UNIVERSITY OF CALIFORNIA

R.F. POWER LOSSES COMPARED

FOR FOUR LINEAR ACCELERATOR

CONFIGURATIONS

BERKELEY, CALIFORNIA

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U C R L . 0 Phys ics Distribution

UNIVERSITY OF CALIFORNIA

Radiation Labora tory Berke ley , California

c o n t r a c t No.. W-7405 -eng-48

RADIOFREQUENC Y POWER LOSSES:COMPARED FOR FOUR LINEAR ACCELERATOR CONFIGURATIONS

Robert A. Weir

September ' 30,. 1955

P r in t ed for the. U. S, Atomic Energy Commiss ion

RADIOFREQUENC Y POWER LOSSES COMPARED FOR FOUR LINEAR ACCELERATOR CONFIGURATIONS

.R'ober't A: Weir

~ a d i a t ' i ~ n ~ a b o r a t & ~ ' . '

Univer s i ty of California Berkeley, . California

September 30, 19.55

ABSTRACT I

The rf power los ses per Mev part ic le energy gain for four possible deuteron-accelerating devices a r e computed over a deuteron energy range of 1 4 IKE . 6 400 Mev. The configurations considered a r e (1) the conven- t ional drift-tube-type l inear acce lera tor , (2) a s e r i e s of ''pillbox" resonators each oscillating 1800 out of phase with i t s successor and each containing two half drift tubes, (3) a s e r i e s of pillbox resonators with no drift tubes, and (4) a helical wave guide.

Values for the frequency, e lec t r ic field strength, and other pa ramete r s of the various configurations considered needed for the calculation of the numer ica l r e su l t s in this repor t were chosen which complare with those which presumably will be employed in fur ther acce lera tor construction a t Livermore.

\

The numerical r e s u l t s indicate that in the deuteron energy range 1 < KE < 40 -Mev Cases (A) and (D) a r e relatively efficient, while in the range 200 < KE < 400 Case (B) i s relatively efficient. Of part icular in te res t i s the point a t which the ascending power los s curve of Case (A) c r o s s e s the . descending curve of Case ,(B). This was found to b e in the neighborhood of 125 Mev. Case .(C) was found to be comparatively inefficient.

RADIOFREQUENCY POWER LOSSES COMPARED FOR FOUR LINEAR ACCELERATOR CONFIGURATIONS

Robert A. Weir

~ a d i a t i o n Laboratory ' . '

University 'of California .Berkeley,. California

September 30, 1955

INTRODUC T1,ON

The object of this repor t is to present a set of calculated r f power los ses for various possible deuteron-accelerating devices. The r e su l t s should give an insight into the relat ive effectiveness of the foll'awing possible machines over a par t ic le energy range of 1 ,< KE,< 400 Mev. As a bas i s of

- comparison each of the various configurations will be assumed t o impar t the same energy gain to a part ic le per unit length of the machine, o r , in other words, as ide f rom s,mall differences due to different t rans i t - t ime fac tors , a length P X r ep resen t s the same part ic le energy gain for each device.

Pour configurations a r e considered (see Fig. 1): ,

(A) Conventional drift -tube -type l inear acce lera tor . Repeat length and g / ~ = 0.25.

(B) A s e r i e s of pillbox resonators , 1/2 P X long and each 180° out of phase with i t s successor . Each pillbox contains two half drift tubes, and the pillboxes a r e separated f rom each other by walls. The distance between gap cen te r s in two adjacent pillboxes r ep resen t s half an r f cycle.

(C) A s e r i e s of separate pillboxes without drift tubes. In,this case a whole resonator ac t s a s a n accelerating gap.

(D) A helical wave guide. The accelerat ing e lec t r ic field is a traveling wave, the phase velocity of which i s equal to the part ic le velocity.

The determination of the power los ses p e r Mev part ic le energy gain in C a s e s (A) and (B) i s based on the method of Walkinshaw, Sabel, and Outram. Use was made of 'the curves presented b y these authors in their r epor t , but additional values needed he re which lay outside the range .of the curves presented were calculated. The curves , when einployed, were scaled to our assumed wave length of 6.2 me te r s . These curves used fo r Cases (A) and (B) a r e F igs . 3a and 6a respect ively in Reference 1. F o r convenience the curves used in Reference 1 a r e included a t the end of this paper a s F igs . 2 and 3.

The various pa ramete r s assumed in this r epor t were chosen a s representative.of those cqntemplated ,for possible acce lera tor constr.uction a t Livermore .

For:Cases (A) and :(B) these a r e : 6 E, = gap gradient = 100 kv/in (3.94 x 10 v o l t s / k e t e i )

X = 6.2 m.eters '(free-space wave length)

' ,

4 =.1/4@X . .

~r

AVERAGE FlELD 25 kv/in. -------

g = 1/4pX GAP FIELD 5 0 kv/in. AVERAGE ' FlELD

.GAP FIEID I00 kv/in. AVERAGE FlELD

' g = ' / 8 p A b 4 M k 4

AXIAL FlELD 25 kv/in,

25 kv/in. - - - - - - - GAP FlELD 100 kv/in.

AXIAL FlELD 25 kv/in.

Fig . 1. Four possible configurations fo r a l inear acce le ra to r (A) conventinnal, $/L = 0,25 ; (B) 1/2 P X pillbox, g / ~ . = 0.25; . . (B ' ) 1/2 P A pillbox, g / ~ = 0.5; (C) separa te pillboxes without drif t tub.es; (D) helix.

Stem diameters = 5 -25 inches (case .A) . . . . .

Tank length = 10 wave lengths .(case A). . , . : . ., . . d / l = .0.1. (d = drift-tube drameter)

' .

. ..

Cos +, = 0.9 = synchronous phase angle) for KE 2 5 Mev.

The numerical results presented here will be,.somewhat on the op- timistic side, a s power dissipation caused by electron loading and imperfections. in the metal surfaces a r e not taken into account.. . ~ x ~ e r i e n c e has shown that al l actual power losses a r e about 30.70 greater than .those obtained theoretically.

. . (A). CONVENT PONAL DRIFT.-T UBE MACHINE

The formula employed to compute the power losses for the velocity range in which ,the shunt ign'pedance curves were ,calcdated a s presented in . . . . . . . Reference 1 i s '

s . . . ,

3 ~ / 2 i S o 2 ~ - l o 2 . i r ' 1 t rd r .

10 X z o L F o,(k$) ;

In this formula Q is the scaling factor. The.shunt impedaice curves in Reference 1 a r e given.for d/h = 0.1 and X = 1.5 meters. The shunt impedance scales to a gqod approximation a s l / . . Sinc our assumed wave length ,is 6 . 2 meters , the scaling factor i s Q = (4.,13j-1fi = 1/2.03.

Also, . .

P = power loss in k w / ~ e v particle energy gain; '

q ,= shunt impedance in megoh.m/meter a s given in Fig. 3 a , j g / ~ = 0.25) of R.eference 1;

b = stem diamete.r;

Zo= impedance of f ree space;

Rs= .characteristic resistance of copper surface = 2.6 1 x,10-'?fi . .... where y .; frequency;

= average gradient = : ~ ~ / 4 ;

k . = 2w/l; . , .

T = transit-time fact-or; , :.

D D Fo(krl = Jo(kr)P,(k 1 - Yo(kr) JoBk :-$,

D = tank diameter. The tank diameter :can . . be 'estirqated by the relation

This i s an approximation to the more exact expression for the tank diameter given in the appendix of Reference 1. It i s valid for small P.

Since the power losses a r e presented'in this report in t e rms of kw per. Mev gain they will depend--especially in the lower-energv end bf the. machlne--on the transit-time factor and on the synchronous phase angle. For the sake of completeness a short resume' on the derivations of these quantities a s used in this paper follows. . . .

Transit Time. Factor ,

The drift tubes a r e approximated by cylinders of radius a , the electr ic field across the gap being assumed constant at r = a. With these boundary conditions the solution of Laplaceqs equation i s

L Tn V(r ,z ) =:E(z + -.-I= TT - n sin pF] I~ b ~ ] 1

and

. . where : E = the design gradient.

We have

- sin (rrng)/L 1 - - sin (srng/L) 1 Tn - io(2nna) / l nng/L - I- ( Z @ n a / ~ ) '

0 . ..

where a i s the drift-tube bore radius. ' ..

The energy change across a gap for the synchronous particle i s

EZ C O S (at + +s)dz,

which, when we set in the expression for E and integrate, becomes Z

r.7 - .:1 sin 1 = /.2wal are

transit .t'ink

Synchronous Phase Angle

The particle energy gain for the synchonous particle i s

AWs = eEgs AT cos + s a

The difference betwe&AW and the e n e i g y g a i i for another in the s

phase-s,table region is

6AW = BEBAT (cos + - cos 0,) " rncL~(ps6p )

F rom this one obtains the phase equation for the particle by replacing A by ' I d/dn;. where n is. the section number, . . .

d - 2 a) 2.w.eEf3,XT dn (Ps a;;) = mcL (cos +s - cos +).

T o find the, l imits of phase stability let Bs be cons,tant and integrate .once, obtaining

2. ' Bs . d+ ZpseEXT .(+ cos +, - sin +) = ,c. .r(& - , m c L '.

The limits of phase. stability a r e the values of + which cause + c.os'+& - sin + to be a ,maximum. These. two values a r e

= - + s 9

= 2 + s 9

which means a total phase acceptance . , of 3 + S a I I As the particle energy increases in an accelerator the phase

o ~ c ~ l l a t i o n s a r e damped in amplitude. This damping a s a function of f3 can be ascertained approximately from the phise oscillation equations a s follows:

d d + ) ? - ps202(0 - + , ) , 0 2 = - (8s aii 2*eEXT sin 4,. mc Ps

Letting

2 .ps.''*.. wehave $'= - $ ( W .=+.),o . . . .

.Ps - . 2 Neglecting ps'*/ps w.ith respect to o , we have the WKB solution:

f l

J . . + = Ps(+ - +,) " [a]-'/' exp [*i d n ] . , '

The amplitude of the phase oscil'lation i s dampkd approximately as.. . .

To get an expression for the synchronous phase angle a s a function of p we' set . . ,

The initial phase acceptance of the A tank of the A-48 linear accelerator at Livermore i s 1350. This section operates in the energy range of 0.9 < KEd 3.69. The synchronous phase angle a t 0.9 Mev i s therefore -45O. Since 0.9 Mev corresponds to P = 0.033 for deuterons, the constant a can be evaluated and the fiollowing equation obtained for the absolute value of the synchronous phase angle a s a function of p:

This estimation of the value of +s will be employed until cos +s = 0,9, which i s the contemplated design parameter for the higher-energy range of the linear accelerator at Livermore.

Setting in the numerical values for the various parameters, we get the following for the power loss in kw per Mev particle energy gain:

0- -

-3/4 +s = 0.0607 P i . , , 1 < KE < 5 Mev;

+s = 0.45, 5,< KE < 100 Mev.

The f i rs t t e rm in the bracket i s the contribution to the power losses f rom the drift tubes and side walls of the tank. The second and third t e rms represent contributions from the drift-tube stems and end walls respectively. If the stem diameters a r e increased by a factor F, the second te rm increases by this same factor. If the tank length i s decreased by a factor F, the third t e rm increases by this same factor.

One will notice on inspection of the power-loss curve for this case that the r f power losses per Mev particle energy gain reach a minimum of

particle 'energies of about 10 Mev. In the low-energy range of the machine the relative contribution to the total power losses from the stems is rather heavy. Moreover the efficiency of the machine i s further reduced a s the transit t ime factor i s relatively low and the synchronous phase angle negatively large. All this rapidly improves, however, a s the particle energy becomes greater and the bunching of the beam becomes sharper. But a s the particle energy becomes greater the drift tubes must be made longer. This increased loading of the tank brings about a reduction in the rat io of the tank diameter to the drift-tube diameter. As a result of both these effects, the power losses in the conventional machine begin to increase for KE > 10 Mev, becoming serious for particle energies over 158 Mev.

A possible alternative to the conventional machine i s a ser ies of resonant cavities, each of length 1/2 P A , 'separated from one-another by walls. Each pillbox would contain a pair of half drift tubes. Two possible arrangements a r e compared with the standard drift-tube machine. The f i rs t i s pillboxes with gap lengths 1/8 phqg/L = 0.25). The second i s pillboxes with gap lengths of 1/4 P A ( ~ / L = 0.5 1. In both cases two pillboxes would be the equivalent of one repeat length of the conventional machine, but the gap gradient. of the second would be half that of the f i rs t and of the conventional machine. The average fields a r e the same, so, apart from differences in transit t ime factors, the over-a11 lengths a r e the same for the same energy gain.

The power losses per Mev deuteron energy gain a r e given by . ~2

P = . . P i 1 3 X - X x,10 k w / ~ e v . an x .lo6 2 ET 9 cos +s

Assuming a gap gradient in the f i r s t case of 100 kv/in. and in the second of 50 kv/in. we get, for A = 6.2 meters andd/A = 0.1,

for g / ~ = 0.25,

for g / ~ = 0.5,

Here sin T& -. .

- 6 . E = 2.5 kv/in. = 0.984 x 1 0 volts/meter fqr both cases.

The ntlmcrjcal. difference i n the numerators between the f i rs t and second case i s due to,the difference in transit-time factors.

. . . . . . . . < . ' . ._. . ... 1

. . , - l o . - . . : b ,- . , . , . . . . . . . ;, ( i - :. . .. . . ..

. , UCRL-3150 .

' The quan t i t , y :~0~zq is 6btained.from g / ~ = ! , 0 i 2 5 and i r o r q g / ~ ' = 0.5 f r o m Fig. 6a ,in.RefeFence 1 6 Values of 43 outside' the' range of' ,given'in :

. . . .. Reference 1 were calcblat&l:: .:' . . . . , . . . . . . . . . . . . ? : . . ( . .. . . . . . . . . . . . . .. . : ,:

As . . . we can see. by examining the. numerical . resu l t s . pi'esented at the . .

end of this p a p e r , t h e 1 / 2 P X configuration i s riot indicated in the low-energ9 > . '

range of a n acce le ra to r a s f a r a s power losses a r e conc.erned. .However, i t s . .

efficiency'inc'reases with highe-r par t ic le energy, surpassing that of the conventional machine a't deuteron energies of about 120 Mev. The g / ~ = 0.5: pillboxes a r e the best , a s f a r .as power is conce-raed, at high energy. . .

. . . ,

, . (C) SEPARATE RESONANT PILLBOXES WITHOUT'DKLF'T T U R E S a .

'111 this case the pillboxes themselve's a r e e q u i v a l e ~ t to ' accelerat ing gaps. Since this' configur'ation i s a possible al ternate t o t h e 1 / 2 ' ~ A pillboxes

'with dr i f t tubes (case B ) we will a s sume the: axial. length of oAe ,of these 'pill.- b o r e s is 1/2 P A and the 'on-axis gradient i s 25 kv/in. Two s u c h pillboxes, as in ' the c a s e of the ~ i ' l l b o x ~ s with dr i f t . tubes, a r e the equivalent -of one repeat .. .

length of the s tandard drift-tube machine apart f rom th'e differenie in t rans i t t i m e fac tors . ~ o c u s i n g . d e v i c , e s in this case would be outside' the pillboxes. , .. . . , .. . . . ,

' . T h e f i e l d . . . . equations a r e a s follows:

, .

F r o m P = -2- tang ds . where P -- powcr loss t o the surface in

/ i watts, we get, for the power loss per' Mev in'kw,

s in (n/2) T = qr'

cos 9, = 0.9

a = radius of pillbox,,

An inser t ion of numer ica l vallres f o r the various pa ramete r s gives

This configuration i s much l e s s efficient than the 1/2 @A pillboxes with drift tubes.

. . . .

, . . , . , .

(D) .HELICAL WAVE GUIDE

In order to obtain an expression for - the power los ses in a helix the following approximations a r e made :2*

The helix is regarded a s a cylindrical surface, infinitely conducting in a direction corresponding to the actual helix, and perfectly nonconducting in a direction normal to the direction of the helix winding.

Under these assumptions the components of the vector .potential a r e ' '

a s follows. .(the components with the subscript 1 a r e ex ter ior to the wave guide):

. . E k Io(ya)

9 i (wt -kz) Kl (yl-1 e 9

. .. . : ~ ~ & n $ Io(ya) i(wt -kz) .. . - - A e l . -.-TI K ~ ( v ~ ) e . y

. . . . . . where k .= 2.rr,/ A = 2 6 , ? / ~ ; v , =. frequency And V = velocity of the

... : , p h a s e - . . . . . traveling :wave ; ,

X = f ree-space wave length; 0 .

a t = radius of the helix; . .

The re.lation between the pitch angle of the helix.(designate'd 4) and the.phase velocity i s

In order to 'c.ompate the power los s in the helix it i s assumed that .the cu r ren t flows along a:ribbon of z e f o thickness arid.of width s cos + oriented in the s a m e direct ion a s the actual helix wire . Here s .is the pitch

. . . , . . . . of - the helix; .s = 2na- tan +. .. . . . . .

. : .

T o cor rec t fo r the uneven cur rent dis t r ibu ion in the actual case an 4 . . ekperimentally obtained form, factor is 'introduced: .. . . . . . . - , . .> . . .

. . .. . . . . ' . . . .

The quantity s$/d is defined a's foilows:

s?j - resis tance '6f a given lengthof helix .w,ird ~

7 - res i s tance of a s t raight wire of same length b i ~ t of diameter'.^ = s.

The res i s tance pe r tu rn of the helix i s then

2aa sec $ d. - wd s . ?

where d i s the d iameter of.the helix wire . The quantity'

2 ~ a s e c + d R s ad

- S

i s the skin res i s tance of a s t raight w i r q ~ f diameter s and of length equal t o one helix turn. Values of the quantit s j / d a s a function of d/s a r e given in Reference 3 . F o r 0.3 < d/s < 0.8, ~ i jTd="4 .~

Since the surface substituted fo r the helix i s considered to be a ribbon of z e r o thickness and of: width s cos +, the cur rent flowing in i t i s equal to the discontinuity in the magnetic f'ield:.in a' direction normal to the helix winding t imes the width ,of the ribbon:

I = [ ~ ~ ( a ) - HZ (a) ] s e & + s:.c&s +. , ,

1 Using the relation P = 1/2 I ~ R , where R. is the res i s tance per unit

axial length ,of the helix and P i s the mean power 10s s pe r unit axial length .in wat ts , we obtain the followin'g expression giving the power. losi in kw per Mev par t ic le energy gain- in this configuration:

t S + ) ~ x Z 1 ~ 3 I , ( Y ~ ) ' p 2 : . . . ., ' .

P = , R s -;T, 0 0

2 4n2zo a cos + 2 " s Il ( ~ a ) K i ha)Ko!ya, . . .,.. I - ? . . .. . . I .

The value of the synchronous phase angle m a y be obtained in a manner analogous t o that in Case A. Assuming the s a m e initial phase acceptance fo r the helix a s for the drift-tube confi%guration, we get

. . 1 < KE < 5 Mev.

F o r comparing this configuration with the other's, the following numer ica l - .. . . values fo r the var ious pa ramete r s w6re'.assuined:

ho = 6.2 m e t e r s , .. . . . . . . . . . '

. . . .. . -3'14' . , . ' 1 .<'!KE '< ,5' . . . + . . . l + & l = 0.06 Ps , , . M ~ V ; ' . . . - ,.! I . ' ! '

14,) _= 0.45, 5 < KE < 100 Mev; . .

a = 1.5 inches. This number for ' the' helix r'adius was ,chosen for . . . . comparable fo'cusing with .Case.A.

Inserting the appropriate numerical values for the various pa ramete r s - - -

into the power-loss equation for this case , one obtains -

1.29 x : 10 3 .Ib h a ) P .=

p2

I-pL , cos 4 . S

Il ( y a w l (ya)Ko(va)

where

A helix appears to be a relat ively efficient device fo r accelerat ing deuterons of energies l e s s than 20 Mev, provided the beam can be focused conveniently. This configuration, however, does not seem to be indicated for high deuteron energies .

ACKNOWLEDGMENTS

The author .would like to tharik Lloyd Smith, who directed this effort , for his valuable help and cr i t ic i sm. He would a lso like t o acknowledge that the expressions for the t rans i t - t ime factor and phase damping used in this repor t were taken f rom R'. L.. 'Gluckstern 's: Engineering Note 43'1 1 - 10 CV-9 A and B.

This work was done under the auspices of the U. S. Atomic Energy Commission.

. . . , . : . . . .

References

1. Walkinshaw, Sabel, and Outram, "Shunt Impedance Calculations for Proton Linear Accelera tors . , P a r t I,.'' AERE-T/M-104, ,H,arwell, England.

2 . J. R. P i e r c e , roc. 1nst. Radio. . E n g r s , - 35, 11 1 (1947,): - . . . . . . " . - . ! ..

' 3 ; K. Johnsen, On the Theory of the Linear Accelerator , Bergen;. : . Norway,. A. S . Johns Gr iegs Boktrykkeri , (1954).

F i g , 2. Power- loss curves fo r four configurations for a l inear acce le ra to r : (A) drif t- tube machine; (B) pillboxes, . g / ~ = 0.25; (B') pillboxes, g / ~ = 0.5; (C) pillb.oxes without dr i f t tubes ; (D) helix.

Fig . 3 . C u r v e s f r o m Walkinshaw, Sabel , and O u t r a m , "Shunt 1,mperlanrc Calcula t ions f o r ' p r o t o n L inca r A c c c l e t a l u r s . P a r t I, " AERE-T/M-104, Harwel l , England. (1) F i g . 3a , g / ~ = 0.25; (2) Fig. 6a , g / ~ = 0.25; (3) Fig'. 6a , g / ~ = 0.5. Dr i f t - tube d i a m e t e r = 15.37 c m , X = 1.5 m . In t h i s r e p o r t r) i s def ined

6 E~ x 10 , T ) =- P

w h e r e E = e l e c t r i c f ie ld i n vol ts p e r m e t e r , p = power l o s s in w a t t s , r) = m e g o h m s p e r m e t e r .