modeling rf feedback for aps-u bunch-lengthening studies · 28 t. berenc ‐5/27/2015 asd seminar...

Modeling RF Feedback forAPS-U Bunch-Lengthening Studies

ASD Seminar 5/27/2014

T. Berenc, M. Borland

Outline

motivation for modeling RF feedback in particle tracking studies– Double RF System overview– review of Dipole Mode Instability (Robinson)

RF Feedback Model

T. Berenc ‐ 5/27/2015 ASD Seminar

2


3


4


5

)(sin)(sin)( harmharmmainmain nVVV


6

Double RF

Single RF


7

Double RF

Single RF


8

Beam Needs to Lose Energy to the Harmonic Cavity

Thus it can drive the cavity without requiring a rf source


9

BIBL j

Bj

GT eIeII ˆˆˆ

ZjeqT eZIV ˆˆˆ

VGI

BI TI

V

cav

Bopt P

P 1 1sinˆtan

VRI BBoptZ

Accelerating Cavity Above Transition=> tan < 0 => fcav < frf

Bunch Lengthening CavityAbove Transition

=> tan > 0 => fcav > frf

222

2

sinˆtan11

8 cav

BB

Zcav

B

o

G P

IV

PP

QQR

VP

Optimum DetuningOptimum Coupling

TI

GI

BI

Beam Loading

Required Generator Power

For a given beam current, accelerating voltage and accelerating phase, there’s an optimum coupling and detuning

If Pb = -Pcav, cavity can closed off, Pg=0 If Pb/Pcav < -1, Pg can be made zero The detuning is of opposite sign for an

accelerating cavity vs a passive bunch-lengthing cavity

Zero Pg needed for proper coupling and detuning for given beam current

Z Z


10

Detuning = 80kHz

Passive Mode

200mAQext = 3.5e5 (optimal:

2.31e5, 15.15kHz)

self-consistent solution


11

Passive Mode

Detuning = 30kHz


2.31e5, 15.15kHz)



12

Detuning = 80kHz

Passive Mode

Detuning = 20kHz


2.31e5, 15.15kHz)



13

Detuning = 80kHz

Passive Mode

Detuning = 15.1kHz


2.31e5, 15.15kHz)



14

Detuning = 80kHz

Passive Mode

Detuning = 14kHz


2.31e5, 15.15kHz)


15


16


titocav

oeeVtV )(

)()sin( EUEeVf

EE

dtd

radso

orev

o

so

orev

o EeVf

EE

dtd cos

ocrev E

Efhdtd

22

rfo

Beam Dynamics

Cavity Dynamics

Q

I

Q

I

Q

I

IIR

VV

VV

1001

1

17


VGI

BITI

Z

V

o

B

o

ZS

o

br

ZS

o

br

crev

Srevo

oSrev

o

oSrev

o

o

V

o

B

o

Vv

EE

VV

VV

fh

fEVf

EVf

EV

Vv

EE

tansin0

tancos0

0002

cossincos0

2

4th order system Small signal model

no IB amp variation (e.g. shape) Stability determined by roots of

characteristic equation

Account for beam current modulations driving the cavity

Zbr

o

S

Z

VV

2sec

costan0

18


Zbr

o

S

Z

VV

2sec

costan0 Low-Current Limit

19


Zbr

o

S

Z

VV

2sec

costan0

totV

GV

BI

BV

Z

aV

P.B. Wilson’s physical interpretation: The high-current stability limit is reached when the beam rides on the crest of the generator voltage

High-Current Limit

20


Zbr

o

S

Z

VV

2sec

costan0

totV

GV

BI

BV

Z

aV


Late arrival time causes a reduction in generator supplied accelerating voltage

High-Current Limit

21


Zbr

o

S

Z

VV

2sec

costan0

totV

GV

BI

BV

Z

aV


As beam current increases….the generator needs to be adjusted for proper Vtot

High-Current Limit

22


Zbr

o

S

Z

VV

2sec

costan0

totV

GV

BI

BV

Z

aV



High-Current Limit

23


Zbr

o

S

Z

VV

2sec

costan0

totV

GV

BI

BV

Z

aV



High-Current Limit

24


Zbr

o

S

Z

VV

2sec

costan0

totV

GV

BI

BV

Z

aV



High-Current Limit

P.B. Wilson’s physical interpretation: The high-current stability limit is reached when the beam rides on the crest of the generator voltage Only the generator voltage can provide the effective restoring force

25


Zbr

o

S

Z

VV

2sec

costan0

totV

GV

BV

Z

aV

BI


But now…Late arrival time causes an increase in generator supplied accelerating voltage

High-Current Limit

26


Zbr

o

S

Z

VV

2sec

costan0

S deg144S

High-Current Limit

27


Not only do we have opposite detuning sign for harmonic vs. main rf But….


VGI

BI TI

V




TI

GI

BI


Z Z

28


Not only do we have opposite detuning sign for harmonic vs. main rf But….we also have RF Feedback in the main rf system which

effectively lowers the impedance it presents to the beam


VGI

BI TI

V




TI

GI

BI


Z Z

29

Storage RingRF Systems in 420


30


Envelope DetectorEnvelope Detector

Klystron

AmplitudeLoop

RF Source

Cavities

Peak Detector

PIDController

HVPSMod Anode

Driver

+ PIDController

Phase Detector

Phase Shifter

Vector Sum

PhaseLoop

Feedback

|Vcav|(Scalar Sum)

φset

Vset

φcontrol

31



Klystron

AmplitudeLoop

RF Source

Cavities

Peak Detector

PIDController

HVPSMod Anode

Driver

+ PIDController

Phase Detector

Phase Shifter

Vector Sum

PhaseLoop

Feedback

|Vcav|(Scalar Sum)

φset

Vset

φcontrol

32



Klystron

AmplitudeLoop

RF Source

Cavities

Peak Detector

PIDController

HVPSMod Anode

Driver

+ PIDController

Phase Detector

Phase Shifter

Vector Sum

PhaseLoop

Feedback

|Vcav|(Scalar Sum)

φset

Vset

φcontrol

31.9 eQL

208

aQR

MHzfrf 352kHzf 6.27

Q

I

Q

I

Q

I

II

kk

VV

VV

00

L

o

Q2

rfo a

o

QRk

4

Envelope Equations: Continuous-Time

TTTT

ee TTAD

cossinsincos

)()()1( nunxnx DD

22

kD

TeTe TT cossin

TeTe TT cossin

Envelope Equations: Discrete-Time

Cavity Model

RF-TN-2014-002ICMS#: aps_1660384

Introduced Phasor NotationCharles Proteus Steinmetz

(1865 – 1923)

HVPS Mod-Anode(Klystron)

)()(

mod

mod

psp

RNZ

vI

sHcav

o

ctrl

klyG

PID Controller

AGC Components

*Proportional & Integral Gain scaledby open loop Vfb / vctrl @resonance

**

AGC Loop

Data From Dec. 2010

)(1)(

)(sH

sHsH

open

openclosed

)(1

1)(sH

sHopen

edisturbanc

Phase Loop

*Detect & Shift scaled by ½ due to Volts/deg gain difference

*

Phase Loop

)(11)(

sHsH

openedisturbanc

Sensitivity to Beam Disturbances

)(1)()(sH

sHsHopen

cavysensitivit

elegant Model

_+φset

+

+

PolarTo

Cartesian

_

+ +

+

|Vcav|

Cavity

IQ

II VI

VQ

Vset

RFCavity IB

AcceleratorPhysics

GeneratorInducedVoltage

Vcav

VG VB

|IG |o

IGoVcav

+

+

RF System Beam Dynamics

Vcav

+ +

AmplitudeFeedback

Filters

PhaseFeedback

Filters

+

++

In-PhaseFeedback

Filters

QuadratureFeedback

FiltersvQ

Vcav

|Vcav|

_

_

_+vI set

vQ set

_

+

RFMODEelement

i.e. Beamline elements

vI

Presently works in Amp/Phase coordinates. I/Q feedback is planned 4 parallel filter blocks for each amplitude and phase

• Unlimited number of filter coefficients• Sample period = any integer number of rf buckets

elegant Model

Example: Direct RF Feedback Simple Proportional Gain=> Controller = , = Loop Gain at resonance

R

R and Q are reduced by (1 + Loop Gain) R/Q stays the same

_

+Setpoint Error

Beam Current

Feedback

VcavRFGenerator CavityController

)(1

)()()()(

iZR

iZiIiViZ

cav

cav

Beam

caveffective

No FeedbackWith Feedback

RF Feedback: Other Types of Feedback

Zbr

o

S

Z

VV

2sec

costan0

Effectively lowers Vbr to increase the high-current stability limit Possibly useful for PAR and/or Booster ??

RF Feedback: Other Types of Feedback

10dB compensation of +/-4 harmonics

Polar (Amplitude / Phase) or Cartesian (in-phase / quadrature): can be narrowband or wideband Comb Filters: reduce impedance & beam-loading at revolution harmonics & synchrotron sidebands Feed-Forward: feed wall-current monitor to generator to cancel the beam-current directly

Theoretical Example of Transient Beam-Loading Compensation for Hybrid Fill(can be achieved with combination of above)

Summary

46


Detuning of (main / harmonic) contributes (damping / growth) of Robinson stability But.... Robinson stability criterion is based upon extremely simplified model

Doesn’t capture non-linearities and mode-coupling Doesn’t take into account frequency spread Modification by RF Feedback explored later by Pedersen et. al., still small signal Later formulation was Sacherer Integral Equation (Robinson is Dipole Mode)

It is essential to include RF Feedback in particle tracking studies to capture the true dynamics between the beam, cavity, and rf system elegant now includes capability for rf feedback Model developed for Storage Ring and used for bunch-lengthening studies

so far no instability found using existing rf feedback model Other types of feedback may be useful

Direct Feedback or Feed-Forward for heavy beam-loading in PAR and Booster Periodic beam-loading compensation for Hybrid Fills

Requires high bandwidth source

modeling rf feedback for aps-u bunch-lengthening studies · 28 t. berenc ‐5/27/2015 asd seminar...

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