InstabilitiesAndFeedback_Lebedev_010329

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					Control of Transverse Multibunch Instabilities in the
              First Stage of the VLHC
                                   Valeri Lebedev
                                      Fermilab

                                                             VLHC mini-workshop
                                                             SLAC, March 21-23, 2001.

Content
  1. Machine Parameters
  2. Multibunch Transverse Instability
  3. Feedback System for Suppression of the Transverse Multibunch Instability
  4. Emittance Growth Suppression
  5. Conclusions
                                   VLHC Parameters
                                                          Stage 1           Stage 2
Beam energy                                 Ep       20 TeV            87.5 TeV
Luminosity                                  L        1034 cm-2s-1      21034 cm-2s-1
Magnetic field                              B0       1.96 T            10 T
Injection energy                            Einj     0.9 TeV           10 TeV
Bunch spacing                               b                    18.9 ns
Circumference                               C=2R                 232 km
Revolution frequency                        f0                   1294 Hz
Number of bunches                           Nb                     36943
RF frequency                                fRF                 477.9 MHz
Betatron tunes                                                  ~214
Momentum compaction                                             2.110-5
Beta-function at IP                            30 cm                50 cm
Head-on beam-beam tune shift per IP             1.810-2             1.810-2
Number of particles per bunch               N                    2.51010
Beam current                                Ib    0.190 A              0.175 A
RF voltage per turn, top/injection energy   V0    50/50 MeV            50/50 MeV
Synchrotron frequency, top/injection energy fs    2.32/10.9 Hz         1.1/3.28 Hz
Rms momentum spread, top/injection energy p     1.5/14.910-4        0.5/2.410-4
Rms bunch length, top/injection energy      s   6.6/14.2 cm          4.5/7.8 cm


                                                                                        2
2. Transverse Impedance for Vacuum Chamber with Thin Wall
Impedance calculation for round vacuum chamber
                     1 i                        c
                k           ,                              ,                                 y
                                             2                                                        d
                        4
                Z0         377  .
                         c
                                                                                         b             a
Solutions for the vector potential in different regions
                                  2 I 0 x0                                                                   x
                                                 C1 r             
                                        cr
                                                                                                   I
                                                                                                         II
                  AI                                                                                    III
                           C 2 e k ( r  a )  C 3 e  k ( r  a )  it
Az r , , t    AII                                            e cos 
                  
                  AIII  
                   
                                                                    
                                              C4                               I0 – the beam current
                                              r                               x0 – amplitude of transverse
                                                                                    beam motion
                                                                   
                                                   dA               dA       dA   dA
Matching solutions at boundaries:                                        ;             yields four linear
                                                   d r d r  0 dr r dr r  0
equations for coefficients Ci . The solution is:


                                                                                                                    3
                  2I x             e ikd bk  1  e ikd bk  1          
              C1  0 2 0      2 ikd
                             e ak  1bk  1  e ikd ak  1bk  1  1 ,
                   ca                                                        
                   4 I 0 x0                e ikd bk  1
              C2                                                         ,
                     ca e ak  1bk  1  e ak  1bk  1
                              ikd                      ikd


                   4 I 0 x0               e ikd bk  1
              C3                                                        ,
                     ca e ikd ak  1bk  1  e ikd ak  1bk  1
                   8I 0 x0              b2k
              C4                                              .
                    ca e ak  1bk  1  e ak  1bk  1
                           ikd                ikd




Taking into account contribution from electric field of the beam we obtain an expression
for the vacuum chamber transverse impedance (case of positive frequency):


         C1    2        Z0           eikd 1  kb   eikd 1  kb          1 
Z   i      2   i         2 ikd                                            1   (1)
         I x ca  
         0 0           2a 2  e 1  ka 1  kb   eikd 1  ka 1  kb   
                               




                                                                                           4
There are following asymptotes for the transverse impedance (  > 0):
             c1  i 
          2a 3 2  ,   d
                       R

               c2
Z  Z0  2                i...     ,    ad    d
          4  Ra d
                      3


          Rd           1
          2 i                    ,   ad
           c a       2a 2 
 There is no simple expression for imaginary part of the impedance in the case
    ad    d and Eq. (1) has to be used
 To simplify the solution the expression for the vector potential inside conductor was
  chosen for the flat geometry (instead of Bessel functions of imaginary argument) and
  therefore the result is not quite accurate if the wall is sufficiently thick.
  In particular the equation yields incorrect value for imaginary part of the impedance
    at very small frequencies,   ad
     The following correction term is applied to fix the problem
                                              Z            d
                                    Z   i 0 2
                                             2a (a  d )1  k 4 d 2 a 2 
     Note that although this term corrects asymptotic behavior of the impedance at low
       frequencies it does not change the impedance at frequencies of interest of VLHC.

                                                                                       5
                               Z0              c ( 1  i)                                                                   Z0              c ( 1  i)
ZtrAH                                                                                   ZtrAH                            
                               2         3                                                                                 2         3
                                          a  2  R                                                                                a  2  R 

                    100                                                                                           100

                                                        f0                                                                                            f0
                                                          f0                                                                                            f0
                     10                                 2                                                          10                                 2
  Ztr [Ohm/cm/cm]




                                                                                                Ztr [Ohm/cm/cm]
                      1                                                                                             1



                     0.1                                                                                           0.1



                    0.01                                                                                          0.01
                                                100 1 10 1 10 1 10 1 10 1 10 1 10                                                       100 1 10 1 10 1 10 1 10 1 10 1 10
                                                               3   4   5   6    7         8                                                                  3   4   5   6    7         8
                           1        10                                                                                   1        10
                                                          f [Hz]                                                                                        f [Hz]

                   d = 0.1 mm                                      d = 1 mm
  Real (red) and imaginary (brown) parts of the transverse impedance and asymptotes for
  the real part of the impedance as function of frequency for round vacuum chamber: Al,
  300 K, = 2.74 cm, a = 9 mm.
      Skin-layer thickness is 3.3 mm at f0/2=646 Hz.
        Peak is achieved when  2  ad and its value does not depend on vacuum
          chamber thickness




                                                                                                                                                                                            6
Transverse Impedance of Elliptic Vacuum Chamber
   Impedance of flat vacuum chamber is about half of the round vacuum chamber
    impedance. For an estimate we will introduce the
    effective radius of the vacuum chamber:
                                                                                                a
                       2a b 
                           3   3   1/ 3                                                                    b
              aeff    3
                       a  b3 
                               
                              
     so that we would get correct result                                   3
     for cases a  b , a  b and a  b .
                                                                                                 Al
   For impedance estimates we will




                                               resistivity[microohm -cm]
    substitute aeff instead of a into Eq.(1)                               2
                                                                                                      Cu                              0.6
Vacuum Chamber Conductivity
   Vacuum chamber cooling




                                                                                                                     Skin-depth[cm]
                                                                           1                          Ag                              0.4
    significantly reduces transverse
    impedances
                                                                           0                                                          0.2
                                                                               0   100          200        300
                                                                                         T[K]

                                                                                                                                        0
                                                                                                                                            0




                                                                                                                 7
Detuning Wake
   For round vacuum chamber the beam excites                                                                s
    dipole field behind him. That corresponds to                        x                      x
                                                                                               0
    normal wake field W(s).
   For non-round vacuum chamber there are
    quadrupole field and higher multipoles.               *A.Burov, V.Danilov “Suppression of transverse
                                                          instabilities by asymmetries in the chamber geometry,”
    Contribution corresponding to quadrupole              PRL, 82, 1999, p.2286.
    field is described by detuning wake D(s):
            Fx ds   e 2  x0Wx ( s )  xD( s) 
           L


              Fy ds   e 2  y 0W y ( s )  yD( s ) 
           L
 Detuning wake changes betatron tunes for tail particles and can cause non-linear
  resonances.
   For round vacuum chamber
                                           D(s) = 0 .
   For two parallel plates
                                                              R c
                          W x s   W x s    D x s   3
                                                             a  Rs




                                                                                                             8
                                      1

                                                               f0
                                                                    f0
                                                               2



                      Attenuation
                                     0.1




                                    0.01
                                                                1 10          1 10
                                                                    3              4
                                           10   100
                                                      f [Hz]

Attenuation of the beam magnetic field by the elliptic aluminum vacuum chamber at 300 K,
a = 9 mm, b = 14 mm; red line - d=1 mm, blue line - d=2 mm
 There is only about 30% attenuation for thickness of 1 mm at the first unstable betatron
  sideband
   That implies that the beam sees everything around vacuum chamber and can acquire additional
    contribution to the impedance from the surroundings if appropriate measures are not taken.
 2 mm vacuum chamber attenuates the beam field by more than 2 times
   That is better but still requires careful attitude to what can be located in near vicinity of
    vacuum chamber

                                                                                                    9
Multibunch Transverse Instability due to Finite Wall Resistivity
Estimate of tune shift and instability increment due to vacuum chamber impedance
                                                                     
                            i                                       i
                           2 
                W ( s)         dZ  ( )e is / v
                                                      , Z  ( )    dsW ( s)e is / v
                                                                   v0
                                                                           i   q  n  t  n 
Let the beam position for mode n of continuous beam be as x(t , )  x0 e                         o


Then, the force acting on the reference particle is
                            eI b s
                                   W s x 0 e i q  n  s / v  ds  ieI b Z   q  n  0 x 0
                             v 
                  F0 (t )                                0


                                 0

                                                    eI b R 2
                               n  i                          Z   q  n  0 
                                                   2 Pv
The exact result for the betatron tune shift of n-th mode for the bunched beam

                                         eI b R 2 
                              n  i             Z    nk  ,
                                         2 Pv k                                               (2)
                              nk   0 [q]  n  Nk 
                 [q] – the fractional part of the betatron tune 
                 N – the number of bunches, n   N / 2, ( N  1) / 2
                 Z    – the transverse impedance per unit length averaged over the ring


                                                                                                          10
The Dimensionless Increment


             n                   eI b R 2                 
                  2 Im             Re  Z    nk        (3)
              f0                   Pv        k           
 The definition of the transverse impedance yields that impedance at a negative
  frequency is related to the impedance at positive frequency through its complex
  conjugated value
                                 Z  ( )   Z  ( )
                                                  *


               or
                                ReZ  ( )    ReZ  ( ) 
                                ImZ  ( )   ImZ  ( ) 
 At low frequencies impedance is completely determined by the wall resistivity.
   For > 0 real part of the impedance is positive  coherent motion is damped
      n   0 [q]  n , n  0...( N  1) / 2
   For < 0 real part of the impedance is negative  coherent motion is unstable
      n   0 n  [q] , n  1...N / 2
   If the impedance grows with frequency decrease preferable fractional tune is in the
  range [0 – 0.5]


                                                                                      11
Asymptotics for the dimensionless increment
                                                   c
                                            2a 3 2        ,  d
                                                        R   n

           n      eI b R 2               
                                                  c2
                           Z 0 sign n  2                  ,  ad    d
            f0      Pv                     4  R  n a d
                                                           3


                                             d
                                            R 2n       ,   ad
                                            c a
                                           




                                                                                12
                                                                              Einj                                                                          Einj  
 rw1  Re  c i  aeff ( .9 1.4)  .1 R                                            rw2  Re  c i  aeff ( .9 1.4)  0.2 R                     
                            i
                                                                              mp                  i
                                                                                                                                                                mp   
                                10                                                                                                                   1

                                                            f0                                                                                                               f0
                                                              f0                                                                                                              f0
  Dimensionless increment



                                    1                       2                                                                                    0.1                         2




                                                                                                                            Tune shift
                                0.1                                                                                                          0.01



                                                                                                                                         1 10
                                                                                                                                                 3
                            0.01



  1 10
                                3
                                                                                                                                         1 10
                                                                                                                                                 4

                                                           1 10       1 10     1 10     1 10   1 10      1 10                                                          1 10     1 10 1 10   1 10   1 10       1 10
                                                                   3        4          5       6          7         8                                                              3       4     5       6           7       8
                                        1       10   100                                                                                                 1   10      100
                                                                       f [Hz]                                                                                                          f [Hz]

Dependence of the dimensionless decrement (left) and the betatron tune shift (right) on the mode
frequency for elliptic aluminum vacuum chamber at 300 K, a = 9 mm, b = 14 mm;
red line - d=1 mm, blue line - d=2 mm; brown dashed line - a = 12 mm, b = 20 mm, d=2 mm.

 9 mm vacuum chamber yields instability growth time about one revolution
   Increase of wall thickness from 1 mm to 2 mm decreases
      the growth time by about 1.5 times
      the coherent tune shift by about 2 times to ~ 0.1
 To decrease decrement to about 2 revolutions one needs to increase vacuum chamber
  size to about 12 by 20 mm
   That also decreases the coherent tune shift to ~ 0.03 (30 deg per turn)


                                                                                                                                                                                                                             13
Coherent and incoherent tune shifts
 The interaction of the beam with round vacuum chamber
   causes instabilities and changes frequencies of coherent beam motion
   but does not change incoherent particle frequencies and cannot cause non-linear resonance
 In the case of elliptic vacuum chamber detuning wake is not zero. That means that the
  interaction of the beam with vacuum chamber creates additional focusing fields which
  changes incoherent particle tunes
   In the case of flat vacuum chamber and uniformly filled ring the frequency of all particles is
                                              eI b R 2
    shifted by the same amount:  ic 0  
                                            mc 3 3 a 2
    That can be compensated by betatron frequency adjustments proportional to the
    beam current
   If the ring is partially filled then there is tune variations proportional to “AC” components in
    the beam current
     For flat vacuum chamber the wake and the detuning wake are equal and therefore tune
        variations can be computed using results obtained for coherent betatron tune shift
                                             eI b R 2             
                     ic        Re       n
                                                     Im  Z   n  ,  n  n 0
                            n
                                               2 Pv    k        
     To compensate tune variations along the beam the modulation of lattice betatron tune on
      revolution frequency is required



                                                                                                       14
 Interaction of the beam DC current with iron of the dipoles causes additional
  incoherent tune shift
                                                          eI b R 2
                                           icB  
                                                      mc 3 2 a d 
                                                                   2


                                        ad – is the dipole half-gap
  That tune shift is close to the incoherent tune shift due to vacuum chamber  ic 0
   If the ring is partially filled then there is tune variations proportional to “AC” components in
     the beam current. Their amplitude will depend on the thickness of vacuum chamber and dipole
     laminations.




                                                                                                       15
4. Feedback System for Suppression of the Transverse
Multibunch Instability
Standard system                                                                     0        0
                T                                                                 80) 90
Vn  V0 exp i n                                                      0   (90+ n1
                N                                                    90
          cos( 0 ) sin(  0 )  1  k ( )     0 
Vn  N                          0         1  k ( )
                                                          Vn
           sin(  0 ) cos( 0 )                     
          
               T         iT 
k ( )   g  j  exp             j
         j 0  N         N 
        N – number of bunches
        T – revolution period
        g(t) – response function
Perturbation theory solution
  exp i T   1  k ( )  exp i
   n                 n
                          ~
                                            n                  0   
         nT   0  2n
        ~

Damping decrement: n   ln  n  T




                                                                                                 16
Integrating type narrow band system
                                                                                                                                                     t
                        10

                                                                                                                                   g (t )  g 0 exp   
                                                                                                                                                     
                                                          Numerical solution
                          1                                                                                                                             g0
                                                                  Perturbation theory                                              k ( ) 
  Decr./f0




                                                                                                                                                      T 1    
                                                                                                                                            1  exp     i  
                        0.1
                                    Resistive w all instability
                                                                                                                                                      N     

                                                                                                                                   g0=0.1, /T=9.48510-5, 0=20.46,
                       0.01
                              0.1   1               10     100
                                                           f/f0
                                                                   1 10
                                                                       3
                                                                                  1 10
                                                                                      4
                                                                                              1 10
                                                                                                      5
                                                                                                                                   N=105
                                                                                          1
                         1


                                                                                                                                                    Perturbation theory
                                                                                                                                0.015
 Eigen-number module




                                                                                                          Eigen-number module




                       0.5                                                                                                       0.01




                                                                                                                                0.005

                                                                                                                                                                  Numerical solution

                         0                                                                                                          0
                         1 10          5 10                         5 10               1 10
                               5                4                             4                   5                                     200   100           0             100          200
                                                          0
                                                         f/f0                                                                                              f/f0




                                                                                                                                                                                             17
 There is no principal limits to reach damping faster than the revolution time
 Narrow band system can be stable in the entire frequency range
 Technical problems
   Signal transfer from BPMs to kickers                   Practical system for the case of
     Forward transfer                               aluminium vacuum chamber at 78 K
       Expensive tunnel for signal transfer                   1

     One turn delay
       Large betatron phase shift after one                 0.1
                                                                                           Feedback system

         turn
       Large delay  digital system




                                                                Decr./f0
                                                            0.01

     Practical system needs subtraction of
                                                                       Resistive w all instability
      orbit offset                                     1 10
                                                                           3



       Slow feedback to nullify BPMs
       Or notch filter at revolution frequency        1 10
                                                                 0.1 1
                                                                           4

                                                                          10       100      1 10
                                                                                                3
                                                                                                   1 10
                                                                                                      4
                                                                                                          1 10
                                                                                                            5
                                                                                   f/f0
         which will reduce maximum
         achievable gain                           g0=0.005, /T=510-4, 0=20.46, N=105
   Limits of the system gain
     Final accuracy of electronics
     Emittance growth due to BPM noise
   Kicker voltage
     5 m, 10 kV kicker suppresses 2.5 mm injection oscillations at 1 turn


                                                                                                          18
System with delayed response
Single system with delayed response
                 T 
Vn  V0 exp  i n 
                 N 
          cos(  0 ) sin(  0 )                 0 0 
Vn N                              Vn  k ( )        Vn N
            sin(  0 ) cos(  0 )
                                                  0 1
                                                       
           
                T         iT
k ( )   g  j  exp               j  j d 
                                                 
          j 0  N         N                   
      N – number of bunches
      T – revolution period
      g(t) – response function
      jd – number of skipped bunches
Perturbation theory solution

                   1  k ( n ) 2                   1  k ( n ) 2 
                                                                     2
                                                                                          1
 n  exp iT                  cos 0        
                                                    1  k ( )     cos 2  0  
                    1  k ( n )                               n                   1  k ( n )
       n T   0  2n



                                                                                                    19
                                          t                                    g0
Integrating type system g (t )  g 0 exp    ,           k ( ) 
                                                                            T 1     
                                                                      1  exp     i  
                                                                               N      
    0.5                                                                                                  1
                                                     1
                               Re(k())
                                                                                               ||


      0
                                                   0.9

                                 Im(k( ))


    0.5                                            0.8
                                                                                             1 10       1 10
          100   50       0           50      100                                                     3        4
                                                         0.1      1        10          100
                        f/f0                                                    f/f0


g0=210-4, /T=.025, 0=20.27, N=105, jd =50 – 50 skipped bunches (105/(4500))

Optimization strategy
   Suppress resistive wall instability at low frequencies
   Achieve minimum increments at higher frequencies



                                                                                                             20
Double system with delayed response                                      0
     cos(  0 ) sin(  0 ) 
                                                                       90
M                          , M  M 1M 2                 ,
     sin(  0 ) cos(  0 )
l  1,2                                                               k2()
               
                    T         i T
   k l ( )   g l  j  exp        j  j d 
                                                                                  M2
                      N         N                          M1
               j 0
                                                               1                  2
           1       0      
   kl                      ,
            0 1  k l ( )
                           
                             1
MV n  k 1  M 2 k 2 M 2 k 1 Vn  N                                    k1()
    N – number of bunches
    T – revolution period                                                    0
                                                                            90
    g(t) – response function
    jd – number of skipped bunches

 n 1  k1 ( n )  k 2 ( n )  k1 ( n )k 2 ( n ) 
   2


                  k ( )  k 2 ( n )                      k ( )k ( ) 
   2 n cos 0 1  1 n                cos1  cos 2  1 n 2 n   1  0
                         2                                      2       



                                                                                        21
Comparison of single (0, g0) and double systems (1=2=0/2, g1=g2=g0)

     1                                               1

                                   =0.2*2                                         =0.4*2

             Single system                                         Single system


   0.5                                             0.5
                                           1                                                 1
                                           e                                                 e


                      Double s ystem      1                                                 1
                                                             Double s ystem
                                           2                                                 2
                                          e                                                 e


     0                                               0
         0     2          4        6           8         0            2         4    6           8
                        gain                                                  gain




Dependence of absolute values of eigen-numbers on system gain, g, for different betatron
tunes.




                                                                                                     22
                 1                                                                             1

                             Single system
                                        Double system                                                    Resistive w all
             0.1                                                                           0.1
                                                                                                                      Single system
  decr/f0




                                                                                decr/f0
                                                                                                                             Double system
                                                  Resistive w all
            0.01                                                                          0.01




   1 10                                                                         1 10
             3                                                                             3

                                                        1 10       1 10                                                            1 10       1 10
                                                                3           4                                                                3           4
                   0.1   1         10            100                                               0.1   1       10           100
                                          f/f0                                                                        f/f0


               Damping decrement                    Residual instability increment
                 N=10 , jd =50; Single system: g0=210-4, /T=2.510-2, 0=20.27,
                       5

                                  Double system: g0=810-8, /T=210-2, 0=20.46.
Conclusions for a system with delayed response for VLHC
   System allows to reduce a requirement for the damping decrement of the bunch-
    bunch dumping system by an order of magnitude
   Installation of multiple systems does not allow to increase the total damping due to
    anti-damping at high frequency.
   Decreasing the number of skipped bunches one can improve performance but it
    increases noise effects due to smaller beta-functions

                                                                                                                                                             23
5. Emittance Growth Suppression

Emittance growth due to noises
                   N /2                         
  d
                                                                                   ,  n   0     n  ,
                               3.3
        f0                              67   2 S noise  n 
            2 2
                         
                     N  2n / f 0   
                                     2    2      
  dt            n   1                         
                         2
                                                             

Spectral density of transverse kicks,                      S  d , for the feedback system
                                                       2
                                                                  noise   n
                                                             
                                                                               2
       is determined by spectral density of BPM noise x  , system gain and beta-function
       at BPM, 1 :
                                     2
                                   x 2
               2 S noise  n       k  n  .
                                     1
              1 – beta-function at BPM
              2 – beta-function at kicker





  Emittance growth due to noise and its suppression with the feedback system in large hadron colliders, V. Lebedev, et.al.
V.V.Parkhomchuk, V.D.Shiltsev, G.V.Stupakov; SSCL-Preprint-188, Dallas, March 1993 and Particle Accelerators, 1994, v.44,
pp.147-164

                                                                                                                             24
Theoretical limit for the standard narrow
band system                                     2 10
                                                        4


   Effective BPM noise of the feedback        d
                                              dt d
    system is determined by the thermal
    noise of amplifiers is                      1 10
                                                        4


                            kT
           x  2a 2
              2

                        I b Z BPM
                            2      2

                                                            0
    Factor of 2 appeared because the                            0   5000   1 10
                                                                               4
                                                                                   f/f0
                                                                                          1.5 10
                                                                                                4
                                                                                                    2 10
                                                                                                            4
                                                                                                                2.5 10
                                                                                                                      4


    system consists of two independent
                                             Relative contribution of different frequencies
    systems shifted by 90 deg in betatron    into emittance growth
    phase
     = 50 , T=300 K, Ib=0.19 A,
    ZBPM=12 
                                           x 2 = 0.02 m (f=25 MHz)
                       2
                    x =10-20 cm2 s,
                           0
                                 1.3105 days - at the injection
                         d / dt
   Required BPM accuracy
    Emittance growth time > 100 hour         
        bunch position measured with        x 2 = 3 m (180 times of theoretical limit)



                                                                                                                          25
Conclusions
 Multi-bunch instabilities can be suppressed
 To suppress the resistive wall instability
   Increase vacuum chamber thickness to 2 mm and vertical size
    to 212 mm will be extremely profitable
   Two feedback systems required
     Low frequency system with high gain, narrow band and
       delayed response to damp low frequencies
          Unstable at high frequencies
     High frequency system to restore stability at high frequencies
        wide band, moderate gain and one turn delay
          Unstable at low frequencies




                                                                       26

				
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