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Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020

Accelerator PhysicsParticle Acceleration

Suba De Silva, G. A. Krafft, S. A. Bogacz,

and B. Dhital

Old Dominion University / Jefferson Lab

Lecture 13


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

• Higher Order Modes

– HOM excitation by a beam

– Wake potential

– HOM damping

• RF Focusing

• Betatron Damping and Anti-damping

2


Terminology

• Use of multi-cell rf cavities to accelerate particles

3

1 CEBAF Cavity

1 DESY Cavity

“Cells”

5 Cell Cavity

9 Cell Cavity


Typical cell longitudinal dimension: λRF/2

Phase shift between cells: π

Cavities 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) are

optimized around a 7 cell design

4

Terminology


On Axis Electric Field

2, cos 0,0, cos 2 /

2 / c.c. = 2 /

2

RF z RF

i

z RF

c z RF

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

eE eV eE


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 /

2 / c.c. = 2 /

2

RF z RF

i

z RF

c z RF

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

eE eV eE

accE MV/m cV

L

6


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 (using “accelerator” definition)

• 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 energy

c

RF

VR

Q

7


• 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 walls

RFQ

Unloaded Quality Factor

8


0 10 20 30 40

Eacc [MV/m]

Q0

AC70AC72AC73AC78AC76AC71AC81Z83Z87

1011

109

1010

Q0 vs. Gradient for Several 1300 MHz

Cavities

Courtesey: Lutz Lilje

9


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

Ea

cc[M

V/m

]

BCP

EP

10 per. Mov. Avg. (BCP)

10 per. Mov. Avg. (EP)

Courtesey: Lutz Lilje

10


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, 1

L

ext

Q QQ

Q

11



12

0

1 1 total power lost 1 1

stored energytot L RF extQ Q Q Q

( ),e ext inputP Q

tot diss e tP P P P

( ),t ext pickupP Q

Transmitted

power

Input power

0,dissP Q

Dissipated

power



• 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

13


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

d) include microphonics

14


RF Cavity Fundamental Quantities

cV

Quality Factor Q0:

Shunt impedance Rsh (accelerator definition);

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

csh

diss

VR

P

00

Energy stored in cavity

Energy dissipated in cavity walls per radiandiss

WQ

P

c cV V

Re

i ti t

c c c cV t V t e V t V e

15


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.

16


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

2cW CV

( ) i tc cV t V e

17


Equivalent Circuit for an RF Cavity

Average power dissipated in resistor R:

From definition of shunt impedance

Quality factor of resonator:

Note:

2

csh

diss

VR

P 2shR R

00 0

diss

WQ CR

P

1

00

0

1Z R iQ

0

1

00

0

For

1 2Z R iQ

,

Wiedemann

19.11

2

2

cdiss

VP

R

18


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

19


Cavity with External Coupling

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

Energy in the cavity decays exponentially with time constant:

tot diss e tP P P P

0L

tot

WQ

P

/ totdW dt P

00

diss

WQ

P

0

00

L

t

Q

L

dW WW W e

dt Q

0/L LQ

20


Decay rate 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

21


Have defined “coupling parameter”:

and therefore

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

1 (1 )

LQ Q

0e

diss e

P Q

P Q

Wiedemann

19.9

22


Cavity Coupled to an RF Source

Z0

Z0

The system we want to model. A generator producing power Pg

transmits power to cavity through transmission line with characteristic impedance Z0

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

23


Transmission Lines

L

Z

L L

C C C

1nVnV

NV…2nI

NI

1nInI

1 1

1

n n n

n n n

V V i LI

I I i CV

Inductor Impedance and Current Conservation

24


Transmission Line Equations

• Standard Difference Equation with Solution

• Continuous Limit ( N → ∞)

0 0

0 0 0 0

/2 /2

,

1 1

2sin /2

/ /

in in

n n

i i i

i i

V V e I I e

V e i LI e I e i CV

LC

V L CI e V L CI e

( ) ( ) ( )

0 0 0 0

( ) ( ) ( )

0 0 0 0

1/

, /

, /

i t kx i t kx i t kx

i t kx i t kx i t kx

LC k L C v L C

V x t V e L C I e Z I e

V x t V e L C I e Z I e

25



1: k

0Z

,V I

,V I

k I Ii

Equivalent Circuit

RF Generator + Circulator Coupler Cavity

Coupling is represented by an ideal (lossless) transformer of turns ratio 1:k

26


From transmission line equations, forward power from RF source

Reflected power to circulator

Transformer relations

Considering zero forward power case and definition of β


2

0/ 2V Z

2

0/ 2V Z

2

0

/ / 2 / /

c k

c k c

V kV k V V

i i k I I k I k V k Z

2 2 2

0 0/ 2 / / 2 /cV Z V R R k Z

27



2( )

Re

g

g

i t

I t

ki t

i e

2

0

g

g

R R RZ

Z k Z

Wiedemann

19.1

Wiedemann

Fig. 19.1

0 0

1 1

2 2

c cc c

V VV kZ i V kZ i

k k

1 1 1 1 2

1

aL eff

eff g

RR R

R R Z R

Loaded cavity looks like

Effective and loaded resistance

Solving transmission line equations

28


Powers Calculated

Forward Power

Reflected Power

Power delivered to cavity is

as it must by energy conservation!

2 22 22 220 0

0 22

0 0 0

1 111 1

8 4 1

c crefl L

a

V VP iQ Q

Z k R

2 222 2

20 002

0 0 0

111 1 1

8 4

c cg L

a

V VP iQ Q

Z k R

2 22 2

2

1 11

4 1

c cg refl diss

a a

V VP P P

R R

29


Some Useful Expressions

Total energy W, in terms of cavity parameters

Total impedance

0

0

2 2

2 0

0

0

2

0 2 2 00

0

00 22

00

0

4 1

11

14

(1 )

4 1

(1 )1 2

diss

g diss

L

g

g

L

QP

W

P PQ

QW P

Q

QW P

Q

1

1

00

0

1 1

(1 )2

TOT

g

aTOT

ZZ Z

RZ iQ

30


When Cavity is Detuned

Define “Tuning angle” :

Note that:

0 00

0 0

0

2 2

0

tan 2 for

4 1=

(1+ ) 1+tan

L L

g

Q Q

QW P

2 2

4 1

(1 ) 1 tandiss gP P

Wiedemann

19.13

31


Optimal β Without Beam

. Optimal coupling: W/Pg maximum or Pdiss = Pg

which implies for Δω = 0, β = 1

This is the case called “critical” coupling

. Reflected power is consistent with 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

0

2

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

32


Equivalent Circuit: Cavity with Beam

Beam through the RF cavity is represented by a current generator

that interacts with the total impedance (including circulator).

Equivalent circuit:

ig the current induced by generator, ib beam current

Differential equation that describes the dynamics of the system:

, , / 2

c c LC R c

L

dV V dii C i V L

dt R dt

2

20 002 2

c c Lc g b

L L

d V dV R dV i i

dt Q dt Q dt

33


Cavity with Beam

Kirchhoff's law:

Total current is a superposition of generator current and beam

current and beam current opposes the generator current.

Assume that voltages and currents are represented by complex

phasors

where is the generator angular frequency and

are complex quantities.

L R C g bi i i i i

Re

Re

Re

i t

c c

i t

g g

i t

b b

V t V e

i t i e

i t i e

, ,c g bV i i

34


Voltage for a Cavity with Beam

Steady state solution

where is the tuning angle.

Generator current

For short bunches: where I0 is the average

beam current.

(1 tan ) ( )2

Lc g b

Ri V i i

0| | 2bi I

Wiedemann 19.19

0

2 22| | 2 2 4

g g g

g

a

P P Pi I

k Z R R

35



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 i

c 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

36



Note that:

0

2| | 2 for large

1

| |

gr g L g L

br L

V P R P R

V R I

Wiedemann

19.16

Wiedemann

19.20

37



As increases, the magnitudes of both Vg and Vb

decrease while their phases rotate by .

cos i

grV e

grV

Im( )gV

Re( )gV

cos

cos

i

g gr

i

b br

V V e

V V e

38


cV

bI

brVbV

gV

decI

accI

b

Example of a Phasor Diagram

Wiedemann

Fig. 19.3

39


Typically linacs operate on resonance and on crest in order

to receive maximum acceleration.

On crest and on resonance

where Vc is the accelerating voltage.

On Crest/On Resonance Operation

grVcV

brVbI

c gr brV V V

40


More Useful Equations

We derive expressions for W, Va, Pdiss, Prefl in terms of and the

loading parameter K, defined by: K=I0Ra/(2 Pg )

From:

0

2| |

1

| |

gr g L

br L

c gr br

V P R

V R I

V V V

0

2

2

0

2

2

0 0

0

2

0 2

21

1

41

(1 )

41

(1 )

22 1

1

( 1) 2

( 1)

c g L

g

diss g

a a diss

c

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

41


Clarifications

On Crest,

Off Crest with Detuning

000

2 2 11

1 1 2 2

aLc g L L g L

g L g

R IR IV P R R I P R K

P R P

2 22

0 0

0

2 222

0 0 0

2

=2 8

2 1 tan 2 cos sin

1 cos tan sin2

g

g

cg b b

L

c L Lg b b

L c c

Z I Z k iP

Vi i I i

R

Z k V I R I RP

R V V

42


More Useful Equations

For large,

For Prefl=0 (condition for matching)

2

0

2

0

1( )

4

1( )

4

g c L

L

refl c L

L

P V I RR

P V I RR

0

2

0 0

0

and

4

M

cL M

M M

c cg M M

c

VR

I

I V V IP

V I

43


Example

For Vc=14 MV, 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

I0Vc 0 W 1.4 kW 14 kW

Prefl 3.62 kW 2.951 kW ~ 4.4 W

44


Off Crest/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 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

i

b

i

c c c

I I e

V V e

2 22

0 0(1 )1 cos tan sin

4

c L Lg b b

L c c

V I R I RP

R V V

Wiedemann

19.30

45


Off Crest/Off Resonance Operation

Condition for optimum tuning:

Condition for optimum coupling:

Minimum generator power:

0tan sinLb

c

I R

V

0opt 1 cosa

b

c

I R

V

2

opt

,min

c

g

a

VP

R

Wiedemann

19.36

46


Bettor Phasor Diagram

Off crest, synchrotron phase

gi

cos

icVe

1 tancos

i

c gc

b

VV i e ii

bi

,g opti

tan sinc b sV i

cVs

s

47


C75 Power Estimates

G. A. Krafft

48


12 GeV Project Specs

7-cell, 1500 MHz, 903 ohms

0

2

4

6

8

10

12

14

16

18

20

1 10 100

Qext (106)

P (

kW

)

21.2 MV/m, 460 uA, 25 Hz, 0 deg

21.2 MV/m, 460 uA, 20 Hz, 0 deg

21.2 MV/m, 460 uA, 15 Hz, 0 deg

21.2 MV/m, 460 uA, 10 Hz, 0 deg

21.2 MV/m, 460 uA, 0 Hz, 0 deg

13 kW

Delayen

and

Krafft

TN-07-29

460 µA*15 MV=6.8 kW49


7-cell, 1500 MHz, 903 ohms

0

2

4

6

8

10

12

14

16

18

20

1 10 100

Qext (106)

P (

kW

)

25.0 MV/m, 460 uA, 25 Hz, 0 deg

24.0 MV/m, 460 uA, 25 Hz, 0 deg

23.0 MV/m, 460 uA, 25 Hz, 0 deg

22.0 MV/m, 460 uA, 25 Hz, 0 deg

21.0 MV/m, 460 uA, 25 Hz, 0 deg

50


Assumptions

• Low Loss R/Q = 903*5/7 = 645 Ω

• Max Current to be accelerated 460 µA

• Compute 0 and 25 Hz detuning power curves

• 75 MV/cryomodule (18.75 MV/m)

• Therefore matched power is 4.3 kW

(Scale increase 7.4 kW tube spec)

• Qext adjustable to 3.18×107(if not need more RF

power!)

51


0

2000

4000

6000

8000

10000

12000

14000

1 10 100

Power vs. Qext

Qext (×106)

Pg (kW)

25 Hz detuning

0 detuning3.2×107

52


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

-2

0

2

4

6

8

10

90 95 100 105 110 115 120

Time (sec)

Fre

qu

en

cy

(H

z)

Probability Density

Medium 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

ba

bilit

y D

en

sit

y

53


Qext Optimization with Microphonics2 2

2 (1 )1 cos tan sin

4

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

2

0

0

( 1) 2 tan

where cos

opt tot

tot atot

c

fb Q b

f

I Rb

V

54


Correct Static Detuning

To minimize generator power with respect to tuning:

The resulting power is

00

0

tan2

tot

ff b

Q

22

2

0

0

1(1 ) 2

4

c mg

a

V fP b Q

R f

22

2 2

0

0

2

11 2 1 2

4

12

c mg opt opt

a opt

copt

a

V fP b b Q

R f

Vb

R

55


Optimal Qext and Power

Condition for optimum coupling:

and

In the absence of beam (b=0):

and

2

2

0

0

22

2

0

0

( 1) 2

1 ( 1) 22

mopt

opt c mg

a

fb Q

f

V fP b b Q

R f

2

0

0

22

0

0

1 2

1 1 22

mopt

opt c mg

a

fQ

f

V fP Q

R f

56


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?

57


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.

58


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

59


RF System Model

60


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.

61


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.

62


Higher Order Modes (HOMs) Excitation

• HOMs are excited due to beam cavity interaction

• As bunch traverses a cavity, it deposits electromagnetic energy,

which is described in terms of wakefields

• Wakefields, in turn, can be presented as a sum of cavity

eigenmodes (fundamental and HOMs)

• Various components such as vacuum chamber, cavities,

bellows, dielectric coated pipes, and other obstacles that beam

passes through generates wakefields

• If a charge passes a cavity exactly on axis, it excites only

monopole modes

• Wakefields can act back on the beam and lead to instabilities

– This may limit the achievable current per bunch, the total

current, or even both

63


Beam Cavity Interaction

64

The charge bunch leave a trail of wakefields that corresponds

to HOMs in the cavity.


Beam Cavity Interaction

• If a charge passes a cavity exactly on axis, it excites only

monopole modes Longitudinal wakefield

– For a point charge this excitation depends only on the

amount of charge and the cavity shape.

– Subsequent bunches may be affected by these fields

and at high beam currents one must consider beam

instabilities and additional heating of accelerator

components.

– Causes energy loss and energy spread within the bunch

• If a charge is off axis, it excites dipole and quadrupole

modes Transverse wakefield

– Deflects particle trajectory

– Leads to beam instabilities

65


Wakefields

66

For a relativistic point

charge (q) that traverses a

cavity parallel to the z-axis


Longitudinal Wake Function and Loss Factor• For a bunch with charge Q:

• Energy loss (gain) by a single particle in the bunch

• Total energy loss by the bunch:

• Loss factor:

67

Gaussian charge distribution

o Can be represented as a sum of individual loss factors of cavity modes

o R/Q is in circuit definition

• Power loss due to HOMs: HOM l avgP k NqI


Impedance

• Impedance is the Fourier transform of the wakefield potential

• Damp impedance with HOM couplers and absorbers to reduce

impedance below threshold

68


HOM Damping Methods

• Antenna couplers

• Waveguide dampers on

beam tubes

• Absorbers in cavity-

interconnecting beam

tubes (or cryomodule-

connecting beam tubes)

• Coupling through

Fundamental Power

Coupler (FPC)

69

Coaxial coupler


RF Focussing

In 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

70


...,

...,

2

10

2

10

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

2

1

22

1,2

2

2

1,

22

2

2

nn

n

nz

nz

zn

cdz

dn

Acdz

AdAn

71


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

72


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

2

0

2

4,0 zzz

z AAcdz

AdzE

c

i

And the potential and vector potential must satisfy

zEc

rirAB zz ,0

2 2 1

73


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,,

tzEc

rtzrB z sin,0

2,,

0 E

74


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

78


The normalized gradient is

2

0,

mc

zeEzG z

and the other quantities are calculated with the integral equations

z

z

zz dzztz

zGzz ''cos

'

'

z

dzztzGz ''cos'

z

zzz cz

dz

c

zzt

'

'lim 0

0

79


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

80


Transfer Matrix

E

EI

L

E

E

TG

G

21

4

21

For position-momentum transfer matrix

dzczzG

dzczzGI

/sinsin

/coscos

222

222

81


Kick Generated by mis-alignment

E

EG

2

82


Damping and Antidamping

By 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

83


Conservation law applied to angles

, 1

/ /

x y z

x x z x y y z y

z

x x

z

z

y y

z

zz z

zz z

84


Phase space area transformation

z

x x

z

z

y 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

z

cavity

z

Mz z

85


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

000

000

y

z

zy

x

z

zx

ddyzz

zddy

ddxzz

zddx

Same type of argument shows that things like orbit fluctuations

are damped/amplified by acceleration/deceleration.

86


Transfer Matrix Non-Unimodular

edeterminat daccumulateby normalize and

)before! (as part" unimodular" track theseparatelycan

detdetdet

!unimodular

det

21

2

2

1

1

21

MPMPM

M

M

M

M

MMP

MP

M

MMP

MMM

tot

tottot

tot

87


LCLS II

Subharmonic

Beam Loading


Subharmonic Beam Loading

• Under condition of constant incident RF power, there is a

voltage fluctuation in the fundamental accelerating mode when

the beam load is sub harmonically related to the cavity

frequency

• Have some old results, from the days when we investigated

FELs in the CEBAF accelerator

• These results can be used to quantify the voltage fluctuations

expected from the subharmonic beam load in LCLS II

89


CEBAF FEL Results

Krafft and Laubach, CEBAF-TN-0153 (1989)

Bunch repetition

time τ1 chosen to

emphasize physics

90


Model

• Single standing wave accelerating mode. Reflected power

absorbed by matched circulator.

• Beam current

• (Constant) Incident RF (β coupler coupling)

2 2 cosg cV ZP t

22

2

bc c c cc c

L c

d ZId V dV dVV

dt Q dt Q dt dt

b

l l

I t q t l I t l

91


Analytic Method of Solution

• Green function

• Geometric series summation

• Excellent approximation

2ˆ ˆexp sin 1 1/ 42

c

c c c L

L

t tG t t t t Q

Q

/22cos cos

1c Lt n Q

c c L c

RP RV t t IQ e t

Q

/2

/ /22ˆ ˆcos cos cos 1

1 2

c L

c L c L

t n Q

Q Qcc c c c

RP Rq eV t t e t n e t n

Q D

92


2

1

RP

L

RIQ

Q

cV

cV

Phasor Diagram of Solution

93


Single Subharmonic Beam

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000 1200

Voltage Deviation1

2

c

c

V

qR

Q

94


Beam Cases

• Case 1

• Case 2

• Case 3

Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µ A)

HXR 1 MHz 145 145

Straight 10 kHz 145 1.45

SXR 1 MHz 145 145


HXR 1 MHz 295 295


SXR 100 kHz 20 2


HXR 100 kHz 295 295


SXR 100 kHz 20 2

95


Case 1

Straight Current

HXR Current

SXR Current

Total Voltage cV t

t2 µsec

100 µsec

96


Case 1Volt

s

t/20 nsec

. . .

ΔE/E=10-4

97


Case 2Volt

s

t/20 nsec

. . .ΔE/E=10-4

98


Case 3

t/100 nsec

Volt

s

ΔE/E=10-4

99


Summaries of Beam Energies

• Case 1

• Case 2

• Case 3

Beam Minimum (kV) Maximum (kV) Form

HXR 0.690 1.814 Linear 10

Straight 1.574 1.574 Constant

SXR 0.753 1.876 Linear 10


HXR 0.631 1.943 Linear 100


SXR 0.788 1.911 Linear 10


HXR 0.615 1.222 Linear 100


SXR 0.618 1.225 Linear 100

For off-crest cavities, multiply by cos φ

100


Summary

• Fluctuations in voltage from constant intensity subharmonic

beams can be computed analytically

• Basic character is a series of steps at bunch arrival, the step

magnitude being (R/Q)πfcq

• Energy offsets were evaluated for some potential operating

scenarios. Spread sheet provided that can be used to

investigate differing current choices

101


Case I’

Zero Crossing

Left Deflection

Right Deflection

Total Voltage

cV t

t20 µsec

100 µsec

102


Case 2 (100 kHz contribution minor)

Zero Crossing

Left Deflection

Right Deflection

Total Voltage

cV t

t2 µsec

100 µsec

103


Case 3

Zero Crossing

Left Deflection

Right Deflection

Total Voltage cV t

t20 µsec

100 µsec

104

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