TRANSVERSE MODE COUPLING WITH FEEDBACK

Ronald D. Ruth* CERN, Geneva, Switzerland

Introduction

All large electron storage rings have been plagued to some extent by a single bunch vertical coherent in­stabi l ity. Calc ulations indicate that this instability (for short bunches ) is due to t he coup l ing and subse­quent degeneracy of modes which evolve from mode ' 0 ' , which ha.s a freq uency close to the betatron freq uency, wp; and mode '-1 ' • which has a freq uency of w~-ws It has been suggested at CERN1 that such an instability might be cured by altering the fre quency of mode '0' rather than by damping si nce damp ing, so far , has yie lded onl y small improvements. In this way the de­generacy mi ght be either avo ided or at l east occur at larger current. This type of feedbac k is called reac­tive feedb ack (damping is termed res ist ive feedback).

This idea has been tested with a two-particle model of a bunch with the result t hat t he threshold for instability can be enhanced by a factor of 2 to 4. 2 In Ref. 2 it ~s demonstrated that there are two distinct methods fop curing the instability. In the 'normal' method the; feedback is used to i ncrease the frequency of mode '0' and thus separate modes 'O' and '-1'; this yields an enhancement of a f actor of 2 in the thres­hold. In the 'abnormal' method the feedback is used to decrease the frequency of mode 'O' so that it has a lower frequency than mode 1 -l'. In this case the two­particle model yields an increase of about a factor of 4 in the threshold current. In both cases the feedback necessary is that which provides a coherent tune shift of dbout vs, the synchrotron tune.

The purpose of th is paper is t o el imin ate the de­ficiencies of t he two part icle model by i ncludi ng reac­tive (and resistive ) feedback in the Vlasov Eq . t reat­ment of this instabi l ity. In t he f irst secti on a flexible feedback system is described which yields either reactive or resistive feedback. In the second section Vlasov mode coupling is treated with feedback included. In the last section the results are applied to LEP using a realistic impedance, and the two differ­ent methods of reactive feedback are compared to resis­tive feedback.

The Feedback System

The feedback system considered here is one that treats each bunch independently. Thus, the pick-up/ kicker must have a response time less than the time be­tween successive bunch passages. This sets a lower bound on the bandwidth of the pick-up/kicker. In addi­tion we only require that the system measure and act on the total dipole moment of the bunch. For the short bunches considered here (less than 8 cm), it would be quite difficult to do otherwise.

In other aspects a react i ve feedback system is also conceptually t he same as a res isti ve one except that the phase adv ance between pickup and kicker must be an even mul t ip le of n/ 2 r ather t han an odd mu lti­ple. However, in practice the pickup and kicker are usually placed physically close so that the bunch must traver se almost t he ent i re ring after being detected before it reaches the ki cker t o be kicked (this is especi all y true for a lar ge machine li ke LEP) . Th is means t hat t he coherent phase adv ance wi 11 be shifted (dependi ng upon t he current ) due to t he transverse impedance of t he ring. Therefore, the resul t ing f eed­back will be shif ted from pure react ive t o react1 ve­plus-r es i st i ve (or anti-r es i sti ve ).

This effect can be compensated by the use of two pickups placed n/2 apart and then combined linearly at the kicker. To see this let the kick at the kfcker be given by

where tiyk is the change in slope at the kicker. and Yl and Y2 are the positions of the bunch measured at the first and second pickup respectively. g1 and g2 can be loosely referred to as the 'gains'; however, since it is the dipole moment rather than the position which is measured, g1 and g2 are implicitly propor­tional to the number of particles in a bunch, N, times an el ectronic gain. Let the second pickup be n/2 in phase after t he firs t. Then t he two important quanti­ties are the determinent and trace of the •coherent• transfer matrix, M. With the kick in (1) it is straightforward to show that

Tr(M)/2 = cosµo

+ l/2[g11iii7ficcos(µ 1k) + g21lf2i!'"ksin(µ1k)]sinµo

+1/2[ -g1llfifksin(µ1k) + g2~kcos(µ1k)]cosµo

(2)

(3)

where the subscripts 1. 2 and k denote pickup 1, 2 and the kicker, µ1k is t he coherent phase advance between pickup 1 and kicker and µ0 i s the coherent phase advance of the entire machine without feedback,

µo = 2nvo • (4)

The condition for reactive .feedback is

Det(M) = 1 (5) or

(6)

When Eq. (6) is satisfied, the phase advance with feed­back is calculated by the inverse cosine of Eq. (3). If the shift is small, we have

2ni'ivFB = L'iµFB = -l/2[g11ii'i]rilcos(µ1k)

+ g21ii2Jfksin(µ 1k)] • ( 7)

or using Eq. (6)

gl -4ntivF

8cos (µ 1 k)

~ ( 8)

-4ntivF6sin(µ1k) (ti vfBsma 11 )

g2 l~2~k

For tiVFB large simply make the replacement

tivFB = [cos(µ 0 )-cos(µ 0+tiµFB)]/[2nsin(µ 0 )] (9)

Equations (8 and 9) indicate the change in the gains which are necessary to maintain reactive feedback of a given ~VFB· Once µ1k as a function of curr ent is either measured or calculated, one can shift t he gai ns accordingly. Also note th at it is, of course. neces­sary to avoid integer and half-integer resonances.

In addition one can adjust the relative gains to obtain a mixture of damping and tune shift. In this case the determinant ts not unity but is related to the damping rate, i:Fs. by

Det -2t hFB e rev (10)

L'iYk = g1Y1 + g2Y2 ~~~~~~~~__;~-

* Visitor from LBL, Berkeley, CA.

(1) In thfs case ff the desired tune shift is tiµfg/2n, g1 and g2 are given by:

-389-

ljITlrkgl =

[2cos(µo+6µFs)loet-cosµ 0 (1+Det)]cos(µ 1k)/sin(µ 0 )

+ (1 - Det)sin(µ1k)

~ g2 = (11)

[2cos(µo+6µFs)loet-cos(µo)(l+Det)]sin(µ 1k)/sinµ 0

- (1 - Det)cos(µ1kl •

This can be useful for setting a small amount of damp­ing in addition to the tune shift in order to deal with instabilities with slow growth rates (e.g. the head­tail effect). In addition one can compare the relative efficiency for pure damping vs. pure reactive feedback by simply tuning g1 and g2 to achieve the desired 6µFB or damping rate.

Vlasov Mode Coupling

The matrix equat ion

It is possible to write down either an integral equation or an (infinite dimensional) matrix equation which is exactly equivalent t o the linearized Vlasov equation for transverse coherent oscil l ations of a bunched particle beam3, ••. Either of these can be sol ved numerically to obtain detailed results; how­ever, t he integral equation form is more useful for bunches long compared to the characteristic wavelength of the wake field while the matrix equation is more useful for bunches shor ter t han or the order of the characteristic wavelength. For short bunches it is necessary to solve the following:

where Zr is the transverse impedance, I is the bunch current, Pave is the average value of the betafunc­tion in the regions that contribute to the impedance. If we think of the impedance as a sum of all the ele­ments in the storage ring, then each factor shou ld be weighted by the local value of the betafunction. E is the energy. Other quantities are: a

0 "' rms bunch

length/radius, wt head-tai l freq uency sh ift , k = (v+vy)/vs, v = coherent tune. The form of Eq. (12) is t hat of an eigenv alue equation with eigenva lue equal to un ity. However, t he matrix is a function of t he coherent tune. This form is the key to mode coupl ing. Notice that the coherent tune is not labeled by any mode numbers as yet. All modes are yielded by the solution of Eq. (12) .

The matrix el ements are given by integrals because the wake field is assumed to decay completely betwen bunches; t hus, t he periodicity is i rre levant. If this is no t true, t hen the integrals become sums. If the bunch to bunch coupling i s strong, coup led bunch modes must be included in the analysis . The head-tai l effect is caused by the frequency shift, wt• in the impe­dance . To simplify the followi ng discussion this i s eliminated by assuming a corrected chromaticity.

If it is possible to truncate t he matr ix in Eq. (12) due to sma ll matrix e l ements for large order, t hen t he coherent frequencies of bunch oscillations are gi ven by:

Det[I - Ntrunc(A)] = 0 . (13)

The convergence (or non-convergence!) of the sequence of truncations is determined by the decrease of Cm/ml with increasing m. This in turn depends upon the bunch length and the impedance.

The matrix elements

To understand the size of the matrix elements in Eq. (12) cons ider the d111gonal el ements. For zero chrornaticity C21 i nvolves only the overlap of the re­active part of the impedance with

( ) 21 - ( a0 p) 2 aop e • (14)

If the impedance is large out to some we where the asymptotic tail begins, then it is clear that the bunch factor in Eq. (12) begins to overlap with the asymptotic region when

(awc) 2 ~ i . (a = oo/wo) (15)

This means that the matrix should not be truncated to a dimension small er than (awc) 2 •

In the asymptotic region let the impedance be given by

(16)

Then for 1 1 arge

C2..l/1! - (awc)a(R.+1)-(l+a)/2 (17)

Th11s for small awe all the high matrix elements are s~all; however, they onl y decrease slowly with 1n­creas1ng 1. The key point is that care must be exer­:ised in using the above analysis for long bunches, and in the case (awe) > 1 a completely different approach is more useful3 :

Adding feedback

Before including the modifi cation of the matrix due to feedback, it is useful to recalculate in terms of a (real) wake function defined here as

WT("') = .!__ /' i~(w)e-iwT dw 21' -

(18)

Thus it is simply the Fourier transform of iZy(w). In addition it is useful to expand this wake function in a series of Hermite polynomials, Hn,

(19)

This yields

Using Eqs. (lZ) and (18) it is straightforward to show that

c "' -n

erp n l in ave 8

2v5w0E n (21)

This gives a nice intuitive picture to the matrix. In addition one can calculate the matrix elements directly from a wake function.

Due to causality the normal wake function obeys

WT("') = 0 , -r ~ 0 (22)

-390-

However for the i ndividua l bunch feedback considered here, 1') the signal arrives at the kicker before the bunch passage, 2) it is constant during the passage, and· 3) it decays before the passage of the next bunch. This means for practical purposes it is a very good approximation to let WfB( -r) = const. This means that the feedback system affects only one of the matrix ele­ments, c0 • In terms of the tunesnift parameters cal­culated in the first section

, C~B = Cm + 6m,0[6vFB + itre/(2uFB) )/vs· (23)

Physically this means the feedback measures and kicks only the rigid bunch mode and thus only changes its frequency (for small currents). For larger current the strong coupling of the modes leads to the feedback affecting the higher modes.

Calculations for LEP

The Impedance

To calculate the instability threshold for a given machine it is necessary to either calculate or estimate the impedance or wake field. There are several methods available now for calculating transverse wakes 5 or im­pedances6. In the present paper a ca lcul at ion 1 of the LEP cavities i s used with an additional scaling factor of 1.4 for the contributions of the rest of the machine.

In the past broad band single resonator models have been used both for estimates and for fitting cal­culations. It is clear that these lack sufficient generality mostly because they have the wrong asympto­tic dependence. In other words they have the wrong smal 1 time dependence of the wake field. However, it is also clear that the wake field can be cut off smoothly after a few a with no change in the physics of the instability. If this yieTds a smooth function with few zeros, then its Fourier transform will also be smooth function. This function used to replace the exact impedance will yield correct results. This is contained in the formalism via the C's which are insen­sitive to details in the structure of Zr(w) on scales < l/a. This means that if Zr is given by poles as for a resonator, then these poles can be pulled away from the real axis with no change in the basic physics so long as the early time wake field is preserved p_re­cisely. In practice one resonator ( two po les) i s prob­ably not enough. Perhaps a more general and useful form is a model impedance with two cuts. This yields asymptotic dependence with fractional powers of w. For actual calculations, however,this is not necessary since the C's can be simply calculated nLDDerically for an arbitrary impedance or wake.field.

Threshold vs. bunchlength

Figure 1 shows the solution of Eq. (12) truncated to a dimension of 4 for LEP with a bunch length of 2 cm. Notice that it is modes 'O' and '-1' which first become degenerate and unstable. For bunch lengths of this order including higher modes does not signifi­cantly change the threshold. Notice also that the so­called radial modes at 0 and ±1 are automatically included. As one truncates at higher and higher dimen­sion, the radial modes appear, as they become impor­tant, automatically. Notice that the next two modes which will become degenerate are mode 1 and a radi a 1 mode which begins at 0.

As the bunch length is changed, the threshold for instability also changes. Figure 2 shows the depend­ence of this threshold on bunch length if all other quantities are held constant. The threshold increases for both shorter bunches as well as for longer bunches. These are due to completely different ef­fects. · For short bunch lengths this is due to the

o ,_o~-0~_2_5~0-5_0~0--1-5~1~. o-0~1~.2-5~1~- -50~1~--1..,----,5 2 ~o Re< w-w, l/w. LEP12

3 . 3

2 . 2

I .

0 . -

-I .. ... ·I

-2 -2 .

- 3 -3

-4 l ;'mh -4 . 0 . 0 0.25 0 . 50 0 75 LOO 1-25 1-50 1-75 2.00

Fig. 1: Transverse Mode Coupling Instability

decrease in the overlap of the wake field with the bunch factor in c0 • More precisely if the imped~nce (wake fiel~) varies at large (small) frequency (t1me) as w-a (t«- ), then

I - 1-« 0 threshold a ' a + •

(24)

For somewhat longer bunches the increase is due to the decrease in the peak current, which yields

1threshold "' a ' a ?. 2 cm · (2S)

Fin ally, for yet longer bunches, other modes become important because the wake function begins to have zero's within the bunch length.

LEP •

• •

0 6 e

BUNCHLENGTH Ccml

Fig. 2: Threshold vs. Bunchlength [Pave = 64 m].

Reactive Feedback

The results of calculations for 'normal' and 'abnormal ' feedback are shown in Figs. 3 and 4 respec­tively. For normal feedback at fixed total gain the frequency of mode zero is increased by vs at smal 1 current. Thi s separates modes '0' and '-1 ' and yields an increase in t he thresho 1 d of about a factor of 2. For fixed electronic the modes move somewhat dif­ferently because the tune shift due to feedback is pro­portional to the current. However, a tune shift due to

-391-

ljITlrkgl =

[2cos(µo+6µFs)loet-cosµ 0 (1+Det)]cos(µ 1k)/sin(µ 0 )

+ (1 - Det)sin(µ1k)

~ g2 = (11)

[2cos(µo+6µFs)loet-cos(µo)(l+Det)]sin(µ 1k)/sinµ 0

- (1 - Det)cos(µ1kl •

This can be useful for setting a small amount of damp­ing in addition to the tune shift in order to deal with instabilities with slow growth rates (e.g. the head­tail effect). In addition one can compare the relative efficiency for pure damping vs. pure reactive feedback by simply tuning g1 and g2 to achieve the desired 6µFB or damping rate.

Vlasov Mode Coupling

The matrix equat ion

It is possible to write down either an integral equation or an (infinite dimensional) matrix equation which is exactly equivalent t o the linearized Vlasov equation for transverse coherent oscil l ations of a bunched particle beam3, ••. Either of these can be sol ved numerically to obtain detailed results; how­ever, t he integral equation form is more useful for bunches long compared to the characteristic wavelength of the wake field while the matrix equation is more useful for bunches shor ter t han or the order of the characteristic wavelength. For short bunches it is necessary to solve the following:

where Zr is the transverse impedance, I is the bunch current, Pave is the average value of the betafunc­tion in the regions that contribute to the impedance. If we think of the impedance as a sum of all the ele­ments in the storage ring, then each factor shou ld be weighted by the local value of the betafunction. E is the energy. Other quantities are: a

0 "' rms bunch

length/radius, wt head-tai l freq uency sh ift , k = (v+vy)/vs, v = coherent tune. The form of Eq. (12) is t hat of an eigenv alue equation with eigenva lue equal to un ity. However, t he matrix is a function of t he coherent tune. This form is the key to mode coupl ing. Notice that the coherent tune is not labeled by any mode numbers as yet. All modes are yielded by the solution of Eq. (12) .

The matrix el ements are given by integrals because the wake field is assumed to decay completely betwen bunches; t hus, t he periodicity is i rre levant. If this is no t true, t hen the integrals become sums. If the bunch to bunch coupling i s strong, coup led bunch modes must be included in the analysis . The head-tai l effect is caused by the frequency shift, wt• in the impe­dance . To simplify the followi ng discussion this i s eliminated by assuming a corrected chromaticity.

If it is possible to truncate t he matr ix in Eq. (12) due to sma ll matrix e l ements for large order, t hen t he coherent frequencies of bunch oscillations are gi ven by:

Det[I - Ntrunc(A)] = 0 . (13)

The convergence (or non-convergence!) of the sequence of truncations is determined by the decrease of Cm/ml with increasing m. This in turn depends upon the bunch length and the impedance.

The matrix elements

To understand the size of the matrix elements in Eq. (12) cons ider the d111gonal el ements. For zero chrornaticity C21 i nvolves only the overlap of the re­active part of the impedance with

( ) 21 - ( a0 p) 2 aop e • (14)

If the impedance is large out to some we where the asymptotic tail begins, then it is clear that the bunch factor in Eq. (12) begins to overlap with the asymptotic region when

(awc) 2 ~ i . (a = oo/wo) (15)

This means that the matrix should not be truncated to a dimension small er than (awc) 2 •

In the asymptotic region let the impedance be given by

(16)

Then for 1 1 arge

C2..l/1! - (awc)a(R.+1)-(l+a)/2 (17)

Th11s for small awe all the high matrix elements are s~all; however, they onl y decrease slowly with 1n­creas1ng 1. The key point is that care must be exer­:ised in using the above analysis for long bunches, and in the case (awe) > 1 a completely different approach is more useful3 :

Adding feedback

Before including the modifi cation of the matrix due to feedback, it is useful to recalculate in terms of a (real) wake function defined here as

WT("') = .!__ /' i~(w)e-iwT dw 21' -

(18)

Thus it is simply the Fourier transform of iZy(w). In addition it is useful to expand this wake function in a series of Hermite polynomials, Hn,

(19)

This yields

Using Eqs. (lZ) and (18) it is straightforward to show that

c "' -n

erp n l in ave 8

2v5w0E n (21)

This gives a nice intuitive picture to the matrix. In addition one can calculate the matrix elements directly from a wake function.

Due to causality the normal wake function obeys

WT("') = 0 , -r ~ 0 (22)

-390-

However for the i ndividua l bunch feedback considered here, 1') the signal arrives at the kicker before the bunch passage, 2) it is constant during the passage, and· 3) it decays before the passage of the next bunch. This means for practical purposes it is a very good approximation to let WfB( -r) = const. This means that the feedback system affects only one of the matrix ele­ments, c0 • In terms of the tunesnift parameters cal­culated in the first section

, C~B = Cm + 6m,0[6vFB + itre/(2uFB) )/vs· (23)

Physically this means the feedback measures and kicks only the rigid bunch mode and thus only changes its frequency (for small currents). For larger current the strong coupling of the modes leads to the feedback affecting the higher modes.

Calculations for LEP

The Impedance

To calculate the instability threshold for a given machine it is necessary to either calculate or estimate the impedance or wake field. There are several methods available now for calculating transverse wakes 5 or im­pedances6. In the present paper a ca lcul at ion 1 of the LEP cavities i s used with an additional scaling factor of 1.4 for the contributions of the rest of the machine.

In the past broad band single resonator models have been used both for estimates and for fitting cal­culations. It is clear that these lack sufficient generality mostly because they have the wrong asympto­tic dependence. In other words they have the wrong smal 1 time dependence of the wake field. However, it is also clear that the wake field can be cut off smoothly after a few a with no change in the physics of the instability. If this yieTds a smooth function with few zeros, then its Fourier transform will also be smooth function. This function used to replace the exact impedance will yield correct results. This is contained in the formalism via the C's which are insen­sitive to details in the structure of Zr(w) on scales < l/a. This means that if Zr is given by poles as for a resonator, then these poles can be pulled away from the real axis with no change in the basic physics so long as the early time wake field is preserved p_re­cisely. In practice one resonator ( two po les) i s prob­ably not enough. Perhaps a more general and useful form is a model impedance with two cuts. This yields asymptotic dependence with fractional powers of w. For actual calculations, however,this is not necessary since the C's can be simply calculated nLDDerically for an arbitrary impedance or wake.field.

Threshold vs. bunchlength

Figure 1 shows the solution of Eq. (12) truncated to a dimension of 4 for LEP with a bunch length of 2 cm. Notice that it is modes 'O' and '-1' which first become degenerate and unstable. For bunch lengths of this order including higher modes does not signifi­cantly change the threshold. Notice also that the so­called radial modes at 0 and ±1 are automatically included. As one truncates at higher and higher dimen­sion, the radial modes appear, as they become impor­tant, automatically. Notice that the next two modes which will become degenerate are mode 1 and a radi a 1 mode which begins at 0.

As the bunch length is changed, the threshold for instability also changes. Figure 2 shows the depend­ence of this threshold on bunch length if all other quantities are held constant. The threshold increases for both shorter bunches as well as for longer bunches. These are due to completely different ef­fects. · For short bunch lengths this is due to the

o ,_o~-0~_2_5~0-5_0~0--1-5~1~. o-0~1~.2-5~1~- -50~1~--1..,----,5 2 ~o Re< w-w, l/w. LEP12

3 . 3

2 . 2

I .

0 . -

-I .. ... ·I

-2 -2 .

- 3 -3

-4 l ;'mh -4 . 0 . 0 0.25 0 . 50 0 75 LOO 1-25 1-50 1-75 2.00

Fig. 1: Transverse Mode Coupling Instability

decrease in the overlap of the wake field with the bunch factor in c0 • More precisely if the imped~nce (wake fiel~) varies at large (small) frequency (t1me) as w-a (t«- ), then

I - 1-« 0 threshold a ' a + •

(24)

For somewhat longer bunches the increase is due to the decrease in the peak current, which yields

1threshold "' a ' a ?. 2 cm · (2S)

Fin ally, for yet longer bunches, other modes become important because the wake function begins to have zero's within the bunch length.

LEP •

• •

0 6 e

BUNCHLENGTH Ccml

Fig. 2: Threshold vs. Bunchlength [Pave = 64 m].

Reactive Feedback

The results of calculations for 'normal' and 'abnormal ' feedback are shown in Figs. 3 and 4 respec­tively. For normal feedback at fixed total gain the frequency of mode zero is increased by vs at smal 1 current. Thi s separates modes '0' and '-1 ' and yields an increase in t he thresho 1 d of about a factor of 2. For fixed electronic the modes move somewhat dif­ferently because the tune shift due to feedback is pro­portional to the current. However, a tune shift due to

-391-

4 . 0 ... 0....,...._o~. _2s~o~·-s .... o~o ._1~5~1 ~. 0_0~1~. 2_5~1~.-50~1 ..... _1.,.5-.2 . ~o LEP

3 . 3.

2 . .. 2

I.

0 .

- I ..... ~ . .. . ....... =~ " ":: .. ·::· - 1.

-2 . - -2 .

- 3 . -3

-4 • .__..._...__...__.__._....._~_.__.__.__.__.___.__r~/mA~__. -4 . 0.0 0.25 0.50 0.75 1.00 1.25 I. S O 1.75 2.00

Fig. 3 'Normal' Reactive Feedback: AvFB =vs.*

feedback of vs/2 at a current of 0.4 ma keeps the modes separated until 1.0 ma. Notice the key ro le played by the radial mode '0'. The shifted mode '0' i nteracts with i ts radi al mode by repulsion rather than attraction. It is ultimately the radial mode which becomes degenerate; however , during t he repu l sion they seem to exchange roles.

0.0 0.25 0 . 50 0.75 1 00 1.25 1.50 1.75 2 ~~ 4. ~~~~-...-~-r--.--r--.--.---.--.--.--..--..-,

Relc.1-.... l/c.10

LEP

3. 3.

2 - 2

I.

o. - - . ··- -·· ... 0 .

·I • I

-2 . -2 .

- 3 . -3 .

I / mil _4

. -4 .

0.0 0.25 0.50 0.75 1.00 t.25 1.50 1 . 7 5 2.00

Fig. 4: Abnormal Reactive Feedback: AvFB = -1.3 vs.*

For abnormal feedback in Fig. 4 the situation is different. In this case the frequency of mode '0' i s decreased by ( 1.J)vs for sma 11 currents. As the cur­rent increases the modes separate because the impedance causes a defocusing force which decreases mode '0' fur ­ther. The shifted mode '0' interacts with mode -2 in a repulsive way and finally becomes degenerate with mode ' -1 ' at a current of 2.0 ma. Thus, the threshold is increased by a factor of about 4 in this case.

This method should not be used at fixed electr on ic gain. Instead, the feedback shou ld be turned on at a finite current between bunches to allow modes 'O' and ' -1' to cross without degeneracy. Then the total gain shou ld be fixed by decreasing the electronic gain in­versely with current. The gain required is that for Avfll = vs at O. 5 ma. However, if the two methods are used sequentially, first normal, then abnormal, the gain required is again AvFe = vs /2 at 0.5 ma.

* Calculated with 2 cm bunch length.

Resistive Feedback

Reactive feedback has been shown to be quite effective in enhancing thresholds, however, given the gain necessary for reactive feedback, would a resistive feedback system perform as well or perhaps even better? In Fig. 5 resistive feedback is shown with a gain sufficient to produce AvFB = vs/2 in the re­active method. As you see in Fig. 5 mode '0' is damped at small current; however, mode 'l' becomes unstable (positive imaginary 'part) at essentially the same cur­rent as the threshold in the absence of feedback. In­creasing the gain by a factor of 2 does not substan­tially change the picture. Thus, resistive feedback seems completely i neffective as a cure for the trans­verse mode coupling instability.

o.o o.2s o . 5.0 o . 75 1.00 1.2s 1.:;o 1.75 2 . no_ 4 . r-~-r--...-..,--r--,---.--,--.--,--.--,--.----,.--..-., ~

LEP

3 . 3

2 2.

1. - I.

o. -······--- · ..... ;;::::::;: :::::::::::::::::: ... .. ..... 1 ...... 4..•-=-··1,.m.. .. J .. 1.u ~ . ... .. .. , · 0.

... ....... .. 4

- 1. - ·-... -1.

-2 . -2.

-3 . -3.

-4 l / mA -4. 0. 0 0 25 0 . 50 0. 75 I. 00 I. 25 I. 50 I. 75 2. 00

Fig. 5: Resistive Feedback: Growth Rate vs . Current.* Conclusions

The purpose of t his report has been to study cures of the transverse coherent mode coupling instability. Since it is real frequency shifts which cause the prob­l em, t hey also form the solution . The conclusions for shor t bunches can therefore be sunmari zed: 1) Reactive feedback can yield increases of up to a fact or of 4 in the t hreshold for the transverse coherent mode coupling instability. This conclusion is independent of the storage ring. The primary restriction is that the bunch be short enough so that it is modes '0' and ' -1' which interact. The result s are similar for larger bunches, independent of the long time wakefield, and insensitive to small changes in the short time wakefield. 2) These results agree qualitatively with those obtained in a two particle model calculation 2 •

3) Resistive feedback fs completely ineffective as a cure for this instability; however, a small amount fs of course useful as a cure for the head-tai 1 effect. Acknowledgements

I would like to thank John Jowett and Monica Gygi for help with figures and computing; Albert Hofman, Daniel Brandt, Steve Meyers, Hatt Sands and Bruno Zotter for many useful discussions, and finally Eberhard Kei 1 and the staff at LEP for their hospitality and support this past year.

1. 2. 3.

4. 5. 6. 7.

References S. Meyers, LEP Note 436, 1983. R.D. Ruth , CERN-LEP-TH/83-22, 1983. R.O. Ruth and J.M. Wang,IEEE Trans. Nucl. Sci. NS-28 (1981) 2405. lr.lJ.'"lfuth, Brookhaven Report, BNL 51425, lg81. T. Weiland, OESY report, DESY 82-015 {1982). D. Brandt, CERN LEP Note 444 (1983). See Ref . 5 and 0. Brandt and B. Zotter, t his conf.

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