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Update on the kicker impedance model and measurements of material properties
V.G. Vaccaro, C. Zannini and G. Rumolo
Thanks to: M. Barnes, N. Biancacci, A. Danisi, G. De Michele, E. Metral, N. Mounet, T. Pieloni, B. Salvant
A simplified EM model of C-magnet for ferrite loaded kickers
Comparing the two models
Using the C-magnet model in the transverse horizontal impedance a high peak at 40MHz appears. At low frequency the Tsutsui model is not able to estimate correctly the kicker impedance. The two models are in good agreement at high frequency (>400MHz). 3
Penetration depth in ferrite[m
]
Simplified model for a CMagnet
Vacuum
Ferrite
PEC
In order to cross check the results of CST and Tsutsui model it is convenient to resort to a simplified devices to which is possible to treat analytically.
The analytical treatment
Vacuum
Ferrite
PEC
1. The model is indefinite in the longitudinal direction
2. The analysis is performed in FD3. All the fields have the same behavior in the
longitudinal direction4. Using the Maxwell equations all the
components of the fields are derived from the longitudinal fields of TE and TM modes.
5. The sources are represented by a discrete number of linear currents placed at the point
6. The longitudinal field is computed at the point
mmmm cosrrrrR 222
m
mmmm
s
z
RkKq
Zkj,,r,rE
0
022
00
2
mmm ,rP
,rP
The analytical treatment
Vacuum
Ferrite
PEC
7. In ferrite EM fields are expanded in TE and TM progressive and regressive radial waves (PW and RW). They can be expressed by Bessel function of order ν=(2/3(2n+1))
8. In vacuo EM fields are expanded in TE and TM cut-off waves expressed by modified Bessel functions of the first kind of integer order.
nn
Fz
nn
Fz
RWPWDH
RWPWCE
4cos
4sin
mmm
V
z
mmm
V
z
msinIBH
mcosIAE
Fields in vacuo Fields in ferrite
Where Im are Bessel functions of order mand PWν and RWν are combinations of Bessel functions of order ν=(2/3(2n+1))
The analytical treatment
Vacuum
Ferrite
PEC
9. The matching conditions are stipulated on the contour between ferrite and vacuum by imposing the continuity of tangential fields and normal induction fields
10. Resorting to the Ritz-Galerkin method, the functional equations are transformed into an infinite set of linear equations.
11. By means of an ad-hoc truncation of matrices and vectors the system can be solved.
Longitudinal and Transverse Impedance
• Longitudinal ImpedanceThe source will consist in only one
linear current placed at the point P0 (0,0)
• Dipolar Transverse ImpedancesThe source will consist in two linear
currents placed in the points P1(r1,0) and P2(r1,π).
Tests of convergence
Longitudinal impedance
Transverse impedance for various gamma
Transverse impedance for various gamma
Comparing analytical model and CST simulations
Future Plans
The kicker loaded by a coaxial cable
An open external cable (length l, propagation constant k and charachteristic impedance ) exhibits an impedance given by the usual formula coming from the transmission line theory.
Resorting to the expansion of the cotangent function as sum of polar singularities we may reproduce the cable behaviour by a lumped constant element circuit.
)klcot(jZZ 00Z
122
0 2n kln
lnjk
jkl
ZZ
The kicker loaded by a coaxial cable
The circuit can be approximated by a finite number of RLC parallel cells connected in series where each cell accounts for a resonance
11
00
11n
nnn LCjGCjGZ
n
lL
;n
lC;
n
YlG
;lC;YlG
rn
rnn
r
0
00
0000
2
2
Simulation models of ferrite loaded kicker and EM characterization of materials
Overview
• Kicker impedance model
• Measurements of material properties (in collaboration with G. De Michele)
The C-Magnet is not symmetric in the horizontal plane
xbaZ xxx
Constant term Dipolar/quadrupolar term
ybaZ yyy For the Frame magnet and the Tsutsui model
0 yx aa
0xa
Frame Magnet model
x
y
Simulation models
C-magnet and Frame magnet
xx bZ
Constant horizontal term comparison with the theory
xx aZ
The effect of the cylindrical approximation
Vacuum
Ferrite
PEC
Round Square
xx bZ
Calculation of the impedance for the C-Magnet model including cablesA theoretical calculation based on the Sacherer TL model approach
TSUNSCMagnet ZZZ
Where is the low frequency impedance in a C-Magnet kicker model calculated using the Sacherer Nassibian formalism and is the impedance calculated using the Tsutsui formalism. The Tsutsui impedance is calculated in H. Tsutsui. Transverse Coupling Impedance of a Simplified Ferrite Kicker Magnet Model. LHC Project Note 234, 2000. Instead we have to spend some word about the Sacherer Nassibian impedance.This impedance is defined as:
NSZTSUZ
gk
k
x||NS
k
||NS
ZLjZ
Zb
lc|Z
a
cZ
Zb
l)ax(Z
2
22
002
2
22
0
2
0
2
4
4
0
Where l is the length of the magnet, x0 the beam position, L the inductance of the magnet and Zg is the impedance seen by the kicker
25
Comparing with theoretical results
The simulations of the C-magnet model are in agreement with a theoretical prediction based on Sacherer-Nassibian and Tsutsui formalism
Horizontal driving impedance calculated at x=1cm: MKP
Frame Magnet modelHorizontal dipolar impedance
xbaZ xxx
xbaZ xxx
External circuits EK PSB
The green curve depends from the cable properties (propagation and attenuation constants)
The kicker loaded by a coaxial cable
An open external cable (length l, propagation constant k and charachteristic impedance ) exhibits an impedance given by the usual formula coming from the transmission line theory.
Resorting to the expansion of the cotangent function as sum of polar singularities we may reproduce the cable behaviour by a lumped constant element circuit.
)klcot(jZZ 00Z
122
0 2n kln
lnjk
jkl
ZZ
11
00
11n
nnn LCjGCjGZ
The kicker loaded by a coaxial cable
xbaZ xxx
The simulation technique has to be improved. Anyway using this technique seems we are able to take into account the effect of external cable in the CST 3D EM time domain Impedance simulation.
Real transverse impedance
Simulating internal circuit for an MKE kicker
The simulations seems to confirm that the simple Tsutsui model is not sufficient to compute the low frequency kicker impedance
xbaZ xxx
Future step
• Loading together internal and external circuit In CST 3D EM time domain Impedance simulations and comparison with the theoretical calculation based on the Sacherer TL model approach.
Overview
• Kicker impedance model
• Measurements of material properties (in collaboration with G. De Michele)
32
Coaxial line method
We characterize the material at high frequency using the waveguide method
Electromagnetic characterization of materials
)'','(GS 21
),(G
),(
),(
Simulations f,tan,S ' 11
tan,'
Measurements
Material fS11TL Model
01111 fSf,tan,S '
Properties of the material
00 tan,'
Z_DUT, k_DUTZ0, k0
l
The coaxial line method
),(G
loadZ
Valid only for TEM propagation
The TL model
The coaxial line method: air gap limitations
Due to mechanical limitations the air gap between the inner conductor and the material is not negligible and has to be take into account.
- To get the reflection Г with 3D EM code- TL model correction (valid only for TEM propagation)- Resort to full wave modal method
We tested the basic TL model with the 3D EM code and
with measurements in the case without air-gap.
G. De Michele BE-RF 35
Coaxial method (Teflon)
Z_DUT, k_DUTZ0, k0
l
O.C. 18 GHz
simulations 2.03-0.032j
model 2.04-0.032j
model(measurements) 2.03 --
S.C. 18 GHz
simulations 2.03-0.032j
model 2.04-0.032j
model(measurements) 2.06 --
36
Coaxial model validation via 3D EM simulations
Simulations f,tan,S ' 11
tan,'
Simulations
Material fS11TL Model
01111 fSf,tan,S '
Properties of the material
00 tan,'
Coaxial model validation via 3D EM simulations
An example of application: EKASiC-F
090
12
.tanr
Coaxial method measurements
Comparing the coaxial and the waveguide method
Coaxial line method Waveguide method
090
12
.tanr
140080
21111
..tan
.r
Good agreement
between the two method
Work in progress
We did measurements for some SiC in the ranges 10 MHz-2GHz and 8 - 40 GHz and for the ferrite 8C11 in the range 10MHz-10GHz
The elaboration of the results is going on. We still have to do a lot of simulation to get the numerical functions and to redone some measurement. We planned to finish all the work on this subject before the end of July.
41
Ferrite Model
The ferrite has an hysteresis loop
The hysteresis effect in this measurements
μ
H
rB
m/A.H 80 TB r610
In this measurements for the coaxial ferrite sample we have:
11
hysteresis
truemeasured
Comparing the two models
Using the C-magnet model in the transverse horizontal impedance a high peak at 40MHz appears. At low frequency the Tsutsui model is not able to estimate correctly the kicker impedance. The two models are in good agreement at high frequency (>400MHz). 43
The effect of the high voltage conductor
xbaZ xxx
45
Comparing the two models
Using the C-magnet model in the transverse horizontal impedance a high peak at 40MHz appears. At low frequency the Tsutsui model is not able to estimate correctly the kicker impedance. The two models are in good agreement at high frequency (>400MHz).
xbaZ xxx
Real horizontal quadrupolar impedance calculated at x=1 cm