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