Bunch lengths between 3 and 12 mm have been measured routinely in LEP in 1997 with a small (7 mm diameter) button electrode. The measurement method is based on the spectral analysis of the electrode signal and relies on the fact that the transfer function of the complete set-up, including the signal cable, can be computed rather exactly thus eliminating the need for external calibration. The information of beam intensity is recovered as a by-product. It provides an interesting internal validation of the measurement by comparison with the normal intensity measurement. The system has been used to detect subtle but real bunch length changes with bunch intensity which can be attributed to the inductive impedance in LEP. A value for the imaginary (inductive) longitudinal impedance is derived from the observations. An indication for the resistive part of the impedance is given as well

Vos, L

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN    SL DIVISIONGeneva, SwitzerlandJune, 1998CERN SL-98-028   APElectro-Magnetic Bunch Length Measurement in LEPL. VosAbstractBunch lengths between 3 and 12 mm have been measured routinely in LEP in 1997with a small (7 mm diameter) button electrode.   The measurement method is based onthe spectral analysis of the electrode signal and relies on the fact that the transfer functionof the complete set-up, including the signal cable, can be computed rather exactly thuseliminating the need for external calibration.  The information of beam intensity isrecovered as a by-product.  It provides an interesting internal validation of themeasurement by comparison with the normal intensity measurement.  The system hasbeen used to detect subtle but real bunch length changes with bunch intensity which canbe attributed to the inductive impedance in LEP.  A value for the imaginary (inductive)longitudinal impedance is derived from the observations.  An indication for the resistivepart of the impedance is given as well.Presented at EPAC-98, 6th European Particle Accelerator ConferenceStockholm22-26 June 1998ELECTRO-MAGNETIC BUNCH LENGTH MEASUREMENT IN LEPL. Vos, CERN, Geneva, SwitzerlandAbstractBunch lengths between 3 and 12 mm have beenmeasured routinely in LEP in 1997 with a small (7mmdiameter) button electrode. The measurement methodis based on the spectral analysis of the electrode signaland relies on the fact that the transfer function of thecomplete set-up, including the signal cable, can becomputed rather exactly thus eliminating the need forexternal calibration. The information of beam intensityis recovered  as a by-product. It provides an interestinginternal validation of the measurement by comparisonwith the normal intensity measurement. The system hasbeen used to detect subtle but real bunch lengthchanges with bunch intensity which can be attributedto the inductive impedance in LEP. A value for theimaginary (inductive) longitudinal impedance isderived from the observations. An indication for theresistive part of the impedance is given as well.1 INTRODUCTIONBunches in LEP are short (a few mm) and Gaussian.The Gaussian time profile also produces a Gaussianfrequency spectrum.  The transfer function of anelectro-magnetic observation system will distort thebare spectrum of the bunch. If the transfer function ofthe elements between the beam and final observationstation are known with sufficient accuracy, then thebare spectrum can be recovered from the measured oneand the determination of the bunch length is trivial.The LEP bunch length measurement consists of abutton electrode, a transmission cable, a spectrumanalyser and an acquisition system. The transferfunction and performance limitations will be discussedin following sections.2 TRANSFER FUNCTION OF BUTTONELECTRODEThe beam monitor in LEP is an electrostatic buttonelectrode. The properties can be derived from thegeometry sketched in Fig. 1. The half aperture A=45mm. The capacitance of the electrode is C. Thetransfer function can be computed as follows.The charge collected on the electrode is :q idttt t=+∫∆ . (1)It should be mentioned thatthe relation between thebeam current ib and thefraction i of it that coupleswith the electrode followsfrom the geometry:i i d Ab = 2pi . The electrode isa current  source and delivers:i dqdti t t i te = = +( ) − ( )∆ . (2)The currents can alsobe expressed as :i t I e j tω ω ω,( ) = ( )  (3)i t I e e jI t ee j t t j tj tω ωωω ωω, sin( ) = ( ) −( ) =+( )∆ ∆∆2 2 2  (4)The current ie flowsthrough a RCparallel circuit asshown in Fig. 2 andproduces a voltageV. The transferimpedance isdefined  as:Z ViiiRjbeb= =+1 ωτ or Z e dAj tjj t=+ωpiτωωτ∆∆22221sin, (5)where R = 50 W , the load impedance, C thecapacitance, τ = RC  the low frequency time constant,∆t d c= pi 4  is the transit time for the (round) electrodeand c the speed of light.For low frequencies ( ω∆t 2 1<< ) this reduces to thestandard low frequency formulaZ dAtCjj Zjj= + = +∞2 1 1piωτωτωτωτ∆. (6)3 THE SIGNAL TRANSMISSION CABLEThe attenuation of a cable is the sum of twocomponents. The first one is caused by the skin effectwhich follows a square root law with frequency andthe second one is caused by the losses in thedielectric which increases linearly with frequency. Thegeneral expression for the attenuation constant can bewritten as :α α α= +skin dielectric . (7)AdreferenceorbitFigure 1 : Geometryof button electrodeie C R VFigure 2 : Equivalent  diagramof button electrode.αµρpiωskingv c a b ba= − + 21 1 1240 ln,  (8)where a and b are the radii of the inner and outerconductor, v is the signal velocity in the cable, r  theresistivity of the conductor and g a geometrical factor.The factor g=1 for smooth coaxial cables but becomes1.1 for the air filled cable that is used in LEP since theouter conductor is undulated.αω δdielectricctg= −2, (9)where tg d  is the loss factor of the dielectric. Takeninto account that only 1/6 of the cable length isoccupied by solid insulation (the rest is air) we find tg d=0.00014. It was found that these values remaintrustworthy up to 9..10 GHz. The equivalent length ofthe cable , including the final 1.6 m flexible connectionto the spectrum analyser, is l= 95 m. All elements inthe measurement chain are now defined and thetransfer impedance can be computed and is shown inFig. 3:Z dAtCjj etlskin dielectric=++( )2 1piωτωτα α∆. (10)0 5 1000.020.040.06frequency GHzZtWFigure 3 : Transfer impedance of button electrode andtransmission cable.4 MEASUREMENTfrequency2s s  ~ Ö slopelog(data)log(I)Figure 4 : Gaussian plot in log/frequency2 coordinates.The measurement is done with a HP8563Aspectrum analyser. The output signal is given by :V Z I fTet csωωσ( ) = −2122∆ , (11)where D f= 2 MHz is the bandwidth used, s s  thebunch length, I the total beam current and T therevolution period. Since the data are available in dB itis straightforward to subtract logarithmically thetransfer impedance and the operational constants.Fitting a Gaussian (which in fact is a linear fit whenthe data are displayed  as function of frequency2) thenyields I and s s  (Fig.4). The value of I is formally onlycorrect when the beam is centered  in the pick-up bodyand is indeed at a distance A from the electrode.5 LIMITATIONSThe first limit is related to the signal to noise ratio.The noise level of the analyser used in LEP is -70 dBV(320 m V) averaged over the operational frequency rangein a bandwidth of 2 MHz. That corresponds with a noisefigure of 48 dB of which 10 dB are due to an inputattenuator  that is switched in with the option of remotecontrol. The most critical part of the spectrum is near10 GHz. For a 10 mm long bunch at least 200 m A totalbeam current is required  to make a propermeasurement. Shorter bunches can be measured withless intensity.The expression for the transfer impedance is knownto be wrong for a certain number of frequency bands.First there is a structure resonance at 5.5 GHz [1]. Thenthree other bands have been identified at 7, 8.5 and 9.2GHz. They correspond to  longitudinal and transversereflections in the air-filled cable caused by the factthat the insulator is not continuous. These frequencybands have been excluded  from the analysis.The finite sample length causes fluctuations in thespectrum which are attenuated by averaging.  It turnsout that the resolution obeys the following simplescaling law: (d s s . s s)~1.4 mm2.6 RESULTS OBTAINED WITH THESYSTEMIt was noticed during the LEP physics runs that thebunch length decreased with decreasing bunch current.The effect is small but significant. A number of datawere collected mainly at 91.5 GeV/c and are shown inFig. 5. The measured synchrotron tune was qs  = 0.117except in 3 cases where it was 0.112, 0.112 and 0.114.A small correction on the measured bunch length wasapplied for these cases in order to keep the parameterspace consistent. The rf frequency shift was 120 Hz,hence Jx =1.58.2 50 3 00 35 0 400 4 50 50 0 5 5010.51111.51212.5bunch length s sin mmbunch intens ity ib in m Alinear fit through datapointscomputed bunch length Z/n=0.2WZ/n=0.25 WZ/n=0.15 WFigure 5: Bunch length as a function of single bunchintensity at 91.5 GeV/c compared with expected valueswhen  Imag(Z/n)=0.2 W .The computed bunch length for    zero    intensity   is11.45 mm. The computed bunch length dependence  onbunch intensity was found as follows. A longitudinalinductive impedance produces a longitudinaldefocusing force. That force lengthens the bunch whichwill decrease  in its turn the defocusing force,...Theequilibrium bunch length can be found with an iterativeprocess. The unknown in that computation is theeffective inductive impedance seen by the basiclongitudinal dipole mode  of the bunch. Its value waschosen with trial and error such that the slope of thecomputed and measured impedance were about thesame. This procedure yielded  Imag(Z/n)= 0.2 W . Thecomputed curves for 0.15 W  and 0.25 W  are shown aswell for comparison. The systematic differencebetween the measured and computed bunch length is~0.7 mm or ~6%. A possible explanation may be thefact that the loss-factor of the dielectric of the signalcable is ~10% lower than the value that was actuallytaken. Published data, only valid for frequencies below~5 GHz,  yield ~1.6 10 -4 for this quantity. Reducingthis to 1.4 10 -4 (see section 3) is sufficient to eliminatethis (small) discrepancy in some circumstancesbearing in mind that the frequency range of themeasurement extends from 4 to    10  Ghz. A slightdependance of tg d  on frequency may not be excluded.Similar observations with the same conclusionswere done at 65 GeV/c. A more interesting case isshown in Fig. 6 which refers to data taken at 22 GeV/cat very low bunch currents. The bunch length iscomputed for Imag(Z/n)=0.2 W .   The synchrotronfrequency was qs =0.12 and Jx =1.0 50 10 0 15 0 2 00 25 0012345678bunch length s sin mmbunch intens ity ib  in m Acomputed bunch lengthZ/n=0.2 Wmeasured bunch length s aturation limit Figure 6 : Computed and measured bunch length at 22GeV/c.The measurement confirms that the inductive partof Z/n = 0.2 W . This value agrees well with transversedetuning measurements. Longitudinal turbulence isprovoked by the resistive part of Z/n. The threshold canbe written as :ℜ( ) = ( )Z n i Vhh sΩσ2, (12)where V is this time the accelerating voltage and hthe harmonic number. That condition can also beexpressed  as follows:ℜ( ) =  Z nhVe R isb213piσ. (13)Assuming that this threshold was reached  at thepoint where the accumulation in the bunch saturated(ib=220 m A) we find Real(Z/n)=0.07 W .AcknowledgmentsK. Rybaltchenko‘s help was essential for dataacquisition and treatment. Many other people madecontributions. I would like to mention M. Albert, J. DeVries and M. Lamont.REFERENCE[1] W.C. Barry, “Broad-band Characteristics ofcircular Button Pickups”, LBL-32939, 1992

Electro-Magnetic Bunch Length Measurement in LEP

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