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

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

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 _.

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

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

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

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

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 ~.

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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