Chapter 4:
RF/Microwave interaction and beam loading in SRF cavity
4.1 RF field in SRF cavity
4.2 Beam loading
4.3 Dynamic detuning (microphonics, Lorentz force detuning, etc)
4.4 Basics on RF control
-develop equivalent circuit for rf system, cavity and beam
-develop equations for steady state and transient
-develop concept for the LLRF control
RF circuit modeling
RF components
RF source; klystrons are the most popular devices for f>300MHz.
tetrode, solid state amplifier for low power and/or low frequency
RF transmission; Waveguides or coaxial cables
Circulator; usually used as an isolator with matched load to protect RF source
Power coupler; feed RF power to a cavity
Cavity; electro-magnetic energy storage device
RF control; control cavity field and phase
Fundamental Power Coupler
HOM
Coupler
(HOMB)
HOM
Coupler
(HOMA)
Field
Probe
High Voltage
Power Supply
Klystron
Waveguide
Network
Waveguide/couplers
Low-level rf
EPICS user interface
Control/monitoring
Transmitter
HVPS
High power RF circuit
Low-level RF circuit
Main RF
Amplifier
Transmission
line
LO
LO
LO
I&Q
or
A&P
I&Q (or A&P)
Timing system &
Synchronization
transmitter
AND…
Protection system
(machine & personal)
Diagnostics &
user interfaces
Load
Resonance detection
& control
~
Main RF
Amplifier
Transmission
line
First, main high power RF circuit and cavity responses without beam.
Equivalent circuit (will use effective quantities for the modeling)
Dummy
Load
Le Ce re
Va
~ Beam, Ib
Z0 1:n
gI
Due to the circulator, this is not an exactly equivalent for generator current.
So introduced
gI twice of equivalent generator current.
Power coupler
Circulator
Transmission
line
(with beam later on)
Covert the model to the cavity side
~ Le Ce re
Va
~ Beam, Ib
n
II
g
g
Zext’=
n2Z0T2
Le Ce re
Va
ΩTR
P
V
P
TV
P
TLEr
W 2R
V
R
V
r
TV
r
V
2r
VP
2
sh
c
2
a
c
2
0
c
2
0sh
2
0
sh
2
0
sh
2
0
sh
2
a
e
2
ac
defined in Chap. 2
L C R
V0
~ L C R
V0
n
II
g
g
Zext=
n2Z0
Remember that equivalent circuit
parameters are defined by
references (V0, Va).
Coupling between cavity and the transmission line through a coupler, b
β
rZ,
β
RZ
Zn
R
Z
R
TZ
RT
Z
rβ e
extext
0
2
ext
2
ext
2
ext
e
Coupling factor b
,U/PωQ ex0ex c00 U/PωQ
impedanceshuntloadedeffective:β1
rr
r
β
r
1
r
1 eL
eeL
QLoaded:β1
Q
1
Q
1
Q
1)PU/(PωQ 0
L
0exL
cex0L
As it should be, coupling factor and Q’s are not function of particle velocity.
It is function of coupler geometry at a given mode (field profile).
That means Qex will be different when there’s a field tilt (field flatness).
impedanceshuntloaded:β1
RR
R
β
R
1
R
1L
L Similarly we can define
Governing equation for RF field in a cavity
~ Le Ce
Va
gI rL
IrL ILe ICe
geLrC LeIIII
eaL
Lar
aeC
dt/L
/r
C
e
L
e
VI
VI
VI
g
e
a
ee
a
eL
aga
e
a
L
aeC
1
CL
1
Cr
1
L
1
r
1C IVVVIVVV
If we use the equivalent circuit with V0,
rL should be replaced with RL
We can eliminate non-practical parameters (Ce, Le, C, L) using the relations:
L0
0
L
e0
LLe0
L
2
a
2
ae
0
cex
0L CRω
Lω
R
Lω
rrCω
r
V
2
1
VC2
1
ωPP
UωQ
g
L
L0a
2
0a
L
0a
Q
rωω
Q
ωIVVV
g
L
L0a
2
0a
L
0a
Q
rωω
Q
ωIVVV
Steady state solution with RF only
Generator current is the only source generator induced voltage Vg=Va
Particular solution in steady state of second order differential equation
ψ)ti(ω
aa eV(t) Vtiω
gg eI(t) Iat
angledetuning:ψ
1 when δ,2Qf
Δf2Q
ω
ωω2Q
ω
ω
ω
ωQtanψ
0)Δfif,eIr(eIψtan1
reI
ω
ω
ω
ωQ1
r(t)
L
0
L
0
0L
0
0L
tiω
gL
ψ)ti(ω
g2
Lψ)ti(ω
g2
0
02
L
La
V
Typical damped driven oscillator equation
Phasor representation
In general, fields can be expressed as
phase:θωtamplitude,:A,Ae )ti( A
To have the total voltage we need to add/subtract generator current/voltage
and beam current/induced voltage. Linear superposition works from the
linearity of Maxwell’s equations. But one should take the relative phase into
account.
If we choose a frame of reference that is rotating at a frequency , the phasor
will be stationary in time.
References can be arbitrary but it is convenient to have:
Reference frequency : operating frequency (rf source frequency) since all
other fields are around operating frequency.
Reference phase: beam arrives at the electrical center of cavity zero phase
(or real axis). How can we represent ‘beam’ at the reference frequency?
1
x
ψcosψtan1
1x
2
tiωiψ
gL
ψ)ti(ω
g2
La eecosψIreI
ψtan1
r(t)
V
common rotating term
relative phase
change due to
detuning
Amplitude
decrease
due to
detuning
iψshiψeiψ
L
g
atot ecosψ
β)2(1
recosψ
β1
recosψr
(t)
(t)Z
I
V
Total impedance of the equivalent model including detuning without beam
0.00E+00
2.00E-01
4.00E-01
6.00E-01
8.00E-01
1.00E+00
1.20E+00
-6000 -4000 -2000 0 2000 4000 6000-1.00E+02
-8.00E+01
-6.00E+01
-4.00E+01
-2.00E+01
0.00E+00
2.00E+01
4.00E+01
6.00E+01
8.00E+01
1.00E+02
-6000 -4000 -2000 0 2000 4000 6000
Va/(rLIg)
f0-f
Ex) QL=7105, f=805 MHz
plot the normalized Va and the detuning angle as a function of cavity detuning
f0-f
Bandwidth at -3bB: 10log10(P/Pref) for power, 20log10(V/Vref) for voltage
20log10(1/sqrt(2))=-3.01 =/4
Half width at -3dB: 1/2= 0/(2QL)=2575 Hz in this example
(time constant of loaded cavity)=1/1/2=2QL/0=277s
f, f/f=QL
RF power without beam loading
As mentioned,
‘due to the circulator, this is not an exactly equivalent for generator current.
So introduced Ig* twice of equivalent generator current.’
To calculate forward power in the transmission line (waveguide or coaxial cable)
8β
rI
β
r
2
I
2
I
2
1P
isgenerator thefrom lineion transmissin thepower forward averaged time theSo,
cavity in the voltageinducedgenerator with confused bet Don'
power. forward actual toscorrespond thisβ),/(r/2)(:Voltage
/2:current Forward
e
2
gegg
g
g
egfor
gfor
V
IV
II
β1
rr,ecosψIreI
ψtan1
rwith e
L
iψ
gL
iψ
g2
La
V
c
2
a0
0
2
ashshe
22
c2
e
22
a
e
2
g
g
a
P
VQ
Uω
Vr,r2r ψ)tan(1
4β
β)(1P
ψcos
1
2r
1
4β
β)(1V
8β
rIP
:is Vget toneeded power' Forward' calculatecan One
This is a useful equation when Pc & b are well defined, as for normal conducting cavity.
loading) beamwithout ( ag VV
forV
refV
phasein,β2
1β
resonanceon voltageinducedgenerator :1β
rr
2β
r
rg,
for
ge
gLrg,
ge
for
V
V
IIV
IV
loading) beamwithout ( ag VV
rg,V forV
refV
rg,V
b>1 b<1
In superconducting cavity, more practical parameters are QL, r/Q, Va (or V0), f1/2
since Q0 is much bigger than Qex, Pc is not well-defined, etc.
ψcos
1
(r/Q)Q
1
4
1V
ψcos
1
r
1
8
1Vψ)tan(1
r
β)(1
8β
β)(1V
8β
rIP
:is Vget toneededpower Forward
2
L
2
a2
L
2
a
2
e
2
a
e
2
g
g
a
relation. same themakes Q ingcorrespond and P ofset Any
)2Q
f(f ,
f
Δf
f
Δf2Q tanψearlier, defined asAnd
r
2
1r
β1
r
Q
r2
Q
1β
1β
2r
Q
2r
Q
r
Uω
P
P
V
Uω
V
Q
r
this.)of validity check the ,assumption thisuses one(when β.1β 1β
xx
L
01/2
1/20
L
LLe
L
L
0
e
0
e
0
shc
c
2
a
2
a
2
1/2L
2
o
2
1/2L
2
22
o
2
1/2L
2
ag
f
Δf1
(R/Q)Q
V
4
1
f
Δf1
Q(R/Q)T
TV
4
1
f
Δf1
(r/Q)Q
V
4
1P
cosψ(r/Q)QP2cosψ(r/Q)QP2cosψIrIψtan1
rxxLggLg
2
La
V
Don’t be confused with other passive couplings.
Quiz) when we measure Ea through field probe, how?
gV
resonanceonissystemthewhen
fieldcavityinducedGeneratorr gLrg, IV forV
refV
When b >>1
reffor
gL
ge
ge
for2
r
1)2(β
r
2β
r
VV
IIIV
iψ
gLag ecosψIrloading) beamwithout ( VV
HOMEWORK 4-1
for f=805 MHz, Ea=10MV/m, L=0.68m, r/Q=279, and
QL=7105, QL=1106, QL=2106,
1. Plot required forward power as a function of detuning (-500 Hz~500 Hz)
using spreadsheet 4_1.xlsx
2. If Q0=11010, What is cavity wall loss, Pc? What does that mean?
Equivalent beam in a RF circuit model
Micro-pulse Bunch spacing=Tb
RF frequency/n
When we say ‘Beam current’, it is an time averaged DC current.
Ex) Ib0=40 mA CW beam at bunch spacing 402.5 MHz
Tb=1/402.5 MHz~2.5ns,
Q (charge per bunch)=Ib0 (C/s) x Tb (s)=0.04 x 2.5e-9 = 100 pC
Temporal distribution of beam can be described by a Gaussian distribution with
standard deviation t
2t
2
2σ
t
t
bunch eσπ2
Q(t)I
t
Ibunch(t)
Ipeak
If t is 1.0 degree of 402.5 MHz (7 ps),
Ipeak~5.7 A t
Ibunch(t)
Ib0
b
b
2
t
2
b
2
b0
2
t
2
b
2
b
tpeakn
1n
bn0
2
t
2
t
bunch
T
2πω
3,2,1,0,n),2
σωnexp(2I)
2
σωnexp(
T
σ2π2Ia
,t)cos(nωa2
a)
2σ
texp(-
σπ2
Q(t)I
Fourier decomposition from –Tb/2 to Tb/2
1)2
σωnexp(, σ
nω
1If
2
t
2
b
2
t
b
Ex) Bunch spacing=402.5 MHz, Operating RF frequency=805 MHz:
So the Fourier component (n=1, 2, 3,…) of bunched beam at operating frequency
is simply 2Ib0.
0
0.2
0.4
0.6
0.8
1
1.2
0.4025 2.415 4.4275 6.44 8.4525 10.465 12.4775 14.49 16.5025 18.515
f (GHz)
Fo
uri
er
Co
mp
on
en
t E
xp
on
en
tial T
erm
Va
Steady state with beam loading
~ Le Ce re ~ Beam, Ib
n
II
g
g
Zext’=
n2Z0T2
- We added ‘beam’ as a current source like the RF generator. The beam
energy effect is in the effective quantities.
- Beam current at the operating frequency is
beamofcurrentDC:I,2I b0b0b I
- The cavity voltage is the sum of generator induced voltage and beam
induced voltage in a cavity.
bgabg
L
L0a
2
0a
L
0a )(
Q
rωω
Q
ωVVVIIVVV
-As mentioned earlier, set the reference phase for beam phase at the center of
electric center beam induced image current sits on negative real axis.
-Beam induced voltage has the same form as generator induced voltage.
ψ)i(π
bL
ψ)i(π
b2
Lb
iπ
bb
ecosψIreIψtan1
r
eIcurrent,inducedbeam
V
I
bIbLr b, r IV
bV
0
L
0
0L
0
0L
f
Δf2Q
ω
ωω2Q
ω
ω
ω
ωQtanψ
In this example is positive,
Which means the resonance
frequency is higher than
generator frequency.
bIbLr b, r IV
bV
Generator induced voltage is about same as before for the RF only case
except relative phase of generator current.
Let’s start with arbitrary phase first.
ψ)i(θ
gLg
iθ
gg
ecosψIr
eI
V
I
gLrg, r IV
gV
aV
The black circle is rotating around the origin
when generator phase (RF phase) changes.
The real component of Va is for acceleration: cosaV
bV
bIbLr b, r IV
bV
gV
bVaV
gLrg, r IV
-To get required accelerating voltage Va and synchronous phase for a beam
current Ib at a fixed cavity detuning () and Loaded Q (QL),
the generator current Ig (RF power and phase) is uniquely determined.
bIbLr b, r IV
bV
gV
bVaV
gLrg, r IV
How about forward voltage (Vfor) and reflected voltage (Vref) of the system with b>>1
forVrefV
bga
refforrefforreffora . and from calculateddirectly becan PandP :
VVV
VVVVV
Generator power
2
a
Lg
2
a
Lg
L
2
ag sinV
rItanψcos
V
rI1
r
1
8β
β)(1VP
For SRF cavities where b>>1, using the relations
2
b
L
1/2
2
b
L
L
2
ag
ab0b
a
b0ab0
b
b0bLL
tanQ
Q
f
Δf
Q
Q1
4(r/Q)Q
VP
cosVIP,cosV
(r/Q)I
ωU
cosVI
Q
1,2II,Q
Q
r
2
1r
Optimum cavity detuning (opt) and Loaded Q (QL),
cosIVP Δfat &QQ If
detuningoptimum:tan2Q
fΔftan
Q
Q-
f
Δf If
boagoptbL
b
0opt
b
L
1/2
No reflected power
statesteady in PPPP balancePower refbcg
HOMEWORK 4-2, Play with the spreadsheet
Ex) Using parameters in the table, (at particle beta=0.61) calculate Va, Qb, and
optimum f, and
generate table of required RF power as a function of f (-1000 Hz ~ 1300 Hz)
for QL1=3e5, QL2=7e5, QL3=1e6.
r/Q(at b=0.61)= 279 Ohm
TTF (at b=0.61)= 0.68
Ib0= 0.04 A
Syn Phase= -15 degree
E0= 15 MV/m
Length= 0.6816 m
QL1= 3.00E+05
QL2= 7.00E+05
QL3= 1.00E+06
f= 8.05E+08 Hz
0
100
200
300
400
500
600
700
-1500 -1000 -500 0 500 1000 1500
delta f (f0-f) in Hz
RF
po
we
r
QL3
QL2
QL1 fopt
Pb
(4_2.xlsx)
change parameters in blue
HOMEWORK 4-3, Play with the spreadsheet
Ex) Using parameters in the table and TTF data,
generate (r/Q), Qb, RF power required, optimum f, required RF power at
optimum f as a function of particle velocity from b=0.5 to b=0.8.
r/Q(at beta=0.61)= 279 Ohm
Ib0= 0.04 A
Syn Phase= -15 degree
E0= 15 MV/m
Length= 0.6816 m
QL= 7.00E+05
delta f= -300 Hz
f= 8.05E+08 Hz
Transit time factor
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.45 0.55 0.65 0.75 0.85
beta
T (from SF)
T(fit)
TTF=1540.22897*b^6 - 6951.52591*b^5 + 12957.19469*b^4 - 12724.40926*b^3 + 6910.19655*b^2 - 1956.643*b + 224.88436
(4_3.xlsx); change parameters in blue
0
50
100
150
200
250
300
350
0.45 0.55 0.65 0.75 0.85
beta
r/Q
(O
hm
)
0.0E+00
5.0E+05
1.0E+06
1.5E+06
2.0E+06
2.5E+06
0.45 0.55 0.65 0.75 0.85
beta
Qb
0
20
40
60
80
100
120
140
160
180
200
0.45 0.55 0.65 0.75 0.85
beta
Op
t. d
etu
nin
g (
Hz)
0
50
100
150
200
250
300
350
0.45 0.55 0.65 0.75 0.85
beta
Po
wer
(kW
)
Required RF
at -300 Hz detuning
Required RF
at optimum detuning
Transient behavior without beam
g
L
L0a
2
0a
L
0a
Q
rωω
Q
ωIVVV
2t/τ
L
2
refref
t/τ
Lgrefareffor
LggLforrg,
t/τ
Lg
t/τ
gLa
tiω
gga
tiω
gg0
)e2(1P2r
(t)V(t)P
)e2(1(r/Q)QP(t)V
(r/Q)QP/2IrVV2
1
)e(1(r/Q)QP2)e(1Ir(t)V
eI RFon turn & 0(0) 0,at t 1)
axis. realon are all ,eIset weIf phase.in are All ω.ω case, resonanceOn
g
VVV
IV
I
Steady state
term
Transient
term
02
L
01
1/20
L
ψ)ti(ω
gL
tiω
2
tiω
1τ
t
a
ω4Q
11ωfrequency) resonance (systemω,
ω
1
ω
2Qconstant) (time τwhere
ecosψIr)ece(ce(t)V 11
General Solution:
initial condition problem
)/τt2(t2/τt
grefemit
a
0
a
0
2
a
0
2
a
ref
c1
brefcg
1
)/τt(
1aa
i/τt
Lg1a1
11
1
1
e)e(14P)P(P
dt
(t)dV
(r/Q)ω
(t)2VdU/dt
(r/Q)ω
VU
Uω
V
Q
r
example in thisdU/dt (t)P
1)(β negligible is P and at t off turnedis RF example, In this
dU/dtPPPP :balancepower General
t t,)e(t(t)
RF off turn & )ee(1(r/Q)QP2)(t ,tat t 2)
tVV
V
The direction of this power after RF off is same as that of reflected power. This
power is a release of stored energy from cavity. The mechanism is different.
This power is called ‘emitted power’.
Some useful information can be taken just from decay curves of the Va and
Pemit after RF off.
shortlyit at look will weamount; detuning
QPemitor Vaeither ofdecay at fit with
property).cavity :(r/QUω
V
Q
r using calculated becan Va U,know when we
,energy)(storedUPemit(t)dt
L
0
2
a
t1
(transient1.xls); change parameters in blue
Pg= 60 kW
r/Q= 279 Ohm
QL= 7.00E+05
t1= 0.0015 s
f= 8.05E+08 Hz
Ex)
0
1
2
3
4
5
6
7
8
0 0.001 0.002 0.003
Time (s)
Va
(M
V)
0
50
100
150
200
250
0 0.001 0.002 0.003
Time (s)
P (
kW
)
Forward P
Reflected P
Emitted P
Area=
stored energy
)/τt2(t 1e
)/τt(t 1e
tiω
tiω
aa
e(t)I(t)
,e(t)V(t)
I
V
Transient behavior with detuning & beam loading
-For general expressions including time-varying detuning and beam loading,
the system equation will be developed.
-Using finite difference method, cavity behaviors will be explored.
-First, separate out fast rotating terms and build up model in vector space (real
& imaginary)
bg
L
L0a
2
0a
L
0a where,
Q
rωω
Q
ωIIIIVVV
Complex envelope
(slowly varying)
RF term
(fast oscillation)
)/(2Qωω,ωωωΔ where
)IiωI(ω2rV)Δω()iωω(Δ2ωV)iω2(ωV
L01/20
1/2a
2
1/2a1/2a
L
IωrV)iωωi(Δ-VV2iω
11/2a1/2aa L
IωrV)iωωi(Δ-V 1/2a1/2a L
arrange and above,equation in the iIII,iVVVinsert iraiara
i1/2ai1/2arai
r1/2aiar1/2ar
IωrVωωVΔdt
dV:Imaginary
IωrωVΔVωdt
dV : termReal
L
L
System Equation for SRF cavity: First order ordinary differential equation.
Initial condition problem
This equation set describes SRF cavity field in complex space.
Using following relations sets of power and voltage can be calculated.
reffora
rirr
fifr
aiar
b0b
L
2
for
gL
for
bg
LL0L01/2
power reflected & phase votage,reflectedV,V
power forward & phase votage,forwardV,V
phase and tagecavity volV,V
2I,r2
P,2
r
:source driving
input too.an becan in time ωΔ yingslowly var
(r/Q)2
Qr ω,-ωωΔ ),/(2Qωω :constantInput
VVV
IV
IV
III
g
ti,1/2t12,tai,1/2t11,tar,t22,
tr,1/2t12,tai,t11,tar,1/2t21,
ti,1/2tai,1/2tar,t12,
tr,1/2tai,tar,1/2t11,
t22,t12,tai,Δttai,
t21,t11,tar,Δttar,
Iωr)K(Vω)Kω(VΔtK
Iωr)Kω(VΔ)K(VωtK
)IωrVωωVΔt(K
)IωrωVΔVωt(K
)/2K(KVV
)/2K(KVV
(FDM) method difference finite using way simple One
L
L
L
L
(filling_test_FDM.xls); change parameters in blue
Ex) RF only RF power 60000
Phase1 0
F 8.05E+08
w 5.06E+09
QL 7.00E+05
r/Q 279
Cl 0.6816
rL 97650000
w1/2 3.61E+03
detuning end of pulse 300
slope Hz/ms 0
filling time us 3.00E+03
dt 2.00E-06
0.0E+00
1.0E+06
2.0E+06
3.0E+06
4.0E+06
5.0E+06
6.0E+06
-4.0E+06 -2.0E+06 0.0E+00 2.0E+06 4.0E+06 6.0E+06
0.00E+00
1.00E+06
2.00E+06
3.00E+06
4.00E+06
5.00E+06
6.00E+06
7.00E+06
0 1000 2000 3000 4000
forV
refV
loading)beamnoV(or gaVaV
0.00E+00
5.00E+00
1.00E+01
1.50E+01
2.00E+01
2.50E+01
3.00E+01
0 1000 2000 3000 4000
aofPhase V
0.00E+00
1.00E+06
2.00E+06
3.00E+06
4.00E+06
5.00E+06
6.00E+06
7.00E+06
8.00E+06
-2.00E+06 0.00E+00 2.00E+06 4.00E+06 6.00E+06 8.00E+06
HOMEWORK 4-4)
Generate these three plots Va at various
detuning using filling_test_FDM.xlsx (RF only)
0.E+00
1.E+06
2.E+06
3.E+06
4.E+06
5.E+06
6.E+06
7.E+06
8.E+06
0 500 1000 1500 2000 2500 3000
amp
litu
de
of V
a
Time (us)
0
20
40
60
80
100
120
0 500 1000 1500 2000 2500 3000
Ph
ase
of V
a
Time (us)
Hz0Δf
Hz300Δf
Hz600Δf
Hz0021Δf
Hz0036Δf
Hz0036Δf
Hz0021Δf
Hz600Δf
Hz300Δf
Hz0Δf
Hz0Δf
Hz300Δf
Hz600Δf
Hz0036Δf
Hz0024Δf
Hz0015Δf
Hz900Δf
aiV
arV
-2.00E+02
-1.50E+02
-1.00E+02
-5.00E+01
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
0 1000 2000 3000 4000
Ex) pulsed operation. RF pulse length 1500 s, detuning -300 Hz
other parameters are same as in the previous example
0.00E+00
1.00E+06
2.00E+06
3.00E+06
4.00E+06
5.00E+06
6.00E+06
7.00E+06
0 1000 2000 3000 4000
0.00E+00
4.00E+04
8.00E+04
1.20E+05
1.60E+05
2.00E+05
0 1000 2000 3000 4000
(filling_test_FDM.xls)
change parameters in blue
aV
aofPhase V
Reflected power
d/dt=
0.00E+00
4.00E+04
8.00E+04
0 1000 2000 3000 4000
-1.60E+02
-1.40E+02
-1.20E+02
-1.00E+02
-8.00E+01
-6.00E+01
-4.00E+01
-2.00E+01
0.00E+00
0 1000 2000 3000 4000
0.00E+00
1.00E+06
2.00E+06
3.00E+06
4.00E+06
5.00E+06
6.00E+06
0 1000 2000 3000 4000
(pulse_beam_test_FDM.xls)
change parameters in blue
Ex) open loop with beam loading
RF power (W) 60000
Vfor phase (degree) 2
f (Hz) 8.05E+08
w 5.06E+09
QL 7.00E+05
r/Q (Ohm) 279
Cl (m) 0.6816
rL 97650000
w1/2 3.61E+03
rL*w1/2 3.53E+11
detuning end of pulse -300
slope Hz/ms 0
rf pulse length us 2.00E+03
Ib0 (A) 2.50E-02
beam enters at t (us) 500
bPhase (degree) 0
dt 2.00E-06 -6.00E-02
-5.00E-02
-4.00E-02
-3.00E-02
-2.00E-02
-1.00E-02
0.00E+00
0 1000 2000 3000 4000
aV
aofPhase V
Reflected power
-Ib=-2Ib0
d/dt=
RF control
To have a beam with a required quality (emittance, energy spread, etc.) cavity
field amplitude and phase should be maintained within a certain (machine
specific) ranges.
For example, <1% in amplitude and <1 degree in phase are typical values.
In modern digital LLRF systems, feedback control is an essential part and feed
forward control becomes more popular.
LLRF control system should provide required cavity field stability against;
Beam loading (transient including jitter and/or CW)
Beam fluctuations
HPRF droop/ripple (mainly from HVPS)
Dynamic cavity detuning (Lorentz force detuning, microphonics)
Loop delay of RF control system
Reference RF phase/amplitude fluctuations
Electron loading conditions in a cavity and/or at around a power coupler
Any changes that affects characteristics of control system
matching condition in HPRF transmission line
thermal drift (electronic board, cables, etc.)
HVPS
Field detection and control: digital I/Q mostly in modern control system
this uses conceptually same as the phasor relations we learned
Main RF
Amplifier
Transmission
line
LO
I&Q
transmitter
Load
Field Reference
RF
IF LPF
+
- errors controller
I: in-phase (corresponds real component),
Q: quadrant (corresponds imaginary component)
-3.00E+06
-2.00E+06
-1.00E+06
0.00E+00
1.00E+06
2.00E+06
3.00E+06
4.00E+06
5.00E+06
6.00E+06
7.00E+06
8.00E+06
0.00E+00 1.00E+03 2.00E+03 3.00E+03 4.00E+03
-3.00E+06
-2.00E+06
-1.00E+06
0.00E+00
1.00E+06
2.00E+06
3.00E+06
4.00E+06
5.00E+06
6.00E+06
7.00E+06
8.00E+06
0.00E+00 1.00E+03 2.00E+03 3.00E+03 4.00E+03
Ex) Feedback control example:
The real world has many complex practical issues. One can develop a
conceptual understandings of the rf control from this example.
(Feed_back_test_FDM.xls) change parameters in blue
Reference cavity field I
Reference cavity field Q
cavity field I
cavity field Q
Parameters used
In this example
Constant detuning
For reference
During filling
Feedback gains
f (Hz) 8.05E+08
w 5.06E+09
QL 7.00E+05
r/Q (Ohm) 279
Cl (m) 0.6816
Vcav MV 6952320
filling time (s) 4.00E-04
(1-exp(-1)) -1 1.5819767
rL 97650000
w1/2 3.61E+03
rL*w1/2 3.53E+11
detuning end of pulse 168
slope Hz/ms 0
rf pulse length us 1.30E+03
Ib0 (A) 2.60E-02
beam enters at t (us) 700
bPhase (degree) -15
proportional gain K 20
integral gain Ki 1
dt 2.00E-06
-5.00E+01
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
0.00E+00 1.00E+03 2.00E+03 3.00E+03 4.00E+03
0.00E+00
5.00E+04
1.00E+05
1.50E+05
2.00E+05
2.50E+05
3.00E+05
3.50E+05
0.00E+00 1.00E+03 2.00E+03 3.00E+03 4.00E+03
0.00E+00
1.00E+06
2.00E+06
3.00E+06
4.00E+06
5.00E+06
6.00E+06
7.00E+06
8.00E+06
0.00E+00 1.00E+03 2.00E+03 3.00E+03 4.00E+03
-4.00E+01
-2.00E+01
0.00E+00
2.00E+01
4.00E+01
6.00E+01
8.00E+01
1.00E+02
0 1000 2000 3000 4000
Va amplitude
Va phase
Forward power
Reflected/emitted
power
Forward power
phase
Forward power
phase
RF control for constant field/phase:
One can say that it forces the system into steady state condition quickly and stably
0.00E+00
1.00E+06
2.00E+06
3.00E+06
4.00E+06
5.00E+06
6.00E+06
7.00E+06
8.00E+06
0.00E+00 1.00E+03 2.00E+03 3.00E+03 4.00E+03
Va amplitude
There are still errors with feedback control from many other sources.
the errors are repetitive, one can generate error tables including loop delay
information and apply to the next pulses. feed forward
Cavity detuning due to the Lorentz force (dynamic) Vibrations, resonances and damping
-Vibration source; RF pulse repetitive hammering by radiation pressure
with frequencies of repetition rate and harmonics
-Mechanical resonance frequencies (wn)
determined by the equivalent mass of each mechanical mode
& equivalent stiffness of the system
-Resonance; source term hits around the mechanical resonance frequency
-Damping; determined by the whole system
energy transfer; sound wave radiation, internal/structural damping,
helium, heat dissipation, transfer to other system thru propagation
-Vibration amplitude; determined by the relation between the damping and the
resonance.
generally amplitude is smaller at higher frequency and higher damping.
This will add initial frequency off-set.
-2000
-1500
-1000
-500
0
500
1000
-50 -40 -30 -20 -10 0 10 20 30 40 50
Rad
iati
on
Pre
ssu
re (
Pa)
axial coordinate (cm)
Radiation Pressure
Impulsive Forcing Function
(time domain)
Transformed Forcing Function
(Frequency Domain)
Radiation Pressure in time and frequency domain
(vibration source)
0 1000 2000 3000 Time (sec)
Cavit
y F
ield
Time
Cavity displacement
or cavity detuning
Cavity field Corresponds static KL
After transient disappear
CW
Pulsed mode
0.5 1 1.5 2
-2?10-9
-1?10 -9
1?10-9
2?10 -9
Static deformation
Dynamic vibrations. Reaches steady state
after transient period.
Mechanical modes of the system
lQ
l
VkQ
l
l
lcavllll
l
ll
modemechanical offactor Quality:
modemechanical offrequency resonance:
222
Resonance system
Mode, damping, & modal mass finding strongly
depends on boundary conditions and whole
mechanical system details.
Mode frequencies & Q of the mode can have
large spread. large error
l:kl
l
ltotal
modeofcoeff.detuningdynamic
,
Vibration due to LF
Measured transfer function by Lorentz force
Once mechanical responses (amplitude and phase) of one selected cavity are
measured, quite accurate prediction or reconstruction is possible for that
cavity.
But due to the sensitivity of mechanical mode characteristics, there is large
scattering. Sometimes unpredicted mode could be found.
-200
0
200
400
600
800
1000
0 500 1000 1500
Time (us)
Dy
na
mic
de
tun
ing
(H
z)
-200
-100
0
100
200
300
400
500
600
0 500 1000 1500
Time (us)
Dy
na
mic
de
tun
ing
(H
z)
Medium beta cavity (installed cavity)
KLDyn: 3~4 Hz/(MV/m)2
17 MV/m
High beta cavity (installed cavity)
KLDyn: 1~2 Hz/(MV/m)2
16.5 MV/m
Ex) SNS cavities
filling flattop
filling flattop
Unpredicted 1.6kHz component sits on nominal low frequency response in
medium beta cavities.
-200
0
200
400
600
800
1000
0 500 1000 1500
Time (us)
Dy
na
mic
de
tun
ing
(H
z)
-200
-100
0
100
200
300
400
500
600
0 500 1000 1500
Time (us)
Dy
na
mic
de
tun
ing
(H
z)
Medium beta cavity (installed cavity)
KL: 3~4 Hz/(MV/m)2
17 MV/m
High beta cavity (installed cavity)
KL: 1~2 Hz/(MV/m)2
16.5 MV/m
Observed detuning agrees with expectations
-200
0
200
400
600
800
1000
0 500 1000 1500
Time (us)
Dy
na
mic
De
tun
ing
(H
z)
15Hz
30Hz
60Hz
filling flattop
-400
-200
0
200
400
600
800
1000
0 300 600 900 1200 1500
Time (us)
Dyn
am
ic D
etu
nin
g (
Hz)
0.0E+00
2.0E+06
4.0E+06
6.0E+06
8.0E+06
1.0E+07
1.2E+07
1.4E+07
1.6E+07
1.8E+07
Ea
cc (
MV
/m)
Dynamic detuning
Eacc
In this example the accelerating gradient is 12.7 MV/m. (high beta cavity)
The 1.6 kHz components shows
resonances at higher repetition rate
in some of medium beta cavities
some cavities show bigger resonance phenomena
repetition rate dependent
Static driving forces are proportional to ‘square of cavity field’ (Lorentz force).
Dynamic responses could be quite different depending modal mass, modal
boundary conditions, driving force spectrums.
Low Qex and high beam loading structure, not a big issue (will need some extra
RF power).
Very important in pulsed machine especially in pulsed high Qex structure.
Generate steady state vibration pattern. Repetitive from pulse to pulse.
Counter vibration (compensation) for the biggest frequency components can
correct quite efficiently. Demonstrations have been done using piezo-electric
actuators.
Vibration due to PT
An input voltage is applied to the piezoelectric
actuator device which make the piezo stack
shrink/expand
piezo Piezo-stack
Piezo-voltage input
Shrink/expand
LFD compensation
Since the forcing mechanisms are
different between LFD and piezo
detuning system responses are not
same.
Building a virtual cavity for dynamic detuning
(complete set of modeling)
tdeIReVVt tdj
L
tdjt
t
t
0
~
2/1
~
0
~~~0
2 2 2mL,m L,m m L,m m L,m
m
L,m
RF 0 mod
SST L 0 mod mod mod
modmod
1/ 2
mod m2
L,m L,m
mod mod m
k VQ
k 0
P P (1 sin t)
V (t) R I (1 cos sin( t ))2
tan
1 cos cos(t) k V
sin( t / 2)
2 2mP,m P,m m P,m m P,m P
m
P osc osc
osc 1/ 2i osc
osc osc
j cos
n n n
n
n
n
i
k VQ
(t) sin t
; ; t
v( ) e P ( )cos cos(n )
P ( ) polynomes of
ntan
General RF eq.
of Cavity field
Dynamic detuning
Due to LF
Dynamic detuning
Due to Piezo tuner
Virtual Cavity
(analytic basis;
very fast and
provide general view
Verification of Virtual Cavity
Many useful and practical tools
; optimization of compensation for LFD
microphonics compensation study in high Qex cavity
ponderomotive oscillation study for RF system…
Ex. LFD compensation (optimization study)
LF only contains harmonics of the repetition rate.
Ideally, perfect compensation is possible.
Decompose the LFD into harmonics find corresponding piezo signal components
(I) Piezo. Tuner input voltage
(II) Generated detuning
(including transient)
(III) Detuning generated by the
Piezo. Tuner in SST
(IV) Its sum with the initially
Targeted portion of the
Lorentz Detuning
(I)
(II)
(III) (IV) < few Hz
Seems to be complex to apply for the real system
Practically applicable and straightforward compensation
scheme (simple harmonic compensation)
-Only the Lorentz
Detuning during the RF
turn-on transient and
the beam pulse is tried
to be compensated
-A single harmonic
seems sufficient to
obtain a satisfying
compensation
-The Piezo. Tuner input
voltage contains only
this harmonic
Simple waveform
Ex. Analysis with dynamic detuning (feed_back_test_FDM_dyn_detuning.xls)
Microphonics
There are always mechanical vibrations from environments.
These vibrations can shake cavities.
Responses of cavities are function of dynamic/modal characteristics of the
system (not only by cavity mechanical properties).
Qex is normally higher in low beam current machines. Cavity bandwidths are
getting narrower as Qex’s get higher.
If HPRF does not have enough margin, a cavity field may not reach its
operating setpoint.
Examples of vibration sources
helium pressure fluctuations
pumps (water, mechanical vacuum)
ground vibration including ocean waves (1/7 Hz)
traffics
etc.
Ex) Microphonics measurement at JLab
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-10.3 -8.9 -7.5 -6.2 -4.8 -3.4 -2.0 -0.7 0.7 2.1 3.5 4.9 6.2 7.6 9.0 10.4 11.7
Occu
ren
ces
Frequency Deviation (Hz)
Beam current in proposed CW superconducting linacs is < several mA.
Qb is mid 107 range
If loaded Q QL is 5x107 (for RF efficiency), 1/2= 0/(2QL) will be in comparable
ranges of microphonics.
f=650 MHz f1/2=6.5 Hz
f=80.5 MHz f1/2=0.8 Hz
If 1/2 is too small,
a few bandwidth of cavity frequency variation will cause large cavity phase
variations, and/or cavity field can not be kept at operating point.
It may need to make stiff cavities to push mechanical frequencies as high as
possible so they don’t couple to the low frequency mechanical noise that has the
largest amplitudes. But required force of mechanical tuner should be in a
reasonable range.
Amplifier may need to have a certain amount margin that can cover detuned
cavity operation.
Active feedback/feed forward control may need to be used.