M02 EPAC June 2008

nal Feedback

Jiajing Xu

Wor -AC02-76SF00515

WEOB

Lessons learned from PEP-II LLRF and Longitudi

John D. Fox.

Themis Mastorides, Claudio Rivetta, Dan Van Winkle,

Stanford Linear Accelerator Center

Dmitry Teytelman

Dimtel, Inc.

k supported by the U.S. Department of Energy under contract #DE

M02 EPAC June 2008

ck

Wh

Ori

• L

• L

• D

Wh

• C

• M

• F

Wh

• L

• L

Imp

3 4 5 6ime, ns

ction from a timing sweep at ALS

28−Oct−96

WEOB

PEP-II LLRF and Broadband Feedba

at Was Expected?

ginal CDRGoals 1991/1993

LRF

ongitudinal Coupled-Bunch Feedback

esign and Methodology

at Happened?

ommissioning Experience

ajor Upgrades

inal PEP-II operations, performance

at wasn’t foreseen (via 2 examples)

FB - Impact of Noise in processing channel

LRF - Impact of Nonlinear Signal Processing

ortant Lessons Learned -Summary

0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

T

Bea

m r

espo

nse,

au

Kicker response reconstru

M02 EPAC June 2008

Twocircuupd

• Gm

• e

• e

RF

• H

• Imfumin

• R ms

Inst planes

• H ies

• H MHz sampling rate)

WEOB

PEP-II

rings of 2.2 kmmference. CDR 1991,

ated 1993

oal of factory e+e- colliderachine L 3E33

- HER 1.5A (0.75/1A 1991)

+ LER 2.14 A (2.14 A 1991)

concerns

igh Beam Loading

pedance of cavityndamental, detuning - low-ode coupled bunchstabilities

eliability -extensive R&D effort in Vacuum, Feedback and RF syste

ability Concerns - operation well beyond stability thresholds in three

OM Impedances of cavities - HOM driven coupled-bunch instabilit

OM Dampers - Still the need for coupled-bunch fast feedback (238

M02 EPAC June 2008

d

Yea ER I LER L199 A 1.0A 1.2E33Run 3.0E33Run 1.7 4.4E33Run 1.9 6.3E33Run 2.5 9.0E33Run 3.0 1.0E34Run 2.9 1.2E34Run 3.0 1.2E34Run 1.2E34

HER tionsThe

HER

LOM damped!)

HOM sign)

The (3.213 A at 3 GeV).

Wer s easy and we overdesigned/ove

WEOB

Where we started, Where we finishe

r/run LER stations LER cavities HER stations HER cavities I H8 2 4 4(+1 parked) 16(+4 parked) 0.6 1 2 4 5 20 0.9 1.5 2 3 6 5 20 1.0 3 3 6 6 22 1.1 4 3 6 8 26 1.5 5a 4 8 9 26 1.7 5b 4 8 9 26 1.9 6 4 8 11 28 1.9 7 4 8 11 28 2.1A 3.2A

reconfigured 4 cavity -> two cavity station in Run 3, subsequently added 2 cavity staoperating configuration, gap voltages, tunes, etc. were constantly changing

current -2x design LER Current -1.8x design Luminosity4X design

Growth rates HER 1.2 ms-1 LER 3.0 ms-1 (design - simulation was

growth rates HER3x design LER growth rates 0.45 ms-1 (5.6x de

PEP-II collider holds the record for stored charge in a storage ring

e we successful in the feedback and LLRF areas because it warestimated things?

M02 EPAC June 2008

pology

Tun

Klys

• Rg

• K

Dire

• Cth

• E

• L

Com

• A the residual impedance

Gap

• R oid saturating the klystron ong

Lon des +/- 10

klystron

direct RF loop

comb filtersbeam

RF

BPMlongitudinal multi-bunch

Σ

feedback system

_

ripple loop

mod.

tunerloop

cavities

HVPSklys sat. loop

or

Loop technology

Klys sat -EPICSGap, (ripple) - digital,(Ripple) -analog

WEOB

PEP-II low-level RF feedback loops: To

er loops - standard tuning for minimum reflected power

tron operating point support

ipple loop adjusts a complex modulator to maintain constantain and phase shift through the klystron/modulator system.

lystron saturation loops maintain constant saturation headroom

ct feedback loop (analog)

auses the station to follow the RF reference adding regulation ofe cavity voltage

xtends the beam-loading Robinson stability limit

owers the effective fundamental impedance seen by the beam

b filter (digital)

dds narrow gain peaks at synchrotron sidebands to further reduce

feedback loop (digital)

emoves revolution harmonics from the feedback error signal to avap synchronous phase transients

gitudinal feedback uses RF as low-frequency “woofer” kicker for mo

RFreference

to widebandkicker

band limitedkick signal

+mod.

gap loop

stationreference

err

Tuner,Comb.Direct

M02 EPAC June 2008

ated?

LLR iver), Rich Tighe (SLAC)

• L

• C

LLR he, Rivetta, Mastorides)

• M n

• 1 s time windows

Des Dynamic range issues, I&Qmism

100

0.005

0.01

0.015

0.02

0.025

0.03

Mod

Pha

se (

deg)

Open LoopPole

g ∆ l xImpedance control

th

l

x

1/t

WEOB

How was required LLRF performance estim

F -Origin and design/modelling F. Pedersen, Stuart Craig (Chalk R

inear frequency domain models

oncerns about non-linear klystron, impact on impedance control

F station model, beam model -nonlinear time domain simulations (Tig

acro-bunch structure, low-mode dynamics with Non-Linear Klystro

994 model - Nocriteria for stability, robustness beyond trajectories in m

ign proceeded based on initial simulations. Little criteria for Noise,atches, other technical imperfections

0

0.2

0.4

0.6

0.8

−10

−5

0

5

Time (ms)

LER ring − modal analysis

e number

−10 −5 0 5 10−2

−1

0

1

2

Gro

wth

rat

es (

ms−

1 )

−10 −5 0 5 103900

4100

4300

4500

Mode number

Osc

illat

ion

freq

uenc

ies

(Hz)

σ

ω

PoleClosed Loop

LGDW dampino

net damping d l

growl

o

rate

j

M02 EPAC June 2008

Eac

• 1

• V

• 2

• H

WEOB

PEP-II RF Station, LLRF

h Station

.2 MW 476 MHz Klystron

XI-based LLRF electronics

or 4 RF cavities, with HOM loads

V power supply, Interlocks, etc.

M02 EPAC June 2008

A D IIR Filters)Dev . Multi-crate VXI/VME

Det

Sca

Sam

Dow te to synchrotron oscillationfreq .

Hold

buffer,

DAC

Power

amplifier

×

Kicker oscillator

locked to 9/4×frf

1071 MHz

QPSK modulator

Low-pass filter

Kicker structure

To RF stations

Woofer link

WEOB

LFB Systems Design

SP based flexible, programmable system (can run arbitrary FIR oreloped for PEP-II, ALS, DAΦNE (later BESSY-II, PLS and SPEAR)

ection at , correction at 9/4 RF (options 11/4, 13/4)

lable VME processing array, up to MAC/sec.

pling, A/D and D/A at 500 MHz (238 MHz PEP-II)

nsampling to reduce computational load (match processing rauency). Original “woofer” taken at DSP farm D/A wideband output

Low-pass filter

AD

C,dow

nsa

mple

r

DSP

Beam

Phase servo

×

BPM

Comb generator

LNA

locked to 6×frf

Master oscillator

2856 MHz

Farm of digitalsignal processors

Timing and control

Low group-delay channel

DSP at 9.81 MHz

6 FRF×

3.2 109⋅

M02 EPAC June 2008

Ho

Line timates, with downsampledFIR

Thre

Bea

Res n)

Req lled power)

Dyn

Bea ate currents

Insu

6000 8000 10000 12000ency (Hz)

Kicker Σ

mb

BPM

−+

Beam

np

kv

|Re(

λ l)| (m

s1 )

WEOB

w was required LFB performance estimated?

ar Growth Rates, GainandTime-domain Simulation(tracking, cavity HOM esfilter)

sholds, Growth Rates - from cavity HOM measurements/estimates

m tests - 1 bunch (SPEAR), ALS 4 processor “Quick Prototype”

olution offront end modelled and lab-tested(noise kept small for high DSP gai

uiredkicker power - estimated from injection error (amp expense,minimize insta

amics estimates from simulation. Filter completely programmable

m-LLRF Simulations predict stable Low-mode behavior - LER issue at ultim

rance policy - design in a low-mode “Woofer” channel

0 2000 4000−3

−2.5

−2

−1.5

−1

−0.5

0

Frequ

Am

plitu

de (d

B)

ε

XOscillatorMaster

Σ Filter

Co

Σns

−

+

+

+φ

0

vm vb

0 20 40 60 80 100 12010

−5

10−4

10−3

10−2

10−1

100

Frequency (MHz)

positive λl

negative λl

Synhrotron Radiation damping dr

M02 EPAC June 2008

d

LFB

prog

500

Gro

mon

Woo

LLR

Soft

Soft

Buil s)

Fau

Woo

The with VxWorks. GUI, etc.

WEOB

Features Anticipated and Implemente

rammable80 processor DSP reconfigurable array

MS/sec. A/D, D/A, Downsampler - table driven2 ns bucket spacing

w-dampdynamics measurements (via dual-port memory, codes)

itoring functions (Signal MUX, RMS detectors front and back-end)

fer output

F

ware controlled broadband (direct - analog) andcomb (IIR digital) loops

ware based low frequencydigital regulators via EPICS

t-in network analyzer (via time domain excitation, response function

lt files

ferinput

days of NIM Modules with Pots - are over.Software intensivesystems,

M02 EPAC June 2008

ALSprot

devmet

VXI

PEP

HOMmea

Thre300

Beafrom

ope

Woo

coopha

400 1600 1800 2000t (mA)

00 1800 1900 2000 2100t (mA)

May 31st 2007June 1st 2007Aug 13th 2007Feb 9th 2008

700 1750 1800 1850 1900nt (mA)

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Commissioning Experience LFB

- extensive experience with the “quickotype” made commissioning fast

eloped control filters, timing/synchronizationhods

system commissioned at ALS 1994

-II (1998)

Thresholds -consistent with cavity HOMsurements - damping rates per simulation

sholds -cavity fundamental driven low modes mA (Simulation had them damped!)

m hasRF power supply noise(very different ALS)

rational issues -

fer required for low mode control

rdination with operations on synchronousse, timing LFB

800 1000 1200 10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Curren

Gro

wth

Rat

es (

ms

−1)

Historical DataRun 4Run 5bRun 6Run 7

1300 1400 1500 1600 17−0.5

−0.4

−0.3

−0.2

−0.1

0

Curren

Dam

ping

Rat

es (

ms

−1)

1500 1550 1600 1650 16650

6700

6750

Curre

Fre

quen

cy (

Hz)

M02 EPAC June 2008

ver

UnaRF

ManLLR

Imp

Qua

4 5 6

.01;|H|var

= 1.0384; Gain ratio = 1

8 10 12

(28.1 at 6.595 kHz)

8 10 12

degrees at 6.595 kHz)

Hz)

1

1

Cou

nts

WEOB

Unexpected Impact of “noise” in Recei

nticipated- amount of “noise” on beam fromsystems (unlike ALS and SPEAR Experience)

y sources, predominantly klystron HVPS,F processing, phase distribution noise

act ofdriven motion vs. HOM instability

ntizing noise in A/D, Rcvr noise insignificant

0 1 2 3

−10

−5

0

5

10

Ratio = 0.58111; ∆Θ = 2.288; Θvar

= 44

0 2 4 6−50

0

50Magnitude of Filter TF

0 2 4 6−200

−100

0

100

200Phase of Filter TF (−90.6

deg

Freq. (k

0 2 4 6 8 10

0−1

00

HER LFB receiver noise (rms A/D counts)

HER 1700 mA − Red

HER A/D 50 Ohm Termination − Green

HER Receiver No Beam − Cyan

Freq (kHz)

M02 EPAC June 2008

Lo ay HOM control

At 1tran

Veryinfre

magsyststat

soluwoo

WEOB

w-Frequency Noise leads to saturation and runaw

900 mA - 2100 mA in HER, unexpectedsient saturation effects, loss of control

hard to diagnose, not a steady state situation,quent transient effects

nitude of 720 Hz constantly changing with RFem configurations, operating points, activeions, maintenance etc.

tions via better 720 Hz control in LLRF andfer ( more kicker amp power would help,too)

M02 EPAC June 2008

IssuSpuIniti inInstmantrad ch faster than anticipated

00

50

100

150

200

250

300

350

400

450

po

we

r −

kW

500 600 700 800 900kHz)

utput Forward PEP−II LLRF tutorial

WEOB

Commissioning Experience LLRF

es with configurations of direct and comb loops -stations oscillatingrious signals in RF output, klystron oscillationsal configuration method - network analyzer no beam,gain/phase margsabilities (Station and/or Beam) at currentual tweaking of the direct and comb loopse-off ofstation stability vs. beam stability -low mode growth rates mu

2 4 6 8 10 12

IQA Module #1 Fault File − LR45 Klystron Output Forward PEP−II LLRF tutorial

time − ms0 100 200 300 400

−40

−20

0

20

40

60

80

Frequency (

Mag

nitu

de (

dB)

IQA Module #1 Fault File − LR45 Klystron O

M02 EPAC June 2008

F

Mod nges with current)

Fau of operational problems

Low . Rapid low-mode growth

Klys ot with expected result

Exte ovedDriver Amplifier-add timal station configurations

−100

−20

−10

0

10

Gai

n (d

B)

Direct

−10−200

−100

0

100

200

Pha

se (

degr

ees)

Connectorboard

EPICS IOC

Parallel portdriver

ard

Linux PC

Ethernet

Front−panel status LEDs,trigger inputs

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Major Developments/upgrades - LLR

el based configuration - reconfigure loops at current. (Dynamics cha

lt file methodology, weekly reports and analysis- understand origins

Group delay Woofer - necessary above 1500 mA (HER), 2A (LER)

tron linearizer - effort to address fast low-mode growth rates - but n

nsive re-investment in LLRF-Beamsimulation models. Led to imprresses limits of impedance control, allows comb rotation, more op

0 −500 0 500 1000

: Fr = 475.9±0.1 MHz; G = 5.268±0.009; T

d = 430.8±0.6 ns; φ = 164.1±0.1 deg

Fit

Data

00 −500 0 500 1000

Frequency (kHz)

Comb: Gc = 0.2042±0.0009; Tc = 5590±2 ns; φ

c = 30.1±0.3 deg

ADC DACFPGA

XC4085XLA

ADC DACFPGA

XC4085XLA

SRAM64Kx16

SRAM64Kx16

Offset&filterboard #1 (HER)

GVA−200 FPGA bo

Offset&filterFro

m H

ER

and

LE

R L

FB

pha

se m

onito

rs

board #2 (LER)

To LER back−end module

To HER back−end module

M02 EPAC June 2008

Un

LLR

Non

Klys

We

very

Ima

Not 0 500 1000 (kHz)

(LR42 Ibeam=1900mA)

−

−

−

−

−

Out

put P

ower

(dB

m)

476 477 478 479 480 481ncy (MHz)

de Response

20W CarrierNo Carrier

WEOB

derstanding the impact of non-linear processing

F Signals - Dynamic range 90 dB!

-linear behavior in loop -imperfections

tron - obviously non-linear

missed - medium power amplifier

significant impact- SS vs. LS gain

ge frequency generation

realized for 7 years - understood via model −1000 −500−120

−100

−80

−60

−40

−20

0

freq

FFT klystron output

471 472 473 474 475 476 477 478 479 480 48150

40

30

20

10

0

10

20

30

40

50

freq (MHz)

Output Signal (single Sideband test)

471 472 473 474 4755

6

7

8

9

10

11

12

13

14

15

freque

Gai

n (d

B)

rela

tive

(+ a

ppro

x 30

dB

)

Amplitu

M02 EPAC June 2008

LER ew LLRF amplifier

Repof “c30%imp

RunLLR

Runrota

This

• N

• N

• Cse

Man(e.gopti

3000 3500

WEOB

Final PEP-II Run April 2008

LLRF and LOM control limit at 3100 mA without comb rotation, n

lacing Driver Amp, useomb rotation” provided

improvement inedance control

5B - old amps, originalF config

6/7 - new amps, combtion

was possible, with

ew Modelling

ew control technique

areful operating pointlection

y Accelerator issues. bunch length, heating,cs, choise of gap voltage, etc.)

1500 2000 25000

0.5

1

1.5

2

2.5

3

3.5

4

Beam Current (mA)

Gro

wth

Rat

es (

ms

−1)

Run 5b configurationRun 5b configurationNew LER42 AmpRun 6, low comb gainEnd of Run 6 configurationApril 2008Run 7 configuration

M02 EPAC June 2008

LLR

• N

With han they should be

Mod l techniques (comb rotation)

This

Fau

PEPjust

who

LFBund

Una

Flexapp

New30 40 50 60

tron Output Forward 14−Jul−2007 16:39:38

e − ms

WEOB

Lessons Learned

F Modelling - key to understanding non-linear effects

on-linear klystron was less significant than drive amplifier

out models - impossible to sort out effects, see if things were worse t

els - prediction of limits, identification of nonlinear amp, new contro

tookyears, but was invaluable

lt Files- so much information

-II Experience - needed a full-time RF expert to understand complexities of faults

is the customer for this information?

- transient domain measurements keytoerstanding dynamics

nticipated - saturation limit from RF system

ibility of DSP architecture key to unexpectedlications

Accelerator Diagnostics developed0 10 20

0

100

200

300

400

500

600IQA Module #1 Fault File − LR42 Klys

tim

pow

er −

kW

M02 EPAC June 2008

s

LFBkick

• k

• C

RF -

ope

• Inst

• In

Imp

com

R&Dpus

How

WEOB

What wasn’t foreseen -more lesson

- thermal problems with beam induced power iners

W power levels -SC/DIN/EIA connectors?

able fires (several systems)

rational task - management of so many stations

dividual station dynamics- unique station toation

dividually configured stations

act ofnon-linear Klystron/Preamp

plexity of fault file analysis

project - continual changes and performanceh in the machine

is this consistent with the operating machine?

M02 EPAC June 2008

Wer s easy and we overdesigned/ove

LLR

• C ompletelyunderestimated

• U dingnonlinear amp

• O s on/off, gap voltages)

• M upported

Bro

• E dous benefit to PEP-II

• V y of output power

Wha

• F ing/measurements

• M hysics/technology skills

• D

WEOB

Summary - Lessons learned

e we successful in the feedback and LLRF areas because it warestimated things?

F and RF dynamics

omplexity of RF system,stability of low modes, operational issues - c

nexpected low-mode instabilities ->2 woofers->klystron linearizer->fin

perational intensity, issues ofconstantly moving configurations (klystron

anpower/skill of operational support woefullyunderestimated/under s

adband (coupled-bunch) longitudinal feedback

ssential techniques developed at ALS and other facilities - tremen

ery lucky (wise?) design choices for detection frequency, scalabilit

t features wereessential for success

lexibility (reprogrammability, modular architecture), close ties tomodell

ost importantelement- Creative, highly curiousgroupwith concurrent p

iverse set of interesting challenges

M02 EPAC June 2008

Tha satko, M. Minty, P. McIntosh,C. L . Young, H. Hindi, I. Linscott(Sta arcellini, M. Serio (LNF-INF d, J. Corlett, G. Lambertson,G. S contributions.

The CERN) and developed by P.Cor rz, A. Hill (SLAC) and manyothe us, K. Krauter, K. Luchini andD. T r LHC-related collaboration.

The fully designed and developedby G are was designed and codedby R d D. Teytelman (SLAC)

The developed by F. Voelker andJ. C ni, A. Gallo, F. Marcellini,et.a

Spe ise and to the ALS, SPEAR,PEP consistent good humor andhelp 3-76SF0051

WEOB

Acknowledgments

nks to D. Andersen, L. Beckman, M. Browne, P. Corredoura, J. Duimborg, S. Prabhakar, W. Ross, J. Sebek, R. Tighe, U. Wienands, Anford), M. Tobiyama, E. Kikutani (KEK), A. Drago, A. Ghigo, F. MN), Shaukat Khan (BESSY), H. Kang ( POSTECH),W. Barry, J. Byrtover and M. Zisman (LBL) for numerous discussions, advice and

PEP-II LLRF Systems were inspired by Flemming Pedersen (redoura, R. Tighe, L. Sapozhnikov,B. Ross, J. Judkins, H. Schwars. System software was designed and coded by S. Allison, R. Claeytelman (SLAC). We thank P. Baudrenghien and J. Tuckmantel fo

PEP-II LFB digital processing architecture and modules were skill. Oxoby, J. Olsen, J. Hoeflich and B. Ross (SLAC) - System softw. Claus (SLAC), I. Linscott (Stanford), K. Krauter, S. Prabhakar an

wideband longitudinal kicker for ALS and PEP-II was designed andorlett (LBL). The kicker for DAFNE (PEP-II) was designed by R. Bol.

cial thanks to Boni Cordova-Grimaldi (SLAC) for fabrication expert-II, DAFNE, KEKB, PLS and BESSY-II operations groups for their. Work supported by U.S. Department of Energy contract DE-AC0

M02 EPAC June 2008

es

Mul EP-II/PLS/SPEARrequ ith up to 500 MHz samplingrate

• D

• V

• D

• H

• T

• F - 600 MHz IF bandwidth

• B ier (1125 MHz, 1071 MHz,1

• S with EPICS-based userin

• D

• L

WEOB

Longitudinal Feedback System Featur

tiprocessor architecture fully implements ALS/BESSY-II/DA NE/Pirements. Scalable, flexible architecture for up to 8192 bunches w

s.

SP processor -VME card,4 AT&T DSP 1610s

ME interface - Bus master for data distribution

ownsampler- 500 MHz A/D and VXI Sequencer

old Buffer -500 MHz D/A and VXI Ring Buffer

iming - VXI oscillators ( , , )

ront-end - Comb filter followed by (3 GHz) phase detector

ack-end - AM modulator transfers baseband kick to QPSK’ed carr196 MHz, 1375 MHz).

oftware - VxWorks operating system for configuration and control terface

ata analysis in Matlab, Automated diagnostics and setup tools

ink error checking, temperature monitoring

Φ

RF 6 RF× 9 4⁄ RF×

6 RF×

M02 EPAC June 2008

m PLS

A 30

AllTransimpin thmodeigetran

A sinstope

A veigecon

WEOB

Grow/damp measurement example fro

ms long data set with 15 ms open-loop section.

filled bunches participate in the modal motion.sformation to the even-fill eigenmode basislifies the picture - there are three strong eigenmodesis transient. Fitting complex exponentials to theal motion we extract estimates of the modalnvalues for both open and closed-loop parts of thesient.

ingle measurement like this only characterizes theabilities and the feedback at a single acceleratorrating point.

ery powerful techniqueis to measure modalnvalues as a function of beam current, RF system

figuration, etc.

M02 EPAC June 2008

trol

DAF

•

•

Wecon

•

•

•

ALS fetime) - unanticipated effectgian lay IIR Filter

80 100 120)

DualNotch

80 100 120)

WEOB

LFB Flexibility -Quadrupole instability con

NE e+/e-collider at LNF

increased operating currents

quadrupole mode longitudinalinstabilities have appeared (theinstalled system suppresses thedipole modes).

implemented a novel quadrupoletrol filter

software programmability ofthe DSP farm

two parallel control paths fordipole and quadrupole modes.

quadrupole control has beensuccessful, allowing a 20%increase in luminosity.

added passive harmonic cavities (to address Tousheck-limited lit tune change with current. Stability required a novelnegative group de

0 20 40 60

−40

−20

0

20

40

Frequency (kHz

Gai

n (d

B)

0 20 40 60−200

−100

0

100

200

Frequency (kHz

Pha

se (

deg)

M02 EPAC June 2008

trol

Las

Plan , optics changes

Lon

Gro z, etc. are a mystery

WEOB

Where We finished - LOM and HOM con

t year of Operations

to push from 1.2E34 to 2E34 - via current increase, bunch length

gitudinal stability- Data from LER 2900 mA

wth/Damping rate isn’t the issue.Strange interfering signals at 1100 H

M02 EPAC June 2008

trol

Wha edance)?

Sev

I). N m several stages -

Fron dynamic range, steady-stateoffs

Pro noise (broadband) is onesyst ntribute. Narrowband filtershelp sensitivity to machine tunes,ope

Pow n expensive way to increasegain

Out oscillation amplitude fromwhic complicated

WEOB

Ultimate/Practical Limits to Instability Con

t Limits theMaximum Gain(e.g. fastest growth rate, or allowed imp

eral Mechanism

oise in feedback filter bandwidth, limits on noise saturation. Gain is fro

t End(BPM to baseband signal) gain limited by required oscillationets (synchronous phase transients, orbit offsets)

cessing Block - gain limited by noise in filter bandwidth. Quantizing em limit - noise from RF system or front-end circuitry may also co with broadband noise. Broad filter bandwidths help with reduced rating point - or variations of dynamics with current

er stages- gain scales with kicker impedance, sqrt(output power). A (more kickers, more output power).

put power (actually maximum kicker voltage) determines maximumh linear (non-saturated) control is possible. Saturated behavior is

M02 EPAC June 2008

Ult

II) S n vs. control frequency)

Rela

For an pickup)

limit over control band

App

Neg ut for causal systems you paythe

WEOB

imate/Practical Limits to Instability Control

tability of the feedback loop itself, (e.g. limits on phase shift and gai

ted to time delay between pickup, processing, and actuator

circular machines (systems with kick signal applied on later turn th

set by revolution time, fastest growth rates, and filter phase slope

ropriate for optimal control theory applications

LQR

Robust Control

Uncertain Systems

ative group delay over a portion of the frequency band is possible, bprice in increased phase slope away from the negative region