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Page 1
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
Page 2
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
Page 3
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= + ¥
Page 4
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
Page 5
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
Page 6
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-∼
Page 7
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=
Page 8
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
Page 9
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
Page 10
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
Page 11
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
= ¥
Page 12
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
Page 13
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
Page 14
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
Page 15
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
Page 16
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= =
Page 17
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∼
Page 18
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
Page 19
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
Page 20
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
Page 21
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
=
Page 22
Cavity with External Coupling
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
∫
Page 23
Cavity with External Coupling
• 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
b ∫
0
1 (1 )
LQ Qb+=
e
diss
PP
b =
Page 24
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
 Â0
0
0
1
efficient: ii
i
Li
Q PQ P
b
b
= =
=+Â
Page 25
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
Page 26
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 ˚ ,
Page 27
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
Page 28
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 ¯
Page 29
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 ˚
Page 30
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
Page 31
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=+
Page 32
Equivalent Circuit for a Cavity with Beam
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
=+
=+
=
Page 33
Equivalent Circuit for a Cavity with Beam
( ) [ ] 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 :
Page 34
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= -
Page 35
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
Page 36
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
Page 37
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 ˚
Page 38
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
Page 39
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
Page 40
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 = -
Page 41
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
Page 42
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
Page 43
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
Page 44
Ponderomotive Effects
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
 ) ( ) ( )
( ) ( ) ( ) ( )
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
 Ú
Page 45
Ponderomotive Effects
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
Page 46
Ponderomotive Effects
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
Page 47
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 +
Page 48
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 - +
Page 49
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
Page 50
GDR and SEL
Page 51
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
Page 52
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= +
Page 53
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
Page 54
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
Page 55
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
Page 56
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 +
Page 57
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
Page 58
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
Page 59
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.
Page 60
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 ˚
-=
Page 61
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
Page 62
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
Page 63
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
Page 64
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
Page 65
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
Page 66
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
Page 67
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
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
Time (sec)
Page 68
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
Normalized Autocorrelation Function
-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
Page 69
Piezo control of microphonics
MSU, 6-cell elliptical 805 MHz, β=0.49
Adaptive feedforward compensation
Page 70
Piezo Control of Microphonics
FNAL, 3-cell 3.9 GHz
Page 71
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
Page 72
TESLA Control System
Page 73
Basic LLRF Block Diagram
Page 74
Low level rf control development
Concept for a LLRF control system
Page 75
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)
Page 76
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