accelerator physics · graduate accelerator physics fall 2015 q ext optimization with microphonics...

Graduate Accelerator Physics Fall 2015

Accelerator Physics Particle Acceleration

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 4


Clarifications from Last Time

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


Off Crest/Off Resonance Power

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

16.31


Optimal Detuning and Coupling

Condition for optimum tuning with beam:

Condition for optimum coupling with beam:

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

16.36


C75 Power Estimates

G. A. Krafft


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 muA*15 MV=6.8 kW


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


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!)


0

2000

4000

6000

8000

10000

12000

14000

1 10 100

Power vs. Qext

Qext (×106)

Pg (kW)

25 Hz detuning

0 detuning 3.2×107


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


Qext Optimization with Microphonics 2 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


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


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


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 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


...,

...,

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


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

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


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


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

z

zz dzztz

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 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


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


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


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.


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


Solenoid Focussing

Can also have continuous focusing in both transverse directions by applying solenoid

magnets:

B z

z


Busch’s Theorem

For cylindrical symmetry magnetic field described by a vector potential:

Conservation of Canonical Momentum gives Busch’s Theorem:

1ˆ, , is nearly constant

0, 0,,

2 2

z

z z

r

A A z r B rA z rr r

B r z r B r z rA z r B

2

22

for particle with 0 where 0, 0

2 2

z

czLarmor

P mr qrA const

B P

qr Bmr

Beam rotates at the Larmor frequency which implies coupling


Radial Equation

2 2

2

2 2

2 2

2 2 22

2

thin lens focal length

1 weak compared to quadrupole for high

4

L z L

L

z

z

z

dmr mr qr B mr

dt

kc

e B dz

f m c

If go to full ¼ oscillation inside the magnetic field in the “thick” lens case, all particles

end up at r = 0! Non-zero emittance spreads out perfect focusing!

x

y


Larmor’s Theorem

This result is a special case of a more general result. If go to frame that rotates with the

local value of Larmor’s frequency, then the transverse dynamics including the

magnetic field are simply those of a harmonic oscillator with frequency equal to the

Larmor frequency. Any force from the magnetic field linear in the field strength is

“transformed away” in the Larmor frame. And the motion in the two transverse

degrees of freedom are now decoupled. Pf: The equations of motion are

2

2

2 2

2

2 2

2

2

2-D Harmonic Oscillator

z

L L z L z

L

dmr mr qr B

dt

mr qA cons P

dmr mr mr mr qr B qr B

dt

mr P

dmr mr mr

dt

mr P

accelerator physics · graduate accelerator physics fall 2015 q ext optimization with microphonics...

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