Fundamentals of Accelerator 2012 - USPASuspas.fnal.gov/materials/12MSU/Lecture Day 3.pdf · o o Energy stored Time average power ... surface! " Q = L C R = #$ $ o % & ’ ( ) * +1

US Particle Accelerator School

Fundamentals of Accelerator2012Day 3

William A. BarlettaDirector, US Particle Accelerator School

Dept. of Physics, MITEconomics Faculty, University of Ljubljana


Lumped circuit analogy of resonant cavity

V(t)

I(t)

CL

R

!

Z(") = j"C + ( j"L + R)#1[ ]#1

!

The resonant frequency is "o = 1LC!

Z(") =1

j"C + ( j"L + R)#1

=( j"L + R)

( j"L + R) j"C +1=

( j"L + R)

(1#" 2LC) + j"RC


Q of the lumped circuit analogy

The width is

!

"#

#0

=R

L /.C

!

Z(") ~ 1#" 2

"o

2

$

% &

'

( )

2

+ ("RC)2

*

+ , ,

-

. / /

#1

Converting the denominator of Z to a real number we see that


More basics from circuits - Q

!

Q = "o o Energy stored

Time average power loss

!

= 2" o Energy stored

Energy per cycle

!

E = 1

2L I

oIo

* and

!

P = i2(t) R =

1

2IoIo

*Rsurface

!

" Q =

LC

R =

#$

$o

%

& '

(

) *

+1


Translate circuit model to a cavity model:Directly driven, re-entrant RF cavity

Outer region: Large, single turn Inductor

Central region: Large plate Capacitor

Beam (Load) current

I

B

EDisplacement current

Wall current

a

Rd

!

L =µo"a

2

2"(R + a)

!

C = "o

#R2

d

!

"o

= 1LC

= c2((R + a)d

#R2a2$

% & '

( )

12

Q – set by resistance in outer region

!

Q =LCR

Expanding outer region raises Q

Narrowing gap raises shunt impedance

Source: Humphries, Charged ParticleAccelerators


Properties of the RF pillbox cavity

We want lowest mode: with only Ez & Bθ

Maxwell’s equations are:

and

Take derivatives

==>

!

1

r

"

"rrB#( ) =

1

c2

"

"tEz

!

"

"rEz =

"

"tB#

!

"

"t

1

r

"

"rrB#( )

$

% & '

( ) ="

"t

"B#"r

+B#

r

$

% & '

( ) =1

c2

" 2Ez"t 2

!

"

"r

"Ez

"r="

"r

"B#

"t

!

" 2Ez

"r 2+1

r

"Ez

"r=1

c2

" 2Ez

"t 2

d

Ez

b

Bθ

!

"walls

=#


For a mode with frequency ω

Therefore,

(Bessel’s equation, 0 order)

Hence,

For conducting walls, Ez(R) = 0, therefore

!

Ez r, t( ) = Ez (r) ei"t

!

" " E z +" E z

r+#

c

$

% &

'

( )

2

Ez = 0

!

Ez (r) = Eo Jo"

cr

#

$ %

&

' (

!

2"f

cb = 2.405


E-fields & equivalent circuit: Ton1o mode

Ez

Bθ

Rel

ativ

e in

tens

ity

r/R

T010

C L


E-fields & equivalent circuitsfor To2o modes

T020


E-fields & equivalent circuitsfor Tono modes

T030

T0n0 has n coupled, resonant

circuits; each L & C reduced by 1/n


Simple consequences of pillbox model

L

Ez

R

Bθ

Increasing R lowers frequency ==> Stored Energy, E ~ ω-2

E ~ Ez2

Beam loading lowers Ez for thenext bunch

Lowering ω lowers the fractionalbeam loading

Raising ω lowers Q ~ ω -1/2

If time between beam pulses,Ts ~ Q/ω

almost all E is lost in the walls


The beam tube makes field modes(& cell design) more complicated

Ez

Bθ

Peak E no longer on axis Epk ~ 2 - 3 x Eacc

FOM = Epk/Eacc

ωo more sensitive to cavitydimensions Mechanical tuning & detuning

Beam tubes add length & €’sw/o acceleration

Beam induced voltages ~ a-3

Instabilities

a


Cavity figures of merit


Figure of Merit: Accelerating voltage

The voltage varies during time that bunch takes to cross gap reduction of the peak voltage by Γ (transt time factor)

!

" =sin #

2( )#

2

where # =$d%c

2TrfFor maximum acceleration ==> Γ = 2/π

dV

t

Epk


Figure of merit from circuits - Q

!

E =µo

2H

v

"2

dv = 1

2L I

oIo

*

!

P = Rsurf

2H

s

"2

ds = 1

2IoIo

*Rsurf

!

" Q =

LC

Rsurf

= #$

$o

%

& '

(

) *

+1

!

Q = "o o Energy stored

Time average power loss=

2# o Energy stored

Energy lost per cycle

!

!

Rsurf =1

Conductivity o Skin depth~ "1/ 2


Measuring the energy stored in the cavityallows us to measure Q

We have computed the field in the fundamental mode

To measure Q we excite the cavity and measure the E fieldas a function of time

Energy lost per half cycle = UπQ

Note: energy can be stored in the higher order modes thatdeflect the beam

!

U = dz0

d

" dr2#r 0

b

" $Eo

2

2

%

& '

(

) * J1

2(2.405r /b)

= b2d $E

o

2/2( ) J1

2(2.405)


Keeping energy out of higher order modes

d/b

ωb/c

0 1 2 3 4

TM020

TM010

TE011

TE111Source bandwidth

(green)

Dependence of mode frequency on cavity geometry

Choose cavity dimensions to stay far from crossovers

10

5

1

TE111 modeEnd view

Side view


Figure of merit for accelerating cavity:Power to produce the accelerating field

Resistive input (shunt) impedance at ωο relates power dissipated in walls toaccelerating voltage

Linac literature commonly defines “shunt impedance” without the “2”

Typical values 25 - 50 MΩ

!

Rin

= V

2(t)

P =

Vo

2

2P = Q L

C

!

Rin = Vo

2

P ~

1

Rsurf


Computing shunt impedance

!

P = Rsurf

2H

s

"2

ds

!

Rin

= Vo

2

P

!

Rsurf =µ"

2# dc

= $Zo

%skin

&rf where Zo =

µo

'o= 377(

The on-axis field E and surface H are generally computed with acomputer code such as SUPERFISH for a complicated cavity shape


To make a linacUse a series of pillbox cavities

Power the cavities so that Ez(z,t) = Ez(z)eiωt


Improve on the array of pillboxes?

Return to the picture of the re-entrant cavity

Nose cones concentrate Ez near beam for fixed stored energy

Optimize nose cone to maximize V2; I.e., maximize Rsh/Q

Make H-field region nearly spherical; raises Q & minimizesP for given stored energy


In warm linacs “nose cones” optimize the voltageper cell with respect to resistive dissipation

Thus, linacs can be considered to bean array of distorted pillbox cavities…

!

Q =LC

R surface

Rf-power in

a

Usually cells are feed in groups not individually…. and


Linacs cells are linked to minimize cost

==> coupled oscillators ==>multiple modes

Zero mode π mode


Modes of a two-cell cavity


9-cavity TESLA cell


Example of 3 coupled cavities

!

x j = i j 2Lo and " = normal mode frequency


Write the coupled circuit equationsin matrix form

Compute eigenvalues & eigenvectors to find the three normal modes

!

Lxq =1

"q

2xq where

!

L =

1/"o

2k /"

o

20

k /2"o

21/"

o

2k /2"

o

2

0 k /"o

21/"

o

2

#

$

% % %

&

'

( ( ( and xq =

x1

x2

x3

#

$

% % %

&

'

( ( (

!

Mode q = 0 : zero mode "0 =#

o

1+ k x

0=

1

1

1

$

%

& & &

'

(

) ) )

!

Mode q =1: "/2 mode #1 =$o x1 =

1

0

%1

&

'

( ( (

)

*

+ + +

!

Mode q = 2 : " mode #2 =$

o

1% k x2 =

1

%1

1

&

'

( ( (

)

*

+ + +


For a structure with N coupled cavities

==> Set of N coupled oscillators N normal modes, N frequencies

From the equivalent circuit with magnetic coupling

where B = bandwidth (frequency difference betweenlowest & high frequency mode)

Typically accelerators run in the π-mode!

"m

="

o

1# Bcosm$

N

%

& '

(

) *

1/ 2+"

o1+ Bcos

m$

N

%

& '

(

) *


Magnetically coupled pillbox cavities


5-cell π-mode cell with magnetic coupling

The tuners change the frequencies by perturbing wall currents ==> changes the inductance==> changes the energy stored in the magnetic field

!

"#o

#o

="U

U


Dispersion diagram for 5-cell structure

!

Emcell n( ) = A

nsin

m" (2n #1)

2N

$

% &

'

( )


Schematic of energy flowin a standing wave structure


What makes SC RF attractive?


Comparison of SC and NC RF

Superconducting RF High gradient

==> 1 GHz, meticulous care

Mid-frequencies==> Large stored energy, Es

Large Es

==> very small ΔE/E

Large Q==> high efficiency

Normal Conductivity RF High gradient

==> high frequency (5 - 17 GHz)

High frequency==> low stored energy

Low Es

==> ~10x larger ΔE/E

Low Q==> reduced efficiency


Recall the circuit analog

V(t)

I(t)

CL

Rsurf

As Rsurf ==> 0, the Q ==>∞.

In practice,

Qnc ~ 104 Qsc ~ 1011


Figure of merit for accelerating cavity:power to produce the accelerating field

Resistive input (shunt) impedance at ωο relates power dissipated in walls toaccelerating voltage

Linac literature more commonly defines “shunt impedance” without the “2”

For SC-rf P is reduced by orders of magnitude

BUT, it is deposited @ 2K

!

Rin

= V

2(t)

P =

Vo

2

2P = Q L

C

!

Rin = Vo

2

P ~

1

Rsurf


Surfing analogy of the traveling waveacceleration mechanism

To “catch” the wave the surfer must besynchronous with the phase velocity of the wave


Typically we need a longitudinal E-field toaccelerate particles in vacuum

Example: the standing wave structure in a pillbox cavity

What about traveling waves? Waves guided by perfectly conducting walls can have Elong

But first, think back to phase stability To get continual acceleration the wave & the particle must stay in

phase Therefore, we can accelerate a charge with a wave with a

synchronous phase velocity, vph ≈ vparticle < c

EE E


Can the accelerating structure be a simple(smooth) waveguide?

Assume the answer is “yes”

Then E = E(r,θ) ei(ωt-kz) with ω/k = vph < c

Transform to the frame co-moving at vph < c

Then, The structure is unchanged (by hypothesis) E is static (vph is zero in this frame)==> By Maxwell’s equations, H =0==> ∇E = 0 and E = -∇φ But φ is constant at the walls (metallic boundary conditions)==> E = 0

The assumption is false, smooth structures have vph > c


To slow the wave, add irises

In a transmission line the irises

a) Increase capacitance, C

b) Leave inductance ~ constant

c) ==> lower impedance, Z

d) ==> lower vph

Similar for TM01 mode in the waveguide

!

"

k=

1

LC

Z =L

C


Ultra-relativistic particles (v ≈ c) can“surf” an rf field traveling at c

RF- in = Po

z

RF-out= PL

Egap


RF-cavities in metal and in plasmaThink back to the string of pillboxes

Plasma cavity100 µm1 m

RF cavity

Courtesy of W. Mori & L. da Silva

G ~ 30 MeV/m G ~ 30 GeV/m


End of unit

Documents

Fundamentals of Accelerator 2012 - USPASuspas.fnal.gov/materials/12MSU/Lecture Day 3.pdf · o o Energy stored Time average power ... surface! " Q = L C R = #$ $ o % & ’ ( ) * +1