Beam loading and coupled-bunch instabilities from a large ...indico.ihep.ac.cn/event/6618/session/12/contribution/96/material/... · Coupled-bunch Instabilities and Beam Loading This

Beam loading and coupled-bunch instabilities froma large ring perspective

D. Teytelman

Dimtel, Inc., San Jose, CA, USA

CEPC Workshop 2017

(Dimtel) Beam loading and instabilities November 7, 2017 1 / 24

Outline

1 IntroductionThe focus of this talk

2 Instabilities, Beam Loading, FeedbackRing Circumference and Coupled-Bunch InstabilitiesBeam Loading in Storage RingsBunch-by-bunch Feedback

3 Beam Loading in CEPC-ZFCC-Z Nominal ParametersPushing FCC-Z Current


Outline





Coupled-bunch Instabilities and Beam Loading

This talk will focus on two important effects in storage rings:I Coupled-bunch instabilities in transverse and longitudinal planes;I Transient beam loading due to non-uniform fill patterns.

Instabilities:I Beam interacts with impedances at betatron or synchrotron

sidebands of revolution harmonicsI Transverse: MωRF ± Nωrev ± ωβ

I Longitudinal: MωRF ± Nωrev ± ωs

Transient beam loading:I Driven by impedances at revolution harmonics MωRF ± Nωrev.






























Outline





Resonant Modes and Revolution Harmonics

−400 −300 −200 −100 0 100 200 300 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (kHz)

Imp

ed

an

ce

(a

rb.

un

its)

A 20 kHz resonance ideally“hidden” between two revolutionharmonics.1 km ring;1.5 km ring;3 km ring;10 km ring;100 km ring;In large rings narrow resonancescannot be “hidden” from thebeam.



−400 −300 −200 −100 0 100 200 300 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (kHz)

Imp

ed

an

ce

(a

rb.

un

its)




−400 −300 −200 −100 0 100 200 300 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (kHz)

Imp

ed

an

ce

(a

rb.

un

its)




−400 −300 −200 −100 0 100 200 300 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (kHz)

Imp

ed

an

ce

(a

rb.

un

its)




−400 −300 −200 −100 0 100 200 300 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (kHz)

Imp

ed

an

ce

(a

rb.

un

its)




−400 −300 −200 −100 0 100 200 300 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (kHz)

Imp

ed

an

ce

(a

rb.

un

its)



Transverse Planes: Resistive Wall

0 100 200 300 400 500−250

−200

−150

−100

−50

0

Mode number

Gro

wth

rate

(s−

1)

Upper sideband

Lower sideband

Resistive wall impedance scaleslinearly with circumference;Impedance vs. frequency is∝ 1/

√ω;

The most unstable line is thelower betatron sideband of thefirst revolution harmonic (mode-1);Overall impedance scaling 1

ω3/2rev

;

RW growth time (in turns) scalesas ω3/2

rev .



0 100 200 300 400 500−250

−200

−150

−100

−50

0

Mode number

Gro

wth

rate

(s−

1)

Upper sideband

Lower sideband


√ω;


ω3/2rev

;


rev .



0 100 200 300 400 500−250

−200

−150

−100

−50

0

Mode number

Gro

wth

rate

(s−

1)

Upper sideband

Lower sideband


√ω;


ω3/2rev

;


rev .



0 100 200 300 400 500−250

−200

−150

−100

−50

0

Mode number

Gro

wth

rate

(s−

1)

Upper sideband

Lower sideband


√ω;


ω3/2rev

;


rev .



0 100 200 300 400 500−250

−200

−150

−100

−50

0

Mode number

Gro

wth

rate

(s−

1)

Upper sideband

Lower sideband


√ω;


ω3/2rev

;


rev .


Outline





Beam/Cavity Interaction

loops

RF feedback

dynamics

Longitudinal

Generator

C R L

Beam

~IG

~VC

~IB

RLC model of the acceleratingcavity with two input currents:generator and beam;Cavity voltage ~VC is defined bythe sum current;Low loading (~IB ~IG) — cavityvoltage is mostly defined by thegenerator current;High loading — cavity voltage isstrongly affected by beam current;“Feedback loop” from cavityvoltage to beam current and backto cavity voltage.



loops

RF feedback

dynamics

Longitudinal

Generator

C R L

Beam

~IG

~VC

~IB




loops

RF feedback

dynamics

Longitudinal

Generator

C R L

Beam

~IG

~VC

~IB




loops

RF feedback

dynamics

Longitudinal

Generator

C R L

Beam

~IG

~VC

~IB




loops

RF feedback

dynamics

Longitudinal

Generator

C R L

Beam

~IG

~VC

~IB



Why Worry about Beam Loading

Two main effects of heavy beam loading in large rings:I Synchronous phase transients;I Longitudinal coupled-bunch instabilities driven by the RF cavity

fundamental impedanceTransient effects depend on

I Total beam loading;I Fill pattern.

Fill patterns can be designed to mitigate transient effects;But longitudinal instabilities due to the fundamental impedanceremain an issue even with completely uniform fills;Reducing beam loading in the RF system design helps bothissues.



Two main effects of heavy beam loading in large rings:I Transient modulation of longitudinal optics;I Longitudinal coupled-bunch instabilities driven by the RF cavity



































Ring Circumference and Beam Loading

Photo/image credit: CERN, SLAC

People don’t build multi-kilometerrings just to spend money;Large circumference — very highenergy;Or very high current;Or both.Large circumference meansheavy beam loading of the RFsystem.


















Cavity Detuning and Longitudinal Stability

−1.5 −1 −0.5 0 0.5 1 1.50

500

1000

1500

2000

Frequency offset from ωrf, revolution harmonics

ℜ(Z

), k

Ω

−1.5 −1 −0.5 0 0.5 1 1.5−2000

−1000

0

1000

2000

Eigenmode numberEffective d

rivin

g im

pedance, kΩ

Growth rate for mode -1 is∝ Z (ωrf − ωrev + ωs)− Z (ωrf +ωrev − ωs);Symmetric on resonance;Growth rates peak whenfundamental crossesrevolution harmonics;Need RF feedback to reducethe effective impedance.



−1.5 −1 −0.5 0 0.5 1 1.50

500

1000

1500

2000


ℜ(Z

), k

Ω

−1.5 −1 −0.5 0 0.5 1 1.5−2000

−1000

0

1000

2000


rivin

g im

pedance, kΩ




−1.5 −1 −0.5 0 0.5 1 1.50

500

1000

1500

2000


ℜ(Z

), k

Ω

−1.5 −1 −0.5 0 0.5 1 1.5−2000

−1000

0

1000

2000


rivin

g im

pedance, kΩ




−1.5 −1 −0.5 0 0.5 1 1.50

500

1000

1500

2000


ℜ(Z

), k

Ω

−1.5 −1 −0.5 0 0.5 1 1.5−2000

−1000

0

1000

2000


rivin

g im

pedance, kΩ




−1.5 −1 −0.5 0 0.5 1 1.50

500

1000

1500

2000


ℜ(Z

), k

Ω

−1.5 −1 −0.5 0 0.5 1 1.5−2000

−1000

0

1000

2000


rivin

g im

pedance, kΩ




−1.5 −1 −0.5 0 0.5 1 1.50

500

1000

1500

2000


ℜ(Z

), k

Ω

−1.5 −1 −0.5 0 0.5 1 1.5−2000

−1000

0

1000

2000


rivin

g im

pedance, kΩ



Mitigating Beam Loading in Design

Cavity detuning

ωd =∣∣∣ωrfI0

Vc

RQ cosφb

∣∣∣Minimize the number of cavities:

I Reduces fundamental impedance interacting with the beam;I Limited by the maximum coupler power and/or the maximum cavity

voltage.Minimize detuning:

I Cavities with low R/Q;I Lower RF frequencies are preferable, especially when coupler

limited.


Mitigating Beam Loading in Design

Cavity detuning

ωd =∣∣∣ωrfI0

Vc

RQ cosφb

∣∣∣Minimize the number of cavities:

I Reduces fundamental impedance interacting with the beam;I Limited by the maximum coupler power and/or the maximum cavity

voltage.Minimize detuning:

I Cavities with low R/Q;I Lower RF frequencies are preferable, especially when coupler

limited.


Outline





Bunch-by-bunch Feedback

DefinitionIn bunch-by-bunch feedback approach the actuator signal for a givenbunch depends only on the past motion of that bunch.

Controller

Beam Kicker structure

Back−endFront−end

SensorBPM Actuator

Bunches are processed sequentially.Correction kicks are applied one turn later.


Feedback Control Limits: Transverse

For single pickup/single kicker topology the maximum growth ratethat can be controlled is limited by the response delay (time frommeasuring bunch position error to correction kick acting on thesame bunch on a later turn).Rule of thumb for robust operation: λcl ≥ −λol.Fast damping in time domain corresponds to wide bandwidth inthe frequency domain→ feedback induced noise can be an issuein the vertical plane.For fractional tunes in 0.2–0.4 range the limit is around 10 turnsgrowth time (with 10 turns closed-loop damping time);Tunes close to integer or half-integer require the feedback withsignals from many past turns, slower damping.











Feedback Control Limits: Longitudinal

Measure longitudinal position (time of arrival), correct energy;To generate required 90 phase shift the feedback must observeat least half synchrotron period;Fastest growth times on the order of 1–2 synchrotron periods.








Longitudinal Example from ANKA

0

0.550100

150

0

0.2

0.4

Time (ms)

a) Osc. Envelopes in Time Domain

Bunch No.

de

g@

RF

0

0.5

0

100

0

0.05

0.1

Time (ms)

b) Evolution of Modes

Mode No.

de

g@

RF

44.5 45 45.536.19

36.2

36.21

36.22

36.23

36.24

Mode No.

Fre

qu

en

cy (

kH

z)

c) Oscillation freqs (pre−brkpt)

44 44.5 45 45.5 46

15.6099

15.6099

15.6099

15.6099

15.6099

15.6099

15.6099

Mode No.

Ra

te

(1

/ms)

d) Growth Rates (pre−brkpt)

44.5 45 45.536.25

36.3

36.35

36.4

36.45

36.5

Mode No.

Fre

qu

en

cy (

kH

z)

e) Oscillation freqs (post−brkpt)

44 44.5 45 45.5 46

−35.4292

−35.4292

−35.4292

−35.4292

Mode No.

Ra

te

(1

/ms)

f) Growth Rates (post−brkpt)

ANKA:mar0516/143812: Io= 138.0587mA, Dsamp= 2, ShifGain= 4, Nbun= 184,At Fs: G1= 119.0572, G2= 0, Ph1= −76.5927, Ph2= 0, Brkpt= 240, Calib= 34.252.

Measured while cavity tuningwalks an HOM onto asynchrotron sideband;Growth time is 2.3Ts, dampingtime is Ts;Filter is 2/3 of a synchrotronperiod.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Time (ms)

deg@

RF

mar0516/143812 Data, Fit and Error for Mode #45

Data

Fit

Error Measured while cavity tuningwalks an HOM onto asynchrotron sideband;Growth time is 2.3Ts, dampingtime is Ts;Filter is 2/3 of a synchrotronperiod.



0 10 20 30 40 50−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (turns)

Coeffic

ient

Measured while cavity tuningwalks an HOM onto asynchrotron sideband;Growth time is 2.3Ts, dampingtime is Ts;Filter is 2/3 of a synchrotronperiod.


ParametersJ. Gao, “CEPC accelerator CDR - status”,J. Zhai, “CEPC SRF system study”

Parameter ValueEnergy 45.5 GeVEnergy loss per turn 35 MeVMomentum compaction 1.14× 10−5

Energy spread 3.7× 10−4

Radiation damping time 433 msGap voltage 53 MVHarmonic number 216664Buckets filled 10725R/Q 106.5 ΩQ0 1010

Coupling factor1 26657

1Optimized for zero reflected power at 83.7 mA(Dimtel) Beam loading and instabilities November 7, 2017 19 / 24

http://indico.ihep.ac.cn/event/6618/session/0/contribution/24/material/slides/

http://indico.ihep.ac.cn/event/6618/session/12/contribution/94/material/slides/

Outline





Beam Loading Effects

0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

Time (µs)

Phase (

deg@

RF

)

Transient is 2.0854 degrees peak−to−peak

0 50 100 150 200 250 300 3500

1

2

3

4

5

6

7

8x 10

−3

Time (µs)

Bunch c

urr

ent (m

A)

CEPC−Z; 12/0 powered/parked cavities; Vgap

= 53 MV; I0 = 0.0837 A; CEPC−Z−1% fill

Nominal CEPC-Z, no parkedcavities, 1 kHz detuning;Small gap transient;Bunch length modulation is alsosmall;Growth rates are moderate (mode-1 at ≈ 3Ts), but mode 0 is tuneshifted nearly to DC, on the vergeof high-current Robinsoninstability;With moderate direct loop gain of10, all modes are stabilized.



0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

Time (µs)

Phase (

deg@

RF

)


0 50 100 150 200 250 300 3500

1

2

3

4

5

6

7

8x 10

−3

Time (µs)

Bunch c

urr

ent (m

A)


= 53 MV; I0 = 0.0837 A; CEPC−Z−1% fill




0 50 100 150 200 250 300 3503.6

3.605

3.61

3.615

3.62

3.625

3.63

3.635

3.64

Time (µs)

Bunch length

(m

m)

Peak−to−peak bunch length spead 0.72%




−10 −5 0 5 10−30

−20

−10

0

10

20

Mode number

Gro

wth

rate

(s−

1)


= 53 MV; I0 = 0.0837 A; CEPC−Z−1% fill

−10 −5 0 5 10−60

−50

−40

−30

−20

−10

0

10

Mode number

Fre

quency s

hift (H

z)

SSM (stable)

SSM (unstable)

Analytical

SSM (stable)

SSM (unstable)

Analytical




−10 −5 0 5 10−30

−20

−10

0

10

20

Mode number

Gro

wth

rate

(s−

1)


= 53 MV; I0 = 0.0837 A; CEPC−Z−1% fill

−10 −5 0 5 10−60

−50

−40

−30

−20

−10

0

10

Mode number

Fre

quency s

hift (H

z)

SSM (stable)

SSM (unstable)

Analytical

SSM (stable)

SSM (unstable)

Analytical



Outline





From 12 to 24 Cavities

0 50 100 150 200 250 300 350−6

−4

−2

0

2

4

Time (µs)

Phase (

deg@

RF

)


0 50 100 150 200 250 300 3500

0.005

0.01

0.015

Time (µs)

Bunch c

urr

ent (m

A)


= 53 MV; I0 = 0.157 A; CEPC−Z−1% fill

157 mA, 3.7 kHz detuning;Gap transient still acceptable;Fastest growth time is 6 ms, athird of synchrotron period;With optimal direct loop gain of 30instabilities are suppressed;Zoom in to see all modes atradiation damping;6.5 kHz open-loop;446 kHz closed-loop;Some small mismatches at higherfrequencies.



0 50 100 150 200 250 300 350−6

−4

−2

0

2

4

Time (µs)

Phase (

deg@

RF

)


0 50 100 150 200 250 300 3500

0.005

0.01

0.015

Time (µs)

Bunch c

urr

ent (m

A)


= 53 MV; I0 = 0.157 A; CEPC−Z−1% fill




−10 −5 0 5 10−200

−150

−100

−50

0

50

100

150

200

Mode number

Gro

wth

ra

te (

s−1)


= 53 MV; I0 = 0.157 A; CEPC−Z−1% fill

−10 −5 0 5 10−60

−50

−40

−30

−20

−10

0

10

Mode number

Fre

qu

en

cy s

hift

(Hz)

SSM (stable)

SSM (unstable)

Analytical

SSM (stable)

SSM (unstable)

Analytical




−10 −5 0 5 10−200

−150

−100

−50

0

50

100

150

200

Mode number

Gro

wth

ra

te (

s−1)


= 53 MV; I0 = 0.157 A; CEPC−Z−1% fill

−10 −5 0 5 10−60

−50

−40

−30

−20

−10

0

10

Mode number

Fre

qu

en

cy s

hift

(Hz)

SSM (stable)

SSM (unstable)

Analytical

SSM (stable)

SSM (unstable)

Analytical




−10 −5 0 5 10−5

−4

−3

−2

−1

0

Mode number

Gro

wth

rate

(s−

1)


= 53 MV; I0 = 0.157 A; CEPC−Z−1% fill

−10 −5 0 5 10−60

−50

−40

−30

−20

−10

0

10

Mode number

Fre

quency s

hift (H

z)

SSM (stable)

SSM (unstable)

Analytical

SSM (stable)

SSM (unstable)

Analytical




−300 −200 −100 0 100 200 3000

2000

4000

6000

8000

10000

12000

Frequency offset (kHz)

ℜ(Z

) (k

Ω)

−300 −200 −100 0 100 200 300−6000

−4000

−2000

0

2000

4000

6000


ℑ(Z

) (k

Ω)




−300 −200 −100 0 100 200 3000

50

100

150

200

250

300

350


ℜ(Z

) (k

Ω)

−300 −200 −100 0 100 200 300−400

−300

−200

−100

0

100

200

300

400


ℑ(Z

) (k

Ω)




−100 −50 0 50 100−2.7

−2.6

−2.5

−2.4

−2.3

−2.2

−2.1

−2

−1.9

Mode number

Gro

wth

ra

te (

s−1)


= 53 MV; I0 = 0.157 A; CEPC−Z−1% fill

−100 −50 0 50 100−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Mode number

Fre

qu

en

cy s

hift

(Hz)



Summary

Even at low currents CEPC-Z is heavily beam loaded;RF system design should be driven by the beam loading andlongitudinal stability considerations;RF feedback loops will be needed to provide beam/cavity stability;Fundamental impedance is large, but very tightly controlled, CBIdriving impedance reduction is straightforward;Cavity HOMs are relatively unpredictable, need to be damped tolevels, manageable by the bunch-by-bunch feedback.Gap transient response cannot be controlled by RF feedback(high peak power), need to manage fill pattern gaps.


Summary



Summary



Summary



Summary



Summary



Documents

Beam loading and coupled-bunch instabilities from a large ...indico.ihep.ac.cn/event/6618/session/12/contribution/96/material/... · Coupled-bunch Instabilities and Beam Loading This