Crab Cavities:Speed of Voltage Change

(a machine protection issue for LHC [and SPS] )

CCinS WG, 27 Nov 2009

J. Tückmantel, CERN-BE-RF

Contents:• The Problem

• Time scales of incidents and equipment

• Cavity and RF basics

– longitudinal

– transversal

• Examples

• (Conclusion)

• When a crab cavity gets out of control and changes its

voltage/phase, the beam may also get out of control:

bunch is ‘banged’ by a single CC passage: Δpt,CC/pt,0 ≈ 1(*)

• If the speed of change is so fast that the beam dump system

– requiring 3 turns (≈ 300 µs) in the worst case –

cannot react in time, severe machine damage is possible.

• Here we consider

only the possible voltage/phase change scenarios

the possible aftermath for the beam is not analyzed.(*) The main RF can change rapidly causing much less problems:

the large longitudinal beam inertia ‘saves the day’: Δp||/p||,0 ≈ eVcav/Ebeam <<<< 1

• The Problem

– Time scales of ‘incidents’+  Mains power cut (anywhere):

RF power supply has enough stored energy to survive many ms (mains 50 … 300 Hz -> 20 … 7 ms) : no problem

+ ‘Short’ or … in low power electronics, controllers:

Develops >> 1 ms : no problem

– RF arcing in high power part (WG, coupler, cavity):

Full arc develops within about 1 µs: rely on τF

– Operator or control-logics error:

‘instant’ change: rely on τF

– Time scales of equipment changes

Any tuner of a (high-powered sc.) cavity is mechanical: it is too slow to change significantly within 300µs

(if foreseen) Qext is changed by mechanical means(stepper motor, ….) generally slower than tuner: it is (much) too slow to change significantly in 300µs

During the total ‘fast’ incident (300 µs):Δω and Qext are what they were at onset

• Some Cavity and RF basics - longitudinal

The proven (longitudinal) model for cavity-klystron-beam

Incident (generator) wave Ig

Reflected wave Ir

Circulator: (1)->(2) Iin=Ig to cavity; (2)->(3) Ir to load; (3)->(1)

If RF switch off (Ig=0):(and no beam IB=0)

Cavity unloadsover R and Z !!!(the coupler sucks)

IBLCRIgIrZIinIr123circulatorklystron(matched) load beam(RF current)Zcoupler

€

Q0 =ωRC ⇒ R = Q0 (R/Q)any resonator:

€

LC =1

ω2 ⇒ L =

(R /Q)

ω

€

C = 1

ω (R/Q)

Dictionary lumped circuit L,C,.. <–––> cavity (R/Q), ..

…. spare you the math …..

€

ΔVDC = qC

carry charge q across capacitor C

€

ΔV = q ω (R/Q)charge q through cavity (R/Q), ω:

(R/Q): Circuit Ω convention: 1 Ωcircuit = 2 linac Ωlinac (or 1 Boussard = 2 Schnell)

€

ω ⋅U st=Pdiss⋅Q0 ⇒ definition of Qext: ω ⋅U st=Pext⋅Qext

equivalent:

€

Z = Qext (R /Q)

€

Ir = V

2(R /Q)

1

Qext

−1

Q0

⎛

⎝ ⎜

⎞

⎠ ⎟− IB ,DC fB sin φ( )

⎛

⎝ ⎜

⎞

⎠ ⎟ − i ⋅ IB ,DC fB cos φ( ) −

V Δω

ω (R /Q)

⎛

⎝ ⎜

⎞

⎠ ⎟

Even if you do not like it, a reflected wave comes for free …€

Ig =V

2(R /Q)

1

Qext

+1

Q0

⎛

⎝ ⎜

⎞

⎠ ⎟+ IB ,DC fB sin φ( )

⎛

⎝ ⎜

⎞

⎠ ⎟ + i ⋅ IB ,DC fB cos φ( ) −

V Δω

ω (R /Q)

⎛

⎝ ⎜

⎞

⎠ ⎟

∝ cos(ωt) ∝ sin(ωt)

€

Ptrans.line = 12 Z I

2 ⇒ Pg,r = 1

2 Z Ig,r

2 = 1

2 (R /Q) Qext Ig,r

2

To get V (steady state = constant quantities)

I g,r are (proportional) model quantities, only P are absolute quantities !!!fB: relative bunch form factor: fB=1 for ‘point bunches’

in ‘proton machine convention’: =90º for beam on top of RF (max. accel.)IDC: DC beam current;   Δω: cavity detuning wrsp. to machine line = RF driveV: cavity voltage (generally considered real)

€

Δωω

= IB ,DC fB ⋅(R /Q)cos φ( )

VReactive beam loadingcompensation: Im(I g,r)=0

Sc. cavity: Qext<<<Q0       1/Qext ± 1/Q0 ≈   1/Qext

€

Ig = V

2(R/Q)⋅Qext

+ IB,DC fB sinφ( )

€

I r = V

2(R/Q)⋅Qext

−IB,DC fB sinφ( )

€

Pg,r = 12 (R/Q) Qext Ig,r

2

Assume sc. cavity + reactive beam-loading compensation

The choice of Qext (for given IB, V, .. ) is not for free:

If too low or too high: reflected power increases in both cases

––> klystron has to deliver this power more

(used to heat coffee-water !!!)

€

Qext,opt = V

2 (R/Q)⋅IB,DC fB sinφ( )

There is a Qext optimum enforcing Ir=0 i.e. Pr=0

Quantities only for steady state, what is it good for ?

Driving ‘force’ jumps(*) from one state to another one:

RF drive Ig suddenly off, Ig jumps in ampl./phase, ….

Linear system: superposition

‘Old’ field decays “exp( )” with natural (field-) time constant τF,

‘new’ field builds up ”1- exp( )” with the same time constant

For any resonator τ ω = Q : τ is the energy decay time !!

When fields decay as A=A0*exp(-t/ τF),

then energy decay as A2=A02 exp(-2t/ τF)= A0

2 exp(-t/ τ),

τF = 2·τCavity (essentially) unloads over coupler: τF = 2·τ=2·Qext/ω

(*) transition ‘much’ faster than τF

Double driving ‘force’: klystron Ig and beam IB,DC:

€

V(t) = VA ⋅exp(−t / τ F ) + VB ⋅(1−exp(−t / τ F ))

€

V (t) = VB − (VB −VA ) ⋅exp(−t /τ F ) = VB − ΔV ⋅exp(−t /τ F )

Assume ‘sudden’(t=0) ( time-scale << τF) change of drive (ΔI):

‘Old’ drive (t < 0): keeps an equilibrium Voltage VA (complex)

‘New’ drive (t > 0): corresponds to new equilibrium Voltage VB

€

Ig = V

2(R/Q)⋅Qext

+ IB,DC fB sinφ( ) ⇒

V =2(R/Q)⋅Qext Ig −IB,DC fB sinφ( )( )

Special case: the klystron is (goes) off, i.e. Ig=0

€

V = − 2 ⋅ IB ,DC fB ⋅(R /Q) ⋅Qext < 0 : decelerating

And Φ is not ‘stabilized’ anymore –> maximum induced voltage Φ –> 90º

€

I r = 2⋅V

2(R/Q)⋅Qext

⇒ Pr = 2⋅ IB,DC fB( )2⋅(R/Q)⋅Qext

Sucked from the beam & dumped into load

€

V =−2(R/Q)⋅Qext⋅IB,DC fB sinφ( )

the so-called RF current IRF (Fourier component)

Examples without change of phase : drive voltage

1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

1.2

Loss of Ig (no beam)1 2 3 4 5

0.5

1.0

1.5

2.0

Step up of Ig to Pg,max

Loss of Ig with strong beam

1 2 3 4 5

- 2.0

- 1.5

- 1.0

- 0.5

0.5

1.0

Feedback action: to peak &back to new equilibrium

1 2 3 4 5

0.5

1.0

1.5

2.0

Speed depends on (same )τf but also on Ig,(max), i.e. Pg,(max)

Ig = 2 a.u. Ig = 4 a.u.

Ig = 10 a.u.Ig = 6 a.u.

1 2 3 4 5

2

4

6

8

10

12

1 2 3 4 5

2

4

6

8

10

12

1 2 3 4 5

2

4

6

8

10

12

1 2 3 4 5

2

4

6

8

10

12

Example of 90º phase jump of drive cos(ωt) -> sin(ωt)

i.e. Real ––> Imag

- 1.0 - 0.5 0.5 1.0real

- 1.0

- 0.5

0.5

1.0

imag

Complex V

Complex V versus time

€

Pmax = 12 (R /Q) Qext ΔIg,max

2 ⇒ ΔIg,max =2Pmax

(R /Q) Qext

Steepest rise by feedback P: 0 –> Pmax

€

dV (t)

dt gap edge

≈ (R /Q) ⋅ IB ⋅ω

Speed of change: feedback versus beam

€

⇒ dV (t)

dt ≈ (R /Q) ⋅ΔI ⋅2 ⋅Qext /τ F = (R /Q) ⋅ΔI ⋅ω

€

V = 2(R /Q) ⋅Qext Ig − IB ,DC fB sin φ( )( ) ⇒ ΔV = 2(R /Q) ⋅QextΔI

€

V (t) = VB − ΔV ⋅exp(−t /τ F ) ⇒ dV (t)

dt ≈ ΔV /τ F

Does not depend on Qext, τF !!!! Same rise for same ΔI

To keep same enforced (FB) speed of change:speed scales as 1/√QextLoop gain scales as √Qext (for same’ hardware gain’)

€

Pcomp = 12 (R /Q) ⋅Qext ⋅ IB

2

main RF doesn' t make it

no longitudinal beam-cavity interaction ( if beam really at x0)

€

Δpx = −i ⋅eω

⋅dVz

dxDeflection requires transverse gradient in longitudinal accelerating voltage (–>Ez)

Generalized Panofsky-Wenzel theorem

- transverse

Δpx, Bunch centre 90º out of phase(set like this since we want only tilt, no kick for bunch center !!)

beam

z

x

y

By

Ez

deflectionEx

Chose field configuration having x0 that Vz(x0) = 0:

beam

z

x

y

By

deflection

Ez

Ex

the same Vz gradient = same deflection !!

Δpx, Vz 90º out of phase

Bunch Center (==Ib), Vz in phase !!!

Bad news (for RF installation):

worst phase angle for parasitic longitudinal interaction

( for x ≠ 0)

Good news (for machine protection):

the beam drives a transverse voltage with phase for

tilting the bunch, NOT kicking the whole bunch !

€

Δp|| ⋅c ≈ ΔE = eV|| ⇔ V|| = Δp|| ⋅c / eFor highly relativistic beam: longitudinal

Analogue definition: transverse voltage

€

V⊥ = Δp⊥⋅c / e ⇔ Δp⊥ = eV⊥/c

€

eV⊥/c = Δp⊥= x = −i ⋅eω

⋅dV||

dx= −

i ⋅eω

⋅V||

x

€

V|| = x ⋅V⊥ω /cBeam passing at offset x sees(only magnitudes, forget 90º phase factor ‘i’ here)

Dipole (=crab) mode:

€

V|| = const ⋅ xxideal beam:V ||=0 zV ||

€

Ig = V||(x)

2(R /Q)(x) ⋅Qext

+ IB ,DC fB sin φ( ); V||(x) = x ⋅V⊥ω /c

(R/Q): Circuit Ω convention: 1 Ωcircuit = 2 linac Ωlinac (or 1 Boussard = 2 Schnell)

€

(R /Q) =V||

2

2 ω Ust

Cavity geometry constant

- indep. of excitation

- indep. of cav. material (Cu, iron, superc., ..)

€

(R /Q)(x) =V||

2(x)

2 ω Ust

=V⊥

2(x)

2 ω Ust

⋅ x 2 ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

= (R /Q)⊥⋅ x 2 ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

€

Ig = V⊥

2(R /Q)⊥⋅ x ⋅Qext

c

ω+ IB ,DC fB sin φ( )

€

Ir = V⊥

2(R /Q)⊥⋅ x ⋅Qext

c

ω− IB ,DC fB sin φ( ) …analogue

€

Qext,opt = V⊥

2(R /Q)⊥⋅ x ⋅ IB ,DC fB sin φ( )⋅

c

ωOnly perfect for a chosen x0

€

Pg,r = 12 (R /Q)(x) Qext Ig,r

2

Factor 1/x in Ig,r ; if x ––> 0 ???

Currents are proportional to real waves, power is ‘absolute’

€

Pg,r = 12 (R /Q)⊥⋅ x 2 ⋅

ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

Qext Ig,r

2

Power finite even for x ––> 0

oufffffff

€

Jg,r = Ig,r ⋅ x = V⊥

2(R /Q)⊥⋅Qext

c

ω ± x ⋅ IB ,DC fB sin φ( )

€

Pg,r = 12 (R /Q)⊥⋅

ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

Qext Jg,r

2

Possibility: Renormalize I and P (J=xI gets dimension [A·m] !)

Transverse impedance: Beam drives Ig=0, ϕ=90º

€

0 = Ig = V⊥

2(R /Q)⊥⋅ x ⋅Qext

c

ω+ IB ,DC fB ⇒

V⊥/ x = − 2 ⋅ IB ,DC fB (R /Q)⊥⋅Qext ⋅ω /c = − IB ,RF ⋅(R /Q)⊥⋅Qext ⋅ω /c

Z⊥ = (R /Q)⊥⋅Qext ⋅ω /c [Ω /m⊥]

Longitudinal impedance of dipole mode at offset x0

€

(R /Q)(x0) = (R /Q)⊥⋅ x02 ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

⇒

Z||(x0) = (R /Q)⊥⋅Qext ⋅ x02 ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

Additional power due to ‘wrong’ frequency: δω – reactive beam loading compensation not perfect – fight a mechanical (*) cavity oscillation = sideband (microphonics, ponderomotive oscillations)

€

δIg,r = ±i ⋅V δω

ω (R /Q)

€

δPg,r = Qext V 2 ⋅ δω( )

2

2 (R /Q) ⋅ω2 ⇒ δPg,r,peak =

Qext δV⊥⋅V⊥⋅ δω( )2

(R /Q)⊥⋅ω2

(*) Perturbations over the RF input or beam and their combat over the same RF are on same ‘footing’ : neutral wrsp. Qext

€

δVresid =1 + i ⋅δω ⋅Qext /ω

gFB +1 + i ⋅δω ⋅Qext /ωδVpert ≈ i ⋅

δω ⋅Qext /ω

gFB

δVpert

Feedback action with gain gFB (not shown explicitly here, delay=0; realistic delay gFB ≤ 100)

for several BW detuning but g still larger

+ assume δV << V

(Intermediate) Summary of facts

€

Qext = τ F ⋅ω /2 ⇔ τ F = 2 ⋅Qext /ω ⇔ BW = f / Qext =1/(π ⋅τ F )

€

δVresid ≈ i ⋅δω ⋅Qext

ω

δVpert

gFB

= i ⋅δf /BW

gFB

⋅δVpert for several BW detuning

€

δPg,r,peak = Qext δV⊥⋅V⊥⋅ δω( )

2

(R /Q)⊥⋅ω2

€

Z⊥ = (R /Q)⊥⋅Qext ⋅ω /c [Ω /m⊥]

€

Z||(x0) = (R /Q)⊥⋅Qext ⋅ x02 ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

… at injection x0 is “not so perfect” …

€

dV⊥

dt max

≈ ω ⋅2Pmax ⋅(R /Q)⊥

Qext

€

dV⊥(t)

dt ge

≈IB ⋅ω

2

c(R /Q)⊥⋅ x

€

Pcomp =IB

2 ⋅ω2

2 c 2(R /Q)⊥⋅Qext ⋅ x 2

€

Qext,opt = V⊥

2(R /Q)⊥⋅ | x |max ⋅IB ,DC fB sin φ( )⋅

c

ω

Qext,opt = 107 (τF=4000µs: field decay to 93% in 300µs) Pmax.opt= 5 kW€

Jg = V⊥

2(R /Q)⊥⋅Qext

c

ω+ x ⋅ IB ,DC fB

Pg,r = 12 (R /Q)⊥⋅

ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

Qext,used Jg,r

2

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

… at injection x is “not so perfect” …(maybe larger than ‘coast’-xmax)

Examples ‘Given’ are ( = LHC 800 MHz ‘test cavity’, others similar) (R/Q)t=60 Ωcircuit (=120 Ωlinac); Vt=2.5 MV; Ib=0.6 A (neglect bunch form factor < 1 at 800 MHz, it helps)

x == 0 not possible in real life: allow (limited) deviation |x|max

Assume: guaranteed |x| ≤ 0.2 mm (=200 µm!)

€

δPg,r,peak = Qext δV⊥V⊥⋅ δω( )

2

(R /Q)⊥⋅ω2

= 15 kW€

BW = f / Qext = 80 Hz

€

δVresid ≈ i ⋅δω ⋅Qext

ω

δVpert

gFB

= i ⋅δf /BW

gFB

⋅δVpert = 0.4 ⋅i ⋅δVpert

€

Z⊥ = (R /Q)⊥⋅Qext ⋅ω /c [Ω /m⊥] =10'000 MΩ /m⊥

€

Z||(x0) = (R /Q)⊥⋅Qext ⋅ x02 ω

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

= 6.8 kΩ; x0 = 0.2 mm

170 kΩ; x0 =1 mm

⎧ ⎨ ⎩

€

τF = 4000 μs (Qext =107, Pmax = 5 kW ); decay ≥ 93%

€

ZmainRF = 100 kΩ /cav; injection

600 kΩ /cav; coast

⎧ ⎨ ⎩

without feedback!!

Assume gFB=100; δf=3 kHz (β-tron f); δVt=2.5kV=10-3 Vt

€

Pcomp =IB

2 ⋅ω2

2 c 2(R /Q)⊥⋅Qext ⋅ x 2 = 2.5 kW (50 kW @1 mm)

€

BW = f / Qext = 1600 Hz

€

δVresid ≈ i ⋅δω ⋅Qext

ω

δVpert

gFB

= i ⋅δf /BW

gFB

⋅δVpert = 0.02 ⋅i ⋅δVpert

€

Z⊥ = (R /Q)⊥⋅Qext ⋅ω /c [Ω /m⊥] = 500 MΩ /m⊥

Assume gFB=100; δf=3 kHz (β-tron f); δVt=2.5kV=10-3 Vt

€

τF = 200 μs (Qext = 5 ⋅105, Pmax = 30 kW ); decay ≥ 22%

Only previously ‘critical’ items:

|x|max = 4 mm <–> Pmax=100kW (P=30 kW @ 0.2 mm)