accelerator physics particle acceleration

USPAS Accelerator Physics June 2013

Accelerator PhysicsParticle Acceleration

G. A. Krafft, Alex Bogacz and Timofey Zolkin

Jefferson Lab

Colorado State University

Lecture 6


RF Acceleration

• Characterizing Superconducting RF (SRF) Accelerating Structures

– Terminology

– Energy Gain, R/Q, Q0, QL and Qext

• RF Equations and Control

– Coupling Ports

– Beam Loading

• RF Focusing

• Betatron Damping and Anti-damping


Terminology

1 CEBAF Cavity

1 DESY Cavity

“Cells”

5 Cell Cavity

9 Cell Cavity


Typical cell longitudinal dimension: λRF/2Phase shift between cells: πCavities usually have, in addition to the resonant structure in picture:

(1) At least 1 input coupler to feed RF into the structure(2) Non-fundamental high order mode (HOM) damping(3) Small output coupler for RF feedback control

Modern Jefferson Lab Cavities (1.497 GHz) areoptimized around a 7 cell design


Some Fundamental Cavity Parameters

• Energy Gain

• For standing wave RF fields and velocity of light particles

• Normalize by the cavity length L for gradient

2

, vd mc

eE x t tdt

2, cos 0,0, cos /

2 / c.c. = 2 /

2

RF z RF

iz RF

c z RF

E x t E x t mc e E z z dz

eE eV eE

accE MV/m cV

L


Shunt Impedance R/Q

• Ratio between the square of the maximum voltage delivered by a cavity and the product of ωRF and the energy stored in a cavity

• Depends only on the cavity geometry, independent of frequency when uniformly scale structure in 3D

• Piel’s rule: R/Q ~100 Ω/cell

CEBAF 5 Cell 480 Ω

CEBAF 7 Cell 760 Ω

DESY 9 Cell 1051 Ω

2

stored energyc

RF

VR

Q


• As is usual in damped harmonic motion define a quality factor by

• Unloaded Quality Factor Q0 of a cavity

• Quantifies heat flow directly into cavity walls from AC resistance of superconductor, and wall heating from other sources.

2 energy stored in oscillation

energy dissipated in 1 cycleQ

0

stored energy

heating power in wallsRFQ

Unloaded Quality Factor


Loaded Quality Factor• When add the input coupling port, must account for the energy

loss through the port on the oscillation

• Coupling Factor

• It’s the loaded quality factor that gives the effective resonance width that the RF system, and its controls, seen from the superconducting cavity

• Chosen to minimize operating RF power: current matching (CEBAF, FEL), rf control performance and microphonics (SNS, ERLs)

0

1 1 total power lost 1 1

stored energytot L RF extQ Q Q Q

0 01 for present day SRF cavities, 1L

ext

Q QQ

Q


0 10 20 30 40Eacc [MV/m]

Q0

AC70AC72AC73AC78AC76AC71AC81Z83Z87

1011

109

1010

Q0 vs. Gradient for Several 1300 MHz Cavities

Courtesey: Lutz Lilje


Eacc vs. time

0

5

10

15

20

25

30

35

40

45

Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06

Eac

c[M

V/m

]

BCP

EP

10 per. Mov. Avg. (BCP)

10 per. Mov. Avg. (EP)

Courtesey: Lutz Lilje


RF Cavity Equations Introduction Cavity Fundamental Parameters RF Cavity as a Parallel LCR Circuit Coupling of Cavity to an rf Generator Equivalent Circuit for a Cavity with Beam Loading

• On Crest and on Resonance Operation • Off Crest and off Resonance Operation

Optimum Tuning Optimum Coupling

RF cavity with Beam and Microphonics Qext Optimization under Beam Loading and Microphonics

RF Modeling Conclusions


Introduction Goal: Ability to predict rf cavity’s steady-state response and

develop a differential equation for the transient response

We will construct an equivalent circuit and analyze it

We will write the quantities that characterize an rf cavity and relate them to the circuit parameters, for

a) a cavity

b) a cavity coupled to an rf generator

c) a cavity with beam


RF Cavity Fundamental Quantities

cV

Quality Factor Q0:

Shunt impedance Ra:

(accelerator definition); Va = accelerating voltage

Note: Voltages and currents will be represented as complex quantities, denoted by a tilde. For example:

where is the magnitude of and is a slowly varying phase.

2

in ohms per cellaa

diss

VR

P

00

Energy stored in cavity

Energy dissipated in cavity walls per radiandiss

WQ

P

c cV V

Re i ti tc c c cV t V t e V t V e


Equivalent Circuit for an rf Cavity

Simple LC circuit representing an accelerating resonator.

Metamorphosis of the LC circuit into an accelerating cavity.

Chain of weakly coupled pillbox cavities representing an accelerating cavity. Chain of coupled pendula as its mechanical analogue.


Equivalent Circuit for an rf Cavity An rf cavity can be represented by a parallel LCR circuit:

Impedance Z of the equivalent circuit:

Resonant frequency of the circuit:

Stored energy W:

11 1

Z iCR iL

0 1/ LC

21

2 cW CV

( ) i tc cV t v e


Equivalent Circuit for an rf Cavity Power dissipated in resistor R:

From definition of shunt impedance

Quality factor of resonator:

Note: For

21

2c

diss

VP

R

2a

adiss

VR

P 2aR R

00 0

diss

WQ CR

P

1

00

0

1Z R iQ

1

00 0

0

1 2Z R iQ

,

Wiedemann16.13


Cavity with External Coupling Consider a cavity connected to an rf

source A coaxial cable carries power from an rf

source to the cavity The strength of the input coupler is

adjusted by changing the penetration of the center conductor

There is a fixed output coupler, the transmitted power probe, which picks up power transmitted through the cavity


Cavity with External Coupling (cont’d)

Consider the rf cavity after the rf is turned off.Stored energy W satisfies the equation:

Total power being lost, Ptot, is:

Pe is the power leaking back out the input coupler. Pt is the power coming out the transmitted power coupler. Typically Pt is very small Ptot Pdiss + Pe

Recall

Similarly define a “loaded” quality factor QL:Energy in the cavity decays exponentially with time constant:

tot diss e tP P P P

0L

tot

WQ

P

tot

dWP

dt

00

diss

WQ

P

0

00

L

t

Q

L

dW WW W e

dt Q

0

LL

Q



Equation

suggests that we can assign a quality factor to each loss mechanism, such that

where, by definition,

Typical values for CEBAF 7-cell cavities: Q0=1x1010, Qe QL=2x107.

0 0

tot diss eP P P

W W

0

1 1 1

L eQ Q Q

0e

e

WQ

P



Define “coupling parameter”:

therefore

is equal to:

It tells us how strongly the couplers interact with the cavity. Large implies that the power leaking out of the coupler is large compared to the power dissipated in the cavity walls.

0

e

Q

Q

0

1 (1 )

LQ Q

e

diss

P

P

Wiedemann16.9


( ) kki tI t i e

Cavity Coupled to an rf Source The system we want to model:

Between the rf generator and the cavity is an isolator – a circulator connected to a load. Circulator ensures that signals coming from the cavity are terminated in a matched load.

Equivalent circuit:

RF Generator + Circulator Coupler Cavity Coupling is represented by an ideal transformer of turn ratio 1:k


Cavity Coupled to an rf Source

20

kg

g

II

k

Z k Z

( ) kki tI t i e

( )g gi tI t i e

By definition,

( )g gi tI t i e

20

gg

R R RZ

Z k Z

Wiedemann

16.1

WiedemannFig. 16.1


Generator Power When the cavity is matched to the input circuit, the power

dissipation in the cavity is maximized.

We define the available generator power Pg at a given generator

current to be equal to Pdissmax .

gI

2

max max 21 1or

2 2 16g

diss g diss a g g

IP Z P R I P

( )g gi tI t i e

Wiedemann16.6


Some Useful Expressions We derive expressions for W, Pdiss, Prefl, in terms of cavity

parameters 20 0

200 0

2 22 2 0

1

1

00

0

02

0 2 2 00

0

0

02

0

161 1

16 16

1 1

(1 )2

14

(1 )

For

4 1

(1 )1

cdiss

ca

g a ga g a g

c g TOT

TOTg

aTOT

g

Q Q VP

W Q VR

P R IR I R I

V I Z

ZZ Z

RZ iQ

QW P

Q

QW

2

0 0

0

2(1 )

gPQ


Some Useful Expressions (cont’d)

Define “Tuning angle” :

Recall:

022

0 0 0

0

4 1

(1 )1 2

(1 )

g

QW P

Q

0 00

0 0

0g2 2

0

tan 2 for

Q4 1= P

(1+ ) 1+tan

L LQ Q

W

0

0

2 2

4 1

(1 ) 1 tan

diss

diss g

WP

Q

P P

Wiedemann16.12


Some Useful Expressions (cont’d). Optimal coupling: W/Pg maximum or Pdiss = Pg

which implies Δω = 0, β = 1

this is the case of critical coupling

. Reflected power is calculated from energy conservation:

. On resonance:

Dissipated and Reflected Power

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

02

0

2

2

4

(1 )

4

(1 )

1

1

g

diss g

refl g

QW P

P P

P P

2 2

4 11

(1 ) 1 tan

refl g diss

refl g

P P P

P P


Equivalent Circuit for a Cavity with Beam Beam in the rf cavity is represented by a current generator.

Equivalent circuit:

Differential equation that describes the dynamics of the system:

RL is the loaded impedance defined as:

, i , / 2

C C LC R C

L

dv v dii C v L

dt R dt

(1 )a

L

RR


Equivalent Circuit for a Cavity with Beam (cont’d) Kirchoff’s law:

Total current is a superposition of generator current and beam current and beam current opposes the generator current.

Assume that have a fast (rf) time-varying component and a slow varying component:

where is the generator angular frequency and are complex quantities.

L R C g bi i i i i

2

20 002 2

c c Lc g b

L L

d v dv R dv i i

dt Q dt Q dt

, ,c g bv i i

i tc c

i tg g

i tb b

v V e

i I e

i I e

, ,c g bV I I


Equivalent Circuit for a Cavity with Beam (cont’d)

Neglecting terms of order we arrive at:

where is the tuning angle.

For short bunches: where I0 is the average beam

current.

2

2

1, ,c c

L

d V dI dV

dt dt Q dt

0 0(1 tan ) ( )2 4

c Lc g b

L L

dV Ri V I I

dt Q Q

0| | 2bI I

Wiedemann 16.19



0 0(1 tan ) ( )2 4

c Lc g b

L L

dV Ri V I I

dt Q Q

At steady-state:

are the generator and beam-loading voltages on resonance and are the generator and beam-loading voltages.

/ 2 / 2

(1 tan ) (1 tan )

or cos cos2 2

or cos cos

or

L Lc g b

i iL Lc g b

i ic gr br

c g b

R RV I I

i i

R RV I e I e

V V e V e

V V V

2

2

Lgr g

Lbr b

RV I

RV I

g

b

V

V



Note that:

0

2| | 2 for large

1

| |

gr g L g L

br L

V P R P R

V R I

Wiedemann16.16

Wiedemann16.20


Equivalent Circuit for a Cavity with Beam

As increases the magnitude of both Vg and Vb decreases while their phases rotate by .

cos igrV e

grV

Im( )gV

Re( )gV

cos

cos

ig gr

ib br

V V e

V V e



Cavity voltage is the superposition of the generator and beam-loading voltage.

This is the basis for the vector diagram analysis.

c g bV V V


cV

bI

brVbV

gV

decI

accI

b

Example of a Phasor Diagram

WiedemannFig. 16.3


On Crest and On Resonance Operation Typically linacs operate on resonance and on crest in order

to receive maximum acceleration. On crest and on resonance

where Va is the accelerating voltage.

grVcVbrV bI

a gr brV V V


More Useful Equations

We derive expressions for W, Va, Pdiss, Prefl in terms of and the loading parameter K, defined by: K=I0/2 Ra/Pg

From:

0

2| |

1

| |

gr g L

br L

a gr br

V P R

V R I

V V V

0

2

20

2

2

0 0

0

2

0 2

21

1

41

(1 )

41

(1 )

22 1

1

( 1) 2

( 1)

a g a

g

diss g

a a diss

a

g

refl g diss a refl g

KV P R

Q KW P

KP P

I V I R P

I V KK

P

KP P P I V P P


More Useful Equations (cont’d)

For large,

For Prefl=0 (condition for matching)

20

20

1( )

4

1( )

4

g a LL

refl a LL

P V I RR

P V I RR

0

2

0 0

0

and

4

Ma

L M

M Ma a

g M Ma

VR

I

I V V IP

V I


Example For Va=20 MV/m, L=0.7 m, QL=2x107 , Q0=1x1010 :

Power

I0 = 0 I0 = 100 A I0 = 1 mA

Pg 3.65 kW 4.38 kW 14.033 kW

Pdiss 29 W 29 W 29 W

I0Va 0 W 1.4 kW 14 kW

Prefl 3.62 kW 2.951 kW ~ 4.4 W


Off Crest and Off Resonance Operation Typically electron storage rings operate off crest in order

to ensure stability against phase oscillations. As a consequence, the rf cavities must be detuned off

resonance in order to minimize the reflected power and the required generator power.

Longitudinal gymnastics may also impose off crest operation operation in recirculating linacs.

We write the beam current and the cavity voltage as

The generator power can then be expressed as:

02

and set 0

b

c

ib

ic c c

I I e

V V e

2 220 0(1 )

1 cos tan sin4

c L Lg b b

L c c

V I R I RP

R V V

Wiedemann16.31


Off Crest and Off Resonance Operation

Condition for optimum tuning:

Condition for optimum coupling:

Minimum generator power:

0tan sinLb

c

I R

V

0opt 1 cosa

bc

I R

V

2opt

,minc

ga

VP

R

Wiedemann

16.36


RF Cavity with Beam and Microphonics

The detuning is now: 0 0

0 0

0

0tan 2 tan 2

where is the static detuning (controllable)

and is the random dynamic detuning (uncontrollable)

m

L L

m

f f fQ Q

f f

f

f

-10

-8-6

-4

-20

2

4

68

10

90 95 100 105 110 115 120

Time (sec)

Fre

quency (Hz)

Probability DensityMedium CM Prototype, Cavity #2, CW @ 6MV/m

400000 samples

0

0.05

0.1

0.15

0.2

0.25

-8 -6 -4 -2 0 2 4 6 8

Peak Frequency Deviation (V)

Pro

bab

ility

Den

sity


Qext Optimization under Beam Loading and Microphonics

Beam loading and microphonics require careful optimization of the external Q of cavities.

Derive expressions for the optimum setting of cavity parameters when operating under

a) heavy beam loading

b) little or no beam loading, as is the case in energy recovery linac cavities

and in the presence of microphonics.


Qext Optimization (cont’d)2 22 (1 )

1 cos tan sin4

c tot L tot Lg tot tot

L c c

V I R I RP

R V V

0

tan 2 L

fQ

f

where f is the total amount of cavity detuning in Hz, including static detuning and microphonics. Optimizing the generator power with respect to coupling gives:

where Itot is the magnitude of the resultant beam current vector in the cavity and tot is the phase of the resultant beam vector with respect to the cavity voltage.

2

20

0

( 1) 2 tan

where cos

opt tot

tot atot

c

fb Q b

f

I Rb

V


Qext Optimization (cont’d)

where:

To minimize generator power with respect to tuning:

0

0

tan 2 mL

f fQ

f

2 22 (1 )1 cos tan sin

4c tot L tot L

g tot totL c c

V I R I RP

R V V

00

0

tan2

ff b

Q

222

00

(1 )(1 ) 2

4c m

gL

V fP b Q

R f


Qext Optimization (cont’d) Condition for optimum coupling:

and In the absence of beam (b=0):

and

2

20

0

222

00

( 1) 2

1 ( 1) 22

mopt

opt c mg

a

fb Q

f

V fP b b Q

R f

2

00

22

00

1 2

1 1 22

mopt

opt c mg

a

fQ

f

V fP Q

R f


Problem for the Reader

Assuming no microphonics, plot opt and Pgopt as function

of b (beam loading), b=-5 to 5, and explain the results.

How do the results change if microphonics is present?


Example

ERL Injector and Linac:

fm=25 Hz, Q0=1x1010 , f0=1300 MHz, I0=100 mA, Vc=20 MV/m, L=1.04 m, Ra/Q0=1036 ohms per cavity

ERL linac: Resultant beam current, Itot = 0 mA (energy recovery)

and opt=385 QL=2.6x107 Pg = 4 kW per cavity.

ERL Injector: I0=100 mA and opt= 5x104 ! QL= 2x105 Pg = 2.08 MW per cavity!

Note: I0Va = 2.08 MW optimization is entirely dominated by beam loading.


RF System Modeling

To include amplitude and phase feedback, nonlinear effects from the klystron and be able to analyze transient response of the system, response to large parameter variations or beam current fluctuations

• we developed a model of the cavity and low level controls using SIMULINK, a MATLAB-based program for simulating dynamic systems.

Model describes the beam-cavity interaction, includes a realistic representation of low level controls, klystron characteristics, microphonic noise, Lorentz force detuning and coupling and excitation of mechanical resonances


RF System Model


RF Modeling: Simulations vs. Experimental Data

Measured and simulated cavity voltage and amplified gradient error signal (GASK) in one of CEBAF’s cavities, when a 65 A, 100 sec beam pulse enters the cavity.


Conclusions

We derived a differential equation that describes to a very good approximation the rf cavity and its interaction with beam.

We derived useful relations among cavity’s parameters and used phasor diagrams to analyze steady-state situations.

We presented formula for the optimization of Qext under beam loading and microphonics.

We showed an example of a Simulink model of the rf control system which can be useful when nonlinearities can not be ignored.


RF FocussingIn any RF cavity that accelerates longitudinally, because of Maxwell Equations there must be additional transverse electromagnetic fields. These fields will act to focus the beam and must be accounted properly in the beam optics, especially in the low energy regions of the accelerator. We will discuss this problem in greater depth in injector lectures. Let A(x,y,z) be the vector potential describing the longitudinal mode (Lorenz gauge)

tcA

1

2

22

2

22

cA

cA


...,

...,2

10

210

rzzzr

rzAzAzrA zzz

For cylindrically symmetrical accelerating mode, functional form can only depend on r and z

Maxwell’s Equations give recurrence formulas for succeeding approximations

12

2

21

22

1,2

2

2

1,2

2

2

2

nn

n

nznz

zn

cdz

dn

Acdz

AdAn


Gauge condition satisfied when

nzn

c

i

dz

dA

in the particular case n = 0

00

c

i

dz

dAz

Electric field is

t

A

cE

�

1


So the magnetic field off axis may be expressed directly in terms of the electric field on axis

00,0 zz A

c

i

dz

dzE

102

2

20

2

4,0 zzz

z AAcdz

AdzE

c

i

And the potential and vector potential must satisfy

zEc

rirAB zz ,0

2 2 1


And likewise for the radial electric field (see also )

dz

zdErzrE z

r

,0

2 2 1

Explicitly, for the time dependence cos(ωt + δ)

tzEtzrE zz cos,0,,

tdz

zdErtzrE z

r cos,0

2,,

tzE

c

rtzrB z sin,0

2,,

0 E


Motion of a particle in this EM field

B

c

VEe

dt

md

V

''

'sin'2

''

'cos'

'

2

'

z

- z

dz

ztzGc

zxz

ztdz

zdGzx

zz

zz

xx


The normalized gradient is

2

0,

mc

zeEzG z

and the other quantities are calculated with the integral equations

z

zzz dzzt

z

zGzz ''cos

'

'

z

dzztzGz ''cos'

z

zzz cz

dz

c

zzt

'

'lim 0

0


These equations may be integrated numerically using the cylindrically symmetric CEBAF field model to form the Douglas model of the cavity focussing. In the high energy limit the expressions simplify.

z

a zz

x

z

a z

x

dzztzz

zGzxazax

dzzz

zzaxzx

''cos''

'

2

'

'''

''

2


Transfer Matrix

E

EI

L

E

E

TG

G

21

4

21

For position-momentum transfer matrix

dzczzG

dzczzGI

/sinsin

/coscos

222

222


Kick Generated by mis-alignment

E

EG

2


Damping and AntidampingBy symmetry, if electron traverses the cavity exactly on axis, there is no transverse deflection of the particle, but there is an energy increase. By conservation of transverse momentum, there must be a decrease of the phase space area. For linacs NEVER use the word “adiabatic”

0

Vtransverse dt

md

x xz z


Conservation law applied to angles

, 1

/ /

x y z

x x z x y y z y

zx x

z

zy y

z

zz z

zz z


Phase space area transformation

zx x

z

zy y

z

dx d z dx dz z

dy d z dy dz z

Therefore, if the beam is accelerating, the phase space area after the cavity is less than that before the cavity and if the beam is decelerating the phase space area is greater than the area before the cavity. The determinate of the transformation carrying the phase space through the cavity has determinate equal to

Det zcavity

z

Mz z


By concatenation of the transfer matrices of all the accelerating or decelerating cavities in the recirculated linac, and by the fact that the determinate of the product of two matrices is the product of the determinates, the phase space area at each location in the linac is

0

00

000

yz

zy

xz

zx

ddyzz

zddy

ddxzz

zddx

Same type of argument shows that things like orbit fluctuations are damped/amplified by acceleration/deceleration.


Transfer Matrix Non-Unimodular

edeterminat daccumulateby normalize and

)before! (as part" unimodular" track theseparatelycan

detdetdet

!unimodular det

212

2

1

1

21

MPMPM

M

M

M

M

MMP

MPM

MMP

MMM

tot

tottot

tot


ENERGY RECOVERY WORKSGradient modulator drive signal in a linac cavity measured without energy recovery (signal level around 2 V) and with energy recovery (signal level around 0).

GASK

-0.5

0

0.5

1

1.5

2

2.5

-1.00E-04 0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04

Time (s)

Vo

lta

ge

(V

)

Courtesy: Lia Merminga

accelerator physics particle acceleration

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