Beam control in a proton synchroton

BEAM CONTROL

IN A PROTON SYNCHROTRON

BEAM CONTROL

IN A PROTON SYNCHROTRON

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BEAM CONTROL

IN A PROTON SYNCHROTRON

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. CJ.D.M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 19 FEBRUARI 1969

TE 16.00 UUR

DOOR

WILLEM ANTOON VAN KAMPEN

l^oi ézöO

Dit proefschrift is goedgekeurd door de promotor

prof.dr.ir. F.A. Eeya.

C O N T E N T S

PREFACE

page

CHAPTER I THE SYNCHROTRON MOTION AND ITS OBSERVATION 11

1.1 Introduction 1.2 Coordinate s y s t e m s 1. 3 Accelerating forces 1.4 The synchrotron motion 1. 5 Beam signals 11 12 16 19 24

CHAPTER II BEAM CONTROL METHODS AND SUBSYSTEMS

2 . 1 Introduction

2. 2 The power amplifier and cavity 2.3 Oscillators

2. 4 Pick up electrodes and associated equipment 2. 5 Phase m e t e r s 27 27 29 35 37 40

CHAPTER i n ANALYSIS AND DESIGN

3.1 Canonical variables 3. 2 Linear analysis 3. 3 F i r s t o r d e r equations 3.4 Stability 45 45 48 54 57

CHAPTER IV CHANGING THE HARMONIC NUMBER DURING

ACCELERATION 63 4 . 1 Introduction 4. 2 Experimental s e t - u p 4. 3 A computer model 63 64 68

CHAPTER V THE DELFT P. S. BEAM CONTROL SYSTEM 5.1 System p a r a m e t e r s 5. 2 P r o p e r t i e s of subsystems 5. 3 Block diagram 5.4 Time sequences 5.5 Observations ACKNOWLEDGMENTS SUMMARY SAMENVATTING REFERENCES

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P R E F A C E

During the acceleration of p a r t i c l e s in a proton synchrotron one t r i e s to keep all injected p a r t i c l e s inside the a c c e l e r a t o r , to obtain a maximum yield and to minimize radiation effects. The first proton synchrotrons showed a steady loss of particles during accelaration due to the high demands made upon the f r e -quency of the accelerating field.

A first attempt to improve this situation was made by the Cosmotron group in Brookhaven (U. S. A . ) . A signal derived from the phase difference be-tween proton bunches and accelerating field was fed back to the oscillator which d e t e r m i n e s the frequency of the accelerating field. Rogers reported that by inserting a phase feed-back loop the beam Intensity was doubled a t 3 GeV. After the construction of the CERN 28 GeV accelerator, where a m o r e complicated beam control system was used, beam control in a proton

synchro-tron has become an established procedure.

It is worth noting that construction and dimensioning of the beam control s y s t e m s were c a r r i e d out before a closed theory was available, a s can be d e

-2 3

duced from the first papers '

It took s e v e r a l y e a r s before a group working with the Russian 7 GeV a c c e -l e r a t o r pub-lished a paper attempting to inc-lude both beam dynamics and the p r o p e r t i e s of the electronic equipment.

In this thesis a description will be given in mathematical t e r m s of beam control s y s t e m s , together with considerations leading to the design and con-struction of them. As such a description depends upon a number of approxi-mations which a r e easily obscured, underlying assumptions will be clearly stated, and all approximations will be mentioned. Since approximations a r e affected by the choice of a particular arrangement, a certain amount of a r b i -t r a r i n e s s canno-t be avoided.

Finally the beam control system of the Delft proton synchrotron will be described, including the new feature of changing the harmonic number of the accelerating field during acceleration.

C H A P T E R I

THE SYNCHROTRON MOTION AND ITS OBSERVATION

1.1 INTRODUCTION

In a proton synchrotron the focusing forces a r e Lorentz f o r c e s . They cause an oscillatory motion around an equilibrium orbit, the betatron motion. What the number of vertical and horizontal betatron oscillations during one particle revolution will be depends upon the shape of the magnetic field. In weak focusing a c c e l e r a t o r s these numbers a r e l e s s than one, in alternating g r a -dient a c c e l e r a t o r s they a r e much l a r g e r than one .

Accelerating forces acting upon the p a r t i c l e s a r e the r e s u l t of an a l t e r -nating e l e c t r i c field in the accelerating stations.

7

The principle of "phase stability", discovered by Veksler and

MacMil-Q

Ian , explains sustained acceleration of p a r t i c l e s on a nearly constant orbit. Phase stability exists only if p a r t i c l e s moving on different o r b i t s have dif-ferent revolution t i m e s . This phenomenon, which is c h a r a c t e r i s t i c of synchro-t r o n s , is called "momensynchro-tum compacsynchro-tion" (cf. 1.4.3).

Every time a particle c r o s s e s an accelerating field, the betatron motion is changed. Successive changes of the betatron motion may shift the betatron equilibrium orbit. Then the orbital frequency, and after some time also the phase a t which a particle c r o s s e s the e l e c t r i c field will change. The combina-tion of changing the phase and shifting the betatron equilibrium orbit a r e cha-r a c t e cha-r i s t i c of the synchcha-rotcha-ron motion. Undecha-r cecha-rtain conditions this motion may be oscillatory around the position of a (fictitious) particle which gains every turn just the right amount of energy to match the change in magnetic flux

density of the guide field. All this holds if the frequency of the betatron motion i s much higher than the frequency of the synchrotron motion and If no resonance between betatron and synchrotron motion o c c u r s . Henceforth, a betatron equi-librium orbit will be called simply an "orbit" and the motion around such an orbit will be neglected. However, there will be a connection between the beta-tron motion and beam control for at least two r e a s o n s :

A) since the synchrotron motion affects the orbit of a particle, the maximum room available for the orbit will be lessened by a horizontal betatron motion, because the total excursion is limited by the useful aperture in the vacuum chamber; and

B) during injection the betatron motion partially determines the number of p a r t i c l e s in the bunches, which affects the p r o p e r t i e s of the beam control s y s t e m .

In most investigations of proton synchrotrons differences in o r d e r s of magnitude a r e tacitly supposed to be known, e. g.

a) the r a t e of change of the flux density of the focusing magnet is slow a s compared with betatron or synchrotron periods,

b) the energy change of a particle p e r revolution a s compared with its kinetic energy is small,

c) the frequencies of the betatron motion a r e l a r g e r than the frequency of the synchrotron motion by at l e a s t one o r d e r of magnitude.

1.2 COORDINATE SYSTEMS

Let there exist a closed equilibrium orbit lying in a plane , and lying within the vacuum chamber. The plane will be r e f e r r e d to as the horizontal plane, the orbit a s "the" equilibrium orbit. Let s be the distance covered by a particle along the equilibrium orbit from a fixed

s t a r t i n g point. If a particle is not on the equilib-rium orbit the n e a r e s t p o i n t on the equilibequilib-rium o r b i t d e t e r m i n e s s. A horizontal departure from the equilibrium orbitwill be designated by x (taken to be in the outward direction), a vertical

d e p a r t u r e by z (taken to be in the upward di- P'g- i- 2.1 rection); see fig. (1.2.1). s is monotonie

in-c r e a s i n g for a partiin-cle moving on the equilibrium orbit.

L e t t h e circumference of the equilibrium orbit be C^j. Let the "azimuth" of a 12

point near the orbit be defined by

and the angular velocity by

e = 7 ^ 2 n , (1.2.1)

") = Ö , (1.2. 2)

the dot denoting differentiation with r e s p e c t to time. In general,

v / c = (1-E V E ^ ) ^ ' ' ^ , ( 1 . 2 . 3 )

where c is the velocity of light, E the r e s t energy of a particle, E the total energy of a particle and v its velocity.

Let

Also let

' " c = ^ 2 n . (1.2.4)

"J = '"- (1.2.5) o C c '

for a particle moving with velocity v^ on the equilibrium orbit. With (1.2.3) we obtain

'"o = '"c ( 1 - E ^ / E ^ ^ ^ ^ (1.2.6)

if E is the total energy of a particle moving on the equilibrium orbit. At every point of the equilibrium orbit

Po = Boer„, (1.2.7)

where Pg is the equilibrium momentum of the p a r t i c l e , B the vertical component of the magnetic flux density, r^ the local radius of curvature of the orbit and e the charge of a proton.

p^ = Bcer = moC, (1.2.8)

where m is the r e s t m a s s of a proton. Then

B^erc = E^. (1.2.9)

With

P = ^ ( E 2 - E 2 ) 1 ' ' 2 , (1.2.10)

and the above expressions

c

"'O = " ' C ^ [ 1 M | ° ) ] ' . (1.2.11)

The angular velocity UUQ will be called the reference angular velocity and the integral

t

e„= Ju)„dt, (1.2.12)

the reference azimuth, t,, being conveniently chosen.

Let a number N^ of accelerating stations o r cavities be located along the orbit. Let at the centre of the first of them s = 0.

The e l e c t r i c field generated by cavity (j, may be represented by the vector

^ = <? (s, z , x , t ) . (1.2.13)

We will call

V^= J ^^.d-S (1.2.14)

turn

the acceleration voltage of cavity |i.. It will be assumed to be independent of x and z, a s this is one of the a i m s of cavity design. Let "^ designate a

devia-1 Ml

tion from the reference angular velocity, and let acceleration take place at a frequency approximately h times the revolution frequency. Then the a c c e l e r -ating frequency of cavity p, may be designated by

h(üu^ +(jü ) / 2 T T , (1.2.15)

h being a positive integer, usually called the harmonic number.

If the cavities do not all have the s a m e harmonic number, h will be the h a r -monic number of cavity n, with accelerating frequency

h ^ ( a ) o + ' « i ^ ) / 2 n . (1.2.16)

The acceleration voltage of cavity \i. will here be given by t

V = V sin h [ I (u)„ + u), )dt + e ] . (1. 2.17)

••o

V may be a function of time or of B, but as a rule will be assumed to be

'^ max

constant. We will call

t

h [ r lu dt+ e ] (1.2.18) u, J 1 u, u, ^

to

the phase of the acceleration voltage of cavity ^ , and set

e j = 0 and c u i i = u ) i . (1.2.19)

We define the phase of a particle relative to the phase of an acceleration voltage by

t

*i = h [-6;+ J (ujo +(i)j )dt + e ] , i = 1,2 N. (1.2.20) to

where N i s the number of p a r t i c l e s and B^ the azimuth of the i p a r t i c l e . This quantity ti equals the phase of the accelaration voltage of the first cavity (9; =0), except for a multiple of 2Tr, at the moment a p a r t i c l e c r o s s e s that cavity.

We define a s the azimuth of a p a r t i c l e relative to the reference azimuth

< P i = e i - e o . (1.2.21)

The quantity cp j i s independent of the deviations of the accelerating frequency and of the harmonic n u m b e r s .

Then t

t = h [ f u) dt - cp + e ] . (1.2.22) i H J 1 i ji,

1.3 ACCELERATING FORCES

Provided its shape is independent of time, the accelerating e l e c t r i c field can be written a s t ^^ ( s , z , x , t ) = ^ ( 9 - 9 ^ , z , x ) s i n h ^ [ J ( ' " o + ' " i ) ' i t + e^,^- (1-3.1) to wllh Sj = 0 and Sj = 0. (1.3.2)

e Is an e l e c t r i c phase shift dependent on the location of the accelerating station, and 9 (0 ^ 9 < 2Tr) the azimuth of cavity ix.

M'

^ i s assumed to be the s a m e for all cavities.

Let

6 - 9 ^ = y (1.3.3)

and # ' ( y , z,x) = # ( y + k2TT,z,x), k an integer. (1.3.4)

It is assumed that

^ ( y , z,x) = ^ ( - y , z , x ) . (1.3.5)

Then é' may be expanded in a F o u r i e r s e r i e s as GO

^ ( y , z,x) = | a o + ^ S a ^ c o s v y . (1.3.6) The F o u r i e r coefficient a will be

V TT

a = — I ^ ( y , z,x) cos y v d y . (1.3.7) - T T

According to (1. 2.21) the azimuth of a particle is t

9 = J (Uodt + * (t). to

Then the e l e c t r i c field of cavity \i, e x p r e s s e d a s a function of the position of a 16

partiele and time will be

t t

é" ( t p , z , x , t ) = [ J a + S a (z,x) cos v ( f («„dt+ cp-9 ) ] s i n h ( [(lu^ + u), )dt+e ),

to to (1.3.8) é' (cp, z , x , t) = i a s i n h ( (*„ + lu dt + e ) to t t + A 2 a (z,x) sin [ ( v + h ) lU dt + v (cp - 9 .) + h ( iu,,,dt+ e ) ] t o to t t - i 2 i. (z, X) s i n [ (v - h ) [ u)„dt + v (cp - 9 )-h ( | lu dt + e ) ] . Always 'o ' o (1.3.9)

IH + hl^,l^'"o' (1.3.10)

so that all t e r m s have angular frequencies of u)o o r higher, except one for

which V = h

t

- è \ ^ ( z , X) sin l^(cp(t) - 9^ - j a.,^dt - e^). (1. 3.11) to

The components of ^ p e r p e n d i c u l a r to the orbit of a particle will not give r i s e to energy transmission between the accelerating field and the p a r t i c l e . We will designate the components along the equilibrium orbit by é' and a . If the e l e c t r i c field does not penetrate far into the vacuum chamber, (1. 3. 7) b e -comes, for the component along the equilibrium orbit,

TT

\ =:^J^(y.z.x)dy (1.3.12)

If the e l e c t r i c a l field extends in the vacuum chamber up to a point where it is not allowed to put c o s y h = 1, which may be the case if h is l a r g e , a and t h e r e -fore the energy gain p e r turn will be a function of z and x.

If (1.3.14) is valid, the e l e c t r i c accelerating field of cavity n^ becomes

V ,ï

^^(cp,t) = ^ ^ ^ s i n l ^ ( J ' " i d t - c p ( t ) - 9 ^ + e^), (1.3.15)

t 0

where all t e r m s with angular velocities of (D^ or g r e a t e r have been neglected. The force on a proton caused by this field is

F^ = e ^ ( c p , t ) . (1.3.16)

If there is m o r e than one accelerating station, one t r i e s to make

and usually h = h . (1.3.18) Then V t F = e " ^ s i n h (JoUjdt-c()(t)), (1.3.19) with """ t '•o Na V „ , , = 2 V . (1.3.20) max |J. n

If different accelerating stations have different harmonic numbers, one has Na V '

^ = 5 J i ^ Sin 1^ (J ^^^ - q.(t)). (1. 3. 21)

provided

e = e . (1.3.22)

1. 4 THE SYNCHROTRON MOTION

If p denotes the momentum of a particle, the synchrotron motion is given by

p = F . (1.4.1)

This equation will be written a s a differential equation in the coordinate s y s t e m s of the p a r t i c l e . If To i s the revolution period of a particle on the equi-librium orbit, Vo i t s velocity, CQ the equiequi-librium orbit length, then, to the f i r s t o r d e r in dC and dv,

^ ' - ^ ^ «^^ (1.4.2)

Let the momentum compaction coefficient be defined by

A d C / C „

This coefficient is determined by the magnetic field.

After expressing the right hand t e r m in p, using (1. 4. 3 and 1. 2.10),

^ = ( C ^ - E / / E ' ) ^ . ( 1 . 4 . 4 ) ' o " Po

Let

Tl 4 a - E o V E ^ (1.4.5)

F r o m the definitions it follows that

dT ^

(1.4.6)

Then the deviation of the momentum from i t s equilibrium value i s

*=-*x^- ^'-^-'^

With

•^ d t

and utilizing (1. 2. 7, 1. 4.1, 7 and 8) we get

F = V r „ - | p ( 9 | - ^ ) . (1.4.9)

Note that at this point the accelerating field can still be chosen. If (1. 3.19)

applies,

1 d • E ®V p BoeroCo

^ ^ ( ^ f ^ = - ^ sin(cp(t)- J ^ d t ) h . ^ ^ ^ . (1.4.10)

<^ t

Let t be determined by

V„,v s i n * o - Bor„Co. • (1.4.11)

and let 9 , according to (1. 2. 22), be defined by

K -- Vnb. (1.4.12)

Then

t

^ " ^ < * f > = " % r [sin(cp(t) - J ^ i d t ) h - s i n 9 „ h ] . (1.4.13)

'"" to

If acceleration takes place simultaneously, with different harmonic numbers

we have (1.3.21, 1.4. 9)

t

i;?dt^'^V^l ^ [ s i n ( c p ( t ) - J ^ ^ ^ d t ) h ^ - s i n 9 o h ^ ] . (1.4.14)

" t„

Eq. (1.4.13) may be written in i|; with the aid of (1. 2. 22), taking e = o, as

hu)

c

hdt ^ < * - ^ ' " i ) ) = - ^ - 2 ^ (Sin* - sinto). (1.4.15)

then

K = - % ^ ' ^ ^ : (1-4.16)

* "" ^ f > E " * + K ( « ^ * - Sin-l-o) = h<i)i + ( ^ | - ) | - h u ) i . • (1.4.17)

In general, the quantity x/dC depends on the azimuth and is determined by the shape of the magnetic field.

F r o m (1. 4. 3, 4 and 6) it follows that

'tu„ a Co Then, defining

(1.4.18)

TldC

aC^K • (1.4.19)

the convenient relation

9/"'o

(1. 4. 20)

is obtained. For a certain azimuth, £ is a p a r a m e t e r slowly varying with time (see 1.4. 5).

Consequently, the quantity x is a m e a s u r e for cp by (1. 4. 20).

C changes sign if Tl changes sign, i . e . when transition occurs, t is connected to X by (1. 2. 22 and 1. 4. 20):

V,

ivc ('"i-h

We cannot solve (1. 4 . 1 3 , 14 or 17) analytically. If u)^ is constant, and if d E

the (small) t e r m s dependent on -77 •ïr a r e ignored for the moment, a f i r s t in-tegral of (1. 4.17) is

• 2 . 2 I t - K(cosi|( - c o s t o ) - K s i n ' t ' o (t - t o ) = i * (1.4.22)

in which u)i does not appear.

> ? ? ^ ^ ^ M , ,

VACUUM CHAMBER WALL

y/A^^y//////-//yAW/y//////4/:///^////.

v/Af//////////////y/////////////////A

I VACUUM CHAMBER WALL >|l=0

^ _ ^ - T

f = Vo

When ^i = 0, then (1.4. 22) gives the particle motion in the fig. 1.4.1).

L e t t ^ , t g . '!'(; and ii^ lie on the s e p a r a t r i x , the boundary d i l a t i o n s .

Then, by equating the a r e a s A and B, we have

* A = " - * o •

cos 1» g + t g sin tp = - c o s %+ (TT - to) sin to ,

and

i^ = [ 2 K (2 cos to + 2t„ sin to - TT SIH to ) ^

Because of the symmetry with r e s p e c t to the line t = o, we

i = ^ c

-( t , t ) plane -(see

curve of stable o s

-1/2

have

Let the limits s e t to free particle motion in the horizontal plane by

X . = X (9) min min ^ '

and

''max =^max (^J'

and l e t the horizontal amplitude of the betatron motion of a

x ^ . , i = l , 2 , N.

Then, for free motion,

X . +x < x < x - x ^ . , min bi i max bi o r with (1. 4. 21), 22 particle be (1.4 (1.4 (1.4 (1.4 23) 24) 25) 26) be given (1.4. (1.4. (1.4. (1.4. 27) 28) 29) 30)

^""oC ( x ^ i „ + X b i ) < - h ^ i + ti < h ^ o £ ( X , , , - x ^ j ) - (1.4.31)

F r o m (1. 4. 25) and (1. 4. 26) for stable oscillations we also have

t n ^ t ; <'l'c . (1.4.32)

and for lij = o, the maximum x-deviation (see fig. 1.4. 2).

The right hand term of (1. 4.14) can be expanded into a Taylor s e r i e s 1 d , A E eV

cu 2 dt ^^ Tl

<'''f>= 2 ^ " (coscpoh)h(cp(t)- J u ^ i d t - c p o ) + . . . . (1.4.33)

c

d E

When the (small) t e r m - r r -=r is neglected, and (cf. 1.4.16)

2 A

u), = K coscpp h = K COS t o ' (1.4,34) the linearised equation of the synchrotron motion

t

tp + 11)2 ^ = u)2 ( J ju^dt + 9o), (1. 4. 35) . . 2 . . „.2

t„

i s obtained. ~^— will be called the synchrotron frequency. A stable solution exists if

U)^ > o. (1.4.36)

s

Eq. (1.4. 35) applies to every particle, 1. e . , t

^. + ml<f. = "),N J ("idt + c p j , 1 = 1 , 2 N. (1.4.37)

'•o

Summing over all p a r t i c l e s and dividing by N yields t

< cpi > + u)^^ < cp. > = u)/( J uJjdt + cp^), (1.4.38) t„

1 '^ where < > stands for -j?- 2 .

N i=i

cp è <9. >, (1.4.39)

M 1

and thus the s a m e equation (1. 4. 38) holds for the azimuth of the centre of m a s s , i . e .

t

^M + '"s%M= '"s^ ( J '"idt + *o )• (1- 4- 40)

1. 5 BEAM SIGNALS

Capacitive pick-up electrodes a r e often used for beam detection and m e a s u r e m e n t . By dividing an electrode into two p a r t s (see fig. 1. 5.1), the induced voltage on each electrode can be made a function of horizontal or v e r t i -cal beam displacement.

Fig. 1 . 5 . 1

Let a point m a s s have coordinates 9, z, x and a charge q. In the e l e c t r i c field of the charge only the Coulomb term is taken into account . Let qk(9-9e, z,x) be the induced charge on an electrode, where B^ is the centre of the e l e c -trode, and where k is a geometrical factor determined by the shape of the electrode and its surroundings. If Ct is the capacitance of the input circuit of the associated equipment and of the electrode against its surroundings, the in-duced voltage is

v = ^ k ( 9 - 9 e , z . x ) . (1.5.1)

One t r i e s to make k / C , l a r g e , in o r d e r to obtain a high voltage, Let

y = 9-9e. (1,5.2)

k is periodic in y with period 2TT. In the event that

k(y, z,x) = k(-y, z,x), (1,5.3) k may be expanded in a F o u r i e r s e r i e s , k(y, z,x) = i a ^ + ^2 a^ cos v y , (1.5.4) with

% ^V J '^^' ^'"^ °°° ^ydy.

After substituting the particle azimuth, cf. (1. 2. 21), t

9; = J u)„dt + 9 i ( t ) , 1 = 1 , 2 , , N, to

the induced voltage can be written a s

N CO *

v = - ^ i = i [ ^ i ^ o + J i a ^ c o s v ( J u ) „ d t + 9 i - 9 e ) ] . (1.5.5) to

If the electrodes a r e short a s compared with —^, rr

% = ^ J k ( y . z , x ) d y . (1.5.6)

Since the azimuth of the electrodes d e t e r m i n e s the phase of the beam signal, the location of the electrodes may be chosen conveniently, e . g . , according to

the type of phase meter used.

If two electrodes have geometrical factors k j and kg, which are linearly

dependent on x and independent of z, so that

^ i " " ^ b ^ k'(9), - b < x < b , (1.5.7)

k 2 = ^ ^ k'(9), (1.5.8)

then the Fourier coefficients of the voltage on the electrodes may be written as

with

TT

a= J k'(y)dy , (1.5.11)

For the voltage difference between the electrodes we obtain

^ z ' ^ i ^ C ^ ' ^ ^ i " ^ ^ i % ? i ^ ° ^ ' ' < J '^odt + 9 i - 9 , ) ] . (1.5.12)

t„

This signal has to be made proportional to l/N in order to be proportional to the

horizontal displacement of the centre of mass

1 ^

XM = ^ i ? i X i . (1.5.13)

From (1.5.12) it follows that any component of the voltage difference may be

used to determine x„,.

C H A P T E R I I

BEAM CONTROL METHODS AND SUBSYSTEMS

2 . 1 INTRODUCTION

The accelerating frequency of the f i r s t proton synchrotrons was p r o -grammed from the magnetic guide field. A " m a s t e r " oscillator was used to drive the radio frequency amplifier and cavity. The oscillator frequency could be accurately determined by a voltage (cf. fig. 2 . 1 . 1 ) , obtained from an integrator connected to a coil in the guide field. Because of the nonlinear r e l a -tionship between magnetic field and accelerating frequency (cf. 1. 2.11), and to compensate for e r r o r s in the system, a function generator was inserted between integrator and oscUlator.

Although the accelerating frequency could be obtained with the required accuracy, it was extremely difficult to keep down the amplitude of the synchro-tron motion. Therefore, the method of damping coherent synchrosynchro-tron oscillations by phasefeedback, resulting in reduced particle l o s s e s , was a g r e a t b n

-12 provement on p r o g r a m m e d acceleration .

Once phasefeedback appeared to be feasible, the accuracy of the a c c e l e -rating frequency still had to be improved for higher beam intensity. A fre-quency deviation causes a horizontal beam deviation. By adding a second control loop, which used the horizontal deviation a s an e r r o r signal (see fig. 2 . 1 . 2), very small xdeviations were obtained. This control loop was intended to c o r -r e c t accele-rating f-requency e -r -r o -r s slowly, i . e . "adiabatically". Because the phase of the accelerating voltage is locked to the phase of the circulating

13 bunches, the CERN group called such a system a "phase lock" system . This

o

- ^

h

t e r m is somewhat misleading because all beam control s y s t e m s show phase lock. We will use the term only if phase lock is obtained by controlling the phase of the accelerating voltage by means of a phase m e t e r .

An essentiallydifferentmethod was employed in Brookhaven for the 33GeV AGSynchrotron, after t r i a l s on the Cosmotron and on an electron analogue (sse fig. 2 . 1 . 3) . The signal from a pick-up electrode d r i v e s the cavities via a voltage variable delay line. When the accelerating frequency component of the electrode signal at the cavity is in phase with the a c c e l e r a -ting frequency voltage, phase lock will automatically be secured, The problem of generating the c o r r e c t frequency by means of an oscillator such a s used in a phase lock system, has now been converted into that of determining the c o r -r e c t delay time. By cont-rolling the delay time by an x-deviation signal, this problem has been solved. The described system is called a "bootstrap s y s t e m " .

Acceleration with a beam control system amounts to determining the c o r -r e c t initial conditions. Once unde-r way, the system stabilizes itself. The f i -r s t p a r t of the acceleration (in the main a c c e l e r a t o r ) is usually c a r r i e d out without beam control.

Various injection methods may be employed. Injection may be single turn or multi turn, it can take place with or without an accelerating field during the injection period, and the injected beam may be prebunched or not.

In beam controlled acceleration, the existence of bunches is used to s e -cure phase lock. F r o m this it follows that before bunching, or at l e a s t before some bunching has taken place, beam controlled acceleration cannot s t a r t .

The simplest way to a s s u r e bunching is to turn on the accelerating field after injection. Then, a s the magnetic field i n c r e a s e s , p a r t i c l e s outside the s e p a r a t r i x will hit the vacuum chamber walls. To s t a r t the bootstrap system in this way, a separate oscillator is needed to determine the initial a c c e l e r a -ting frequency.

Another method to a s s u r e bunching consists of injecting a prebunched beam. With single turn injection it may be possible to prebunch with constant frequency. With m u l t i - t u r n injection it is almost certain that the prebunching frequency h a s to follow the magnetic field with the high accuracy required for

an accelerating frequency p r o g r a m ,

Injection energy and final energy determine the range of orbital frequent c i e s . Since protons of 2GeV already have a speed of more than . 9 times the speed of light, this range will in p r a c t i c e be determined by the injection energy.

Therefore, the injection energy d e t e r m i n e s to a large extent the difficulties which a r e encountered in the dimensioning and construction of the accelerating s y s t e m .

The stability of the large number of control loops, p r e s e n t in a beam con-trol system, many of which a r e mutually dependent, is an important problem. It is good policy, to make s y s t e m s independent of each other a s far a s possible. F o r example, one t r i e s to design the beam control loops so that their p e r f o r -mance is independent of the number of accelerated p a r t i c l e s , even when sudden jumps in intensity occur.

We will often encounter frequency modulated signals of the form jt

sin J ('«o + "'i)dt, (2.1.1).

to

U)^ + U)^ may be F o u r i e r transformed (cf. 1. 2. 6 and 19), where «0 is the r e f e r -ence angular velocity (cf. 1.2.6), anduJi a departure from UJQ (cf. 1.2.19). The frequency components of cu^^ + w^ will be called modulating frequencies, and signals determining uuo + u)i will be called modulating signals. Note that the reference angular velocity, multiplied by h, is not the c o r r e c t accelerating frequency if the r a t e of change of the magnetic field v a r i e s with time. Therefore, the term "ideal frequency" will not be used for hou^.

The frequency spectrum of tUo depends in our case on B(t) only, and t h e r e -fore on the repetition r a t e of the a c c e l e r a t o r , the self of the magnet and the shape of the voltage of the magnet power supply.

The frequency spectrum of ooi depends on B(t), but also on the noise i n t r o -duced into the control loops, including the disturbances generated by the chang-ing particle distribution of the accelerated bunches.

< " l

During acceleration, ——has to be small and is determined by the h o r i

-"' o

zontal dimensions of the vacuum chamber.

2. 2 THE POWER AMPLIFIER AND CAVITY

The dimensions of the particle orbit and the rate of change of the magnetic flux density of the guide field determine the equilibrium gain p e r turn, (cf. 1.4.10). To ensure phase stability, the amplitude of the accelerating voltage, V , h a s to be l a r g e r than the equilibrium acceleration voltage. The choice

of Vj^^jj determines the length and the width of the stable regions (cf. 1.4.10, 23, 24 and 1. 4.14, 21, 25, 26). Values of to usually range from 10 to 30 d e g r e e s .

The cavities a r e f e r r i t e loaded. They a r e tuned to cut down the power consumption of the driving c i r c u i t s . Tuning is accomplished by polarizing the f e r r i t e by means of an additional magnetic circuit. Fig. 2. 2 . 1 and 2 show the Delft situation. FERRITE 6AP, \ K

w

m

COIL POLARIZING YOKE

Fig. 2.2.1 Fig. 2.2.2

Automatic tuning of a cavity has to be p r e f e r -red to programmed tuning because of the high t e m p e r a t u r e d e pendence of the r e l a -tive permeability of the f e r r i t e , and the a c -curacy required for the c u r r e n t in the po-larizing windings. A method to accomplish automatic tuning is illustrated by fig. 2. 2. 3. PHASE METER POWER AMPLIFIER POLARIZ. AMPLIFIER F i g . 2. 2.3 TUNED CAVITY

The phase of the driving signal and the gap voltage a r e compared in the p h a s e -m e t e r , which produces z e r o output when its two input voltages a r e in phase. The output of the phase m e t e r is fed into an amplifier, producing a c u r r e n t proportional to its input signal. The polarizing current determines the r e s o n ance frequency of the cavity. In this way the cavity is kept close to its r e s o n -ance frequency provided the loop gain is large enough.

A tuned cavity can be regarded as a parallel LCR circuit with a time constant

T = 2Qor^ , (2.2.1)

where Q is the quality factor of the c i r c u i t and (Uf/2n its resonance frequency. As long a s T " is large compared to the modulating frequencies of cuo + cuj, the steady state solution of the differential equation governing the LCR circuit may be used to describe the behaviour of the cavity. Let

a sin C J (cDo + cuj )dt ] h, (2. 2. 2)

be the input of the power amplifier. Then the accelerating voltage may be written a s

b sin C J (u)o + ("i )dt h + X ] . (2. 2. 3)

Provided X "^ TT ,

X « ! - 2QC0r"^ (h"'o + buij - U)J. (2.2.4)

The polarizing c u r r e n t I determines the resonance frequency of the cavity:

"^r = "'t (I)- (2. 2. 5) If I is linearly dependent on the phase difference as m e a s u r e d by the phase m e t e r we have

1 = AX , (2.2.6) 32

where A is dependent on the modulating frequency. Let

b'»'o=%(Io)- (2. 2. 7)

Then, for accelerating frequencies close to the cavity resonance frequency, and for s m a l l d e p a r t u r e s from IQ,

X = - 2 Q ( h . , - ^ A I ) .

hiuo ' ^ -'T d l (2. 2. 8)

where

I = I o + A I (2. 2. 9)

Hence for the phase difference with closed loop is obtained: dcu 2Q h ^ " d ^ ï ° hüÜQ 2 Q du)r o 1 -hcuo h i U g d i A (2. 2.10)

Eq. (2. 2.10) shows that

(a) since for a p a r a l l e l LCR c i r c u i t

Q = u ) , R C ^ , d*

the open loop gain is proportional to - ^ A (see fig. 2. 2. 4)

Fig. 2.2.4

(b) for signals with a modulating frequency below the cut off frequency of the control loop, the phase shift can be reduced substantially.

(c) in a system without p r e p r o g r a m m e d polarizing c u r r e n t a s described above, a slowly increasing phase e r r o r is introduced during acceleration. In boot-s t r a p boot-s y boot-s t e m boot-s thiboot-s e r r o r m u boot-s t be compenboot-sated by the variable delay line.

A typical problem encountered In a tuning system is due to the t r a n s -fer c h a r a c t e r i s t i c s of some phase m e t e r s . Although the phase dif-ference

is monotonie d e c r e a s i n g with increasing driving frequency (see fig. 2. 2. 5), the output of the phase m e t e r

usually d e c r e a s e s for l a r g e phase e x c u r s i o n s . Then three stable regions occur: A, B and C. Region B is the control region. In region A the polarizing c u r r e n t will r e m a i n low when the driving frequency in-c r e a s e s . In region C the polarizing c u r r e n t will r e m a i n too high when the driving frequency i n c r e a s e s . Therefore, when cavities a r e turned on and off during a c c e l e r a -tion, one must take c a r e to r e a c h region B. This may be difficult because of the high t e m p e r a t u r e dependence of the relative p e r -meability of the f e r r i t e .

Fig. 2.2.5

h(u).+ü),')-o),

intensities

The power consumption of the beam may be considerable for high beam 17

^ ~ "C ® ^niax Sin t o (2. 2.11) 12

F o r example, the CERN P. S., when accelerating 2.10 p a r t i c l e s , d e l i v e r s 7kW to the beam if OJQ «S U)^ . The load of the accelerated bunches on the cavi-ties may given r i s e to a considerable distortion of the shape of the accelerating voltage. If this o c c u r s , the differential equation governing the synchrotron m o -tion h a s be modified. Then it becomes much more difficult to prove the stability of the beam control s y s t e m .

2 . 3 OSCILLATORS

TABLE I

for + 0 . 1 of vacuum chamber width

CERN DELFT P . S . COSMOTRON NIMROD BEVATRON

1 - ^

To control the acceleration frequency by means of an oscillator, the o s -cillator output frequency has to be made dependent on an input voltage. Usually no frequency multiplication is employed between the oscillator and the cavities.

During programmed acceleration the oscillator frequency must follow the guide field accurately (cf. 1. 2.11 and 1.4. 31). Table I gives, for several a c -c e l e r a t o r s , the relative a -c -c u r a -c i e s with whi-ch the beam deviation a t inje-ction is kept within . 1 of the horizontal a p e r

ture of the vacuum chamber. The r e -quired accuracy can only be m e t if the modulating frequency passband covers a sufficiently large p a r t of the spectrum of the guide field. Even then, the time lag between guide field m e a s u r e m e n t and accel-erating frequency may r e n d e r it impossible to follow the guide field with sufficient accuracy.

During beam controlled accel-eration, the demands on accuracy

a r e much l e s s stringent, although the performance of the system is improved by a high accuracy and a low noise level. However, in this case the p r o p e r t i e s pertinent to stability a r e of paramount importance. We think it to be good policy to design the oscillator

i n s u c h a w a y thatthe transfer c h a r a c t e r i s -tics a r e independent of modulating f r e quency over the f r e -quency range which e n t e r s stability considerations. This means that the modulating frequency p a s s -band m u s t extend from D. C. up to frequen-LOW PASS FILTER POLARIZ. AMPLIFIER

IK

t I

cies an o r d e r of magnitude higher than the synchrotron frequency. Then the c h a r a c t e r i s t i c s of the control loops can be chosen freely.

The oscillator frequency Is controlled by varying one or m o r e circuit pa-r a m e t e pa-r s . The inductance of a coil with a fepa-rpa-rite copa-re can be changed ovepa-r a wide range by polarizing the f e r r i t e . If the polarizing circuit is too slow, a voltage dependent capacitor may be added, which leads to a large output f r e -quency range a s well as a large modulating fre-quency bandwidth (see fig. 2.3.1). A substantial d e c r e a s e in the required frequency sweep may be obtained with a heterodyne system. Then the noise level is higher, although not prohibitive.

The introduction of variable p a r a m e t e r s impairs the stability of the o s -cillator. F u r t h e r m o r e , it is difficult to obtain a linear input-oulput relation. By placing the oscillator in a control loop together with an amplifier and f r e -quency m e t e r , the stability, accuracy, and linearity can be much improved. With

- 4

such a system a long t e r m stability of the output frequency of 2.10 (relative) and a modulating frequency bandwidth of 0-100 kHz may be realised. However,

''Xy

n^

^ ^

the noise level at the ouput of the oscillator will be increased due to the low output signal level of the frequency m e t e r .

The frequency m e t e r may consist of the following components: shaper, differentiator, rectifier, andlow p a s s filter (see fig. 2. 3. 2). In the range of 1-10 MHz the rectifier is the critical component. If the rectifier is reasonably temperature independent, the amplitude of the shaper output determines the stability. To obtain good linearity, the time constant associated with the differ-entiator m u s t be small compared to the s m a l l e s t oscillation period occurring, which brings about the low output voltage of this frequency m e t e r .

Hereafter a control system a s described above will be called an "oscillator"

for short. To simplify block d i a g r a m s the p r o p e r t i e s of the power amplifier and cavity will be assumed to be incorporated in the oscillator, i. e. the out-put of the oscillator will be equal to the accelerating voltage.

2.4 PICK-UP ELECTRODES AND ASSOCIATED EQUIPMENT

Pick-up electrodes a r e used

(a) to establish the phase relationship between the circulating bunches and the accelerating field,

(b) to determine the horizontal deviation of the beam, (c) to m e a s u r e beam intensity,

(d) to obtain information about the p a r t i c l e distribution within a bunch, and (e) to check betatron p a r a m e t e r s and magnet alignment, with both horizontally

and vertically sensing e l e c t r o d e s .

Fig. 2.4.1 Fig. 2.4.2 Fig. 2.4.3

The differential electrodes originally employed in the Cosmotron (see fig. 2. 4.1) produced equal but small signals when the beam was centered b e -tween the electrodes. Since a low signal level i m p a i r s the performance of the diode c i r c u i t s of the associated equipment, these electrodes a r e no longer used. Fig. 2. 4. 2 shows e l e c t r o d e s which produce, under s i m i l a r c i r c u m -stances, a voltage which is about half of that obtained on one electrode when the beam is in an e x t r e m e position. A further advantage over the e l e c trodes of fig. 2. 4 . 1 is the independence of the sum signal of horizontal d i s -placement, so that the sum signal need not be determined with the aid of a s e p a r a t e electrode. However, the voltages on the electrodes a r e slightly out of phase, because the beam always e n t e r s one of the electrodes first. E l e c -trodes devised by a French group of the "Saturne" a c c e l e r a t o r show complete

18 azimuthal s y m m e t r y (see fig. 2. 4.3) .

The cross section of the electrodes is made rectangular or elliptical.

El-19

liptical electrodes can be made to weigh the centre of mass almost ideally,

i.e. the induced voltage is accurately linearly dependent upon the horizontal

dis-placement of a particle. However, electrodes with a rectangular cross

sec-tion show only a negligible small error.

A suitable set up for a beam detection and measurement system is

illus-trated by fig. 2.4. 4. The inner and outer electrode are each connected to an

amplifier. The amplifier outputs are fed into voltage controllable attenuators.

The sum of the attenuator outputs is fed into a zero level restorer followed by

a low pass filter. The filter output is added to a reference voltage and

ampli-fied. The amplifier output controls the attenuators. In this way the attenuator

outputs are made independent of beam intensity.

The attenuator outputs are also fed into zero level restorers followed by

low pass filters. After subtraction of the filter outputs, a signal proportional

to the horizontal deviation of the centre of mass of the bunches is obtained,

provided thatthe electrodes weigh the horizontal deviation of a particle linearly.

By feeding the sum of the electrode amplifier outputs into a zero level

restorer followed by a low pass filter, a signal proportional to the beam

in-tensity is obtained.

The cut-off frequency of the filters used to determine the horizontal

de-viation is limited by the lowest frequencies contained in the beam signals.

In practice, therefore, this cut-off frequency may be chosen between 10 and 100

kHz. Similar considerations apply to the filter in the intensity control loop.

Special attention has to be paid to the attenuators. If they show differences

in attenuation with various control voltages, an x-error is introduced. By

using capacitive bridge attenuators containing varactor diodes, tracking of the

attenuators is no problem. At the expense of introducing a synchronous

detec-tor, the tracking problem may be avoided (see fig. 2.4. 5).

The synchrotron motion within the bunches may bring about variations in

the shape of the beam signals. If the beam signals are intensity controlled, very

short bunches may produce amplitude overload of the electronic circuits.

There-fore, the electronic circuits must be designed for a reasonable amplitude range.

If an additional maximum amplitude control is employed, the loop gains of the

beam control system will decrease with increasing amplitude of the sum signal.

Usually one tries to avoid such a coupling between bunch shape and control

parameters.

AMPLIFIER ATTENUATOR

éhéh

f]

(ih

© — < ï i >

DIFFERENCE SUM DIFFERENCE SUM (AS ON ELECTRODES) (NORMALIZED)

AMPLIFIER -INTENSITY F i g . 2 . 4 . 4 AMPLIFIER

ttl

DIFFERENCE SUM DIFFERENCE SUM (AS ON ELECTRODES) (NORMALIZED)

Finally, we l i s t the important design p a r a m e t e r s concerning a beam detection and m e a s u r e m e n t system:

(a) the fundamental frequency range of the observed particle bunches. This frequency range equals the accelerating frequency range if all stability r e -gions a r e equally filled

(b) the p a s s band of the observation channels and the phase channel, i. e. the constant intensity sum channel. To obtain " t r u e " signals, this p a s s band has to be much l a r g e r than the frequency range mentioned sub (a) (c) the intensity range which can be handled by the intensity control

(d) the attenuation of spurious signals in the phase and x-channel due to sudden particle l o s s e s

(e) the maximum e r r o r (in m e t e r s ) which can be tolerated in the x-channel (f) the maximum e r r o r (in degrees) which can be tolerated in the phase signal (g) the p a s s band of the x-channel.

Since these p a r a m e t e r s cannot all be chosen entirely satisfactorily, and since they a r e also mutually dependent, in each p a r t i c u l a r case an optimal solution has to be sought. In general, one may say that the beam detection and m e a s u r e m e n t system m u s t provide a passive transmission of beam information, i. e. the dynamics of the system must not become evident in the beam control loops, the signals m u s t be " t r u e " , and m u s t have a low signal to noise r a t i o .

2 . 5 PHASE METERS

Phase m e t e r s a r e used to syn-chronize the accelerating field and the circulating bunches. The choice of a p a r t i c u l a r phase m e t e r has a strong influence on the system p e r -formance.

It will be assumed that the beam signals a r e intensity controlled (cf. section 2.4). Then the mean a r e a of the beam pulses fed into the phase m e t e r will be constant. This input signal may be written a s (cf. 1.5.4).

P('P)i

— 9

Fig. 2 . 5 . 1

N co

:i-.2 S a cos V (e„ + 9i - 9„)

N 1=1 v = l V ^ o ^ i e ' (2. 5. 1)

If p (t, 9, ^) is the density distribution function of p a r t i c l e s in the (9,'?') plane, the beam signal can also be written a s

C" ƒ Jp('^'*'^)'^(Öo '^^ ~ ^e^'^*^'''- (2.5.2)

(cp.cf)) plane

cp and 9 a r e governed by the synchrotron differential equation (1.4.10). The equivalent of (2. 5.1), expressed in the density distribution function, is

J J p(t,9,9)k(9^+9 - 9^)d9d9

JJp(t,9,9)d9d9

(2. 5. 3)

If u) = constant, p a r t i c l e s in the (9,9) plane move in approximately closed paths. However, the bunch signal will have a constant shape only if the pai> t i d e distribution function is independent of time, i. e. if - ^ = o, (see fig. 2.5.1).

at

Two types of phase m e t e r s will be considered h e r e : the switching phase m e t e r and the multiplicative phase m e t e r .

The switching phase m e t e r (see fig. 2. 5. 2) transforms the accelerating

ACCELERATING VOLTAGE BEAM SIGNAL FLIP-FLOP A FLIP-FLOP B LOGIC CIRCUIT SWITCHED CURRENT], SOURCE '

voltage and the beam signal into square wave voltages by means of flip-flops. The o u ^ u t s of the flip-flops a r e fed into a logic circuit, which t r i g g e r s two switched c u r r e n t s o u r c e s I j and l^- The outputs of the c u r r e n t s o u r c e s a r e

I^(A. B + A.B) (2. 5. 4)

and

I2(A. B + A. B), (2. 5. 5)

where A and B a r e Boolean symbols expressing the state of the flip-flops, and where

I^(true) = l2(true) = Ij,, Ii(false) = l2(false) = o. (2.5.6)

(Several other switching schemes a r e possible). The two c u r r e n t s a r e alge-braically subtracted, and the signal obtained led through a low p a s s filter. F r o m fig. 2. 5. 3 it can be seen that, for a signal A with duty factor J and a signal B with constant duty factor, the output at the low p a s s f i l t e r will be z e r o

BEAM B ACCELERATING VOLTAGE A A.B + A . i - ^ I | A.B+A.B-^l2 I1-I2 Fig. 2.5.3

if the fundamental frequency components of A and B have a phase difference of n / 2 . F u r t h e r m o r e , the output will be linearly dependent on deviations from n / 2 within a limited region.

If the beam flip-flop changes its state a t 9 „ , „ a n d 9 „ , , , the bunch azimuth a s determined by this flip-flop is (cf. fig. 2. 5.1)

è (* . +9 ).

" min max

(2. 5. 7)

This expression is difficult to use in a differential equation. In addition, if the density of the bunches v a r i e s with time, and the azimuth of the bunches, a s given by (2. 5. 7), is kept constant with r e s p e c t to the accelerating field, the centre of m a s s and the phase of the fundamental component of the bunches will vary with r e s p e c t to the phase of the accelerating voltage.

The multiplicative phase m e t e r consists of a circuit giving an output which contains the product of its two input signals. Such an electronic circuit may be very simple if use is made of components with a quadratic c h a r a c t e r i s t i c . The signal (2. 5.1) multiplied by the accelerating voltage (cf. 1. 2.17) contains f r e -quencies of u)o/2n and higher, and one t e r m proportional to

1 N r

J a ^ V ^ 2 sin ( U , d t - 9 . + 9 - 9 + e )h, (2.5.8) ^ h max N 1=1 ^J 1 i e \i, \i.

which can be obtained by feeding the multiplicator output into a low p a s s filter. If the location of the phase m e t e r is chosen so that

6 ^ - 9 + 6 = 0 or a multiple of n, (2.5.9)

then the phase m e t e r output is proportional to (cf. 1. 2. 22)

a h V m a x < s i n * i > • (2.5.10) This output signal is also proportional to the average energy gain of the p a r

-ticles; cf. (1. 4.15). If a t e r m proportional to the equilibrium energy gain eVsin ^^ is subtracted from (2. 5.10), we obtain

a V < sin il; - sin >|/ > . (2. 5.11) h max i o

A control loop using (2. 5.11) a s e r r o r signal keeps the mean deviation in e n e r -gy gain from the equilibrium value small. This s e e m s to be exactly the r e s u l t one wishes to obtain with a phase m e t e r .

The same output (2. 5.11) is obtained if the beam signal is led through a low p a s s filter first, so that only t h e h harmonic

1 ^

is fed into the phase m e t e r . This can be seen by multiplying this signal by the accelerating voltage. Eq. (2.5.12) may also be written a s

a(9^ , 9 2 \ )cos(e^ + a (9^ ,92, 9j^))h, (2. 5.13)

where both a and a depend on all cp.. Therefore, the multiplicative phase m e t e r can also be regarded as giving a signal proportional to

aV sin (feu dt - Q()h. (2. 5.14) max J 1

The switching phase m e t e r with the same low p a s s filter as mentioned above produces an output signal proportional to

Ju) d t - a . (2.5.15)

F r o m (2.5.14 and 15) the differences between the multiplicative phase m e t e r and the switching phase m e t e r with a low p a s s filter can be seen. If no low p a s s filter is used, a comparison is much m o r e difficult.

The phase m e t e r s described above show a zero output if the phase differ-ence between the (sinusoidal) input voltages is equal to n / 2 . To obtain an output signal which is independent of the reference angular velocity, it is n e c e s -s a r y to connect the pha-se m e t e r input-s with equal length of cable to the accel-erating stations and to the pick-up electrodes equipment. However, some ad-ditional delay is introduced in the equipment associated with the accelerating stations and the pick-up electrodes, which may change when other equipment is used. Therefore, a variable delay line which can be s e t manually is a con-venient device to obtain balance of delays. The delay lines a r e readily avail-able commercially and have the additional advantage that they can be used for testing the phase loop of the beam control system.

C H A P T E R I I I

ANALYSIS AND DESIGN

3.1 CANONICAL VARIABLES

The use of canonical variables enables us to apply the theorem of

Liou-•11 20

ville . written a s

Eq. (1.4.13), describingthe synchrotron motion in the (9,9) plane, can be 21 with and 9 = au , (3,1,1) Ü = b [ s i n (9 - J ' " i d t ) h - s i n 9 o h ] , ( 3 , 1 , 2) T] 2 a = ^ ("c . ( 3 . 1 , 3) eV , max b = - ^ ^ . (3,1,4) Eqs. ( 3 . 1 . 1 and 2) can be derived from the Hamiltonian

9

s i m i l a r l y , (1.4.15), describing the synchrotron motion in the (Y,'!') plane, can be written a s

i(r = ahw + hu) , (3.1. 6)

w = b (sin + - sin ijip) , (3.1.7)

which can be derived from the Hamiltonian

H = i h a w ^ + huJjW-b J (sino; - sin t^) dc^ . (3.1.8)

' o

Therefore, both in the (9,u) plane and in the (i|t,w) plane, a r e a is conserved independently of variations in any p a r a m e t e r of H.

F r o m the existence of a r e a conservation a number of important p r o p e r -ties of the accelerated beam can be derived. Using ( 3 . 1 . 6), (1.2.22), (1.4.19 and 20), x is related to w by

x = c«o^ QfCo(gJ) (u)o E ) " ^ w . (3.1.9)

Hence, a r e a in the (1',x) plane is proportional to (UOQE) .

In general, the p a r a m e t e r s in ( 3 . 1 . 8) change slowly a s compared with a synchrotron period. Therefore, w h e n * i is constant during one or m o r e synchrotron periods, paths exist in the (i|(,w) plane, which can be regarded a s closed curves, s i m i l a r to the curves in fig. 1 . 4 . 1 .

Let p ( t , t , w ) be a particle distribution in the ('l',w) plane, extending over a surface A, 1. e. outside A we have p = o.

The integral

J j H p dtdw, (3.1.10)

may be minimized, observing the a r e a conservation theorem and with

J J p d t d w = constant, (3.1.11) A

and i ^ = o . (3.1.12) 46

Fig. 3.1.2 Fig. 3.1.3

Starting with surface A (cf. fig. 3 . 1 . 1 ) , where the dotted lines connect points for which p is constant, the regions inside the dotted lines may vary in shape but not in a r e a . The distribution for which (3.1.10) is minimal will be called p . , and the associated surface An,in: see fig. 3 . 1 . 2 . Some p r o p e r t i e s of p . can be derived directly with the aid of this figure. F i r s t , the maximum of the p . distribution (if a maximum exists) lies in (i(t , o ) . For, if a region containing this maximum is interchanged with a region outside (41^ , o ) , the in-tegral i n c r e a s e s . Also, A^i^ must be limited by a curve for which H is con-stant. Any other curve yields a higher value of the Integral (see P in fig. 3.1.2). Finally, p^j^^ is unique, because if two minimum distributions would exist, a t l e a s t two domains of different density would be interchangeable. By bring-ing the domain with the highest density n e a r e r to the centre, a lower value of the integral would be obtained.

Therefore, A ^^^ is a convenient surface for comparing different situations of an accelerated beam. Two such situations a r e shown in fig. 3.1.3. Both a r e a s -sumed to be minimum surfaces obtained from the same Initial distribution, which may be identified with one of them. Let all p a r a m e t e r s belonging to situation 1 have a suffix 1 and those belonging to situation 2 a suffix 2. Then

a r e a A a r e a A

2 ' (3,1,13)

and, since the boundary curves connect points for which H is constant

t < *

min '^max

2 ' ^ ^ W m a x = - b J (slnof- s i n t o ) d a = - b J ( s i n a - sin •o)daf . (3.1.14) o ' o

When Aj^ and A ^ a r e small and can be approximated by e l l i p s e s , from (3.1.13) it follows that

^1*1 ='^2*2 • (3,1,15) Similarly, (3,1,14) becomes

i ha w^ = - è b cos +0 l»^ . (3.1.16)

F r o m the above relations the following r a t i o s can be obtained:

w, ^„ ^V, cos ill , E,fl„ h, . i _ i = _2 = ( li^ax 01 1 2 2 y

and

^2 E , % W2

i r = ^ i ; r - i r - <3.i.i8)

* 1 ^ 2 02 "^1

It follows from (3.1.16) and (3.1.17) that varying Vj^^x has but little influence on bunch shape and phase excursions.

The minimization of integral (3.1.10) may bring about an increase of the dimensions of the surface A. It may happen that a p a r t of the surface moves outside the s e p a r a t r i x . This may also occur when two unconnected surfaces a r e weighted with the s a m e H. Then the "heaviest" surface may move toward

(t , o) and the other surface move outside the stable region. In section 3.4 we will r e t u r n to this point.

3.2 LINEAR ANALYSIS

One of the first steps in the design of a beam control system is the l i n e a r i -zation of the differential equations. The s y s t e m then obtained contains slowly changing p a r a m e t e r s . After dimensioning the components, the performance of the system can be evaluated for all values which the p a r a m e t e r s take.

When the equations a r e linearized, the particle motion may be described by the motion of the centre of gravity cf. (1. 4. 37) and (1.4.40),

9 . +'" ^ ^x. = ' " ^ ( f ' ^ i dt + 9 ), (1.4.40)

M s M s *J 1 o " ' '

and by a motion around the centre of gravity,

q. +("ƒ q. = 0, q = 1 . 2 ,N, (3.2.1) N

where q. = 9 - 9 j ^ , and where one equation is superfluous, since .^ qj = o, Note that (1.4.40) and (3. 2.1) a r e independent. Let the deviation from the azimuth of the centre of gravity be

(3. 2. 2)

By

fro using

m (1.4

the Laplace transform

f = P .40) we obtained, with V = cp - 9 . •^ ^M o' 00 ' f ( t ) e - P ' d t , 0 constant 9o , 2 (U C U , s 1 2 ' ' + iu^ p (3. 2. 3) ( 3 . 2 . 4 ) putting y = y = o a t t = o.

The Bode diagram of the 9 excursions with '"idt a s input signal shows a flat modulating frequency response below (u^ , resonance at lUj , and a slope of -12dB/octave above cu^ . Let the oscillator input be

V. = V + V + V , (3.2.5) 1 o 1 n * '

and let the oscillator output angular velocity be

("o + " ' i = " o ( V o ) + V l " l ' (3-2-6) with

"o(Vo)=">o a n d n j ( V i + v „ ) = c u i . (3.2.7)

and noise in the system. It is possible to design the oscillator so, that Hj^ is a linear transfer function. When Laplace transforms a r e used, Ci may be a meromorphic function in p, with the degree of the numerator equal to o r lower than the degree of the denominator.

The output phase of the oscillator

ƒ ( u ) o + ' " i ) d t , (3.2.8)

will be regarded a s its output signal. Then the oscillator output can be written a s

" i

(u^ = - i - (Vj + v_^). (3. 2. 9)

Linearization of the phase m e t e r signal (2. 5.11) yields

d cos tj, <+j -^o^ ' (3-2-10) with

d = a , V ^ ^ ^ . (3.2.11) h max * '

In 9-coordinates (cf. 1. 2. 22 and 1.4.12), (3, 2.10) becomes

d ( c o s t ^ ) h ( J " J i d t - < 9 j - 9 o > ) , (3,2,12) in the p domain, using (3, 2, 2) and after expressing y iniUj,

p2 CU

d(cos t ) h — . - ^ (3. 2.13)

p 2 + u , 2 P

The phase m e t e r signal may be led to a modulating frequency dependent net-work Y. The output of this netnet-work may be written a s

2

P *,

v^f= Y d ( c o s t ) h - - _, (3.2.14) p +CU p

where t is a meromorphic function like fi j in the p domain. The horizontal 50

deviation of the centre of m a s s and cp a r e related by

x = 9 (cUoC)'^ • (1.4. 20)

If X denotes a network c o r r e c t i n g the x m e a s u r e m e n t output voltage, we may write Vjj( - X X ^ . (3.2.15) With (1.4.20) and (3.2.2), Vix=x(iu<,C)"^ p y (3. 2.16) Finally, by putting V = V + V , 1 If Ix

the loops a r e closed. The above relations a r e shown in fig. 3. 2 . 1 .

(3. 2.17)

r

.

J

X

«1

rt

—

" ;

With X = o, the open loop gain of the phase loop i s P

n, f d h c o s t .

p <= + IJÜJ '^

(3.2.18) 51

If n and Y a r e independent of the modulating frequency, the loop gain shows a slope of 6dB/octave below the synchrotron frequency, and above the synchro-tron frequency a slope of -6dB/octave (cf. fig. 3. 2. 2).

LOG 5AIN I PHASE LOOP +6dB/0CTAVE 6dB/0CTAVE

| - ( ^ J

The closed phase loop response, also with X = o, is

o

p p - n^Y dh (cos t o ) P + ("8

(3. 2.19)

The phase loop is critically damped if

n , f dh cos t = 1 o

2(U (3. 2. 20)

F r o m (3. 2.19), it follows, that in the case of critical damping, the resonance of y at the synchrotron frequency d i s a p p e a r s . By lowering or increasing the open loop gain, the y response can be made to have a peak at the synchro-tronfrequency o r to have two break points lying at equal distances on either side

22

of the synchrotron frequency (cf. fig. 3 . 2 . 3 ) .

(JU

The open loop gain (3. 2.18) is l e s s than unity outside an interval with ^ a s centre frequency (cf. fig. 3. 2. 2). For these modulating frequencies the f r e -quency response may fall off without affecting the phase loop. In particular, Y is often made to fall off a t low frequencies in order to suppress slowly changing spurious signals. This facilitates the phase m e t e r design, since then 52

slowly changing signals do not affect the x-deviation.

The x-response with closed phase loop and open x-loop is (cf. 1. 4. 20 and

3.2.19),

x=(<u C)"' " j - ^ ^ v (3.2.21)

° p^ - n ^ f dh(cos<i'o)P + '"s

With (3. 2. 21) a stable x-loop can be designed by applying the Nyquist stability

criterion. For example, if the phase loop is critically damped, an X-filter,

falling off with -6dB/octave from a frequency below the synchrotron frequency,

yields a stable x-loop, provided that

I n^X(U)^C)-^ I < 1 , (3.2.22)

for modulating frequencies sufficiently below the synchrotron frequency and for

the lowest lu Q occurring.

The phase loop shows constant gain when V and t are constant.

Therefore, it is easy to maintain nearly critical damping over the whole

acce-leration period. The x-loop shows a variable gain during acceacce-leration. It

de-pends upon the spectrum of v whether this changing loop gain can be accepted

or not.

WhenO Y, and X are chosen, the response of the system must be

evalua-ted. In particular, one must be sure that for all v which occur, the bunches will

be phase locked and will stay free from the vacuum chamber walls. It must be

remembered however, that the linear analysis shows an important difference

with respect to the real system. Because of the "soft spring" characteristic of

the reset force in the synchrotron differential equation, large lUjdt deviations

must be avoided while correcting an error.

In particular, closing the phase loop and the x-loop has to be done with

care, When the system is far removed from its closed loop equilibrium

posi-tion, particle losses may occur. Because the phase loop is fast, the point at

which the phase loop is closed is not critical. But the x-loop must be closed at

a point between the x-measurement and the slow X-filter, to keep the transients

small.