ps booster resonance compensation...

PS Booster resonance compensation measurements

Meghan McAteer 29 April 2014

Outline

Review of some basics

Resonant behavior in synchrotrons

Multipole expansion of magnetic fields

Hamiltonian formalism for nonlinear

optics

Results of first measurements in the

PSB

28/04/2014 Meghan McAteer 2

Part I

Review of some basics


Resonant behavior in synchrotrons

In general, resonant

behavior can occur

whenever

order of resonance:

nth order multipole

magnet perturbation can

excite resonances up to

nth order


integerQnQn y2x1

21 nnn

3Qy = 13

3Q

x =

13

2Q

x =

9

2Qy = 9

Qy = 4

Qx =

4

Complex potential of magnetic

multipoles Multipole magnets can be described by complex magnetic potential:

Hamiltonian is proportional real part of ψ

Example: normal sextupole (n=3, A3=0)

Vector equipotentials=field lines, scalar equipotential=pole face contour


32

3

3

33

23

3

3

33s

3yy3xB3

1

iyxiAB3

1Imy)V(x,

xyxB3

1

iyxiAB3

1Rey)(x,A

3

y)iV(x,y)(x,A

iyxiABn

1Ψ

s

n

n

nn

Part II

Hamiltonian formalism for

resonance measurement


The general approach

Step 1: define a method of mapping nonlinear particle motion

in an accelerator (Taylor maps)

Step 2: define the relationship between map and machine

observables (Fourier spectra of transverse beam trajectories)

Step 3: identify resonance driving terms

Trajectory through a multipole can be mapped using the

Hamiltonian of the multipole magnet:


Taylor maps for

nonlinear magnet elements Maps for nonlinear lattice elements can’t be written in terms of transfer

matrices; need new approach (exponential Lie operators)

Definition of exponential Lie operator acting on a function g:

Particle’s coordinates after passage through a multipole can

be mapped using the Hamiltonian of the multipole magnet:


x

g

p

f

p

g

x

fgf,

...g]][f,[f,gf,gge :f:

2

1

:f:e

0:H:

f XeX ],[ Hg

d

d

t

g

Recall from classical

mechanics: time evolution

of a function g(x,px) from

Poisson bracket with

Hamiltonian

...!!

211

21f ff

e

Recall Taylor series

expansion of

exponential function:

(Poisson

bracket)

X

Example: Taylor map for a thin-lens

normal sextupole kick (1 of 2)

Hamiltonian for an nth order multipole kick is proportional to the

vector potential for the multipole:

So for a thin-lens normal sextupole (n=3, A3=0), the Hamiltonian is

And the map relating trajectory before and after the sextupole kick is


n

nn

0

iyxiABn

1Re

p

qLh

23

0

3 xyx3p

qLBh 3

0y

x:xyx

3p

qLB-:

fy

x

p

y

p

x

e

p

y

p

x23

0

3

3

L is magnet length,

q is particle charge,

p0 is momentum

Example: Taylor map for a thin-lens

normal sextupole kick (2 of 2)

Hamiltonian for normal sextupole lens:

The derivatives of h (for the Poisson bracket ) are

and so the new coordinates after the sextupole kick are


00

0

3y0

0

2

0

2

0

0

3x0

0

y0

y

0

x0

x

0

fy

x

y2xp

qLBp

y

yxp

qLBp

x

y

hp

p

hy

x

hp

p

hx

p

y

p

x

22

0

3 yxp

qLB

x

h3

0

p

h

p

h

yx

6xy

p

qLB

y

h

0

3

1

yx p

X

y

X

p

X

x

X

],[ Xh

23

0

3 xyx3p

qLBh 3

The Hamiltonian for

many multipole kicks (1 of 2)

To first order, the Hamiltonian for all multipole elements in the ring can be

expressed as a sum over i individual multipole elements:

Insert expression for x and y (solutions to unperturbed equations of motion):


n

n

iiinin

0

i

i

i )iy(s)x(s)(siA)(sBn

1Re

p

qLhh

2

ee)(sβ2J)φ)(sCos(φ)(sβ2J)y(s

2

ee)(sβ2J)φ)(sCos(φ)(sβ2J)x(s

)φ)(si(φ)φ)(si(φ

iyyy0iyiyyi

)φ)(si(φ)φ)(si(φ

ixxx0ixixxi

y0iyy0iy

x0ixx0ix

The Hamiltonian for

many multipole kicks (2 of 2) Recall: Multinomial expansion of polynomials

Using multipole expansion, arrive at a general expression for the perturbative Hamiltonian representing i multipole kicks:

(Vni = Ani (skew coefficient) if l+m is odd;

Vni = Bni (normal coefficient) if l+m is even)


mlkj

nmlkj

ndcba

m!l!k!j!

n!dcba

yx φmlφkji2

ml

x

jklm

2

kj

xjklm e2J2Jhh

yixi φmlφkji

ni2

ml

yi

i

2

kj

xiin

0

jklm eVββLm!l!k!j!

n

n2p

qh

Relation between Hamiltonian driving terms

hjklm and observable spectrum

Frequencies excited by multipole perturbations are visible in the Fourier spectrum

of the beam trajectory

The Hamiltonian term hjklm excites the resonance (j-k)Qx+(l-m)Qy=integer

the horizontal spectrum line (1-j+k)Qx+(m-l)Qy (if l ≠ 0)

the vertical spectrum line (k-j)Qx+(1-l+m)Qy (if l≠0)

Example: tracking simulation with normal sextupole errors; Qx=4.238, Qy=4.389


Qy 2Qy Qx+Qy

Qx-Qy

Qx

Summary of resonances and spectral lines

excited by driving terms hjklm(up to n=3)


Normal Quadrupole

Normal Sextupole

Skew Quadrupole

Skew Sextupole

Terms with l+m=even

correspond to normal

multipoles, l+m=odd to

skew multipoles

A single line in the

spectrum can be

excited by several

Hamiltonian driving

terms

Theory predicts

amplitudes and phase

of spectral line from

each driving term

Amplitude and phase of Hamiltonian

driving terms hjklm


Amplitude and phase of a resonance driving term are identified via

comparison w/ amplitude and phase of spectral lines

Once driving terms are known, can find settings for a pair of corrector

magnets that will compensate for each driving term

Part III

Spectrum measurements


Turn-by-turn trajectory measurements

Trial of trajectory measurements was done

with three BPMs

Tune kicker and transverse damper used to

cause transverse oscillations

Oscillation amplitude from tune kicker or

damper was smaller than desired (~1 mm

peak-to-peak)

Taking advantage of transverse instability

gives better horizontal oscillation amplitude


Spectra of

measured

trajectories Orange - measured x spectrum

blue - measuredy spectrum

red - tracking w/ normal sext errors

Peaks visible near (but not exactly on)

resonance frequencies

Possible appearance of skew octupole

term h0121:

H line (2,-1), V line (1,0)

Amplitude of peaks is small relative to

noise floor; phase and amplitude

inconsistent on repeated pulses

Spectrum also shows noise peaks,

always visible at ~263 and 297 KHz


Qx-2Qy

Qy

3Qx

2Qx

2Qx

Qx

2Qx-Qy Qx

noise

noise

Spectra of measured trajectories


Summary

Hamiltonian driving terms describing nonlinear imperfections can be

determined from measured beam trajectory spectra

Driving terms can then be compensated with corrector magnets

Trial measurements were made in PSB before LS1, but spectral

analysis was complicated by several factors: low oscillation amplitude/signal-to-noise ratio

large tune ripple

“noise” peaks which are always present at ~263 and 297 KHz

Nonetheless, first measurements show some hints of higher-order

frequency components

After LS1, measurements will be repeated with abovementioned

problems (hopefully) resolved, and we’ll try to compensate whatever

resonance driving terms we observe


Thank you for your attention.


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