Blas, A

Comblin, J F

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Updating the BTFM of the PSB

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHORGANISATION EUROPEENNE POUR LA RECHERCHE NUCLEAIREPS/RF/Note 2001-013Updating the BTFM of the PSBA. Blas,  J-F. ComblinRef.  PS/RF-LLGeneva, Switzerland18 Sept 20011. PreambleThe aim here is to reconsider a measurement tool (Beam Transfer FunctionMeasurement) developed by H.Schönauer and M. Sjöström [1] that was using obsoletehardware. The software and hardware interfaces need to be modified to make use of anew instrument (vector analyzer) that is now available.2. Some definitions:The BTF (Beam Transfer Function) = RFBeamVIis the beam current response over voltageexcitation (longitudinal or transversal) ratio. This is the primary calculation carried out bythe equipment.The LSD (Longitudinal Spectral Density) is a 2D graph showing particle density versusrevolution frequency or momentum. This plot is obtained by integrating the BTF usingVlasov equation.The Shottky power spectrum gives a direct measurement of the LSD, but at PSBinjection, the remaining rf structure inherited from the Linac (bubbles in phase space)makes it unreliable to use it as a tool.3. New measurement set:C04CavityPick-upVectorAnalyzerSourceReferenceTestPSBHPIB toEthernetinterfaceSome useful parametersPhysical parameter Symbol PSB value unitParticle charge e 1.60219 . 10-19 AsProton rest energy E0 938.2796 MeVMean radius R 25 MMean kinetic energy (inj) Ek 49.89 MeVMean revolution frequencywrev 2p.598.73 Krad/sRelative velocity factor b 0.3137Relative mass factor g 1.0532Off-momentum parameterh -0.8412Relative momentum spreadDps/p0 » 1.5 . 10-3Number of particles Np fi 1013Excitation harmonic numberh 6Cavity – PU azimuth qm 249.2 degreeCoupling impedance Zlong » 50 – i 1500 W4. Basic measurements4.1  Beam Transfer Function (BTF):BTF(ù) = ( )( ) referencetestVIRFBeam=wwAs the BTF is the ratio of two complex rf signals, it has an amplitude and a phase. Toobtain proper results, care has to be taken to equalize delays on both signal and tocompensate the phase lag of beam pick-up with respect to cavity pick-up.4.2  Longitudinal Spectral Density (LSD):Vlasov gives an expression linking BTF to Density Distribution F0(w) in the longitudinalcase:( ) ( )( ) ( )( )ò¥¥---×××××××××-== revrevrevrevrevprevrfBeamlong dhddFhENeiVIBTF wwwwwwgbphwwww000223022ˆˆwrev0 is the mean revolution frequencywrev  is the revolution frequency of each individual particleThe integral is calculated on an infinite domain, but is practically limited to where theresponse is significant.To simplify the notation the first factor will be gathered in a constant Cm.( ) hENeC prevm×××××××=0223022 gbphwThe integral can be split into principal value (PV) and residue at the pole. This gives:( ) ( )( )( )÷÷÷÷øöççççèæ--×+÷øöçèæ -×±×== ò revrevrevrevrevrevrevmrfBeamlong dhddFPVihddFCVIBTF wwwwwwwwwpwww0000ˆˆThe + or – sign in front of the residue depends on the initial conditions. Withoutexcitation prior to the measurement it should be +. Else it has to be -.( ){ } ÷÷øöççèæ÷øöçèæ -××=Þ 00Re revrevmlong hddFCBTF wwwpwWhen working at base band zero we take w  = h . w rev0 + e( ){ } ÷÷øöççèæ÷øöçèæ××=ÞhddFCBTF mlongewpe 0ReThis equation means that the BTF response is wider when h increases.By integrating this relation we get the density distribution:( ){ }ò¥-××=Ûweepw dBTFCFmRe1)(0This formula will be used to obtain the density distribution.As the measurements are made at harmonic h of the revolution, they need to betransposed. The measured frequency spread needs thus to be divided by h.The momentum spread can be obtained by applying the following relation:hww 100-×D=Drevppw rev0 is the mean revolution frequencyp0 is the mean momentumWhen the F0(w) distribution (LSD) is obtained, Dp/p0 can be expressed in terms ofstandard deviation or Full Width at Half Maximum value (FWHM).4.3  Schottky SpectrumThe coasting beam is composed of a set of particles (delta pulses) with differentrevolution frequencies.( ) ( )å×-×=in irevirevschottky TTntqti, __dThe Fourier transform is:( ) ( )å×-×=in irevirevschottky Tnqi, __wwdwSpectral (frequency) lines can be transformed into momentum lines by applying:imeanmeanrevirev pp××-= hww __In each spectral band, the power spectrum is equal to the mean square current ( )ti2  overthis band.Power per Shottky band = ( ) 02 022 2 prev Nfeti =frev0 is the central frequency of the Shottky band.Np0  number of particle affecting the Shottky band.This formula is used to obtain the spectral density from the beam current measurement.5.  Computation operated by the system (until now)5.1  Empirical Transfer Function Estimate (ETFE)The aim is to divide the Discrete Fourier Transform (DFT) of beam response andexcitation.)()()(wwwrfbeamjVIeETFE =Property: its variance doesn’t decay with the number of samplesSome hintThe momentum spread of particles leads to a revolution frequency spread:The initial BTF measurement was made using a tracking filter; it is now carried out usinga FFT.5.2 Black-box Prediction Error Method (PEM)In this method the transfer function is assumed to be:( ) ( )( )zAzBzazazbzbbzzBTFmmnnk =×++×+×++×+×= -------...1...)(1111110where k is the pipeline delay in terms of clock periodsAssuming a noise model W(z) driven by a white noise process w(t) the beam response issupposed such as:( ) ( ) ( ) ( )twzWtVzBTFtI rfBeam ×+×=)(More generally, a black box model can be expressed such as:( ) ( )( ) ( )( )( ) ( )twzDzCtVzFzBtIzA rfBeam ×+×=× )(A loss function is then defined and a minimization algorithm is applied to obtain thevector containing all the coefficients of the z polynomials to be estimated:[ ]Tfdcba 11101 ...,...,...,...,=J5.3  Coupling impedance compensation and stability diagramThe Beam experiences the cavity voltage but also the self-induced voltages in the beampipe. Both Schottky and BTF measurements are modified by the effect of these couplingimpedances. They should be compensated for.( )BeamBeaminducedCoupling IdssEIVZ ò== ,wThere are four major contributors to the coupling impedance:1. Space charge forces2. Vacuum wall impedance3. Vacuum chamber discontinuities (resonators)4. RF cavities (resonators)Space charge is the most dominating effect in low energy rings as the PSB. It acts as apure negative inductance.bgww2000gZjZrevlongSC××-=-g0 = 1+2loge(b/a) : coupling coefficient with a : beam radius and b : vacuum tube radiusZ0 = m 0.c = 377 W = free space impedanceAs the beam response is normally in a narrow bandwidth compared to the revolutionfrequency w / w rev0 can be approximated by h.At 50 MeV in the PSB |ZSC-long| is in the order of 1-1.6 kW/hVacuum chamber impedance can be expressed as:wlongVCI bRZds ××=-  at DCR = machine mean radiuss , b, d w are the vacuum pipe conductivity, radius and thicknessVRF IBeamBut due to skin effect the impedance at high frequencies has to be noted:( )000 21revrevlongVCIcZbjZswbww××+×=-ZVCI-long is in the order of a few tens of W/h.The two other resonating effects are described by:÷÷øöççèæ-+=-wwww resresSlongsjQRZ1ReRS is the shunt impedanceQ the quality factorThe order of magnitude of these impedances is 0.2 to 50 W/h.The coupling impedance effect can be described in the following block-diagram:Analytically this gives:CouplingMeasured ZBTFBTFBTF×-=1This arrangement becomes instable when BTFMeasured tends to infinity, or in other wordswhen ÷øöçèæ - CouplingZBTF1 tends to zero1/BTFmeas plotted in the complex plane is called the stability diagram. The curveshouldn’t approach the origin.BTFZCouplingThe Shottky spectrum is also influenced by collective effects:CouplingSmeasuredS ZBTFPP×-=- 16  BTFM working set (used until now)FileSave settingsLoad settingsSave raw dataLoad raw dataViewRing 1Ring 2Ring 3Ring 4HorizontalVerticalLongitudinalTable(the following parametersappear on specialist modeonly)DAC contentRaw data BeamReferenceComplex FFT BeamReferenceCheck denominatorBTF: RawBTF: AveragedBTF: angle comp.BTF: time comp.Time gated signalBTF: time gatedBTF: Coupling impedanceMagnitude/PhaseStability DiagramTableOptionsShow ETFEShow PEMDp/p DefinitionFWHMGaussianParabolicQuadratic5-QuadraticEnergy axisSmoothLog Scale (Magn)Zoom X allZoom X noneZoom Y allZoom Y noneSpecialist modeControlRing 1Ring 2Ring 3Ring 4HorizontalVerticalLongitudinalSettings (Spec)Measure ADC OffsetsReset ADC OffsetsPulseWhen opening Control-Setting (Spec):SettingsMeasure: Beam/test (with ccv)Noise: Max lengthRandom phaseSineSweepNo noiseExcitation: On / Off (with ccv)Sine harmonic (using 4 digits)Damp. Width (using4 digits)Damp. Min: (0. followed by 2 digits`)Cavity detuning (R1,2,3,4) with ccvStart train: C / B with 4 digits aqnStart time (pulses) using 5 digitsInterburst time (pulses) using 5 digitsInterburst train B / C / fastrf with 4 digits ccvOverhead time (Smpl) using 5 digits andaqnBurst phase (smpl) using 5 digits and aqnAverages: 1 / 2/ 4/ 8/ 16 / 32 with aqnBurst length (Smpl) using 4 digits andaqnFlat top (MeV) 3digits +_2 after comaWindow: RectSineHammHannPLS line with aqnAttenuation (dB):3 times: one for each planeExcitation using 2 digits + 1 after comaReference using 2 digits + 1 after comaBeam using 2 digits + 1 after comaDown mixer gains (dB):3 times: one for each planeReference Cos using 2 digits + 1 after comaReference Sin using 2 digits + 1 after comaBeam Cos using 2 digits + 1 after comaBeam Sin using 2 digits + 1 after comaFrequency, Bandwidth3 times: one for each planeSpectral line using 5 digitsBandwidth (kHz):50/100/200/400 with aqnSynt. Frequency (kHz): 4 digits with aqnQ value: 2 digits + 2 after comaMeasure time (ms) using 9 digits displaySignal processing:3 times: one for each planeDT ring 1,2,3,4 (ns) using 5 digitsqm (degrees) using 5 digitsTime gate win using 4 digitsRe{Z/n}(Ohm) using 5 digitsIm{Z/n}(Ohm) using 5 digitsParametric estimation:3 times: one for each planeData windowNa using 2 digitsNb using 2 digitsNc using 2 digitsNd using 2 digitsNf using 2 digitsNk using 2 digits7  Implementation7.1  General environmentDuring BTFM measurement:· Beam and C04 cavity signals have to be connected to the instrument – they must notbe during other pulses because the induced voltage may kill the instrument.· The rf excitation voltage should be applied to the summing point at cavity amplifierlevel.· A DC voltage should be applied to the tuning offset control input of C04 so to set theresonant frequency at the proper harmonic number.· The control voltage – or voltage program - of C02, C04, C16 should remain at zeroVolt during the measurement. If unused, C02 and C16 should even be short-circuitedby applying “rf act OFF”.During Shottky measurement:· All cavities should be programmed to zero volt or short-circuited.· The rf excitation should be disconnected from the cavity.7.2  Timing6 pulses have to be supplied (TG8):· One pulse to reset the equipment that should disconnect all BTF signals fromcavities and from the vector analyzer. This pulse should come by default at thebeginning of each cycle.· Four different pulses for ring selection: BAXi.SBTFM· One pulse to disable the rf excitation in case of a Shottky measurement:BAX.SSHOTTKY.7.3  Tuning offsetFour VME DACs (one for each ring) should be supplied to set the tuning frequency7.4  Measurement dataThe vector analyzer will sample the cavity and beam pick-ups voltages. These samplescould be transferred to any computing tool but as it represents a big data flow it seemmore reasonable to use the FFT tool supplied with the instrument and send only thecomputed complex values (only a few tens points).At this point we would need some mathematical tools:· A phase addition to compensate any phase offset (mainly cavity – beam PU)· A delay compensation7.5  Hardware setupVectorAnalyserTestReferenceSourceData andControlBusFromBeamPick-upsCavityReturnToCavityAmplifierVMEDACToTuningOffsetControl7.6  Instrument setupThe HP vector analyzer (HP 89410A) has to be properly configured to supply the FFT ofthe two main signals: beam and rf. About 400 points per shot, with frequency, real andimaginary parts have to be sent through the network. We need to be able to set the startand stop frequencies of the analysis and also the amplitude of the excitation signal. Allthe computation will be operated on a dedicated PC using an adequate mathematicalpackage (mathematicaTM).7.7  Very basic computation to be supplied· Cavity transfer function  FFT of “cavity return/VA source”· Schottky Spectrum  FFT of Beam PU· BTF  FFT of “Beam/Cavity return”· LSD  ( ){ }ò¥-×wee dBTFRe· Delay difference (beam return-cavity return) compensation algorithm· Phase difference (Beam PU – cavity) compensation algorithmThe results should appear as a Bode or Nyquist plot and also optionally as a table.

Updating the BTFM of the PSB

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