.c. and a.c. components given by
V= Vo(r) + V,(r,t)
(3.22) I Io I1 10
^ =r fo) + 'r,)
The radial force equation is derived in Appendix II,
equation (11.8), and is
(3.23) r = e + rie e Br6
Jf
and when the value of e from equation (11.9) is substi
tuted in (3.23), the force equation becomes
(3.24) nr l"=. t' r r w t f ,J
dr 4 (& ;Lr
We may now define an apparent potential P such that
SP
(3.25) Yv r = e r
Then from (3.24) we see that P V V., where V. is the
contribution of the magnetic field to the apparent poten
tial and is given by
(3.26) Vc  r
The force equation now becomes
d CG d_ V eSV .
(3.27) L d *S+ rr r
dfL d* c r / r adr
But
Vo.) = V.) + r, ,vo
and
Vc(r) = Vc(r.) r, j 1
J r
Therefore (3.27) becomes
(3,28) 4 d'r e /d(r) d V(r.) p ~ V, d~ V e V,
(5.28) t (r 7rF / ;;r 5W r
Equating the a.c. terms of (3.28) gives
(3.29) drr _ _^ e\.
( 3 H<9 {  dt *' dr
The radial current per unit length, including displace
.ment current, is
I =27~rrr( p + )=27rrr2T ,
dt/ it
where D is the electric displacement and vr is the radial
component of electron velocity. We now note that
(5.31) (r (D) () + (r ),
di,)t d
and from the divergence theorem in one dimension
(3.32) ( ) =
We see that (3.31) may be written
(3.33) dLrt ) = (r) + r .
Comparison of (3.33) and (3.30) gives
(3.34)
(3.35)
r T
di ; TT
t
dV1 (jAt
(3.50)
where the substitution
be JV
b=E 5
has been made.
We may now substitute the expressions of (3.22) in
(3.35) and obtain
t
(3.36) (ro+r) V. +r rA)= rf(rL*rI,)d .
If we now neglect the products of a.c. terms (small signal
approximation) and equate the a.c. terms of (3.36), the
result is
(3.37) d t 5 rd L T 1
We first solve the homogeneous equation, i.e., equation
(3.37) with the righthand member set equal to zero, and
obtain the natural vibration frequency uj. of r, The
solution to the homogeneous equation is of the form
(3.58) r,= A 6'' B 6i"
where W%. is given by
(3.39) 4)V =_ d Le dro/ I
d r J drl
The value of Vc is given in equation (3.26), and Vo is
given by equation (11.19) in Appendix II. These two poten
tial expressions are equal, so when the differentiations of
(3.39) are performed, the natural radian frequency becomes
(3.40) W0. 
V 7 T
In arriving at this result the approximation >> has
been made.
We now look for forced oscillations and assume that I1
is of the form
(3.41) I, = I E
and that rl is of the form
(3.42) r, = r4 L .
Then equation (3.37) becomes
(5.45) ) ( z)=_ T ( L r
where Z'= o is the transit time of an electron from
the cathode to the radius ro.
From (5.29), under cutoff conditions, we have
(3.44) C = ~ r,
and substituting the value of rl from (3.43) this becomes
(3.45) ( (A e )
Equation (3.45) indicates that the radial electric
field becomes infinite when uJ~= We recognize that
this is not a physically realizable situation and add a
damping factor S to the equation. Thus
(3.46) =  / : E
dr z 716 r. w uw } \ws
The d.c. components of equation (3.36) may be equated
to give
(3.47) = T 2.
But Vo as given by equation (II.19) in Appendix II is
(3.48) Vo(r) = ~ ')
where a is the cathode radius. This expression was derived
for cutoff conditions, but should hold approximately for
. __
small values of Io. Then
(3.49)
d V / . )
 = ; r I I
dr 4e.
Substituting (3.49) into (3.47) and solving for T we have
7T i'^ I aSo \
(3.50) r= 6 e I r
Now substituting (3.50) and (3.40) into (3.46) we
obtain
d X l E )
rrC r" (e t ) w,'f+ 1"
(3.51) =
dr
The internal impedance of the magnetron is given by
(3.52)
where ra is the anode radius. This becomes
rc. ;s
.dr
(3.53) o ,
/ "d
The integration of the impedance expression for the
insidecathode magnetron, which is quite similar to
V, tr
I;
equation (3.53), is the subject of a lengthy analysis by
Brillouin. We shall adapt his result to the present case
and obtain the approximate result
(3.54) T I
O1
o "
Analysis of (3.54) shows that the condition of physical
significance for which the real part of 2 may be negative
heresmall. Then the internal resistance is
and se e M 5 n b n ir, a.
Analysis of ((.54) shows that the condition of physical
sigenifica since for which the real part of neg may be negative
alueis that ) r large, and c negligible
small. Then the internal resistance is
(3.55) R
We see that equation (3.55.) can be negative for all
values of Wx which make co.f( 5. r.j < 0 ii
110" r, 36
r __ I_
) <  This means that for values of a) which make
R negative the tube is capable of sustaining oscillations
in an external tuned circuit. Equation (3.55) shows that
the value of R will be greatest when = or when
S= ( 7r The first of these conditions gives
(3.56) U)
The second condition yields
(3.57) 1) , .
When the anode radius is quite small compared to the cathode
radius, equation (3.57) reduces to (3.56).
Thus we see that the outsidecathode magnetron is
capable of sustaining oscillations in an external resonant
circuit. Oscillations are possible for frequencies which
make coS fo r ~ ') but will be strongest
for W in the vicinity of 'o/.
CHAPTER IV
ANALYSIS OF THE SPIRAL BEAM
TRAVELINGWAVE MAGNETRON
A variation of the travelingwave magnetron is shown
in Figure 4.1. This tube,first suggested by Harris during
the course of his work on hollow cylindrical electron beams,
operates in the following manner:
A hollow cylindrical electron beam is formed in the
electron gun. This beam is passed through a radial magnetic
field which gives the beam some tangential velocity. The
beam then enters the space between two concentric cylinders
between which there is a difference of potential as shown
in Figure 4.1. This potential is adjusted so that the in
ward electric field force acting on the beam balances the
outward centrifugal and spacecharge forces. The result is
a hollow beam spiraling along the axis of the tube.
A slowwave structure is now wound on the inner con
centric cylinder with a pitch which is equal to the pitch
of the spiraling electrons. The physical dimensions of this
guiding structure are such that a radiofrequency wave ap
plied to the input will travel down the slowwave structure
6vw
WMBAIw.UAK !RAVLMINGLVAUX KUM"=
IPG JI'.1
6d
with a phase velocity which is approximately equal to the
average linear velocity of the beam. This gives a wave and
an electron stream moving along in synchronism in a spiral
path.
This appears at first glance to be just a traveling
wave tube wrapped into a spiral, since there is no magnetic
field present in the interaction space. However, the sig
nificant difference is that here there is a force field,
the centrifugal force, which acts at right angles to the
path of the electrons just as the magnetic field force does
in the travelingwave magnetron described in Chapter II.
The result is travelingwave magnetron action. Electrons
give up potential energy to the wave by falling toward the
center conductor, rather than giving up kinetic energy as
in the travelingwave tube.
The advantages of this arrangement over the conven
tional travelingwave magnetron are threefold. First, the
interaction space can be made long,as in the traveling wave
tube,without making the tube physically long. Second, no
static magnetic field is required in the interaction space;
true, a magnetic field is required to give the required
tangential velocity, but this radial field is relatively
small. And third, analysis8 of the spacecharge conditions
in the beam shows that the charge density varies as ,
which means that the majority of the electrons will be
\
concentrated near the wave being amplified, where they will
do the most good.
The tube will be analyzed using the velocitypotential
approach as in Chapter III, but in this case both the geom
etry and the spacecharge conditions will be different from
those of the cylindrical magnetron. We shall consider the
particular case in which the slowwave structure is a helix
of rectangular crosssection. It will also be assumed that
the actual structure of the interaction portion of the tube
can be approximated by a linear structure as shown in Fig
ure 4.2. The problem then becomes one in rectangular geom
etry, but one in which the charge density and electron
velocity of the actual spiral structure will be used.
Since there is no centrifugal force acting on the elec
trons in the linear system, a static magnetic field, B is
added to the system. This field is assumed to vary with y
in such a manner that it produces a force on the electrons
equivalent to the actual centrifugal force.
The derivation of the differential equation for Fl,
the a.c. part of the velocity potential, proceeds in a man
ner exactly analogous to that given in Appendix I. The only
difference is in the geometry. In the present case, Fl is
a function of y and is assumed to vary as exp j( w P" ).
Since this differential equation has been derived by Mac
farlane and Hay25 and by Bohm,26 it will be used as the
 i7I
I~ ~~~ t a H 
I I*t14
7
IA I.
 . iT tr 1 ::
. : i :  2 L  :
AL I'
444. 1
TIM
~1ciL. I I .1. 7.
 IL
1 F 7 I I .T :i .~i
~ irV  N. L:4JI LI_ t~
I., .j .
4i 4 ll H _
 : i ,.  t ti 5
rj7E41{ _ ___
S  ~  _ ~
_i _
starting point for our analysis. Thus,
Here up is the plasma resonance frequency, and is given
by
(4.2) V ,
where vo is the d.c. electron velocity. From equation (1.1),
the value of UJ; is
(4.3) u *
Thus we see that the velocity potential is dependent
upon vo, the d.c. electron velocity, and upon 9, the d.c.
charge density. All electrons are accelerated to the same
velocity in the electron gun, and since the passage of the
electrons through the radial magnetic field does not alter
the linear velocity, vo will be a constant. As mentioned
previously, the equilibrium conditions established by Harris
for the beam show that the charge density varies inversely
with the fourth power of the radial position. If this re
sult is translated to our rectangular system, the charge
density variation becomes
(4.4) P = V
Here, Y) is the magnetic flux linked by the electrons at
the cathode.
The expressions for the potential of the two cylin
8
drical electrodes are also derived by Harris. These are
necessary in the design of the tube, but since they are not
used in the gain analysis, they will not be given here.
The complexity of the differential equation which re
sults when (4.3) and (4.4) are substituted into (4.1) makes
it desirable to seek some simplifying assumptions. We re
call from the discussion of the travelingwave magnetron in
Chapter II that maximum gain occurred when wave velocity was
equal to electron velocity. Under these conditions F is
equal to I plus a small imaginary part. From (4.2) we
see that is quite small for r near and that under
such conditions it is permissible to neglect 1 in E7/'
This results in considerable simplification of the differ
ential equation, so we shall consider only the case where
the phase velocity of the wave is equal to the electron
velocity. For this condition, substitution of (4.2) and
(4.3) into (4.1) results in the differential equation
(4.5)  d.
J. d
The dnd result which we are seeking is a solution for
r from which we can determine the gain of the tube. The
approach will be the same as that used for the outside
cathode magnetron in Chapter III. An admittance, defined
as is found at ys, the lower surface of the beam
Ex
(see Figure 4.2). This admittance is calculated in two ways;
one expression is derived from a consideration of the dynam
ics of the beam, and the other from a solution of the wave
equation on the slowwave structure. The two admittance ex
pressions are equated, and the result is an equation in
which r is the only unknown.
The admittance, Y1, derived from the beam dynamics has
25 26
been determined by Macfarlane and Hay and by Bor6 for
the general case of a linear beam. We shall make use of
the result obtained by these authors. It is
(4.6) /:F, d j .t Fa o r M
An exact solution to the wave equation in which the
boundary is a helix of rectangular crosssection is pro
hibitive. However, this particular problem has been solved
22
approximately by Brossart and Doehler in their analysis
of the linear travelingwave magnetron, which was discussed
in Chapter II. These authors simplified the problem by as
suming that the highfrequency fields in the interaction
space are due entirely to highfrequency current on the face
of the helix nearest the beam, the influence of the farther
face and the lateral faces being negligible. The values of
E and Hz obtained in this manner are
(4.7) EA= s.' d. d^^
and
(4.8) H = co'" I, Al coAk(d) e
In these equations A1 is an arbitrary constant, d is the
spacing between the two electrodes bounding the interaction
space, 9l is the pitch angle of the helix, and oC is given
by
(4.9) 0( z 4 )
The factors p and s are, respectively, the pitch of the
helix and the length of the portion of a turn lying in the
face of the helix nearest the beam.
The admittance, Y2, determined from (4.7) and (4.8) is
(4.10) ,4 a d )
YZ Exco jY)
The two admittance expressions, equations (4.6) and (4.10)
are now matched at y,, the lower surface of the beam, to
give the desired equation for r It is
(4.11) C5 r 5 C [i Ca3 c +CrL C,,r P Co =o 0
where the coefficients are given by
IMM
)coJt'
(4.12) CC = k 4~) ed)\et)
(4.15) C,= I d jdo' 3S 6 o i. A c t o
(4.16) C j = y 27 ^ e oW0 3 a o. a ar
and
(4.17) Co '= e/ w' + uJ4 .
The factor R which appears in C2, C3, C4, and 05,
i dF
is the ratio, ., evaluated at y ys, Tt is apparent,
therefore, that the differential enuation for F1. erua
tion (45), must be solved before r, can be evaluated, The
solution of this equation is found in Apprndix TII,
The fifth degree equation in f, euation (&.11), must
now be solved. Let us denote rbv
(4.1) r : a +' b
From the assumed vpripti)n of the wave as pxi i(wtrx)
we see that a is the phase constant And r the attenuation
constant. A positive value for b will indicate a wave
which is increasing with x.
To solve equation (I.11) we try the value r'=
This proves to be a solution to the e'iation and indicates
an unattenuated wave traveling in the x direction with
phase velocity, c When the root,r= = is fsctorpd out
of equation (4.11), the result is
f.19) r4 +B,Pr ?B r + 3 r + B,=o .
. r
The cieffi~ rt ntsq PrP p ?,n bv
(L.20) 8, ro
w w R's 1f
(4.21) B y o'cfcod4(^) '
3o
(Z .22) 03 Y v s+o
tr, a ckK i/ coit%0Cd )
and
(..23) Bo = ~ / coV cot a(dy)
"e Ise thp metmhoe of ePsumTfle, uadratio factors to
solve the quartic fauation, (,.19). That is, we express
equation (4,19) in the form
(4.24) (r aL, r + )(r b, r + b.)=
For a wave traveling in the +x direction with Dhase velo
city, ITo i either aI or bI nust ePual Y Tf w choose
a,= and eauete coefficients of like powers of f
in enuations (4.19) and (6.24), we find that b,
This inAicatrs a Dair of waves traveling in the x direct
ion with a phase velocity enual to onrhalf the electron
velocity
I
The four values of r obtained from the solution of
equation (4.24) are
2 t (+ ZJ e) ,o
(4.25) w,,4 Ur U b wcot.^coi4iCtWyf)
and
^Z(L) 4j '4ww6. p. e) P., "
(4.26) '3 =o 7 ;"coi'eo/4( fJ .
9, is the value of interest, since it indicates an in
creasing wave traveling in the +x direction. For this
wave, the value of the ettenuation constant, $, in e~uat
ion (4.18) is
(h.27) b '" 4 
The value of b can thus be celculpted for a particular
tube, mh ,gaining wavP is amnlified at the rate of 6
nepers Drr meter, or 8.7 b decibels Der meter.
.It should be remembered that the smallsignal
approximation was made in the derivation of the differ
ential equation for,Fl, Therefore, if the interaction
space of a tube is made long enough to allow highlevel
bunching, equation (4.27) may renutre modification.
CHAPTER V
EXPERIMENTAL RESULTS
Three cylindrical smoothanode magnetrons were con
structed and data were taken on each to determine the fre
quency of observed oscillations as a function of magnetic
field strength and anode potential. The tubes tested were
(1) an insidecathode magnetron with large anode to cathode
radius ratio, (2) an insidecathode magnetron with inter
mediate anode to cathode radius ratio, and (3) an outside
cathode magnetron with large cathode to anode radius ratio.
InsideCathode Tube With Small Cathode
This tube was designed to check the theoretical pre
dictions made by Harris for a tube with "vanishingly small"
cathode.
The method of testing is shown in the block diagram of
Figure 5.1. The d.c. anode voltage supply was modulated at
a 60 cycle rate by the output of the transformer, T. Oscil
lations which occurred were detected by the crystal, X, and
the amplified r.f. envelope was applied to the vertical de
flection plates of the cathoderay oscilloscope. The hori
zontal deflection voltage was a 60 cycle sine wave which was
/I
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