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THEORY OF THE CONSTANT GRADIENT LINEAR
ELECTRON ACCELERATOR
by
R.B.Neal
Technical Report
A.E.C. Contract T(04-3)-21 (Project ment o. 1)
M.L. ReI rl N • 513 1\Iay, 1958
ftficrowave Laboratory • • lliNSEN L..uJORATOJll.ES 0 PJJy I
5'1: RD U frY TA! ORD CA:uro
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THEORY OF THE CONSTANT GRADI ENT LINEAR
ELECTRON ACCELERATOR
by
R. Bo N~a1
A.E.C , Contract AT(04-3)-21 (Project Agreement No.1)
TECHNICAL REPORT
M. L. Report No . 513
Ma,yo 1958
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TABLE OF CONTENTS
Page I. Introduct i on....... 10 • • • • • • • • ••
II. Design philosophy . . . . . 0 . 0 , . 0 20 . · III. Required variation of t.he attenuation
coeff i.cient q I . . 0 . . . . · . . 30 0 0 IV. Effect of varying electron beam current
on power flow. . . . . . • • < . . . . 50 V. Electric field strength and electron energy 6·
VI. Filling time. o • • 7
VII. Stored energy • 8
VIII. Effective shunt impedance per unit length 90
IX.. R-F to beam conversion effic i ency. 12
X. Example of constant gradient accelerator . 150 0 0 0
Appendix Power flow in the constant gradient
accelerator 17D 0 0 0
- ii _.
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LIST OF PIGURES Page
1. Variation of the r-f power with the distance alo~ 22
t he accelerator for various values of the beam loading
paramete r 0 m 0
2. Effective shunt impedance -r per unit length 23 versus T = wtp/2Q for various values of beam load i ng
paramet e r 0 m. -3. Effect!ve shunt impedance r per unit length 24
versus beam loading parameter 0 m for various values
of 't':;:: wtpf2Q
4. T = wtF/2Q versus beam loading parameter 11 25
m 0 for t hree conciitions (with Ina c:: 0 ) 0 5. Conversion ef fici encyo ~ v versus 26
.or = wtp/2Q for various values of beam loading parameter 0 m.
6. Conver sion eff1.ciency 0 ~ 11 versu' beam loading 27
paramet eru m 0 f or various values of T ~ ootF/2Q •
7 . Comparison of t he maximum conversion ef f i c i encies 28
~ max. for t he constant gradient structure (with0
mO : 0 ) and the unifo~ st ruc ture.
8. Comparison of the maximum conversion effic ency 29 -~ max. a nd effective shunt ilnpedance o r of the0
constant gradi ent st ructure versus T ~ ~tp/2Q for
various design val ues of t he beam load. ng parameter Q roo; also shown are values f or the unifo~ structur e .
9. Variation of t he group veloc 1t~ v g f or various 30 Q
lengths of consta nt gradi ent structure (for nu = 0 ) wi t h certain assumed parameters.
10 . Elect ron beam energy 11 V , versus peal:: beam 31
current 0 i f or v.arious l engths of constant gradient
structur e (wit h ~ ~ 0 ). Also shown is the conversion
ef f i c i encyo ~ 11 when PL c:: 0 for each length.
~ iii - .
-
32
page
011. Relative values of r shunt impedance per unit length and Q versus group veloc i tyo v g " for the di sk
loaded st ructure operating i n t he TI/2 mode ; andTO Oa are the values at v Ic =0.01.
g
- i v
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THEORY OF THE CONSl'ANT GRAD IENT LINEAR
ELECTRON ACCELERATOR.
I. I NTRODUCTION
The linear e l ectron accelerator with the usual unifozm modular'
dimensions has the property that the fractional dissipative loss i n
r -f power per unit length in the conducting walls of he structure
is constant over t he entire 1 ngtho This means that the magnitude
of t he r-f power a nd the e1 etria field strength in the structure
decrease exponentially th lengtho Thus " the peak field is higher
than the average field in the structure " In an accelerator of
opt:lJ:num designq it may be shownl that the ratio of peal:: to average
f i e l ds is appro:dmat ly 10 75 ( neqligible beam loading case) u
The maximum electron nezvy Mch may be achieved in a given
l e rgth is limited by the peak fields which may be produced i n t he
accelerat or structure without arcing, excessive field emissionq or
gassinqo It is c lear that a structure designed to have a constant
f ield throughout can produce electron energies about 1 015 times as
high a s an optimized unifozm structure when both are operatil'g' at
t he l imit of electric f 1 ld strength o Of courseo a larqer amount
of r-f power is needed to obtain the incr ased electron energie
but even with the same power input the constant gradient structure
produce s s lightly h i gher electron energies t han t he uniform
structure 0 1 The increased accelera ion is not large enoug to
wa rrant the mbre complied ed design of the constant gradient
s t ruoture , Such a desig n is i ndi cated howeve r where the electric
fields expected in the struoture are near the critical value at
the operating f requency .
The constant gradient linear lectron accelerator has
pre i ously been tud i edl under the Simplifying assumption of zero
beam l oadiIW;1 0 In t h is r eport II the effect of beam loading upon the
design and perfollllance of the accelerator will be examined . In
-
addit iono the results will be given i n dimensionless form so that
t he various equations and graphs can be used at any operating
f r equency.
Another advan'age of the constant gradient accelerator which
will be shown in this study is that it i s somewhat less subj ect to
beam loading t han t he uniform accelerator structure . Thus 0 f or an
acceler ator of given f illing time and lengtho it shoul d be possible
to obtain approximatel y 5 to 20 per cent more beam power with the
constant gradient machine.
II. DESI GN PHILOSOPHY
It is possible t o des i gn the accelerator struc ure i n the
following wayS3
.1 a To achi eve constant electric field gra ent i n the
struct ure i n t he absence of the elect ron beam~ in this case o the
steady-state electric field will decrease wi h distance after the
beam is turned on ,
2. To achieve const ant electric f ield gradient in t he
structure in the presence of the electron beamg i n this case o t he
steady~state electric f ield wil l increase with distance before t he
beam s t urned on . It will become uniform over the accelerator
length when an elect ron beam of the design value i s t urned ona It
wi ll i ncrease (o r decrease) with distance if the elect ron beam is
less (or re) than the design value o
3 0 To compromise betwee ca ses (1) and (2) by causing the
electric field to i nc rease with distance before the beam i s
turned on i n the same relative amoun that it decr eases with
d stance after a beam curre ' of the desi gn value is t urned on.
Usually 0 case (1) will be the favored design as it is the
most oonservative from the r-f breakdown viewpoint o Ther e may be
cases o however 0 when case (2) may be desirable. Suppose q for
exampleo t hat the f illing time of he accelerator is short
compared to the t otal beam time . In this event Q t he desi gner may
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choose the des i gn of case (2 ) in the expectation that t here i s les s
danger of r -f breakdown before the beam is turned on. When the
beam is present. the probability of breakdown may be enhanced by
the presence of stray elect rons and secondary gamma radiat ion .
Case (3 j may be chosen if the designer is not qUite so confident
of the valid.i.ty of t hese argument s and wants t o compromise between
cases (1 ) and (2) .
In the discussion which f oll ows u we wil l. s pecifically treat
case (2) but the results may eadlly be applied to either case (1 )
or case (3j. The graphs wi ll mainly apply t o ca se (I ) .
III. REQUIRED VARIATION OF THE ATl'ENUATION
COEFFICIENT I .
For constant electric f ield strength. t he r~f power dissipated
per unit length in the accelerator structure ~lst be constant over
the accelerator enqtho In other words q the same amount of r-f
energy must be made ava. labl e t o every ca vity of the s t ructure in
order to obtai n a constant el ect ric field gradient . Tlms o the r - f
power must decay l inearly with d istanceo i oeo 0 2
where
PI '''' inpUt r -f power
p 2 "'" r-f power rema,ini ng at the end of the accelerator section
L ,~ length of acceler ator section
z "" d istance from inpllt end
y - (P1 - P 2 ) / L a verage power expenditure per un!ta:;: length in the accelerator.
6If the power P is plotted aga.i.nst distance Z the result is a
straight line of slope
http:valid.i.ty
-
In the presence of electron beam oadinqo the t otal r-f power
loss per unit l ength is gi ven a; -7
dP/dz = (dP/dz wall losses + (dP/dz)to electron beam
2 1 /2 ~ -21P - (2IPri ) (3 .3)
where I is the voltage attenuation coeffic ient i n em- a.nd r 8
is the shunt impedance per unit length .
Equating Eq. (3 .2 ) and Eq. (3. 3) :
(3 . 4)
Substit uting for P f rom Eq . (3 . l) ~
(3 . 5)
Eq. (3.5) i s a quadratic equa ion which may be so ved for I
_. 2 2 -J 1 /21 + I (1 +!.L.) - 1 .J
- '\ 2Y , (3.6 ) 2(P = yz )
1
0For a given value of Y t he r-f attenuation coef ficient
I must be ess with beam loading than in t he case of no-load~ therefore" the negative sign must be al::en in Eq. (3 06) . It is cl ear
fram Eq. (3 .6 ) that for constant electric field str ength the
attenuation coefficient I must be increased a l ong t he accelerator
-
length to compensate for the loss in power to the walls and to t he
beam.
IV. EFFECT OF VARYING ELECTRON BEAM CURRENT
ON PCMER FLOW
After a constant gradient accelerator has been designed for
particular values of electron current i and entrance and exit
powers P and P u how do the r - f power q electric field q andl 2 electron energy vary as the electron current is changed? We must
return to the basic Eq. (3 . 3). We will designate t he des i gn
values of the parameters i and I as i O and 10 The same
convent i on will be followed with other parameters to be i ntro
duced later. Then g from Eq. (3.6)0 the design va lue of t he
attenuation coefficient I(z ) is given by~
(4. 1)
where
[( . r.i~ ) 2 1 /2
1.,. · - 1 ] (4. 2 ) . 2Y O
Substituting the design value TO from Eq. (4. 1) into Eq. (3 . 3) ,
the result is
(4.3)
Equation (4 . 3) may be integrated ( see Appendix) to find the
r-f power f low as a funct i on of di s tance z . The result is~
- 5 -.
-
- \
When iO = a (case (1 ) of Sect i on 11 )0 KO = 1 and Eq. (4. 4 )
reduces tog
. II i o ' 0 (4.5)
The var iati on of P with distance z (for KQ = 1) is shown in F,i.g. 1 where P IP is plott ed agai nst YOz/P for ~arious
1 l
I .2) lf2rl values of m == ~~ • The physical meaning of t he parameter
o
In will be discussed i.n a l ater section.
V. ELECTRIC FIELD STRENGTH AND ELECTRON ENERGY
The electric field strength may be obtained from the power
f low Eqs. (4.4) and (4 .5 ) 11 and fram Eq. (4. l) .
( S . l)
E == 12-1 Pr o I (. . 2 1 / 2
I _n \ ::: (Yor) , 1/2 -L1 - 1/2 J
\ YO /
-
Equat ions (5.1) and (5.2) can be i ntegrated to f' nd the
electron energy as a f unct i on of he peak beam c r rent i.
(5 .3 )
V :: (Y r)1/2 · 0
j = 0 ( 5 .4)o
VI. FILLING TIME
In the incremental t ime dt u the energy i n the .r - f wave
will move through the distance
dz ::.; v dt (6. ) g
where v 1s the group veloci ty i n the accelerator structure. But 0 g 1 it . . 1-.10the group ve oc y 1S g1ven ~3
j
where w is the angula r frequency in radians/second.
- 7
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Vlir 0 EFFECTIVE SHUNr IMPEDA CE PER UNIT LENGTH
To obt ain a better understan nq of t he const ant gradient
accelerator" it is hel pful to r ewrit e the electron energy
equations i n dimensionl es s terms wh i ch. are suitah~e at any
frequency. The follow.ing t erms will be used
(a) }f where :r i s t he ef f ect ve shu nt impedance per unit(I l e ngth. Us i ng r t he e act ron energy is g i ven by
so t hat
c V = I !.
V r Ip Lr
v 1
(b) m ~ the rat i o of ther~f powe r transferred to the electron
beam per unit length t o the power d issipat ed in t he walls of t he
st ruc t ure per un ' t l ength ,. bo measured at the i nput end of
the accelerat or (z = 0 ). Thus,
where 1 is the design value of I a t the i nput end of t he 10
accelerator and the othe r t erms are as previously defi neda
From Eq. {4 .l)
( 8.4)
- 9
-
Subst i tuting Eq. (S.4 ) into Eq. (S.3) and rear ranqi nqu we obtain
, \1/2 m '~ (Y~o) (8 05)
(c) mO " the design value of m perla! i nq to a particular
design value of the beam cu r rent, iO
Substitutinl;1 Eq. (8 05 ) (using 10 and nul into Eq. (4.2) and solving for KO ~ we obtain
K ' ~l;:...._0'- + Ino
In order to make use of the energy equations already obtained
(Eq. (5 .3 ) and (5 .4 »), it is convenient to rewrite the relation for
if as {8. 7 )
Equation (6.4) may be rearranged to yie d t he following
expressions
(8 . 8)
or
(S o9)
- 10 =
-
Using the expressions devel oped n t his sect onu t he
no r.mal zed effective shunt tropedance factor~/ r may be obtained fram the energy Eqs. (5. 3 ) and (5.4). The results are
j~ ~ [1
x
_2T\1/2 - e )
where 20
The more general aquati·on (8. 10) does not lend itself s mply to
graphical r epr esentation. Howeveru the more customary case
(~ ~ 0 ) may be easily presented. This has been done i n Fig. 2
where Jr/ r is plot edl 2 aga inst T = wtp /2Q for various values
of m ,-The quanti y /r/ r may also be plo t ed against the beam
loadi ng parameter m wi th T as a parameter a.s shown in Pi g. 3 .
In t his presentationo it is not ed that 'he value of !r/ r decreases
linearly with increasing beam loading , The slope of each curve
is given by
- 11 ~
-
As the b eam loading parame er m is ncreased. q the max imum
val u e of jr/-; occurs f or lower va.lues of -r By di f ferentiati ng Eq. (8.ll ) w th respect to 't" Q the oondition for DlaJtimum
$/r is found to be~
2
m :~ ------------~---- !80 l3)
2-r0 · 1 ~ ~ -2-i) - 1 Th is conditi on 1s plotted as curve A of Fi go 40 The value of
m at which the r-.Jf power is reduoed. to zero at z =: L is given
by
m ,~.: nul '1 - e-'t1DQ) (8 . 14) oro
-1m :: .t" o ( 8.15 )
These poi.nt s are marked by dot s on the curves of Fig. 2 and Fig, 3 0
IX. R-F TO BEAM CONVERSI ON EFFIe ENCY
We wi l def in the r -f to beam conversion eff'c iency as
Vi n (9 1)0P
This is obviously the frac tion of e inci de t r-f p e r which is
converted to b eam power. Using Eqs o ( 802)g (8 0 5 ) # a nd ( 8.9 )u we
may write Eq . ( .1 as
~2'r(l+mo )/1 - e n m 9.2I
I -ff 1 "11- nu Then 0 fram Eqs. ( 8 .10) and ( 8 11) we obtain
- 12
-
T] = 1 : mO (1 - e - 2T ( 1 +mo)j r!L ~ 1mo l
1 O
- -2T (1+mo/ 2) ( s .3)m ~ ,l e____~~______ ~lJrl~~~
( in
-
Equation (9 . 6 ) is seen to be i dentical with Eqo (a .lS) ~ the
latter equation was shown to be t he condition for P ~ 0 0L
These relationso together with the condition for maximum
1~/ r (F.qo (8 . 1:3 ) ) are shown plotted together in Fi g o 40 It is observed t hat for a given beam loadi ng m 0 the optimum
value ot the quant ity r/2Q becomes successively larger i n
0the order ~ (1) max . Y Ilr (2) max 0 n f rom a'fJ /aT; :::: 0 IJ and (3) max . TJ f rom a'llam iii 0 In the latt ei' caseo insertion of the condition of Eq. (9 0 5) into Eqo (8 , 11) 0 i t is
found that .JIlr is reduced to one-half of the no--load valueo i .e . o the electron energy in every case drops to one-half when
the current is increased to g i ve maximum conversion effici encyo 7
This general behavior was also found for the unifoII11 accelerator
st ructure 0
Substituting Eq . ( 9. S) a nd Eq. (9 . 6) .into Eq . (9 04) we fi nd
f or the maxtmum conversion efficiencies
. )2 (1 - 2T 1 - e IFrom ~ . 0 TJ max . = - [ ) '-2T ( 9 0 7) am 2 1 - (1 + 2T) e .J
_ 1 _ e-2T 1 r rom 2.!J. 0 1 (9 08) osaT 1'] max . 2T I*r The maxtmum efficienci es given by Eqso (90 7) and ( 90 8) are
shown plotted versus 't' in curves A and B of Fig . 70 For
comparisonu the corresponding' conversion efficiencies f or the 7uniform accelerator structure are plotted. as curves C and D 0
The advantage of the constant gradient accelerator over t he
unifonn accelerator is seen t o increase as T becomes larger 0
t should be recalled that T = IL for the uniform accelerator and T ~ IL for the constant g radient accelerator where I is the average value of the attenuation coeff icient over the length
of the accelerator section. The f illing t imes of e two t ypes
- 14 ~,
-
of acce lerator are i dentical f or equal ~
The conversion eff ciency of the consta,nt g radient a ccelerator
can be increased even further by designing t he stIucture such that
II\) > 0 Th sis shown :1.n i g 0 8 where '\ax and j r / r are plotted ve r sus T for severa val ues of mO f or the case where
P = 0 Also q s own f or compar son a r e the c orresponding curvesL for the uniform s t ructure D
X. EXAMPLE OF CONSTANT GRADI ENT ACCELERATOR
To make us of he resul s of thi s study i n working out a
spec fie example q the f ol owing values of the pert i nent acc e lerator
parameter s a re taken~
f "" 2856 Mo /sec
r = O ~ 473 megohms/em 4Q :- 10
6 15 megawat 's
L(ClIl) X 10seconds
305
The design value of t he a, tenuation coef fic i ent gi.ven by
Eq. (4. 1 ) may be wr! t en i n t er.ms of the f illing ime t F as
- 2 't' 1 - e
6Substitut i ng L ::::: -.:/ (2935 X 10-. ) a.nd recall ng tha.t
5w/2I Qc :.:. 2.98 X , 0- /10 we obtain f rom Equ (10 0.1)O
~. 15
-
-2 I Z (, - 2.,:'\v 2 .03 X 10 or 1 _. 1: ~ ~ e )-'1 =
-2-r c 1 - e
Using Eq. { 0. 2 )0 curves of v /c vs z/L have been plot ed in g
Fig. 9 for va.rio'lls lengths of accelerator section from 2 to 12
feet . This shows t he manner i n which the group veloc ity must vary
over the accelerat or l ength to obtai n constant electric f ield
gradient.
The beam energy i n Mev is g i ven in Fig . 10 vs the peak beam
current i n amperes for each l ength of accelerator section, The
end point on each curve 1S the beam current which will cause t he
r- f power t o be reduced to zer o a· the end of he secti on. The
resulting conversi on eff i c i ency is also given at each termi 1
poi nt. By compari son with the similar set of curves f or the
unHorm accelerator structure given i n Ref a 7 it is noted that the
constant gradient accelerator exhibit s superior performance both
n beam energy and in terminal conversion efficiency,
Thr oughout t his study. it has been assumed that the shunt
impedance and Q are constant over the range of group veloc ities
requi red t o achieve a constant electri c field gr dient o This
assumption does not lead to seriou discrepancies as shown in
Fig. 11 wher e (r/r ) 12 and QIQ are plot ed against theO O nomal .zed group veloci ty over t he range required in the previous
example. Oa dnd rO are the values at vg /c ~ 0. 010 , In the dis l oaded acceler t or wi phase velocity equal to
c the group velocity varies approximately as the f ourth power of
the disk aperture. Therefo re q a change in group veloc ity by a
factor of 5 or more over the accelerator l ength as in the example
of Fig. 9 does not requi re an excessive var ation of the aper t ure.
To reduce the number of cavi y sizes i t sho Id be f easible to
approxi mate the const.ant gradient case by adjusting the d.imensions
stepwise with several cavt i es in each step.
- 1.6 w
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APPENDIX
The power f l ow 1n t he accelera or in the presence of beam
l oading was g iven in Eq. (4.3) ~
dP -= d z
(A-I)
It is desired to s 1ve this e ation for t he power P at any point
z in the accelera.tor. Substitute~
(A-2 )
or
so t hat
D
, 2dP 1 YOKOPri
-
or
dv 1 dx3-- (A-6)( cf-o ~, YOh /YcfOvd2 YO"I' x
let
2v;'; w (A-7)
dv = 2w dw 0
then
2dw 1 dx :-- (A-a)
Yo(K - l)w +- \ 'YoKuri Yo x o
Integrating Eqo (A- B) between the limits w - ! to
( p \1/2 W =(PI - Yoi)
af er simpl ificat on ~
where
1
- 1 ;
- 18 "~
-
0When i O =0 (! oeo o KO 1} u it is most simple return to Eq • (A-a ) to stat the solutiono Thi& becames g
{A-IO)
Integrating Eq 0 (A-IO) 0
o (A-l) )o
Equations (A-,9) and (A-ll ) gi ve the power flow at any point z
in the accelerator o These equations can be used to calculate
the accelerating f ields and the electron energy ,
~ 19 ~.
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REFERENCES
IR. B. Neal 0 Report No. 185 u Appendix Aw Mi c rowave Laboratory.
Stanford UniversitYIl St a nford q California .
2Since the power dissipa.ted per unit l ength is constant over the
ent i r e accelerator sect ion, the t emperature r i se should be t he
same at all points. Thus, the constant gradient accelerator
should be l ess t r oubled with phase shif t duri ng war.m-up than t he
unifor.m accele rator which has an exponential power variation,
.3K• Johnsen Proc . Phys . Soc. (London)g B, 64, 1062 (l95l).q
4G• Saxong Proc. Phys. Soc. (London) . B~ 670 705 (1954 ).
SE. L. Chu and E. L. Ginzton. Appendi x B. Report No. 274, Microwave Labora t ory u Stanf ord Uni versity . Stanford g Ca lifornia,
(1 955).
6N. C. Chango Report No. 203 0 Microwave Laboratory 9 Stanfo,rd
UniversitYIl Stanford, California q (1953) .
7R. B. Nea , Report No. 379, Mi crowave Laboratory . Stanford
Uni.vers i t y, Stanford o California~ (1957 ) .
8The shunt impedance ( r ) per unit length and the Q of the
cavity vary slowly as the cavity shape is chanqed. e. g ., as the
attenuat ion I is varied . Since the variations in r and Q
are small over t he range in I wi th which we are dealing. r
and Q wi l l be considered as constant i n thi s discuss i on. This
will enable us to obtai n general solut ions whi ch will reveal ' the
important features of the constant gradient accelera to.r. Fr om
a detai led knowledge of the variation of rand Q wi 'h
the same principles which we shall use may be applied to obtain
exact solut i ons but numerical methods are r equired and the over
all pi cture is obscured,
= 20 ~
I
-
-----"------------------------------------------------------- "- ~ --~ -~-
9This result may be obtained directly from Eq. (4.4) but it is
perhaps simpler to start afresh from. Eq. (4 03 ) with ICO :' 1
10J. C. Slater Q Revs. Modo Physo 3.Q.g 473 (1948).
HIt is pertinent to note that the filling time as given by Eq.
i6. 4 ) (with KO 1 ) is i dentical with the filli ng time of the
unifo~ accelerator structure with the same ratio P /P21
12The te~ T = wtp/2Q has been used as the abscissa instead of"
wt p/Q since the for.mer expression 1s the value of the integral
of the attenuation l over the length L of the accelerator
section. i. e oq IL ~ wtF/2Q where I is the average value of I .
- .21
-
p = (1- y._OZ) II - _m 21}'\ ~I- ] 2 ~ ~ L 2 (1_Yoz )
)- ~ ( r m = Yo 2 i mo=0
N N
a.. Ia..
O 8 1 ' "lo ......... .. .... ~ ~
O 6 ' .. 1 ... I..... ~.. -.. -.... ~
O 4 1 I' I "....... "'-.'.. -... -- ~
0 2 1 "I ---. 1" ' -.J.. -.... - -..::
o1 1 ............ 1 --.. ! 1 ! 1 ..........1 o 0.1 0.2 0.3 0 .4 0.5 0.6 0.7 0.8 0.9 1.0
Yo z
~
FI G. l--Va riat ion of t he r-f power with the distance along the accelerator for various val ues of t he beam loadi ng parameter, m
-
JT = (I - e- 2T)~ [1- ; ( 1 - e;T~ 1)] m=Oo • =BEAM LOADING FOR WHICH R-F POWER IS REDUCED
TO ZERO AT Z = L
N c,.,
rl'-I'~
\.O! I m=0 m=O. m= 0 .4
0.81 I~ 1 I m = 0.6 ~ I I..._m= O.S
0 .6 1-1---+.
I .,. _ ;0 .41 , I - I- ___ _ I
0.211 ~,m, 0 ' ! ! , I ! I ! o 0.2 0 .4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
T=-wtF 2Q
FI G. 2--Ef f ective shunt impedanoe r per unit length versus T ~ wtr/2Q f or various values of beam loading parameter, m
-
Jf= (I - e- 2T )"2 I
[I -2 (I - e~{ , )] mo= 0
• = BEAM LOADI NG FOR WHICH R-F POWER IS REDUCED TO ZER O AT Z =L
I.O[.-----
.=0.10 ---____ " 1'":;:...... l 0 ~
, .vO'" ,..... ~
----,.----~-------r-----......,...---------,
G.aL:: ~ N ~
C.SI==--- -= ' " ~
r , "
II~I~
0 " ','........
---,. 4 , '............ , " ..... , ,,', " 2 1 , " l- ..... L ____-+___~l'-:'\ ~ "/~..... l"'...
, ".-.:') " '"' ... 0O. ,,~o ,00 "1',.,>~O ', I 'v'
00 234 BEAM LOADING PARAMETER, m
FIG. 3--Ef fective shunt impedance r per unit length versus beam loading paramete r, m f or various values of ~ = wtF/2Q
5
http:I.O[.-----.=0.10
-
(Al CONDITION FOR MAX J ~ ': m = 2 I . 2T(I+ -2,)-1
I-e m =0 Io (8) CONDITION FOR MAX TJ (FROM ~TJ =0): m =raT
0): m= (1- i~ )-1am e -I
2,0
1.61
~ CJ1
1.2 ~
0 .8
o.4r I ~ I ~ ~ I J 00 2 3 4 5 m
FIG. 4--. = wtr/ 2Q versus beam loadi ng parameter , m , for three cond itions (with mo = 0 ).
(C) CONDITION FOR MAX 'T} (FROM ~TJ =
-
0.2 1/ #£ - ,... I ....... T..... .......
7J = m(1 - e-2T) [I -~ (1- e;T:1 )] mo = 0
0= BEAM LOADING FOR WHICH R-F POWER IS REDUCED TO ZERO AT Z = L
I.O~I--~~~~~~~--~~~~~~~~--~~--~--~
0.8
. ~ N en ~0.6
u Z w u 0.4 -~
~
0 """ , o 0.2 0 .4 0 .6 0 .8 1.0 1. 2 I. 4 I. 6 1.8 2.0
WtF T = 2Q
FI G. S--Conver si on eff iciency, ~ , versus T = wtr/2Q for various values of beam l oadi ng parameter, m
-
IS REDU CED TO ZERO AT Z = L 1.0,.---------.----------,---------~------~~--------~
r= 0.2 ,
...... ' ........ ..:(= 0.4
\ I \ \ ,\ \ \ ,
" '" " MJJ /' \ r= 2 .0 r =0.8 \r=0.6 .1
\ \ \\ \ r =I~ O\
o r '" ,~------~
"7 = m(I - e- 2 T) [I - m (I - 2 r )]e2T2 _ 1
m=Oo
• = BEAM LOADING FOR WHICH R- F POWER
~0.8 .. >U
N Z-...J wO. 6
u u. u. 0.4 w
0.2 I
o 2 3 4 5 BEAM LOADING PARAMETER, m
FIG. 6--Conversion efficiency. ~ , versus beam loadi ng paramet er, m f or various values of T; wt /2Q
F
-
(A) CONSTANT GRADIENT ACCELERATOR (FROM : ~ =0) : 7JMAX=~ ~ (1- e-ZT_):Tl (m = 0) LI-(1+2T)e J o
(S)CONSTANT GRADIENT ACCELERATOR(FROM : ~ =o): 7JMAX=+~ _ I- e- 2TJ (mo= 0) ~ 2 1"
I- e_T )2 a7J \ 1" ( - T
((C) UNI FORM ACCELERATOR FROM a m=O/ : 71M AX =2" I 1_ e-T)
1- - T
1- iT)]I-a . -T -T ~ T(D)UNIFORM ACCELERATOR (FROM rt= O). '7MAX=2e I-Te l l -e-T)z~
~
E==""0 .8 1r: u
~0 . 61 LL LL wO.4 1
0 . 2
00
~IO~ I I ~
~ C A
I~
1~--~---4----4----+----+----r--~r
0 .2 0.4 0.6 0.8 1.0 1.2 1.4 1. 6 1.8 2 .0 w tF
T=--2Q
FIG. 7--Cornparison of the maximum conversion efficiencies ~ for the
constant gradient structure (with rnO:: 0) and the uniform struc ture.
- 28
-
m CON STAN T GR AD lENT S T R UCT URE - C ONDI T ION FO R P =0 : m = .0
L , - e -,
UNIFORM STRUCTURE-----COND'ITION FOR PL
=0: m=(eT- I)
1.0k IT] /' ,
,---m =4-~ o m =2o m = Io UNIFORM
STRUCTURE
0.81 I~ [ f II ! \ \ \ \ I II \ I N
-
c" 0 0.012
0.028 I
I 0 - 179 10 ' RADIANS W - . X SEC
0.024~~J_ Q= 10
4
- 6 X0.0201 ~fl ~ t = L(C;65 10 SECONDS
F
O.OI6~~T= ;~F = 2935 X 10-6 Lg'lu L- d. & ...
0 .0081 ~
rno= 0 0.004. ~
0' , I , I , o 0.2 0.4 0.6 0.8 1.0 z
L
FIG. 9--Var iation of the group velocity, Vg , for various lengths of constant gradient structure (for mo = 0) with certain assumed parameters.
-
- -
m = 0o P, = 15 Mw t = L (ftJ 11 SEC
F 10 r
M.n.. r = 0 .473 em
10 4Q = TERMINAL POINT ON EACH CURVE IS CURRENT FOR WHICH P =0 AT Z =L
> 1]=0.55 L =10ft.
1] =0 .60 L = 8 f t.
1] =
IOr =2 ft( L
,..... 50r I I > w
C-' ~ ..... - 401--.3to,;'\.,-l'("fr------I-----+------l---+---l------+---t----t----l
0 .66 1] = 0 .72
=r-=t='7:O.BO I I '7 =~.BB jI
o o 0.25 0 .50 0.75 1.00 1.25 1.50 1.75 2.00 2 .25 2.50 PEAK BEAM CURREN T, i (AMP )
FIG. 10--Electron beam energy, V , versus peak beam current, i for various l engths of constant gradient structure (with mO = 0). Also shown is the conversion efficiency, ~ , when PL. 0 for each length.
http:r-=t='7:O.BO
-
1.2 1.2
1.01.0 ,(I )"2
0.8 -IC\J ro..---. 0 8 . ~ I ~o """'--"
0.6 0.6
0 10 o
(.0)
t.:l
0 .4 0 .4
0 .2 0.2
0 1 1 I I I I 10 o .004 .008 .012 .016 .020 .024 .028 .032 .036
Vg c
FI G. ll--Relative values of r , Shunt impedance per unit length and Q versus group velocity, v , for the disk loaded structure operatinggi n t he n/2 mode; rO and Q are the values at vqlc = 0.01.o
-
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