1 Trying to understand the routine that Bruno is using to get the impedance from the wake Many thanks for their help to: Bruno S. (of course), Alexej, Bruno Z., Elias, Erk, Gianluigi, Giovanni, Massimo, Rama, Rogelio

2 Context Simulation campaign to gather the impedance of SPS elements in ZBASE. Collaboration with Bruno Spataro (INFN). Time domain and frequency domain MAFIA simulations (1 st version, ~ 1988, now unavailable) to obtain the wakes and impedance (longitudinal and transverse). The main objective here is to understand what results Bruno is giving us, so that we know how to compile these results in ZBASE and how to use them in a relevant way in HEADTAIL.

3 Study triggered by strong differences between the impedance computed by Bruno from the wake and the plain FFT of the wake computed by myself Both signals share common features, but : - Plain FFT of the wake yields negative longitudinal real impedance for certain frequencies  not physical - Bruno’s sampling is ~8 times higher than mine - Peaks appear around the same frequencies but their shapes are very different - in addition, suspicions with the given negative sign of the imaginary part of the transverse impedance  Bruno provided us with his routine to obtain the impedance.

4 Bruno’s routine Bruno took this small routine from the ABCI Fortran source code (Yong Ho Chin) to transform the computed wake into impedance. What does this routine do? Let’s see…

5 Initial longitudinal wake obtained from Mafia time domain simulations

6 Obtaining a Matlab code that behaves like the Fortran code I will use the Matlab routine from now on.

7 After some work, Bruno (Yong Ho)’s “FFT” of the Wake W seems to be: Amplitude factor due to Gaussian distribution “Half” Blackman Harris Window Zero padding Time delay to account for centering the gaussian source k: frequency iterator j: time iterator i: imaginary unit N: number of wake samples  f: Impedance sampling frequency  t: Wake sampling time  l : Bunch length (in sec) A: Zero padding factor c: speed of light T: time delay between the wake start and the center of the gaussian distribution For comparison, plain DFT of the wake W

8 The source is not a point charge… but a gaussian bunch

9 We need the impedance of a point charge Z pc, but MAFIA gives the impedance of a gaussian beam Z GB Longitudinal Impedance for a bunch at frequency f=k.  f=k/(N  t) : For a Gaussian beam (  t rms): and in frequency domain (TF = Fourier Transform) Discretizing for f=: Therefore: Finally, the impedance of a point charge is: For a bunch, we have [Chou and Jin ANL LS-115 (1988)]

10 In addition, the gaussian distribution is not centered with respect to the wake start T Source  (t) Wake W GB (t) tt 0 In this case, For Matlab and Fortran, origin of time is at j=1 and origin of frequency is at k=1, so: then, so, Finally

11 …continued With andwe get Remarks: if the source bunch is too long, then the exponential factor increases very rapidly with k, and the signal to noise ratio to estimate Z GB deteriorates, and. (ex of Bruno:  l = 1.5 cm, N=1001, and c  t = 3 mm). The cutoff frequency due to sampling in time domain is f c =1/(2  t)~50 GHz - At f=4 GHz (k= f.N.  t ~ 40), the exponential correction is a factor 2 - At f=8 GHz (k= f.N.  t ~ 80), the exponential correction is a factor 23 Therefore, need for short source bunches. But shorter source bunches require smaller mesh, i.e. longer CPU time for the same wake length. Therefore there is a tradeoff between CPU time and accuracy.

12 Plain DFT of the wake Bruno’s DFT Plain DFT of the wake

13 DFT of the wake deconvoluted by the gaussian source bunch (only time delay) Bruno’s DFT DFT of the wake deconvoluted by the delay between the gaussian bunch center and the wake (the gaussian factor is not yet taken into account)

14 DFT of the wake deconvoluted by the gaussian source bunch (time delay and frequency dependant amplitude correction) Bruno’s DFT DFT of the wake deconvoluted by the gaussian bunch source

15 Remaining questions Why is Yong Ho’s exponential correction offset by 1 index?

16 Windowing

17 Half Blackman Harris window With H(j) = *cos((j-1)  /N) *cos(2(j-1)  /N) *cos(3(j-1)  /N) One of the 4 terms Blackman Harris windows High dynamic bandwidth - low resolution window to increase the high frequency cutoff due to finite length wake field

18 DFT of the windowed wake deconvoluted by the gaussian source bunch Bruno’s DFT DFT of the wake deconvoluted by the gaussian bunch source, and windowed

19 Zero padding From Bruno’s file, it seems a strong zero padding is used. The factor is around 8 (NN/(FREQ.DT.NWW) in Yong Ho’s notation, in clear: 624/(f max.  t. N), where 624 is an “arbitrarily” chosen number smaller than N (1001 in our case), f max is the lowest of the Nyquist frequency and the frequency at which the exponential factor gets bigger than 20.

20 Effect of Zero padding Bruno’s DFT Zero padded windowed gaussian-source-deconvoluted DFT

21 Conclusions To be used in Headtail, the wake should be first deconvoluted by the source beam gaussian distribution. The windowing leads to significant changes in the impedance spectrum. Is it legitimate to use it? The Zero padding does not change much but gives the fake impression of a higher resolution The current resolution (0.1 GHz) seems very poor to resolve the SPS impedance peaks.

22 Ongoing work: Studies on resonator (see also Bruno Zotter’s book)

23 Parallel RLC circuit RL C Impedance: Laplace transform (variable s): u(t) i(t) With Impulse response (t>0): With

24 If  > 1 If  ~  0 and Q >> 1 If  >>  0 and Q >> 1

25 If Q>>1 and Q >>  0 T simulated ~ 100

26 Effect of truncating the wake (Q=10)

27  Widening due to truncation (convolution with rectangular window in frequency domain)  resolution in frequency domain is clearly missing Re(Z) Im(Z) Effect of truncating the wake (Q=1000) normalised

28 Using the truncated wake as input into Bruno’s routine theory

29 Therefore windowing seems a good idea? Signal looks ‘nicer”, but the peak is strongly widened. With Bruno, we find: R=3.3 Ohm, and Q=14.5  R/Q=0.22 Ohm theoretical = 0.15 Ohm (150 Ohm/1000)