feedback 101

S. Henderson, IU e-p meeting ORNLMarch 15-19, 2004

Feedback 101

Stuart Henderson

March 15-18, 2004

S. Henderson, IU e-p meeting ORNL2

March 15-19, 2004

Outline

• Introduction to Feedback– Block diagram

– Uses of feedback systems (dampers, instabilities, longitudinal, transverse

– System requirements

– Resources (paper)

• Simplest feedback system scheme– Ideal conditions

– Eigenvalue problem and solution

– Loop delay, delayed kick

• Closed-orbit problem– Filtering schemes (analog/digital)

– Two turn filtering scheme

– Type of digital filters (FIR, IIR)

• Kickers– Concepts

– Dp and dtheta calculation

– Figures of merit

– Plots of freq response, etc.

• Complete System Response

• Estimates for damping e-p

• RF amplifiers– Parameters, cost, etc.

• Feedback in the ORBIT code


March 15-19, 2004

Resources

• Several good overviews and papers on feedback systems and kickers:– Pickups and Kickers:

Goldberg and Lambertson, AIP Conf. Proc. 249, (1992) p.537

– Feedback Systems: F. Pedersen, AIP Conf. Proc. 214 (1990) 246, or CERN PS/90-49

(AR) D. Boussard, Proc. 5th Adv. Acc. Phys. Course, CERN 95-06, vol. 1

(1995) p.391 J. Rogers, in Handbook of Accelerator Physics and Technology, eds.

Chao and Tigner, p. 494.


March 15-19, 2004

Why Feedback Systems?

• High intensity circular accelerators eventually encounter collective beam instabilities that limit their performance

• Once natural damping mechanisms (radiation damping for e+e- machines, or Landau damping for hadron machines) are insufficient to maintain beam stability, the beam intensity can no longer be increased

• There are two potential solutions:– Reduce the offending impedance in the ring– Provide active damping with a Feedback System

• A Feedback System uses a beam position monitor to generate an error signal that drives a kicker to minimize the error signal

• If the damping rate provided by the feedback system is larger than the growth rate of the instability, then the beam is stable.

• The beam intensity can be increased until the growth rate reaches the feedback damping rate


March 15-19, 2004

Types of Feedback Systems

• Feedback systems are used to damp instabilities– Typical applications are bunch-by-bunch feedback in e+e-

colliders, hadron colliders to damp multi-bunch instabilities

• Dampers are used to damp injection transients, and are functionally identical to feedback systems– These are common in circular hadron machines (Tevatron, Main

Injector, RHIC, AGS, …)

• Feedback systems and Dampers are used in all three planes:– Transverse feedback systems use BPMs and transverse

deflectors…

– Longitudinal feedback systems use summed BPM signals to detect beam phase, and correct with RF cavities, symmetrically powered striplines,…


March 15-19, 2004

Elements of a Feedback System

• Basic elements:– Pickup– Signal Processing – RF Power Amplifier– Kicker

• Pickup is BPM for transverse, phase detector for longitudinal

• Processing scheme can be analog or digital, depending on needs

• Transverse Kicker:– Low-frequency: ferrite-yoke

magnet– High-frequency: stripline kicker

• Longitudinal Kicker can be RF cavity or symmetrically powered striplines

Kicker

Pickup

RF ampSignal Processing

Beam


March 15-19, 2004

Specifying a Feedback System

• Feedback systems are characterized by – Bandwidth (range of relevant mode frequencies)

– Gain (factor relating a measured error signal to output corrective deflection)

– Damping rate

• In order to specify a feedback system for damping an instability, we must know– Which plane is unstable– Mode frequencies– Growth rates

• RF power amplifier is chosen based on required bandwidth and damping rate. Typical systems use amplifiers with 10-100 MHz bandwidth, and 100-1000W output power.


March 15-19, 2004

Simple picture of feedback

• Take simple (but not very realistic) situation: -functions at pickup and

kicker are equal

– 90 phase advance between kicker & pickup

– Integer tune

X

X

Position measurement (coordinates x, x’)

Kick (coordinates y, y’)• System produces a kick

proportional to the measured displacement:

• At the kicker:

• At the BPM after 1 turn:

x

G

/

0

00/1

0

0

0'0

0

x

x

y

y

0/

0

0/1

0 0

0'1

1

x

xx

x


March 15-19, 2004

Simple picture of feedback, continued

• So x-amplitude after 1 turn has been reduced by

• Giving a rate of change in amplitude:

• Giving a damping rate:• But, we don’t really operate with integer tune. Averaging over all arrival

phases gives a factor of two reduction:

• In real life, we may not be able to place the BPM and kicker 90 degrees apart in phase, and the locations will not have equal beta functions. We need a realistic calculation.

ttGf AeAetx

xGfdt

dx

0)(

0

0/1 Gf

20Gf

opt

Gxx


March 15-19, 2004

Realistic damping rate calculation for simple processing

• Follow Koscielniak and Tran

• Coordinates at pickup are (xn,xn) on turn n

• Coordinates at kicker are (yn,yn) on turn n• Transport between pickup and kicker has

2x2 matrix M1 and phase 1

• Transport between kicker and pickup has 2x2 matrix M2 and phase 2

• Give a kick on turn n proportional to the position measured on the same turn:

• Where G is the feedback gain

Pickup (x,x)

Kicker (y,y)

M1, 1

M2, 2

pk

nnn

xGkxy

'


March 15-19, 2004

Simple processing, cont’d

nnnn

nn

n

n

nn

nn

xKMMxk

xMMx

kxy

yM

yy

yMx

xMy

)(0

00

0

12121

'2''21

1

• The coordinates one turn later are given by:


March 15-19, 2004

More realistic damping rate calculation, cont’d

• After n turns the coordinates are

• This is an eigenvalue problem with solution

• The eigenvalues can be obtained from

012 )]([ xKMMx nn

ex nn

0)(det 12 IKMM

0

00)( '

2'2

22'0

'0

0012 kSC

SC

SC

SCKMM

One-turn matrix

0det '0

'2

'0

020

SkSC

SkSC


March 15-19, 2004

General solution for 2x2 real matrix

0det

dc

ba

sincos)(4)(2

1)(

2

1 2/12 ieecbaddada

Giving, cbade 2

e

da

2

)(cos

• Since we have a 2x2 real matrix, we expect two eigenvalues which are complex conjugate pairs. Writing

• Where we can identify as the damping rate (per turn), and as the tune, which in general will be modified by the feedback system

• Solution:

ie


March 15-19, 2004

Damping rate and tune shift for simple processing

• We have

• With p, p the twiss parameters at the pickup, k, k at the kicker, the tune, 1 the phase advance between pickup and kicker, 2 the phase advance from kicker around the ring to pickup:

• Finally,

)()( '2

'00

'020

2 kSCSSkSCcbade

))sin(cos/sin1

(sin

)sinsin)(cossin(cos

22

2

22

ppkp

pp

kppp

k

ke

1112 sin1sin1 Gke

2/112

2/1

11

21

sin12

sincos2

sin12

sincos2cos

G

G

k

k


March 15-19, 2004

Damping rate and tuneshift for small damping

• For weak damping,

• And

• Optimal damping rate results for 1=90 degrees

10

1

sin2

sin2

Gf

G

turns-1

sec-1

1cos2

G radians

10 cos2

Gf


March 15-19, 2004

Damping vs. Gain for 1=90 degrees

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.05 0.1 0.15 0.2 0.25 0.3

Gain

Dam

pin

g R

ate

(tu

rn-1

)

Exact Result Weak Damping Approximation


March 15-19, 2004

Tuneshift vs. Gain for 1=90 degrees


March 15-19, 2004

Finite Loop Delay

• Up to this point we have ignored the fact that it takes time to “decide” on the kick strength in the processing electronics

• It is not necessary to kick on the same turn

• We can kick m turns later:

• In this way we can “wait around” for the optimum turn to provide the optimum phase

)2cos(2

)2sin(2

10

10

mQGf

mQGf


March 15-19, 2004

Closed-Orbit Problem: the 2-turn filter

• Our simplification ignores another problem:– A closed orbit error in the BPM will cause the feedback system to

try to correct this closed orbit error, using up the dynamic range of the system

• Solution:– Analog: a self-balanced front-end

– Digital: Filter out the closed-orbit by using an error signal that is the difference between successive turns

• 2-turn filter constructs an error signal:

1 nnn xxu


March 15-19, 2004

2-turn filter, cont’d

• With

• The transfer function of the filter is:

• This gives a “notch” filter at all the rotation harmonics, which are the harmonics that result from a closed orbit error

000

0

1TjTjnTjn

nnn

Tjntjn

eAeAexxu

AeAex n

01 Tj

n

n ex

u


March 15-19, 2004

2-turn Filter Frequency Response

Two-turn Filter Amplitude

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6

f/f0

Am

plit

ud

e


March 15-19, 2004

2-turn Filter Phase

2-turn filter phase

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2 3 4 5 6

f/f0

degr

ees


March 15-19, 2004

Kickers for Transverse Feedback Systems

• For low frequencies (< 10 MHz), it is possible to use ferrite-yoke magnets, but the inductance limits their bandwidth

• Broadband transverse kickers usually employ stripline electrodes

• Stripline electrode and chamber wall form transmission line with characteristic impedance ZL


March 15-19, 2004

Stripline Kicker Layout

+VLZL

ZL

-VL

ZL

ZL

Beaml

d


March 15-19, 2004

Stripline Kicker Schematic Model

VK

Zc

Beam In

p p+ p

Beam Out


March 15-19, 2004

Stripline Kicker Analysis

• Deflection from parallel plates of length l, separated by distance d, at opposite DC voltages, +/- V is:

• We need to account for the finite size of the plates (width w, separation d). A geometry factor g 1 is introduced:

• Because we want to damp instabilities that have a range of frequencies, we will apply a time-varying potential to the plates V().

• We need to calculate the deflection as a function of frequency and beam velocity.

+V

-Vcl

d

eVteEtFp L2

clg

d

eVp L

2

d

wg

2tanh


March 15-19, 2004

Stripline g

Transverse Stripline Geometry Factor

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Stripline Width/Separation

G-p

erp


March 15-19, 2004

Deflection by Stripline Kicker

• Stripline kicker terminated in a matched load produces plane wave propagating in +z direction between the plates.

• For beam traveling in +z direction:

• For beam traveling in –z direction:

• For relativistic beams, we need the beam traveling opposite the wave propagation!

xy

kztjLx

EcB

ed

VgE

)(2

)1( xyxx eEBeceEF

)1( xyxx eEBeceEF


March 15-19, 2004

Deflection by Stripline Kicker

• Where

• This can be written in phase/amplitude form as:

0

/

)1(0

/

)1(2

)(

cl

tj

cl

L dted

eVgdttFp

lkkjL LBejd

eVgp )(1

2

ck

ck

L

B

jL ed

eVgp sin

4 2/lkk LB


March 15-19, 2004

Powering the Stripline Kicker

• For transverse deflection, one could – Independently power each stripline with its own source

– Power the pair of striplines from a single RF power source by splitting (e.g. with a 180 degree hybrid to drive electrodes differentially)

• Using a matched splitting arrangement, the delivered power is:

• Which equals the power dissipated on the two stripline terminations:

• So that the input voltage is:

c

K

Z

VP

2

2

L

L

Z

VP

22

2

LLcK VZZV /2


March 15-19, 2004

Figures of Merit for Stripline Kickers

• One common figure of merit seen in the literature is the Kicker Sensitivity.

• From which we get:

• Which can be written in the form

• Important points:– Deflection has a phase shift relative to the voltage pulse

– sin/ shows the typical transit-time factor response

Kc

eVp K

j

C

L

BC

L

L

eZ

Z

dk

g

Z

Z

eV

cpK

sin2

4

2

j

C

L ed

lg

Z

ZK

sin)1(

2

2


March 15-19, 2004

Transverse Shunt Impedance

• In analogy with RF cavities, one can define an effective shunt impedance that relates the transverse “voltage” to the kicker power:

•

• So after all this, what’s the kick?

22

2

222

sin2

2

222

BLC

K

C

K

dk

gZKZR

R

VK

R

V

Z

VP

PRE

e

E

eVK

p

p k

22

2


March 15-19, 2004

Transverse Shunt Impedance (w=d, =0.85, 50, d=15cm)

Stripline Frequency Response

-2.00E+03

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

1.40E+04

0 100 200 300 400 500 600

Frequency (MHz)

Tra

nsve

rse

Shu

nt Im

peda

nce

(Ohm

s)

1 meter 0.5 meter 0.25 meter 0.125 meter


March 15-19, 2004

Transverse Shunt Impedance (w=d, =0.85, 50, d=15cm)

Stripline Frequency Response

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

0 100 200 300 400 500 600

Frequency (MHz)

Tra

nsve

rse

Shu

nt Im

peda

nce

(a.u

.)

1 meter 0.5 meter 0.25 meter 0.125 meter


March 15-19, 2004

Multiple Kickers

• For N kickers, each driven with power P,

• Where PT=NP is the total installed power

• To achieve the same deflection (damping rate) with N kickers requires only

• Example: One kicker with P1=1000W gives same kick as two kickers each driven at 250 W

RNPE

ePRN

E

eT22

22

N

PPT

1


March 15-19, 2004

Putting it all together

• The RF power amplifier puts out full strength for a certain maximum error signal

• The system produces the maximum deflection max for a maximum amplitude xmax

• For optimal BPM/Kicker phase, the optimal damping rate is

• For a Damper systems, xmax is large enough to accommodate the injection transient

• For a Feedback system, xmax is many times the noise floor

RNPE

e

x

f

x

fGfT

pkpk

opt 2222 2

max

0

max

max00


March 15-19, 2004

Parameters for an e-p feedback system

• Bandwidth:– Treat longitudinal slices of the beam as independent bunches

– Ensure sufficient bandwidth to cover coherent spectrum

– Choose 200 MHz

• Damping time:– To completely damp instability, we need 200 turns

– To influence instability, and realize some increase in threshold, perhaps 400 turns is sufficient

• Input parameters: y = 7 meters

– Xmax = 2mm

– Stripline length = 0.5 m, separation d = 0.10, w/d = 1.0

feedback 101

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