C Zannini Indirect space charge wake function A nalytical solution of the Fourier transform does not exist Fourier transform needs to be solved numerically A pproximation ID: 787037
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Slide1
Approximate formula for the indirect space charge wake function
C. Zannini
Slide2Indirect space charge wake function
A
nalytical
solution of the Fourier transform
does
not existFourier transform needs to be solved numerically
Approximation of the impedance to get analytical solution of the Fourier transformSimple formulae for the calculation of the indirect space charge wake function
Indirect space charge impedance is defined as the beam coupling impedance of a perfect electrical conductive pipe
"Resistive-Wall Impedance of an Infinitely long Multi-Layer Cylindrical Beam Pipe" E.
Métral
, B. Salvant, B.
Zotter
proc. PAC 2007
Slide3Impedance formula approximation
Approximation for small argument of Bessel functions
Useful in frequency domain
Useless in time domain. Independently of beta the wake function results a Dirac Delta
Slide4Impedance formula approximation
Approximation for small argument of
I
1Useful in frequency domainUseful in time domain.Good approximation of the ISC wake function
Slide5Indirect space charge wake function
Approximated ISC impedance
The approximated equation allows to account for the frequency windowing as function of the relativistic beta factor
L = 1 m
b=0.03 m
Slide6Indirect space charge wake function
Compute the wake function with the analytical equation and comparing with numerical FFT
L = 1 m
b=0.03 mThe approximated formula allows to account for the behavior of the wake function versus the relativistic beta factor
Slide7Indirect space charge wake function versus relativistic beta
As expected the width of the indirect space charge wake become larger and larger when beta decreases and vanishes when beta approaches one
Slide8Impedance formula approximation:
improved formula
Valid for small arguments of the Bessel functions but diverging in opposite direction with respect to the previous approximation
Useful in frequency domainUseful to get an improved formula
=
Slide9Second formula
First formula
Averaging the wake function
gives
a very good
approximation of the
The divergence from the real equation in the two approximations has opposite sign
Impedance formula approximation:
improved formula
Slide10Improved formula
Impedance formula approximation:
improved formula
Slide11Impedance formula approximation:
improved formula
The improved equation gives a very good
approximation of the ISC wake functionThe formula can be applied for any relativistic beta
Slide12Impedance formula approximation:
exact formula
Comparing the transverse wake function obtained from the equation on the left with the one obtained with the Fourier transform can be observed that the discrepancy factor
F (plotted below) is a function of τ aUsing the peculiarity of the discrepancy function we can obtain an exact formula for the indirect space charge wake function. We only need to compute once a reference discrepancy function
Slide13Impedance formula approximation:
exact formula
The formula and the DFT results are perfectly superimposed
Slide14Impedance formula approximation:
exact formula
The formula and the DFT results are perfectly superimposed
β=0.5
Slide15Summary
A simple analytical equation has been derived for the calculation of the indirect space charge wake function
An improved formula which gives a very good approximation of the indirect space charge wake function has been also derived
An exact formula (not discussed during the presentation) could also be obtainedThe formula could potentially be implemented in PyHEADTAIL to speed up simulations when including the indirect space charge