Update on the kicker impedance model and measurements of material properties

1. 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

2. A simplified EM model of C-magnet for ferrite loaded kickers

3. 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).

4. Penetration depth in ferrite [m]

5. 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.

6. The analytical treatment The model is indefinite in the longitudinal direction The analysis is performed in FD All the fields have the same behavior in the longitudinal direction Using the Maxwell equations all the components of the fields are derived from the longitudinal fields of TE and TM modes. The sources are represented by a discrete number of linear currents placed at the point The longitudinal field is computed at the point Vacuum Ferrite PEC

7. The analytical treatment 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. Vacuum Ferrite PEC Fields in vacuo Fields in ferrite Where Im are Bessel functions of order m and PWν and RWνare combinations of Bessel functions of order ν=(2/3(2n+1))

8. The analytical treatment 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. Vacuum Ferrite PEC

9. Longitudinal and Transverse Impedance • Longitudinal Impedance The source will consist in only one linear current placed at the point P0 (0,0) • Dipolar Transverse Impedances The source will consist in two linear currents placed in the points P1(r1,0) and P2(r1,π).

10. Tests of convergence

11. Longitudinal impedance

12. Transverse impedance for various gamma

13. Transverse impedance for various gamma

14. Comparing analytical model and CST simulations

15. Future Plans

16. The kicker loaded by a coaxial cable An open external cable (lengthl, 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.

17. 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

18. Simulation models of ferrite loaded kicker and EM characterization of materials

19. Overview • Kicker impedance model • Measurements of material properties (in collaboration with G. De Michele)

20. Simulation models The C-Magnet is not symmetric in the horizontal plane Frame Magnet model y x For the Frame magnet and the Tsutsui model Constant term Dipolar/quadrupolar term

21. C-magnet and Frame magnet

22. Constant horizontal term comparison with the theory

23. The effect of the cylindrical approximation Square Round Vacuum Ferrite PEC

24. Calculation of the impedance for the C-Magnet model including cablesA theoretical calculation based on the Sacherer TL model approach Where is the low frequency impedance in a C-Magnet kicker model calculated using the SachererNassibian 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 SachererNassibian impedance. This impedance is defined as: 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 Horizontal driving impedance calculated at x=1cm: MKP The simulations of the C-magnet model are in agreement with a theoretical prediction based on Sacherer-Nassibian and Tsutsui formalism Frame Magnet model Horizontal dipolar impedance

26. External circuits EK PSB The green curve depends from the cable properties (propagation and attenuation constants)

27. The kicker loaded by a coaxial cable An open external cable (lengthl, 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.

28. The kicker loaded by a coaxial cable Real transverse impedance 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.

29. 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

30. 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.

31. Overview • Kicker impedance model • Measurements of material properties (in collaboration with G. De Michele)

32. Electromagnetic characterization of materials We characterize the material at high frequency using the waveguide method Coaxial line method

33. The coaxial line method The TL model Valid only for TEM propagation Z0, k0 Z_DUT, k_DUT l Properties of the material Simulations Material TL Model Measurements

34. 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.

35. Coaxial method (Teflon) Z0, k0 Z_DUT, k_DUT l G. De Michele BE-RF

36. Coaxial model validation via 3D EM simulations Properties of the material Material Simulations TL Model Simulations

37. Coaxial model validation via 3D EM simulations

38. An example of application: EKASiC-F Coaxial method measurements

39. Comparing the coaxial and the waveguide method Coaxial line method Waveguide method Good agreement between the two method

40. 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

42. The hysteresis effect in this measurements μ H In this measurements for the coaxial ferrite sample we have:

43. 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).

44. The effect of the high voltage conductor

45. Comparing the two models Real horizontal quadrupolar impedance calculated at x=1 cm 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).