Download freqdomainbeamloading

frequency-domain study of
acceleration & beam
loading based on
a circuit model
by raquel fandos
Outline
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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
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–
–
–
Phase advance
Power and electric field
S parameters
Group delay
RF response
Beam loading
Motivation
Low vg
structures
Dispersion is
not negligible
Accurate model
Beam Loading
& Acceleration
Introduction
An accelerating structure
cell 1
A series of coupled resonant circuits
cell 2
cell 3
…
…
Matching elements: Zin=Zmatch
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)
Scheme of the analysis
Beam
&
RF pulse
params.
Struct.
params.
f0

vg
Q
RQ
Circuit
params.
R
L
C
k (coupling )
PSPICE
Cell to Cell
Transfer
Function
Signal proc.
RF response
Beam loading
H ij
Rt , Lt (match.
elements)
S-params.
Filling time
Power & Grad
Due to insufficient accuracy in PSPICE the analysis was performed using scripts that
work with all the signals in the frequency domain.
From structure parameters to
circuit elements
Cell i
2R
L
0Q
Q
C
0 R
 , f 0 , Q, v g  k
(circuit differential equations)
(from PhD thesis of C.D. Nantista, SLAC)
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.
Information extracted from the
circuit model
…
…
Directly in PSPICE we can measure:
- Filling time
- Voltage (prop. to electric field) and power flow along the structure
- S parameters
Vin
 Rt
I
s11  20 log 10 in
Vin
 Rt
I in
s21  20 log 10
Vout
Vg
Rt
: matching impedance
Information extracted from the
circuit model
…
…
Voltage Amplitude & Phase as functions of frequency at
the output of every cell n
Vn  Vampn  exp jVphasen 
Hij(f)
1.46
1.00
Transfer Functions from input cell i to output cell j
H ij 
Vj
0.50
Vi
0
11.88GHz
11.90GHz
V(R27:2)/v(r01:2)
11.92GHz
11.94GHz
11.96GHz
11.98GHz
f(GHz)
Frequency
12.00GHz
12.02GHz
12.04GHz
12.06G
Working in the frequency domain
V(f)
v(t )  FT 1 (V ( f ))
 f 0  f min
 f0
f0
f 0  f max
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 
AV(f)
1
v(t )  Re( FT ( AV ( f )))
 f min
f
f max
RF pulse response calculation
Envelope of the
Input RF pulse
FT
RFIN
cell1
RF1
cell2
RF2
cell3
RF3
…
t
RFIN
RF1
H 21
RFi  RFIN  H n1
V
H n1  n
V1
cell1
RFIN
cell1
cell2
Transfer function from the input to cell n
Vn  Vampn  exp jVphasen 
H 31
Voltage signal at the output of cell n
RF2
Beam response. The beam signal.
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 ( Nbunches ) and the bunch spacing T0
- 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
N  qe
I0 
Vbeam
BEAM
Beam signal (v)
…
T
FT
T0
t
T0
f(Hz)

1
T0  Nbunches
Beam response
BEAM1
BEAM 3
BEAM 2
cell1
cell2
cell3
RESP3
RESP2
BEAM1
BEAM 5
BEAM 4
cell4
…
RESP5
RESP4
cell1 cell2 cell3
p

BEAM i  BEAM i 1 exp  j 
c

i
RESPi   BEAM n  H n i
BEAM 2
cell2 cell3
n 1
BEAM 3
cell3
RESP4
Example: G241
Parameters:
– 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
G241 phase advance
Phase advance (degrees)
Phase advance (degrees)
Phase value @ nominal frequency
Cell number
f(GHz)
Very sensitive to changes in matching
elements.
Example: 0.01% change in the output Lt
Phase advance (degrees)
Nominal  =120deg.
Matched to 119deg.
Phase value @ nominal frequency
Cell number
G241 S-parameters
s11
s21
s22
0
-5
-10
S params (dB)
-15
-20
-25
-30
-35
-40
-45
-50
11.8
11.85
11.9
11.95
f(GHz)
12
12.05
12.1
Hz
G241. Group delay
11.850GHz
11.900GHz
11.950GHz
12.000GHz
12.050GHz
12.100GHz
12.15
Frequency
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.
Circuit Model
Difference model based on HFSS results
150
127.7
118
100
63.8
50
29.5
0
0
5
10
15
cell number
20
25
cell number
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.
G241.Beam loading
300 bunches separated
by 6 cycles
G241.Beam loading
Loaded and unloaded electric field along the structure
Circuit model
Electric field (MV/m)
Difference model based on HFSS results
cell number
cell number
G241.Beam loading compensation
70ns
7ns
Filling time=65.15ns
G241.Beam loading
RF pulse and beam response along the structure
Thanks
¿ ?