1. Na Wang and Qing Qin Institute of High Energy Physics, Beijing
Extension of Tsutsui model for kicker impedance calculations with beta<1
Na Wang and Qing Qin
Institute of High Energy Physics, Beijing

2. Outline Context Tsutsui’s model Longitudinal impedance
Transverse impedance
Applications

3. Context
Extraction kickers are known as the major contributor to the impedance budget.
Non-ultrarelativistic effects
In the beam pipe, the EM field generated by the beam will be further modulated by the surroundings.
In free space, the EM field carried by a point charge q is contracted in a thin disk with a angular spread of 1/. In the limit v = c, the disk shrinks into a  thickness. (A. Chao)
v < c v = c

4. Context
Longitudinal and transverse resistive wall wake (Frank Zimmerman and Katsunobu Oide)
The differences between the ultra-relativistic limit are significant for  < 3.

5. Context
Real and imaginary part of the longitudinal impedance of a pillbox cavity (S.A. Heifets and S.A. Kheifets)
(1)  = 100
(2)  = 10
(3)  = 5
(4)  = 2
(5)  = 1.4
The non-ultrarelativistic effect should be considered, more general formulae are needed!

6. Tsutsui’s model Vacuum: a < x < a, b < y < b
Ferrite
Vacuum: a < x < a, b < y < b
Ferrite: a < x < a, b < |y| < d
Perfect conductor: |x| > a or |y| > d
The kicker has infinite length.
Beam passes at x = y = 0 with constant velocity v = c.
Frequency Domain - Field matching method
Impedances are expressed as an integration of the EM field experienced by a test particle over a finite length.
Impedance calculation  electro-dynamic problem of finding the fields in the vacuum chamber by a given beam current.

7. Field matching Field in the vacuum region Field in the ferrite region
Synchrotron part (source fields)
Fulfill the vacuum boundary conditions
Convergent at z   
Radiated part (waveguide modes)
Field in the ferrite region
Continuity of tangential field at |x| < a, y = b

8. Longitudinal impedance
Beam source
A point charge q moving on the axis of the beam pipe with a velocity c. The current density in frequency domain is
k = /c is the longitudinal wave number.
To meet the boundary condition at the perfect conducting edge, we add image currents at (x, y) = (2na, 0), n=1, 2, …

9. Longitudinal impedance
By solving the Maxwell equations, we obtain the nonzero source fields in the vacuum region
n = 0: self-field
n = 1, 2, …: field generated by image currents
kr = k/ = /c, K0, K1 are modified Bessel functions

10. Longitudinal impedance
We solve the wave equations in Cartesian coordinates, and obtain the EM field in the vacuum region
An and Bn are unknown coeffiecients, and

11. Longitudinal impedance
Electromagnetic field in the upper ferrite block
Cn and Dn are unknown coeffiecients,

12. Longitudinal impedance
Unknown coefficients An, Bn, Cn and Dn are determined by field matching at the interface y = b and |x| < a
here, we have introduced the Fourier transformations

13. Longitudinal impedance
Longitudinal impedance per unit length
with

14. Transverse impedance
The method used to calculated the transverse impedance of a ferrite is similar to the one developed for longitudinal impedance calculation.
Both horizontal and vertical impedances are considered.
Current density
Horizontal
Image current densities at (x, y) = (2ma, 0), m = 1, 2, …
Vertical

15. Transverse impedance
Transverse impedances have similar forms as the longitudinal one.
with Nxn, Nyn and M5n defined as

16. Comparion with Hsutsui’s results
In the ultra-relativistic limit of v  c
Longitudinal
Horizontal
Vertical
The results agree with to Tsutsui’s expressions!

17. Applications CSNS: Einj/Eext = 0.08/1.6 GeV, inj/ ext = 1.1/2.7
a = 100 mm, b = 60 mm, d = 100 mm, L = 1 m
r = 12, r= rjr
CSNS extraction kicker model

18. real part imaginary part
Longitudinal
Vertical
The differences between different  are significant for  < 2.7.

19. Conclusion
Both longitudinal and transverse coupling impedance of Hsutsui’s model is extended to more general case of beta < 1, and compact expressions are obtained.
The results are benchmarked with Tsutsui’s formula at the relativistic limit.
Numerical result shows significant differences for  < 2.7

20. Thank you!