Room temperature RFPart 2.1: Strong beam-cavity coupling

(beam loading)

30/10/2010A.Grudiev

5th IASLC, Villars-sur-Ollon, CH

Outline1. Linear collider and rf-to-beam efficiency2. Beam loading effect: what is good and what is bad about it

a. Beam-cavity couplingi. Steady-state power flowii. Efficiency in steady-state and in pulsed regimeiii. Compensation of the transient effect on the acceleration

b. Standing wave structures (SWS)c. Travelling wave structure (TWS)

i. Steady-state power flowii. Efficiency in steady-state and in pulsed regimeiii. Compensation of the transient effect on the acceleration

3. Examples:a. CLIC main linac accelerating structureb. CTF3 drive beam accelerating structure

Linear collider and rf-to-beam efficiency

Modulator

RF power source

Linac

AC power from the grid: PAC

High voltage DC power: PDC

RF power: Prf

Beam power: Pb=IbEE

Ib

AC-to-DC conversion efficiency: ηAC-DC = PDC/PAC

DC-to-rf conversion efficiency: ηDC-rf = Prf/PDC

rf-to-beam efficiency: ηrf-to-beam = Pb/Prf

Two main parameters of a collider are center of mass collision energy E and luminosity L

For a fixed E: L ~ Ib ~ Pb ~ PACηAC-DCηDC-rfηrf-to-beam

All efficiencies are equally important. In this lecture, we will focus on the ηrf-to-beam

Beam loading in steady-state (power flow)

W

V

Pin

P0Cavity parameters w/o beam (reminder):

1;

;:condition Matching

;;;

00

0

00

2

0

extext

in

inext

Q

QQQ

PP

P

WQ

P

WQ

R

VP

W

V

Pin

P0

Cavity parameters with beam:

loading beam relativewhere

111;111

:condition Matching

;

0

0

0

0

V

RIY

YQ

Q

QQQ

PPP

constRI

QV

P

WQIVP

bb

bbbext

bin

bbbbb

PbIb

satisfied at any V

satisfied only if V = IbR

Beam loading and efficiency

In steady-state regime which correspond to CW (Continues Wave) operation V = const:

b

b

b

ext

in

bSWSbeamtorf Y

Y

Q

Q

P

P

1

W

V

Pin

P0

PbIb

The higher is the beam loading the higher is the rf-to-beam efficiency

In pulse regime, V ≠ const is a function of time:

V(t)

t

Pin

tf tp

V

p

fpbeamtorf

pin

bbpulsedbeamtorf t

tt

tP

tP

Ib

Transient beam loading effectEnergy conservation in a cavity without beam (reminder):

factor loaded 1

:where

;11

2)(ˆ

;0,1

0,01)(ˆ

:excitationfunction -step afor

ˆ2ˆ11

1ˆ2

in resultswhat

ˆ;

ˆ

ˆˆ;)ˆˆ(

;ˆ

where

0

2

0

2

0

2

0

22

0

QQ

Q

etV

t

t

Z

RtV

dt

VdQV

Z

RV

Z

V

Q

W

Q

W

R

VP

VVZ

VVP

Z

VP

dt

dWPPP

L

Q

t

in

in

rad

ext

refinrefrad

outin

in

outin

L

W

V

Pin

P0

Pout

IinVin

IrefVref

Vrad

Irad

·

Pin

Pout

Short-CircuitBoundaryCondition:

Matching condition: Vrad = Vin, only if β=1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

/2

V

=10=2=1=1/2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.2

0.4

0.6

0.8

1

1.2

1.4

/2

V

tinj

=1000

tinj

=TinjM

tinj

=2000

tinj

=4000

tinj

=6000

no beam

TinjM

Transient beam loading effectEnergy conservation in a cavity with beam:

inj

injbin

bin

bb

outbin

tt

ttRtI

t

t

Z

RtV

dt

VdQV

RI

Z

RV

IVP

dt

dWPPPP

,1

,0)(ˆ;

0,1

0,01)(ˆ

:excitationfunction -step afor

ˆ2ˆ11ˆ1ˆ2

in resultswhat

ˆˆwhere

0

0

Depending on the tinj, there is transient beam loading which is variation of the voltage gained in the cavity along the upstream part of the beam. It is compensated only when:

W

V

Pin

P0

PbIb

Pout

fLM

injinj tQ

Tt

1

2ln

2

Solution for β=2 and different beam injection times tinj

From a cavity to a linacFor obvious reasons multi cell accelerating structures are used in linacs instead of single cell cavitiesWe will distinguish two types of accelerating structures:

Standing Wave Structures (SWS) Travelling Wave Structures (TWS)

In linacs, it useful to use some parameters normalized to a unit of length:

Voltage V [V] -> Gradient G [V/m] = V/Lc

Shunt impedance: R [Ω] -> R’ [Ω/m] = R/Lc

Stored energy: W [J] -> W’ [J/m] = W/Lc

Power loss: P [W] -> P’ [W/m] = P/Lc

where Lc – cell length

SWS

TWS

Beam loading in SWS

Here it is assumed that the coupling between cells is much stronger than the input coupling and the coupling to the beam then the beam loading behavior in SWS is identical to the beam loading behavior in a single cell cavity.

SWS parameters w/o beam (reminder):

1;

;:condition Matching

;;;

00

0

00

00

2

0

extext

L

in

inext

Q

QQQ

dlPPP

P

WQ

P

WQ

R

GP

s

SWS parameters with beam:

loading beam relativewhere

111;111

:condition Matching

;

0

0

0

0

G

RIY

YQ

Q

QQQ

PPP

constRI

QG

P

WQIGP

bb

bbbext

bin

bbbbb

satisfied at any G

satisfied only if G = IbR’

z

g

bl

dzzvzQ

g

g

g

inz

g

b

g

g

g

bb

dzzQzv

zR

zG

IzGzG

eR

zR

zQ

Q

zv

vGzG

Qv

RPGG

Qv

RI

Qvdz

Rd

Rdz

dQ

Qdz

dv

v

G

dz

dG

GIR

GPP

dz

dP

z

g

0 0

)()(2

1

0

00

000

00

0

0

2

0

)()(2

)(

)(1)()(

)0(

)(

)(

)0(

)(

)0()(

:[*] form closed ain obtained isSolution

conditionboundary input ;with

2

111

2

0 0

Beam loading in TWS in steady-state

velocitygroup;:where

TWS along flowpower ;

losses wall;;

0

2

00

2

0

W

Pv

QR

GvP

P

WQ

R

GP

g

g

In a cavity or SWS field amplitude is function of time only f(t). In TWS, it is function of both time and longitudinal coordinate f(z,t). Let’s consider steady-state f(z).

z

TWS parameters are function of z:

Energy conservation law in steady-state yields:(it is assumed hereafter that power flow is matched and there are no reflections)

[*] – A. Lunin, V. Yakovlev, unpublished

Efficiency in steady-state:

Pin Pout

sL

in

b

in

bCTWSbeamtorf dzzG

P

I

P

P

0

)(

constantn attenuatio TWS;2

where

1)(

0

0

g

bl

vQ

zeRIzeGzG

TWS example 1: constant impedance TWSIn constant impedance TWS, geometry of the cells is identical and group velocity, shunt impedance and Q-factor are constant along the structure.

This simplifies a lot the equations:

z

Efficiency in steady-state:

Pin Pout

loading beam relative TWS;

where

1212

00

200

G

RIY

sL

eLYsL

eY

bb

sbbCITWSbeamtorf

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z/Ls

G/G

0

= 0.45

unloaded

loaded

Gradient unloaded and loaded for Yb0=1

•The higher is the beam loading the higher is the rf-to-beam efficiency•Beam loading reduces the loaded gradient compared to unloaded

Full beam loadingIn TWS, depending on the beam current, beam can absorb all available rf power so that Pout = Pin – P0 – Pb = 0 This is the case of full beam loading where rf-to-beam efficiency is closed to maximum.

z

Pin Pout=0

Pb

Ib Ib

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

z/Ls

G/G

0

Yb0

= 0

Yb0

= 1

Yb0

= 2

Yb0

= 3

Yb0

= 4

Yb0

= 5

Yb0

= 6

unloaded

loaded

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Yb0

Structure is overloaded if the beam current is even higher

00

00

000

2

1

2

10

02

1

0

21;)0(:TWS aperedStronger t

)0(:where;1ln)(;2;)(:TWSgradient Const

loaded, ;111

)()( unloaded; ;1)(00

aLGzG

aza

RIGzGaGzG

aza

RIzGzGazGzG

s

bl

a

a

bla

TWS example 2: tapered TWSIn tapered TWS, geometry is different in each cell. If cell-to-cell difference is small, it is a weakly tapered structure and smooth approximation (no reflections) can still be applied:

Let’s consider a case of linear vg-tapering such that Q0=const, R’=const but vg = vg0(1+az):

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

z/Ls

G/G

0

= 0.5

unloaded

loaded

vg/v

g0

Yb0=1, a=-2α

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

z/Ls

G/G

0

= 0.53

unloadedloaded

vg/v

g0

Yb0=1, a=-3α

W’ = P/ vg ≈ const

;2:for 111ln12

12

1111

11

2111

2

02000

2

1

02

102

1

202

02

1

02

1

00

00

aaLaLYLY

aLaa

YaLa

Y

ssbsbCGTWSbeamtorf

as

aa

bas

a

bTWS

beamtorf

Tapering and rf-to-beam efficiencyFor the case of linear vg-tapering such that Q0=const, R’=const but vg = vg0(1+az)Efficiency in steady-state:

For low beam loading tapering helps to increase rf-to-beam efficiency

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Yb0

SWS

TWS-CI: a = 0TWS-CG: a = -2

0

TWS: a = -30

The stronger is the tapering the higher is the efficiency at low beam loading The stronger is the tapering the low is the Yb0 for the maximum efficiency (full beam loading)SWS has higher efficiency than TWS at low beam loading BUT lower efficiency than fully beam loaded TWS

TWS efficiency in pulsed regime

timefilling;)(

:where0

sL

gf

p

fpbeamtorf

pin

bbpulsedbeamtorf

zv

dzt

t

tt

tP

tP

V(t)=∫G(z,t)dz

t

Pin

tf tp

V

In pulse regime, V ≠ const is a function of time. In simplest case of vg=const, R’/Q0=const, Q0=∞

G(z,t)

zLs

G0G(z,t1) G(z,t3)G(z,t2) G(z,tf)

z3z1 z2

tn = zn/vg; tf = Ls/vg In general, in TWS:

Transient beam loading in TWSLet’s continue with the simplest case of vg=const, R’/Q0=const, Q0=∞

tftp

G(z,t)

zLs

G0 G(z,t1) G(z,t3)G(z,t2) G(z,tf)

z3z1 z2

Ib = 0; 0 < t ≤ tf

0)( GzG zRIGzG bl 0)(

tinj tinj+tf

Ib

Ib > 0; tinj < t ≤ tinj+tfG(z,t)

zLs

G0G(z,t1)

G(z,t3)G(z,t2)

G(z,t4)

z3z1 z2

G1=G0-IbR’αLs

Transient beam loading result in different voltage gained by the bunches during tinj < t ≤ tinj+tf

t

Pin

V

Ib

Vl

tp+tf

Compensation of the transient beam loading in TWS

Let’s continue with the simplest case of vg=const, R’/Q0=const, Q0=∞

tf tp

tinj=tf

Two conditions:1. Ramping the input power

during the structure filling in such a way that the loaded gradient profile along the structure is formed.

2. The beam is injected right at the moment the structure is filled. (same as for SWS)

t

Pin

V

Ib

VL

tp+tf

G(z,t)

zLs

G0

G(z,t1) G(z,t3)G(z,t2) G(z,tf)

z3z1 z2

Structure filling process:Ib = 0; 0 < t ≤ tf

G1

G(z,t)

zLs

G0G(z,t1)

G(z,t3)G(z,t2)

G(z,tf)

z3z1 z2

Injection right after filling:Ib > 0; t > tinj = tf

G1

Pin-ramp

Ib

Transient beam loading in TWSLet’s consider general case of vg(z), R’(z), Q0(z)

),(),(),(

toarrive to time;)(

)(:

)(

)()()(),(

)()(),(

)();(:modeldependent 2.Time

)(

)()()()()()(

)()(

;:solution StateSteady 1.

0

0

0

00

0

0

0

0

tzGtzGtzG

zzv

dzzwhere

dzzg

zfztIzgtzG

zgztGtzG

tIItGG

dzzg

zfIzgzgGzGzGzG

zgGzG

constIconstG

bl

z

g

z

bb

bb

z

bbl

b

Model approximations:1. Smooth propagation of

energy along the structure (no reflections)

2. No effects related to the finite bandwidth of the signal and the structure (TWS bandwidth >> signal bandwidth)

3. Time of flight of the beam through the structure is much shorter than the filling time (Ls/c << tf)

Transient beam loading in TWS (cont.)

where:

voltagebeam;),()(

voltageunloaded;),()(

voltageloaded);()()(

0

0

dztzGtV

dztzGtV

tVtVtV

z

bb

z

bl

Compensation of the transient beam loading in TWS

In general case of vg=const, R’=const, Q0=const

;)(

)(ofsolution theis )(:where

)(

)()(

0;0);(

: timefilling duringgradient input Then

solution loaded is )(

andsolution unloaded is )(

:state-steadyin If

0

0

00

0

z

g

f

fl

fb

l

zv

dzzttz

ttzg

ttzGtG

ttItGG

zG

g(z)GzG

Compensation of the transient beam loading in constant impedance TWS

Let’s consider the case of constant impedance TWS: vg=const, R’=const, Q0=const

Uncompensated case

1)(

)()(:0for function -ramp when the

)()(andsolution statesteady ;1)(;)(

00

0

00

ttveRItGtG

v

Ltt

v

z

zv

dzztzeRIzeGzGzeGzG

fgbf

g

sf

z

ggbl

Compensated case

Compensation of the transient beam loading in constant gradient TWS

Constant gradient: a=-2α0

Let’s consider a case of linear vg-tapering such that Q0=const, R’=const but vg = vg0(1+az):

a=-3α0

CLIC main linac accelerating structure in steady-state

Parameters: input – outputf [ GHz] 12Ls [m] 0.25

vg/c [%] 1.7 – 0.8

Q0 6100 – 6300

R’ [MΩ/m] 90 – 110Ib [A] 1.3

tb [ns] 156

<G> [MV/m] 100

In reality, vg ≠ const, R’ ≠ const, Q0 ≠ const and general expressions must be applied but often a good estimate can be done by averaging R’ and Q0 and assuming linear variation of vg

In this case: <Q0>=6200; <R’>=100 MΩ/m; and vg/c = 1.7 - 0.9/0.25·z

Parameters calculated

Pin [MW] 67.6

ηCW [%] 48

tf [ns] 70

η [%] 33

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

14x 10

7

z[m]

G[V

/m];

vg[m

/s]

CW = 0.48

GG

l

vg

CLIC main linac accelerating structure during transient

Parameters: input – outputf [ GHz] 3Ls [m] 1.22

vg/c [%] 5.2 – 2.3

Q0 14000 – 11000

R’ [MΩ/m] 42 – 33Ib [A] 4

Pin [MW] 33

SICA - CTF3 3GHz accelerating structure in steady-state

Parameters calculatedηCW [%] 95

This structure is designed to operate in full beam loading regime

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2x 10

7

z [m]

G [

V/m

], v

g[m

/s]

CW = 0.95GG

l

vg

Experimental demonstration of full beam loading