rf fundamentals - universidad de guanajuatomepas2011/wp-content/uploads/... · rf fundamentals...

Jean Delayen

Center for Accelerator ScienceOld Dominion University

andThomas Jefferson National Accelerator Facility

RF FUNDAMENTALS

SURFACE RESISTANCERF and CAVITIESMICROPHONICS

First Mexican Particle Accelerator SchoolGuanajuato

26 Sept – 3 Oct 2011

RF Cavity

• Mode transformer (TEM→TM)

• Impedance transformer (Low Z→High Z)

• Space enclosed by conducting walls that can sustain an infinite number of resonant electromagnetic modes

• Shape is selected so that a particular mode can efficiently transfer its energy to a charged particle

• An isolated mode can be modeled by an LRC circuit

RF Cavity

Lorentz force

An accelerating cavity needs to provide an electric field E longitudinal with the velocity of the particle

Magnetic fields provide deflection but no acceleration

DC electric fields can provide energies of only a few MeV

Higher energies can be obtained only by transfer of energy from traveling waves →resonant circuits

Transfer of energy from a wave to a particle is efficient only is both propagate at the same velocity

( )F q E v B= + ¥

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 module

Chain of coupled pendula as its mechanical analog

Electromagnetic Modes

Electromagnetic modes satisfy Maxwell equations

With the boundary conditions (assuming the walls are made of a material of low surface resistance)

no tangential electric field

no normal magnetic field

22

2 2

1 0E

c t HÏ ¸Ê ˆ∂— - =Ì ˝Á ˜∂Ë ¯ Ó ˛

0

0

n E

n H

¥ =

=i

Electromagnetic Modes

Assume everything

For a given cavity geometry, Maxwell equations have an infinite number of solutions with a sinusoidal time dependence

For efficient acceleration, choose a cavity geometry and a mode where:

Electric field is along particle trajectory

Magnetic field is 0 along particle trajectory

Velocity of the electromagnetic field is matched to particle velocity

22

2 0E

c Hw Ï ¸Ê ˆ

— + =Ì ˝Á ˜Ë ¯ Ó ˛

i te w-∼

Accelerating Field (gradient)

Voltage gained by a particle divided by a reference length

For velocity-of-light particles

For less-than-velocity-of-light cavities, there is no universally adopted definition of the reference length

1 ( )cos( / )zE E z z c dzL

w b= Ú

2NL l=

Design Considerations ,max

,max

2

2

2

2

2

minimum critical field

minimum field emission

minimum shunt impedance, current losses

minimum dielectric losses

minimum control of microphonics maximum

s

acc

s

acc

s

acc

s

acc

acc

HE

EE

HE

EE

UE

< >

< >

voltage drop for high charge per bunch

Surface Impedance - Definitions

• The electromagnetic response of a metal, whether normal or superconducting, is described by a complex surface impedance, Z=R+iX

R : Surface resistanceX : Surface reactance

Both R and X are real

Definitions

For a semi- infinite slab:

0

00

(0)

( )

(0) (0)(0) ( ) /

Definition

From Maxwell

x

x

x x

y x z

EZJ z dz

E EiH E z z

w m+

•

=

=

= =∂ ∂

Ú

Definitions

The surface resistance is also related to the power flow into the conductor

and to the power dissipated inside the conductor

212 (0 )P R H -=

( )0

0

0

1/2

0

(0 ) / (0 )

377 Impedance of vacuum

Poynting vector

Z Z S S

Z

S E H

me

+ -=

= W

= ¥

Energy Content

Energy density in electromagnetic field:

Because of the sinusoidal time dependence and the 90º phase shift, he energy oscillates back and forth between the electric and magnetic field

Total energy content in the cavity:

( )2 20 0

12

u e m= +E H

2 20 0

2 2V VU dV dV

e m= =Ú ÚE H

Power Dissipation

Power dissipation per unit area

Total power dissipation in the cavity walls

2 20

4 2sRdP

dam wd

= =H H

2

2s

A

RP da= Ú H

Quality Factor

Quality Factor Q0:

00

00 0

0

Energy stored in cavityEnergy dissipated in cavity walls per radian diss

UQ

Pw

ww tw

∫ =

= =D

2

00 2

V

s

A

dVQ

R da

wm= Ú

ÚH

H

Geometrical Factor

Geometrical Factor QRs (Ω)Product of the Quality Factor and the surface resistanceIndependent of size and materialDepends only on shape of cavity and electromagnetic mode

2 2 2

00 2 2 2

0

1 22

377 Impedance of vacuum

V V Vs

A A A

dV dV dVG QR

da da da

m phwm pe l l

h

= = = =

ª W

Ú Ú ÚÚ Ú Ú

H H H

H H H

Shunt Impedance, R/Q

Shunt impedance Rsh:

Vc = accelerating voltage

Note: Sometimes the shunt impedance is defined as or quoted as impedance per unit length (ohm/m)

R/Q (in Ω)

2

in csh

diss

VR

P∫ W

2

2c

diss

VP

2 2 2R V P E LQ P U Uw w= =

Q – Geometrical Factor (Q Rs)

32 20 0

00 0

2 2 20

0

0

2 2 1 12 2 2 2 6

1 1 12 2 2

2006

Energy contentQ:Energy disspated during one radian

Rough estimate (factor of 2) for fundamental mode

is si

s s

s

s

UP

c LU H dv HL

P R H dA R H L

QR

G QR

ww wtw

m mp p pwl e m

p

mpe

= = =D

= =

= =

~ = W

=

Ú

Ú

275

ze (frequency) and material independent. It depends only on the mode geometry It is independent of number of cellsFor superconducting elliptical cavities sQR W∼

Shunt Impedance (Rsh), Rsh Rs, R/Q

( )

2 22

2 20

2

1 12 2

33,000

/ 100

/

In practice for elliptical cavities

per cell

per cell

and Independent of size (frequency) and material

Depends on mode geometr

zsh

s

sh s

sh

sh s sh

E LVRP R H L

R R

R Q

R R R Q

p=

W

W

yProportional to number of cells

Power Dissipated per Unit Length or Unit Area 2

12

12

12

2

2

1

For normal conductors

For superconductors

S

S

S

S

E RPRL QRQ

R

PLPA

RPLPA

w

w

w

w

w

w

w

-

μ

μ

μ

μ

μ

μ

μ

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. This is usually very weakly coupled

Cavity with External Coupling

Consider the rf cavity after the rf is turned off.Stored energy U 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

Similarly define a “loaded” quality factor QL:

Now

∴ energy in the cavity decays exponentially with time constant:

tot diss e tP P P P= + +

0L

tot

UQPw

∫

totdU Pdt

= -

00

diss

UQPw

∫

0

00

L

tQ

L

UdU U U edt Q

ww -= - fi =

0

LL

Qtw

=


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 PU Uw w

+=

0

1 1 1

L eQ Q Q= +

0e

e

UQ

Pw

∫


• Define “coupling parameter”:

therefore

β is equal to:

• It tells us how strongly the couplers interact with the cavity. Large βimplies that the power leaking out of the coupler is large compared to the power dissipated in the cavity walls.

0

e

QQ

b ∫

0

1 (1 )

LQ Qb+=

e

diss

PP

b =

Several Loss Mechanisms

(

1 1L

i

-wall losses -power absorbed by beam -coupling to outside world

Associate Q will each loss mechanism

index 0 is reserved for wall losses)

Loaded Q: Q

Coupling co

i

ii

i

L

P P

UQP

PQ U Q

w

w

=

=

= =

Â

Â Â0

0

0

1

efficient: ii

i

Li

Q PQ P

QQ

b

b

= =

=+Â

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

Equivalent Circuit for an rf Cavity

Parallel Circuit Model of an Electromagnetic Mode

• Power dissipated in resistor R:

• Shunt impedance:

• Quality factor of resonator:

212

cdiss

VPR

=

2c

shdiss

VRP

∫ 2shR Rfi =

1/20

0 0diss c

U R CQ CR RP L Lw w

wÊ ˆ∫ = = = Á ˜Ë ¯

1

00

0

1Z R iQww

w w

-È ˘Ê ˆ

= + -Í ˙Á ˜Ë ¯Î ˚1

00 0

0

1 2Z R iQw ww ww

-È ˘Ê ˆ-

ª ª +Í ˙Á ˜Ë ¯Î ˚ ,

1-Port System

2 2 00 0

0 0

0

1 21 2

gg

kV RI V kVR Qk Z R k Z iQiw

www

= =Ê ˆ+ + + DÁ ˜Ë ¯+ D

20

0

0

1 2Total impedance: Rk Z Qi w

w

++ D

1-Port System

( )

( )

2 20

22 20

222 4 2 2

0 0 00

2

0

20 0

02 22

0 0

0

1 12 212

4

8

4 11 21

1

Energy content

Incident power:

Define coupling coefficient:

g

ginc

inc

QU CV VR

Q Rk VR

R k Z k Z Q

VP

ZR

k ZQU

P Q

w

w ww

b

bw b w

b w

= =

=Ê ˆD+ + Á ˜Ë ¯

=

=

=+ Ê ˆÊ ˆ D+ Á ˜ Á ˜+Ë ¯ Ë ¯

1-Port System

( )

( )

2 220 0

0

2

4 11 21

1

0, 1 :

4 111 21

Power dissipated

Optimal coupling: maximum or

critical coupling

Reflected power

diss inc

diss incinc

ref inc diss mc

UP PQ Q

U P PP

P P P P

w bb w

b w

w b

bb

= =+ Ê ˆÊ ˆ D+ Á ˜ Á ˜+Ë ¯ Ë ¯

=

fi D = =

= - = -+

+2

0

01Q wb w

È ˘Í ˙Í ˙Í ˙Ê ˆDÍ ˙Á ˜Í ˙+Ë ¯Î ˚

1-Port System

( )

( )

02

0

2

2

414

1

11

At resonance

inc

diss inc

ref inc

QU P

P P

P P

bw b

bb

bb

=+

=+

Ê ˆ-=Á ˜+Ë ¯

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

Equivalent Circuit for a Cavity with Beam

• Beam in the rf cavity is represented by a current generator.

• Equivalent circuit:

(1 )sh

LR

Rb

=+

0

0

tan -21

produces with phase (detuning angle)

produces with phase

b b

g g

c g b

i V

i V

V V V

Q

yy

wyb w

= -

D=+


1/21/2

0

0

2( ) cos1

cos2(1 )

sin22

2: beam rf current: beam dc current: beam bunch length

g g sh

b shb

b

bb

b

b

V P R

i RV

i i

ii

b yb

ybq

q

q

=+

=+

=


( ) [ ] 2

2211 (1 ) tan tan

4c

gsh

VP b b

Rb b y f

b= + + + + -

0 cosPower absorbed by the beam = Power dissipated in the cavity

sh

c

R ib

Vf

=

2

(1 ) tan tan

1

1 (1 )2

opt opt

opt

opt cg

sh

b

b

b bVP

R

b y f

b

+ =

= +

+ + +=

Minimize Pg :

Frequency Control

Energy gain

Energy gain error

The fluctuations in cavity field amplitude and phase come mostly from the fluctuations in cavity frequency

Need for fast frequency control

Minimization of rf power requires matching of average cavity frequency to reference frequency

Need for slow frequency tuners

cosW qV f=

tanW VW Vd d df f= -

Some Definitions

• Ponderomotive effects: changes in frequency caused by the electromagnetic field (radiation pressure)– Static Lorentz detuning (cw operation)– Dynamic Lorentz detuning (pulsed operation)

• Microphonics: changes in frequency caused by connections to the external world– Vibrations– Pressure fluctuations

Note: The two are not completely independent.When phase and amplitude feedbacks are active, ponderomotive effects can change the response to external disturbances

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)

mL L

m

Q Qdw dw dw

y yw w

dwdw

±= - = -

-10-8-6-4-202468

10

90 95 100 105 110 115 120

Time (sec)

Freq

uenc

y (H

z)

Probability DensityMedium 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)

Prob

abili

ty D

ensi

ty

Qext Optimization with Microphonics

22

00

222

00

( 1) 2

( 1) ( 1) 22

mopt

opt c mg

sh

b Q

VP b b QR

dwbw

dww

Ê ˆ= + + Á ˜Ë ¯

È ˘Ê ˆÍ ˙= + + + + Á ˜Í ˙Ë ¯Î ˚

Condition for optimum coupling:

and

In the absence of beam (b=0):

and

2

00

22

00

1 2

1 1 22

If is very large

mopt

opt c mg

sh

m m

Q

VP QR

U

dwbw

dww

dw dw

Ê ˆ= + Á ˜Ë ¯

È ˘Ê ˆÍ ˙= + + Á ˜Í ˙Ë ¯Î ˚

Example7-cell, 1500 MHz

0

2

4

6

8

10

12

14

16

18

20

1 10 100

Qext (106)

P (k

W)

21.0 MV/m, 460 uA, 50 Hz 0 deg

21.0 MV/m, 460 uA, 38 Hz 0 deg

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

21.0 MV/m, 460 uA, 13 Hz 0 deg

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

Example

3-spoke, 345 MHz, =0.62

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

1.0 10.0 100.0

Qext (10^6)

P (k

W)

10.5 MV/m, 400 uA, 10 Hz 20 deg

10.5 MV/m, 300 uA, 10 Hz 20 deg

10.5 MV/m, 200 uA, 10 Hz 20 deg

10.5 MV/m, 100 uA, 10 Hz 20 deg

10.5 MV/m, 0 uA, 10 Hz 20 deg

Lorentz Detuning

Pressure deforms the cavity wall:

Outward pressure at the equator

Inward pressure at the iris

2 20 0

2

4

RF power produces radiation pressure:

Deformation produces a frequency shift:

L acc

H EP

f k E

m e-=

D = -

Lorentz Detuning

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1,000 -800 -600 -400 -200 0 200

Detuning (Hz)

Ener

gy C

onte

nt (N

orm

aliz

ed)

CEBAF 6 GeV

CEBAF Upgrade

Microphonics• Total detuning

0

0

where is the static detuning (controllable)

and is the random dynamic detuning (uncontrollable)m

m

dwdw

dw dw+

-10-8-6-4-202468

10

90 95 100 105 110 115 120

Time (sec)

Freq

uenc

y (H

z)

Probability DensityMedium 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)

Prob

abili

ty D

ensi

ty

Ponderomotive Effects

• Adiabatic theorem applied to harmonic oscillators (Boltzmann-Ehrenfest)

2

/

2 20 0

1 1

( )

( ) (4 4

If = , then is an adiabatic invariant to all orders

(Slater)

Quantum mechanical picture: the number of photons is constant:

d

VV

d Udt

UU U o eU

U N

U d H r E

e

wew w

ww w w

w

m e

- D DÊ ˆ Ê ˆD fi =Á ˜ Á ˜Ë ¯ Ë ¯

=

= +Ú

∼

2 20 0

)

( ) ( ) ( ) ( )4 4

(energy content)

(work done by radiation pressure)S

r

U dS n r r H r E rm ex

È ˘Í ˙Î ˚

È ˘D = - ◊ -Í ˙Î ˚Ú


2 20 0

2 20 0

( ) ( ) ( ) ( )4 4

( ) ( )4 4

( )

( ) ( )

( ) (

Expand wall displacements and forces in normal modes of vibration of the resonator

S

V

vS

V

dS n r r H r E r

d H r E r

r

dS r r

r q r

m

m mn

m mm

m exw

m ew

f

f f d

x f

È ˘◊ -Í ˙D Î ˚= -È ˘+Í ˙Î ˚

=

=

Ú

Ú

Ú

Â ) ( ) ( )

( ) ( ) ( ) ( )

S

S

q r r dS

F r F r F F r r dS

m m

m m m mm

x f

f f

=

= =

Ú

Â Ú


2

2

2

12

12

Equation of motion of mechanical mode

(Euler-Lagrange)

(elastic potential energy) : elastic constant

= (kinetic energy)

d L L F L T Udt q q q

U c q c

qT c

mm m m

m m mm

mm

m m

m

∂ ∂ ∂F- + = = -∂ ∂ ∂

=

WW

Â

Â2

2

222

: frequency

= (power loss) : decay timec q

q q q Fc

m

m mm

m m m

mm m m m m

m m

tt

t

FW

W+ +W =

Â


22 2 2 20

0 0

2

( ) ( )

The frequency shift caused by the mechanical mode is proportional to

Total frequency shift:

Static frequency shift:

q

FU k V

c U

t t

m m

mm m m m m m m

m m

mm

mm

w m

ww w wt

w w

w w

D

Ê ˆD + D +W D = - W = - WÁ ˜Ë ¯

D = D

D = D

Â2

Static Lorentz coefficient:

V k

k k

mm

mm

= -

=

Â Â

Â

Ponderomotive Effects – Mechanical Modes

Fluctuations around steady state:0 (1 )

o

V V vm m mw w dw

dD = D +

= +Linearized equation of motion for mechanical mode:

2 2 20

2 2 k V vm m m m m mm

dw dw dw dt

+ +W = - W

The mechanical mode is driven by fluctuations in the electromagnetic mode amplitude.

Variations in the mechanical mode amplitude causes a variation of the electromagnetic mode frequency, which can cause a variation of its amplitude.

→Closed feedback system between electromagnetic and mechanical modes, that can lead to instabilities.

( )2 2 20

2 k V n tm m m m m mm

w w wt

D + D +W D = -W +

Lorentz Transfer Function

TEM-class cavities ANL, single-spoke, 354 MHz, β=0.4

simple spectrum with few modes

( )

2 2 20

2 20

2 2

2 2

2( ) ( )2

k V v

k Vv

i

m m m m m mm

m mm

mm

dw dw dw dt

dw w d ww w

t

+ +W = - W

- W=

W - +

Lorentz Transfer Function

SNS Med β Cryomodule 3, Cavity Position 1, Lorentz Transfer Function (5MV/m CW)

25

30

35

40

45

50

55

60

65

70

0 60 120 180 240 300 360 420

AM Drive Frequency (Hz)

Cav

ity D

etun

ing,

Res

pons

e R

atio

(dB

)

0

45

90

135

180

225

270

315

360

Rel

ativ

e Ph

ase

(deg

)

TM-class cavities (Jlab, 6-cell elliptical, 805 MHz, β=0.61)Rich frequency spectrum from low to high frequencies

Large variations between cavities

GDR and SEL

Generator-Driven Resonator

• In a generator-driven resonator the coupling between the electromagnetic and mechanical modes can lead to two ponderomotive instabilities

• Monotonic instability : Jump phenomenon where the amplitudes of the electromagnetic and mechanical modes increase or decrease exponentially until limited by non-linear effects

• Oscillatory instability : The amplitudes of both modes oscillate and increase at an exponential rate until limited by non-linear effects

Self-Excited Loop-Principle of StabilizationControlling the external phase shift θl can compensate for the fluctuations in the cavity frequency ωc so the loop is phase locked to an external frequency reference ωr.

Instead of introducing an additional external controllable phase shifter, this is usually done by adding a signal in quadrature

→ The cavity field amplitude is unaffected by the phase stabilization even in the absence of amplitude feedback. Aq A

Ap

φ

tan2

cc lQ

ww w q= +

Self-Excited Loop

• Resonators operated in self-excited loops in the absence of feedback are free of ponderomotive instabilities. An SEL is equivalent to the ideal VCO.– Amplitude is stable – Frequency of the loop tracks the frequency of the cavity

• Phase stabilization can reintroduce instabilities, but they are easily controlled with small amount of amplitude feedback

Input-Output Variables

• Generator - driven cavity

• Cavity in a self-excited loop

Detuning (ω - ωc)

Field amplitude (Vo)Generator amplitude (Vg)

Cavity phase shift (θl)

Ponderomotiveeffects

Loop phase shift (θl)

Field amplitude (Vo)Limiter output (Vg)

Loop frequency (ω)

Pondermotiveeffects

Lorentz Detuning

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-6 -5 -4 -3 -2 -1 0 1 2

Detuning (Norm.)

Am

plitu

de (N

orm

.)

During transient operation (rise time and decay time) the loop frequency automatically tracts the resonator frequency. Lorentz detuning has no effect and is automatically compensated

Microphonics

• Microphonics: changes in frequency caused by connections to the external world– Vibrations– Pressure fluctuations

When phase and amplitude feedbacks are active, ponderomotive effects can change the response to external disturbances

( )2 2 20

2 2 k V v n tm m m m m mm

dw dw dw dt

+ +W = - W +

Microphonics

Two extreme classes of driving terms:

• Deterministic, monochromatic– Constant, well defined frequency– Constant amplitude

• Stochastic– Broadband (compared to bandwidth of mechanical

mode)– Will be modeled by gaussian stationary white noise

process

Microphonics (probability density)

SNS M02, CAVITY 3 BACKGROUND MICROPHONICS HISTOGRAM

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-20 -15 -10 -5 0 5 10 15 20

Cavity Detuning (Hz)

Prob

abili

ty o

f Occ

urre

nce

Std Dev = 2.2 Hz

SNS M03, CAVITY 1 BACKGROUND MICROPHONICS HISTOGRAM

0

0.01

0.02

0.03

0.04

0.05

0.06

-20 -15 -10 -5 0 5 10 15 20Cavity Detuning (Hz)

Prob

abili

ty o

f Occ

urre

nce

Std Dev = 5.6 Hz

Single gaussian

Noise driven

Bimodal

Single-frequency driven

Multi-gaussian

Non-stationary noise

805 MHz TM 805 MHz TM 172 MHz TEM

Microphonics (frequency spectrum)

TM-class cavities (JLab, 6-cell elliptical, 805 MHz, β=0.61)Rich frequency spectrum from

low to high frequenciesLarge variations between

cavities

SNS M02, Cavity 3, Bkgnd Microphonics Spectrum, 1W

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0 60 120 180 240 300 360

Frequency (Hz)

RM

S A

mpl

itude

(Hz

rms)

TEM-class cavities (ANL, single-spoke, 354 MHz, β=0.4)Dominated by low frequency (<10 Hz) from pressure fluctuations Few high frequency mechanical modes that contribute little to microphonics level.

Probability Density (histogram)

0

0.5

1

1.5

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

δω/δωmax

p(δω

)

0

0.1

0.2

0.3

0.4

0.5

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

δω/δωσ

p(δω

)

Harmonic oscillator (Ωμ,τμ) driven by:

Single frequency, constant amplitude White noise, gaussian

2 2max

1( )p dwp dw dw

=-

212

1( ) exp2

pw

dwsw

dws p

È ˘Ê ˆÍ ˙Á ˜Í ˙Á ˜Ë ¯Í ˙Î ˚

-=

Autocorrelation Function

-1

-0.5

0

0.5

1

0 2 4 6 8 10

τ

r(τ)

-1

-0.5

0

0.5

1

0 2 4 6 8 10

τ

r(τ)

( )( ) cos( )

(0) dR

rRdw

dwdw

tt w t= =

0

1( ) ( ) ( ) lim ( ) ( )T

x TR x t x t x t x t dt

Tt t t

Æ•= + = +Ú

Single frequency, constant amplitude White noise, gaussianHarmonic oscillator (Ωμ,τμ) driven by:

/( )( ) cos( )

(0)R

r eR

mt tdwdw m

dw

tt t -= = W

Stationary Stochastic Processes

x(t): stationary random variable

Autocorrelation function:

Spectral Density Sx(ω): Amount of power between ω and dω

Sx(ω) and Rx(τ) are related through the Fourier Transform (Wiener-Khintchine)

Mean square value:

0

1( ) ( ) ( ) lim ( ) ( )T

x TR x t x t x t x t dt

Tt t t

Æ•= + = +Ú

1( ) ( ) ( ) ( )2

i ix x x xS R e d R S e dwt wtw t t t w w

p• •-

-• -•= =Ú Ú

2 (0) ( )x xx R S dw w•

-•= = Ú

Stationary Stochastic Processes

For a stationary random process driving a linear system

( ) ( ) ( ) ( )

( ) ( ) [ ][ ] [ ]

( )

2 2

2

22

0 0

: ( ) ( )

( ) ( ) : ( ) ( )

( ) ( ) ( )

( )

auto correlation function of

spectral density of

y y x x

y x

y x

y x

x

y R S d x R S d

R R y t x t

S S y t x t

S S T i

y S dT i

w w w w

t t

w w

w w w

w ww

+• +•

-• -•

+•

-•

= = = =

È ˘Î ˚

=

=

Ú Ú

Ú

( )x t ( )y t( )T iw

Performance of Control System

Residual phase and amplitude errors caused by microphonics

Can also be done for beam current amplitude and phase fluctuations

2

Assume a single mechanical oscillator of frequency and decay time

excited by white noise of spectral density Am mtW

Performance of Control System

( )

( ) ( )( )

( ) ( )( )

2

22 2 2 222 2 2

22

22 2 22

22

22 2 22 2

2

2

2

2

2

2

ex

a

a ex

ex

d

G i di

G i

G i G i d di

G i

G i G i d di

A A A

v A

A

mm

m mm

mm

mmm

jm

m jmm

m

ww w

w wt

w

w w w ww wt

w

w w w ww w

t

ptdw

d dwpt

dj dwpt

+• +•

-• -•

+• +•

-• -•

+• +•

-• -•

- + + W

- + + W

- + + W

< >= = =W

W< >= = < >

W< >= = < >

Ú Ú

Ú Ú

Ú Ú

The Real WorldProbability Density

0.00001

0.0001

0.001

0.01

0.1

-15 -10 -5 0 5 10 15

Hz

Hz-1

Probability Density

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-15 -10 -5 0 5 10 15

Hz

Hz-1

Microphonics

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 0.02 0.04 0.06 0.08 0.1

Time (s)

Hz

Normalized Autocorrelation Function

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

Sec

The Real WorldProbability Density

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-10 -5 0 5 10

Hz

Hz-1

Microphonics

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.02 0.04 0.06 0.08 0.1

Time (sec)

Hz

Probability Density

0.00001

0.0001

0.001

0.01

0.1

-10 -5 0 5 10

Hz

Hz-1


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

Time (sec)

The Real WorldProbability density

0.00001

0.0001

0.001

0.01

0.1

-20 -15 -10 -5 0 5 10 15 20

Hz

Hz-1

Microphonics

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.02 0.04 0.06 0.08 0.1

Time (sec)

Hz


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Time (sec)

Probability density

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-20 -15 -10 -5 0 5 10 15 20

Hz

Hz-1

Piezo control of microphonics

MSU, 6-cell elliptical 805 MHz, β=0.49

Adaptive feedforward compensation

Piezo Control of Microphonics

FNAL, 3-cell 3.9 GHz

SEL and GDR

• SEL are best suited for high gradient, high-loaded Q cavities operated cw.– Well behaved with respect to ponderomotive instabilities– Unaffected by Lorentz detuning at power up– Able to run independently of external rf source– Rise time can be random and slow (starts from noise)

• GDR are best suited for low-Q cavities operated for short pulse length.– Fast predictable rise time– Power up can be hampered by Lorentz detuning

TESLA Control System

Basic LLRF Block Diagram

Low level rf control development

Concept for a LLRF control system

Pulsed Operation

• Under pulsed operation Lorentz detuning can have a complicated dynamic behavior

Cavity Pos. 3, Pulsed Power Response60Hz, 1.3ms, 12.7MV/m

0

100

200

300

400

500

600

700

800

0.000 0.001 0.002 0.003 0.004 0.005

Time (sec)

Det

unin

g (H

z)

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Tran

smitt

ed P

ower

Det

(V)

Pulsed Operation

• Fast piezoelectric tuners can be used to compensate the dynamic Lorentz detuning

Cavity # 2 @ 10 MV/m, with and without piezo compensation

-200

-100

0

100

200

300

400

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Time (sec)

-0.2

0

0.2

0.4

0.6

0.8

1

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