Frequency-domain study of acceleration & beam loading based on a circuit model by raquel fandos.
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Transcript of Frequency-domain study of acceleration & beam loading based on a circuit model by raquel fandos.
frequency-domain study of acceleration & beam
loading based on a circuit model
by raquel fandos
Outline• Motivation• Introduction• Scheme of the analysis• From structure parameters to circuit elements• Information extracted from the circuit model• RF response calculation• Beam response calculation• Example: G241
– Phase advance– Power and electric field– S parameters– Group delay– RF response– Beam loading
Motivation
Low vgstructures
Dispersion isnot negligible
Accurate modelBeam Loading& Acceleration
IntroductionAn accelerating structure
Matching elements: Zin=Zmatch
cell 1 cell 2 cell 3
…
Input matching cell Output matching cell Tapered structure
- vg, Q & R/Q vary R, C, L & k vary.
- Input & output have different matching
parameters (Rt & Lt)
…
A series of coupled resonant circuits
Scheme of the analysis
Due to insufficient accuracy in PSPICE the analysis was performed using scripts that work with all the signals in the frequency domain.
S-params.Filling time
Power & Grad
RF responseBeam loading
Struct.params.
Circuitparams.
PSPICE
Cell to CellTransferFunction
ijH
)
.(,
)(
elements
matchLR
couplingk
C
L
R
ttQR
Q
v
f
g
0
Signal proc.
Beam&
RF pulseparams.
From structure parameters to circuit elements
R
QC
Q
RL
0
0
2
kvQf g ,,, 0
When the structure is tapered, vg, Q and R/Q vary along the structure, and so do R, C, L and k from cell to cell.
(circuit differential equations)
Cell i
(from PhD thesis of C.D. Nantista, SLAC)
Information extracted from the circuit model
g
tin
in
tin
in
V
Vouts
RIV
RIV
s
1021
1011
log20
log20
…
Directly in PSPICE we can measure:
- Filling time- Voltage (prop. to electric field) and power flow along the structure- S parameters
…
tR : matching impedance
Information extracted from the circuit model
Voltage Amplitude & Phase as functions of frequency at the output of every cell n
Transfer Functions from input cell i to output cell j
… …
i
jij V
VH
nnn jVphaseVampV expHij(f)
Frequency
11.88GHz 11.90GHz 11.92GHz 11.94GHz 11.96GHz 11.98GHz 12.00GHz 12.02GHz 12.04GHz 12.06GHz 12.08GHzV(R27:2)/v(r01:2)
0
0.50
1.00
1.46
f(GHz)
Working in the frequency domain
0f0f
))(()( 1 fVFTtv
f
V(f)
In order to have a reasonable number of samples in the pass band, we need to store a lot of zeros
Solution: Work in baseband
f
AV(f)
)))((Re()( 1 fAVFTtv
min0 ff max0 ff
minf maxf
RF pulse response calculation
1nINi HRFRF
11 V
VH n
n
1RFcell1 cell2 cell3
t
Envelope of the Input RF pulse
…
Transfer function from the input to cell n
2RF 3RFINRF
nnn jVphaseVampV exp
FT
Voltage signal at the output of cell n
1RFcell1INRF
cell1 cell22RFINRF
21H
31H
The beam in the time domain can be assumed to be a Dirac train, therefore its FT is a sinc signal centered in f0 and with
- a width that depends on the number of bunches ( ) and the bunch spacing
- an amplitude Vbeam that depends on the charge per bunch. The voltage amplitude that corresponds to a certain bunch charge is estimated in the PSPICE circuit model from the response in voltage to a current Dirac signal of value
Beam response. The beam signal.
0T
0T
0T
…t
Beam signal (v)
FT
NbunchesT
0
1
Nbunches
Te
0
qNI
VbeamBEAM
f(Hz)
Beam response
in
i
nni
ii
HBEAMRESP
c
pjBEAMBEAM
1
1 exp
3RESP2RESP
1BEAM
cell1 cell2 cell3 …2BEAM
3BEAM
4RESP
4BEAM
1BEAM cell1 cell2 cell3
cell2 cell3
cell3
2BEAM 4RESP
3BEAM
cell4
5RESP
5BEAM
Example: G241
– f0 = 11.994GHz
– = 120deg
– 26 cells
Cell First Middle Last
vg/c[%] 1.66 1.19 0.83
Q 6100 6177 6265
R’/Q[Linac kOhm/m]
14.6 16.2 17.9
Parameters:
G241 phase advance
Nominal =120deg. Matched to 119deg.
Very sensitive to changes in matching elements. Example: 0.01% change in the output Lt
f(GHz)
Ph
ase
ad
van
ce (
de
gre
es)
Cell number
Cell number
Phase value @ nominal frequency
Phase value @ nominal frequency
Ph
ase
ad
van
ce (
de
gre
es)
Ph
ase
ad
van
ce (
de
gre
es)
G241 S-parameters
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
11.8 11.85 11.9 11.95 12 12.05 12.1
s11 s21 s22
f(GHz)
S p
aram
s (d
B)
G241. Group delay
Frequency
11.800GHz 11.850GHz 11.900GHz 11.950GHz 12.000GHz 12.050GHz 12.100GHz 12.150GHz11.752GHzVG(R27:2)
0s
40.0ns
80.0ns
120.0ns
158.4ns
65.15ns @ f0 (62 ns from difference model)
G241. RF pulse response
RF pulse at cell
70ns
40ns
G241. Unloaded Power and Electric Field along the structure.
0
50
100
150
0 5 10 15 20 25
63.8
29.5
127.7118
Electric field (MV/m)
Power (MW) HFSS data for the first, last and middle cell were available. 2nd order polynomial interpolation used for the rest.
cell number
Circuit Model Difference model based on HFSS results
cell number
300 bunches separated by 6 cycles
G241.Beam loading
Difference model based on HFSS results Circuit model
G241.Beam loading
cell number cell number
Ele
ctric
fie
ld (
MV
/m)
Ele
ctric
fie
ld (
MV
/m)
Loaded and unloaded electric field along the structure
70ns
7ns
Filling time=65.15ns
G241.Beam loading compensation
G241.Beam loadingRF pulse and beam response along the structure
Thanks
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