Citation
On space-charge wave propagation in crossed electric, magnetic, and centrifugal force fields

Material Information

Title:
On space-charge wave propagation in crossed electric, magnetic, and centrifugal force fields
Added title page title:
Space-change wave
Creator:
Lear, William Edward, 1918- ( Dissertant )
Larsen, M. J. ( Thesis advisor )
Chen, Wayne H. ( Reviewer )
Smith, C. B. ( Reviewer )
Fyner, Mack ( Reviewer )
Smith, Alex G. ( Reviewer )
Hanson, Harold P. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1953
Language:
English
Physical Description:
147 leaves. ; 28 cm.

Subjects

Subjects / Keywords:
Anodes ( jstor )
Cathodes ( jstor )
Charge density ( jstor )
Electric fields ( jstor )
Electrons ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Magnets ( jstor )
Velocity ( jstor )
Waves ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Electronics ( lcsh )
Vacuum tubes ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Abbreviated introduction: Space-charge 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.
Thesis:
Dissertation (Ph.D.) -- University of Florida, 1953.
Bibliography:
Bibliography: leaves 145-146.
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
000577387 ( ALEPH )
13977482 ( OCLC )
ADA5082 ( NOTIS )

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Full Text









ON SPACE-CHARGE 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 Space-Charge Condition
in a Plasma ..

Klystron Amplifier .

Traveling-Wave Tube .

Electric Field of Traveling-Wave in
Moving Coordinate System .

Outside-Cathode Magnetron .

Comparison of Theoretical Cutoff
Curves .

Spiral-Beam Traveling-Wave Magnetron

Linear Traveling-Wave Magnetron .

Block Diagram of Circuit Used to
Determine Oscillation Frequencies .

Observed Oscillations, Tube No. 1 .

High-Frequency 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 ILLUSTRATIONS--Continued

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. SPACE-CHARGE WAVES INTRODUCTION


II.


ANALYSES OF SPACE-CHARGE WAVE TUBES


* *


The Klystron
The Electron Wave Tube
The Traveling-Wave Tube
The Traveling-Wave Magnetron
The Magnetron Oscillator


:I. THE CYLINDRICAL DIODE MAGNETRON WITH
OUTSIDE CATHODE .

V. ANALYSIS OF THE SPIRAL BEAM
TRAVELING-WAVE MAGNETRON .

V. EXPERIMENTAL RESULTS .


Inside-Cathode Tube With Small
Cathode
Inside-Cathode.Magnetron With
Intermediate Cathode Radius
The Outside-Cathode 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 CONTENTS--Continued



SOLUTION OF THE DIFFERENTIAL
EQUATION FOR F ...


BIBLIOGRAPHY . .*.

BIOGRAPHY . .


Page


142

145

147


vii
















CHAPTER I


SPACE-CHARGE WAVES INTRODUCTION

Space-charge 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 10-6 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 space-charge

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 time-varying 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 ---,+- s-4 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 eigen-equation

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 two-velocity 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 space-charge 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 natural-period 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 time-varying 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 electron-wave 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

time-varying electric field of angular frequency, uJ to

the beam and determine the velocity of propagation of the

space-charge 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


V-V = -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 small-signal approximation, we have

neglected the second-order 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 one-dimensional 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 waves-having phase velocities of and L

Wp _-ro respectively.. That is, when u >7A p

the two space-charge 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


traveling-wave tube, electron-wave tube, and traveling-wave

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 space-charge wave tubes rather than as a group of unre-

lated devices. This will be done in Chapter II.
















CHAPTER II


ANALYSES OF SPACE-CHARGE 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 two-cavity 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\H-a.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) space-charge 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 ;(uij-ATTTN)



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 space-charge

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'= I-f ^- ( 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 small-argument 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 space-charge 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 space-charge 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 two-cavity 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 space-charge waves propa-
gated in the single-velocity 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) V-r. = 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 space-charge wave point of view. Suppose that the, two

space-charge waves with which we are now familiar have been

set up in each of the two streams of the two-velocity 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 Traveling-Wave 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 space-charge waves but between

a space-charge 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 "slow-wave structure".

Analyses of the traveling-wave tube have been of two







------ -- .T 4



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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 t-r ).

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 (tJt-Pj) 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 g-p sVo

where E is the peak magnitude of the electric field acting

on the electrons, and P is the power flow.
The field-theory approach to the traveling-wave 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 small-signal

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 charge-free 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 space-charge 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 small-argument 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.] P-P. E PPI-
(2.52) Yr= -(P -P P) P-. (P'P P-FP'


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 traveling-wave tube operates at a much lower level of

bunching than this, however.






40


It is apparent that the traveling-wave 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 slow-wave structure is one (such as the helix)

for which the phase velocity of a wave is relatively inde-

pendent of frequency.


The Traveling-Wave Magnetron

This tube is similar to the traveling-wave tube in that

there is a wave guided along a slow-wave 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 traveling-wave 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

traveling-wave tube. The traveling-wave 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 traveling-wave 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
traveling-wave 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 traveling-wave 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 slow-wave 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 traveling-wave 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 traveling-wave 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 traveling-wave

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 traveling-wave tube.

The wave solution for the traveling-wave 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. Space-charge

forces are neglected in this case. In the other system the

inner electrode is the cathode. Here the electron trajec-

tories are epicycloids, and space-charge 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 slow-wave structure is a flat helix with its axis
along the y-coordinate 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 -.

ii-sink (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 time-varying 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 small-signal










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 slow-wave 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 (V-r + 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 ) 1-t .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 traveling-wave 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 traveling-wave 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 --4-I--

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 space-charge wave terminology, and returning to a

stationary frame of reference, we have a traveling wave and

a space-charge 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 traveling-wave tube and of the

electron-wave 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 traveling-wave 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 broad-band.

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 large-signal, rather than small-signal, behavior, and

it is also apparent that space-charge 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 self-consistent 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 space-charge 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 small-signal 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 outside-cathode 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 smooth-anode tube is not so apparent since there is no

slow-wave 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 electron-wave 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

outside-cathode 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 two-beam 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 two-beam tube behavior contin-

ues up to a frequency Ir Above this frequency

the tube has a low gain of a traveling-wave tube nature.

For oC 0.42 the tube behaves primarily like a

traveling-wave 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 slow-wave structure.

This traveling-wave 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 slow-wave structure.

The case analyzed by Bohm is a limiting case of the

slipping-stream tube where IV- =--0 and it is shown for

this case that no amplification will result.

In addition to the traveling-wave 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 traveling-wave type,

are the negative resistance oscillator and the cyclotron

frequency oscillator. Since our primary concern is with

traveling space-charge waves, we shall not discuss these

oscillators here. However, an analysis for the negative

resistance characteristics of the outside-cathode magnetron

will be carried out in Chapter III.


J -*.















CHAPTER III


THE CYLINDRICAL DIODE MAGNETRON

WITH OUTSIDE CATHODE


The cylindrical smooth-anode 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 -' '




-- --___ .-_4-K .-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(wk-4e ).

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 L-40


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 smooth-anode


z ++fT











magnetron. The admittance expression is


dF,
0dr
(3.10) y, -





From (3.9),



dr


and


Sdf,
F; di-


Atv-f 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
r-rc














d Law-t',o ^<~ <44 Wo J
(3.12) y, = -- *
,/ r -_ -



We now direct our attention to the charge-free region

in order that we might determine a radial admittance for

that region at the edge of the space-charge 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 space-charge 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(wd-t 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 smooth-anode mag-

netron with outside-cathode 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 outside-cathode 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 10-3 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 inside-cathode 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 outside-cathode 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 outside-cathode 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-- 2-L | 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-










outside-cathode 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 inside-cathode 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- { -- d-t *' dr


The radial current per unit length, including displace-

.ment current, is













I =-27~rrr( p- + )=--27rrr-2T- ,
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)= r-f(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 right-hand 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
inside-cathode 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 outside-cathode 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

TRAVELING-WAVE MAGNETRON


A variation of the traveling-wave 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 space-charge forces. The result is

a hollow beam spiraling along the axis of the tube.

A slow-wave 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 radio-frequency wave ap-

plied to the input will travel down the slow-wave structure




6vw


WMBAIw.UAK !RAVLMINGLVAUX KUM"=


IPG JI'.1


6-d











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 traveling-wave magnetron described in Chapter II.

The result is traveling-wave 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 traveling-wave tube.

The advantages of this arrangement over the conven-

tional traveling-wave 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 space-charge 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 velocity-potential

approach as in Chapter III, but in this case both the geom-

etry and the space-charge conditions will be different from

those of the cylindrical magnetron. We shall consider the

particular case in which the slow-wave structure is a helix

of rectangular cross-section. 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*-------t------1------4





-7-
IA I.
-- -. iT t-r 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 --l-l-- -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 traveling-wave 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 E-7/'
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 slow-wave 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 cross-section is pro-

hibitive. However, this particular problem has been solved
22
approximately by Brossart and Doehler in their analysis

of the linear traveling-wave magnetron, which was discussed

in Chapter II. These authors simplified the problem by as-

suming that the high-frequency fields in the interaction










space are due entirely to high-frequency 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


IM-M





)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 (4-5), must be solved before r, can be evaluated, The

solution of this equation is found in Apprndix TII,

The fifth degree equation in f, e-uation (&.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(wt-rx)

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(^-) '
3-o


(Z .22) 03 Y- v s+o
tr, a ckK i/ coit%0Cd- )


and


(..23) Bo = ~ / co-V cot a(d-y)


"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 onr-half 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.^coi4iCtW-yf)


and


^Z(L) 4j '4-ww-6. 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 small-signal
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 high-level
bunching, equation (4.27) may renutre modification.

















CHAPTER V


EXPERIMENTAL RESULTS


Three cylindrical smooth-anode 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 inside-cathode magnetron with large anode to cathode

radius ratio, (2) an inside-cathode magnetron with inter-

mediate anode to cathode radius ratio, and (3) an outside

cathode magnetron with large cathode to anode radius ratio.


Inside-Cathode 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 cathode-ray oscilloscope. The hori-

zontal deflection voltage was a 60 cycle sine wave which was


/I




w


- :: ( I r
'1 I... .1. ..
-: I~i.. -- -- -
____ .1




-4. -- -- .- -- t-
L i I .I.
-t`! .

f 1 -I--
-.i I-I:. ;. .
-7~ tt --- 77 A- iV- Op~ 7 K K

---Th2-- ~- -----I


-r _-
* -i --- -i _i- -
i-~~ -- .

__LL'I *1Ii __II'--
__ __ L i w




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