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Longitudinal impedance of a prototype kicker magnet system Tran, Hy J. 1993

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LONGITUDINAL IMPEDANCE OFA PROTOTYPE KICKER MAGNET SYSTEMbyHy J. TranB.Sc., The University of British Columbia, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEINTHE FACULTY OF GRADUATE STUDIES(Department of Physics)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1993© Hy J. Tran, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of^PhysicsThe University of British ColumbiaVancouver, CanadaDate  October 4th, 1993DE-6 (2/88)AbstractThe longitudinal impedance of a prototype kicker magnet system for the proposedKAON Factory was measured from 0.3 to 200 MHz using the coaxial wire method.The method consists of transforming the aperture of the kicker magnet under test intoa coaxial structure by inserting a central conductor so that the transmission coefficientcan be measured. The longitudinal impedance was calculated from the transmissioncoefficient. The measurement was performed in two frequency ranges. From 0.3 to 50MHz the magnet was transformed into a 50 it coaxial line, and from 45 to 200 MHzinto a 180 St coaxial line. TSD calibration of the measurement assembly was performedin the higher frequency range while HP calibration of the test cables was performed inthe lower frequency range.Resonances in the longitudinal impedance are present below the cut—off frequency(30 MHz) of the magnet. The effects on the impedance of a speed-up network and asaturating inductor, installed on the input to the kicker magnet to improve its kickperformance, were determined. It was found that a speed-up network damped only thehigh—frequency resonances whereas a saturating inductor reduced their number. Abovethe cut—off frequency, other components of the magnet system, such as the speed-upnetwork, the saturating inductor and the cables, were found to have negligible influenceon the impedance.The maximum real longitudinal impedance of the magnet was measured to be 32R. The total contribution to the Booster ring is 425 Si and to the Driver ring is 1184ft, respectively. These contributions are small compared with that of rf cavities whichhave typical values of 10 kSZ. Hence the contribution of kicker magnets is negligible andfurthermore any longitudinal instabilities due to kicker magnets can be damped withexisting damping systems for the rf cavities except, perhaps, for short bunch (1 ns)operations.ii3345671113ContentsAbstractTable of ContentsList of FiguresList of TablesAcknowledgementsIntroduction1 Impedance and Kicker Magnets1.1 Impedance and Beam Instability ^1.2 Wake Potential and Longitudinal Impedance ^1.3 Wakefield of a Beam Pipe ^1.4 Wakefields of Kicker Magnets 1.5 Kicker Magnets for KAON ^1.6 Performance of Kicker Magnet Systems ^1.7 Longitudinal Impedance of a Prototype Kicker Magnet ^2 Principles of the Coaxial Wire Method^ 152.1 Principles of the Method ^  152.2 Energy Loss by a Beam Bunch  162.3 Method of Transmission Measurement ^  192.4 Longitudinal Impedance and Transmission Coefficient ^ 213 Experimental Set—up and Experiment^ 243.1 Overview^  243.2 Experimental Set-up ^  253.2.1 Magnet Tank  253.2.2 Reference Pipe ^  25iii3.2.3 Matching Sections ^  273.2.4 Central Conductors  273.3 Calibrations ^  283.3.1 Full Two—Port Calibration ^  283.3.2 HP Calibration for the 50 SI Line  293.3.3 TSD Calibration for the 180 11 line ^  303.4 Digital Noise Reduction ^  323.5 Computer Calibration Program  333.6 Longitudinal Impedance Calculations ^  334 Measurement Results and Discussions 354.1 Longitudinal Impedance of the Reference pipe ^ 354.2 Electrical Resonant Modes ^  394.3 Magnet Configurations  514.4 Effect of a Speed—up Network and a Saturating Inductor ^ 574.5 Gaps Between the Beam Pipe and Kicker Magnets^ 634.6 Kicker Magnet Contribution to the Total Longitudinal Impedance^665 Conclusions^ 68References 69Appendix A^ 71Effects of a Central Conductor and Transit Time ^  71Appendix B^ 74The Cut-off Frequency of the Prototype Kicker Magnet ^ 74Appendix C^ 77The Scattering Matrix Definitions ^  77Appendix D^ 79Input Impedance of the Prototype Kicker Magnet ^  79ivList of Tables1^Resonant frequencies of the 30 ns quarter—wavelength resonator and the30 ns half—wavelength resonator. ^  422^Resonant frequencies of the 60 ns quarter—wavelength resonator and the60 ns half—wavelength resonator. ^  48vList of Figures1 Wakefield of a non-resistive beam pipe^ 52 Wakefield of a resistive beam pipe. 53 An equivalent circuit diagram for an ideal kicker magnet^ 84 Cross-section of a kicker magnet. ^ 95 The prototype kicker magnet. 106 Components of a typical kicker magnet system. ^ 127 Effects of a speed-up network and a saturating inductor on the predictedkick strength of a magnet. ^ 138 Experimental set-up for sending and recording a current pulse. ^ 179 Experimental set-up for measuring S21^ 2010 Two localized impedances^ 2211 The prototype kicker magnet in the ground tank ^ 2612 The 50 I/ coaxial line. ^ 2713 The 180 S2 coaxial line 2814 Calibration planes. ^ 3115 Longitudinal Impedance of the Reference Pipe from 0.3-50 MHz. ^. 3716 Longitudinal Impedance of the Reference Pipe from 45-200 MHz. ^. 3817 Resonant structures of the prototype kicker magnet. ^ 4018 Longitudinal impedance of the prototype kicker magnet 4319 Resonant structure of the kicker magnet system with a feed cable. . . ^ 4420 Longitudinal impedance of the prototype kicker magnet with a 30 nsfeed cable^ 4621 Longitudinal impedance of the prototype kicker magnet with a 193 nsfeed cable^ 4722 Equivalent diagram of the prototype kicker magnet with a 30 ns outputcable ^ 4923 Longitudinal impedance of the prototype kicker magnet with a 30 nsoutput cable resistively terminated^ 50vi24 Schematic diagram of matched-termination configuration. ^ 5125 Schematic diagram of short-circuit configuration. ^ 5226 Longitudinal impedance of short-circuit and matched-termination con-figurations (0.3-50 MHz). ^  5327 Longitudinal impedance of short-circuit and matched-termination con-figurations (45-200 MHz). ^  5428 Effect of shielding the ferrite of the prototype kicker magnet^ 5629 Schematic diagram of magnet system with saturating inductor. ^ 5730 Effect of saturating inductor. ^  5831 Schematic diagram of magnet system with speed-up network (matched-termination configuration. ^  6032 Schematic diagram of magnet system with speed-up network (Short-circuit configuration)^  6033 Effect of speed-up network (matched-termination configuration).^6134 Effect of speed-up network (short-circuit configuration). ^ 6235 Resonances due to gaps (0.3-50 MHz). ^  6436 Resonances due to gaps (45-200 MHz)  65J. An improved model of the prototype kicker magnet ^ 7438 Magnitude of transmission coefficient of the prototype kicker magnet asa low pass filter. ^  7639 Parameters defining the scattering matrix. ^  7740 Magnitude of input impedance of the short-circuited prototype kickermagnet. ^  8041 Magnitude of input impedance of the terminated prototype kicker magnet 81viiAcknowledgementsI am particularly grateful to my project supervisors Mike Barnes and Gary Wait fortheir continuous support and guidance during my thesis project. I have learnt a greatdeal from many discussions with them and have benefited a lot from their experience.I wish to thank Mike Craddock, my thesis supervisor, and Shane Koscielniak for theirassistance and guidance throughout the preparation of this thesis.I take this opportunity to express my deepest thanks to my parents who have madegreat sacrifices to make a new life and home here for us.viiiIntroductionMajor goals of accelerator physics today are to increase beam intensity and energy.With the proposed KAON Factory, a high beam intensity is needed to improve theproduction of secondary beams, since cross—sections decrease as beam energy is in-creased. Beam intensity in a high energy accelerator such as the proposed KAONFactory is limited by beam stability. For example, space—charge tune spread at injec-tion restricts the intensity of the beam because some particles are carried near unstablebetatron resonances. A beam also interacts electromagnetically with its environmentand coherent instabilities in the motion of the beam in the longitudinal and transversedirections can be induced by the interaction. If these motions are undamped, theycan grow quickly as they are being driven repeatedly. In an extreme case, when theamplitude of the driven motions is sufficiently large, control of the beam could be lost,resulting in beam blow—up or even loss.The interaction of the beam with its environment is characterized by a wake poten-tial. It describes the electromagnetic field, usually called the wakefield, produced by thebeam and its image currents in surrounding materials. Wake potentials are classifiedinto two categories according to the type of motion they induce on a beam: longitudinaland transverse. The wake potentials are time dependent as well as source dependent.A more convenient concept to characterize the same interaction is impedance since itis is a property of the accelerator component but not of the beam. The longitudinaland transverse impedances are the Fourier transforms of the respective wakefields of apoint charge.In the proposed KAON Factory, a beam of 100 izA is accelerated to 30 GeV by twosynchrotron rings in series with three storage rings. Kicker magnets are used to extractand inject the beam from ring to ring. A large number of magnet modules is neededto achieve a sufficient "kick" because the momentum of the beam is high, especiallyin the Driver ring (30 GeV). The wakefields of the kicker magnets could greatly affectthe motion of the beam. Of particular interest here are the longitudinal wakefields.Instability thresholds and energy loss can be calculated from the measured longitudinal1impedance. Comparing the measured longitudinal impedance of the magnets with thatof other accelerator components in a ring, such as the rf cavities, one can determinewhether the magnets play a major role in causing coherent instabilities.This thesis describes the measurement of the longitudinal impedance of a prototypekicker magnet for the proposed KAON Factory. Chapter 1 deals with the definitionof a wakefield and the concept of longitudinal impedance. It also describes the role ofthe kicker magnets in the KAON Factory and the configurations in which they wouldoperate. The principles and the method of measurement are outlined in Chapter 2.The chapter ends with a discussion on transmission—coefficient measurements. Chapter3 begins with an overview of the experimental set—up and the measurement. Theremainder of the chapter describes the set—up, the calibration of the measurementassembly, and the measurement procedures. The measured longitudinal impedancesalong with discussions of the results are presented in Chapter 4. The thesis ends withsome conclusions.21 Impedance and Kicker Magnets1.1 Impedance and Beam InstabilityThe electromagnetic field induced by a point charge in an accelerator component isusually called the wakefield. Subsequent charges following the point charge will see thewakefield and interact with it. The interaction can be described by a wake potentialwhich is given in terms of a Green's function. In order to calculate the wake potentialin a given accelerator component due to the passage of a bunch, firstly, the wakefieldexcited by a point charge must be found by solution of Maxwell's equations. Then,the bunch wake potential is found by the convolution of the point wake potential withthe bunch charge distribution. The same interaction can be described in the frequencydomain by the Fourier transforms of the wakefields. Wakefields are divided into twotypes depending on their effects on the beam motion: longitudinal and transverse. Thetransformed wakefields are the longitudinal and transverse impedances, respectively.The impedance is a property of an accelerator component but not of the beam itself.This is the main advantage of the impedance concept. The real part of the longitudinalimpedance is responsible for the energy loss or gain of the bunch. The imaginary partof the longitudinal impedance is associated with the incoherent tune shift and bunchlengthening of the beam. Instabilities, such as the microwave longitudinal instability,depend approximately on the absolute value of the longitudinal impedance as givenby the Keil-Schnell criterion [20]. In general, single-bunch instabilities are due to thehigh-frequency broad-band impedance, while multi-bunch instabilities depend on thelow-frequency narrow-band impedance. Narrow-band impedance is a term used todescribe the narrow resonances in the impedance spectrum at low frequencies. Eachresonance is produced by a local slow-decaying wakefield whose frequency is below ornot much above the wave-guide cut-off frequency of the accelerator component. Inthe high frequency region well above the cut-off frequency, the resonances overlap,producing a smooth frequency dependence in the impedance.The high-frequency narrow-band impedance describes the interaction of the beam3with abrupt changes of the beam pipe cross-sectional area as well as the high-frequencyresonant modes of accelerator components such as rf cavities, bellows, and vacuumports. For bunch lengths larger than the beam pipe radius, the detailed behavior ofthe high frequency impedance is not a major concern. It is approximated by a singlebroad-band impedance. For short bunches, the high-frequency impedance plays amajor role in determining the single-particle motion.1.2 Wake Potential and Longitudinal ImpedanceIn the frequency domain, the counterpart to the wakefield is the impedance. Supposethe beam current I has a component of single frequency co travelling in the longitudinaldirection z, described by the real part ofI Z i = i eikz-iwt ( , )where i is the complex beam current amplitude and k the wave number. The longi-tudinal component of the wakefield EE due to the beam can be obtained by addingup the contributions from all the charges q preceding the field point in the form of awakefield [17]EE(z,t)= - 1- f°00 ds I (z,t - -sv--)W(s), (2)v - where the wakefield W'(s) is the first derivative of the wake potential produced by aunit point charge with respect to s, its distance behind, and v is the group velocity ofthe beam. Substituting Equation 1, the above equation can be written aso^dsEz (z,t) = -I(z,t) I e i'lv W'(s) —v.-0.(3)A bunch traversing some length L of the wake potential will then experience an energyloss due to the voltage drop EEL. Borrowing the impedance concept from circuittheory, the voltage drop V(z, t) can be expressed as a product of the beam current andan impedance Zil(co) asV(z,t) = -/(z,t)Zii(w),^ (4)(1)45Figure 1: Wakefield of a non-resistive beam pipe.fieldV=CFigure 2: Wakefield of a resistive beam pipe.where V(z, t) = V e1 (kz-i4 and the impedance is in Ohms (f2). Finally the impedancecan be written asZil(w) _1 f° eiwaivwf (s ) ds.^ (5)L o f-00The longitudinal impedance is just the one-sided Fourier transform of the wakefieldW'(s).1.3 Wakefield of a Beam PipeLet us consider the wakefield of an component with a simple geometry. The simplestaccelerator component is the beam pipe through which the beam circulates under5vacuum. Its cross—section is usually circular or rectangular and it is made of metal toprovide electromagnetic shielding and a path for image currents. The particle has acharge e and is travelling with velocity v along the axis of a cylindrical pipe of radiusb as shown in Figure 1. First, assuming that the walls of the beam pipe have zeroresistance, then the wakefield at the walls will only have a radial component. Theresultant wakefield is plotted in Figure 1. The solution is worked out in detail inReference [11]. The impedance per unit length L isZI1(w) _ .  Zowg L — " zirc-y 2 ,32 'where # and 7 are the relativistic parameters of the bunch, Zo L'-' 377 ft the free spaceimpedance and g a geometric factor defined by the beam radius a and beam pipe radiusb as g = 21n(b I a) + 1. As expected, the real part of the impedance is zero since noenergy loss can occur in walls of zero resistance. If the walls of the beam pipe havesome resistance, then the axial field f',, will not be zero. The wakefield of a beam pipewith non—zero wall resistance is plotted in Figure 2 for the case where v approachesthe velocity c of light. The longitudinal impedance then contains a real term which isresponsible for the energy loss of the beam [11]Z1(w) i  Zowg  + 0 + j )L = -- 4,71-c72 02 cr(Sr,where 0 is the conductivity of the walls and 6" =1.4 Wakefields of Kicker MagnetsFor kicker magnets, analytic solutions of the wakefields excited by a beam are verydifficult due to their complicated geometry and electrical properties. Estimates ofimpedance can be obtained for kicker magnet systems with simple electrical configu-rations from equivalent electrical circuit analysis. The excitation of the wakefield ismodelled as mutual coupling between the kicker magnet and the beam. This methodof impedance calculation was pioneered by G. Nassibian [12]. It is limited to a fewideal special cases where the input and output cables which connect the magnet to the(6)(7)(2/wpo-) 1 /2 is the skin depth.6rest of the system and the magnet itself have zero attenuation and are non-dispersive.Such strict and unrealistic conditions would mean that calculated values are very farfrom actual values.In the KAON Factory, kicker magnets will be used to extract and inject the beam.The wakefields of the magnets can greatly affect the beam if the magnitudes of thewakefields are large. Of particular interest here are the longitudinal wakefields ofthe kicker magnet systems, which can give rise to beam instabilities. Such concernsnecessitate an accurate method of measuring the longitudinal impedance of kickermagnet.1.5 Kicker Magnets for KAONIn the proposed KAON Factory a high intensity beam of 100 pA is accelerated to30 GeV step by step in a series of five rings. The five rings are: the Accumulatorring, which accumulates beam from the cyclotron; the Booster ring, which initiallyaccelerates the beam to 3 GeV; the Collector ring, which collects beam from 5 consec-utive Booster cycles; the Driver ring, which accelerates the beam to 30 GeV, and theExtender ring, which stores the beam for slow extraction.Kicker magnets will be used to inject and extract the beam from the acceleratorrings. A 1 MHz beam chopper installed in the transport line from the TRIUMFcyclotron will be used to create gaps. The duration of a gap is approximately 108 ns.During the gap, the field in the kicker magnet must rise or fall from 1% to 99% inorder to kick the next segment of the beam with minimum beam loss. The requiredrise (fall) -time of the magnetic field in each kicker magnet varies from ring to ring.The fastest rise (fall) -time will be 82 ns.The design of the kicker magnets for the KAON Factory is based on those of theCERN PS Division, which are of the transmission line type. Each kicker magnetconsists of usually ten LC cells connected together in series to form approximatelya transmisssion line. Figure 3 shows an equivalent circuit diagram of an ideal kickermagnet with four cells. Each cell consists of a ferrite C-core sandwiched between high-7$ 1 1 1 56 1 1 1 $One cellFerite inductorL L LcirLOutput$ 1 1 1Input^C•^Pan?llel platecappcitorFigure 3: An equivalent circuit diagram for an ideal kicker magnet.voltage capacitance plates. The ferrite C-core provides the inductance and shapes themagnetic field which kicks a beam. A long kicker magnet is divided into several smallermodules to improve its kick performance. A typical cross—section of a kicker magnetis shown in Figure 4 and a photo of the prototype kicker magnet with ten cells withcapacitance plates visible is shown in Figure 5.The characteristic impedance of a kicker magnet can be defined in terms of theinductance L and capacitance C of each of its LC cells as /L/C. The injectionand extraction kicker magnets in the KAON Factory will have a design character-istic impedance of 25 0 and will be terminated with matched resistors. However, inthe Booster ring, the magnets will be short—circuited due to space limitations. Thesekicker magnets will have an input impedance of 16.7 SI. Short—circuiting the magnetshas the effect of doubling the kick strength for a given driving voltage. To ensure max-imum power transmission, the characteristic impedance of transmission cables whichdeliver power is matched to that of the kicker magnets.C8Magnet aperturePositive conductorCapacitor plateC—core ferritesGround conductorFigure 4: Cross—section of a kicker magnet.9Figure 5: The prototype kicker magnet.101.6 Performance of Kicker Magnet SystemsA kicker magnet system consists of a pulse-forming network, power-transmission ca-bles, a kicker magnet perhaps with a speed-up network and a saturating inductor. ASpeed-up network and a saturating inductor are components which improve the per-formance of a kicker magnet system. Pulse-forming networks supply current pulseswhich energize the kicker magnets. To avoid reflections in a system, the characteristicimpedances of all the components are matched as closely as possible. The input cableof a magnet (Tx in Figure 6) is connected to the main-switch thyratron of a pulse-forming network. Thus when the main-switch thyratron is in the off state, this end ofthe cable is effectively an open-circuit (see Figure 6). The output cable (T s in Figure6) is usually connected to a resistive load. For some applications, the output of themagnet may be short-circuited and no output cables are needed. For example, themagnets in the Booster ring will be short-circuited magnets.The performance of a kicker magnet is characterized by the rise-time of the magneticfield which kicks the beam and the uniformity of the field before and after the initialbuild-up. To a first approximation, the kick rise-time is the pulse rise-time (PT) plusthe transit time (TT) through the magnet. The pulse rise-time is determined by thepulse-forming network. The transit time is given approximately by the propagationtime of the leading edge of the pulse through the magnet. The propagation time interms of the total inductance L and capacitance C of the magnet is given by AIL -C. Inorder to get a satisfactory kick rise-time, a long magnet is split into several identicalmodules, each driven by its own pulse-forming network. In this way the propagationtime is reduced since the length of each module is shorter than the combined length.A kicker magnet does not behave exactly like a transmission line because each LCcell consists of lumped elements instead of distributed ones. This difference makesthe magnet behave more like . a low pass network which attenuates the high frequencycomponents of an applied pulse as it propagates through a magnet. As a result, the rise-time of the pulse is increased and hence the kick rise-time of the magnet is increasedas well.11^c 0^ThyratronswitchOpen—circuitMatchedresistorT ^Speed—upnetwork1Input cable T.SaturatinginductorKicker magnet Output cable T,Pulse—formingnetworksystemFigure 6: Components of a typical kicker magnet system.External circuits, called speed—up networks, are designed to control the overshoot(or undershoot) of a driving current pulse (Figure 7). These networks, which consistsof series capacitance and resistance, are connected adjacent to a magnet (Figure 6).In Figure 7 an improvement in the rise—time due to a speed—up network which com-pensates the undershoot of a pulse is illustrated. Another effect to be considered isthe network cut—off frequency of a magnet , which can be attributed to its low—passbehavior (Appendix B). High frequency components above the cut—off frequency arestrongly attenuated.Three—gap thyratrons are used as switches in pulse—forming networks to energizekicker magnets [6]. A three—gap thyratron turns on in three stages. Because of parasiticcapacitance across the gaps, a flow of displacement currents occurs before a maincurrent. A thyratron switch becomes conducting when the dielectric of the gap isionized to form a conducting plasma. When this happens, the main current can thenflow through the plasma to a magnet. In Figure 7, displacement currents are presentbefore a main current pulse. These displacement currents in turn flow through the120 %1 %induc1 01 %99%Undershoot80%With a saturating inductorC an Optimized Speed —up Network48ns rise —timeNo Saturating Inductors orSpeed —up Networksa)N0E 40% -Disp ocementZ currents64ns fall —tim eUndershoot- 1 %O. jus O. ius^O. ius^1.OusTime1.2us^1.4us^1.6usFigure 7: Effects of a speed—up network and a saturating inductor on the predictedkick strength of a magnet.magnet, exciting its prematurely and thus effectively increasing the rise—time of apulse. An increase in the rise—time of a pulse will result in an increase in the kickrise—time of the magnet system.One proposed method of eliminating displacement currents involves the use of a fer-rite saturating inductor at the input of a magnet. Because of its non—linear response tocurrent amplitudes, it has a high impedance for small currents and a low impedance fora large current pulse. Such an inductor will allow a main current pulse to pass throughbut stop small displacement currents. The effectiveness of a saturating inductor ineliminating displacement current is shown in Figure 7.1.7 Longitudinal Impedance of a Prototype Kicker MagnetIn order to achieve high beam intensity and energy, collective instabilities driven bylongitudinal impedance must be minimized or carefully controlled. Thus all accelerator13components which can give a large contribution to the total longitudinal impedancehave to be carefully designed and measured before installation. Due to the large numberof kicker magnet modules required in the KAON Factory, their contribution to thetotal longitudinal impedance could be significant and must be determined. How othercomponents of a kicker magnet system affect the longitudinal impedance is describedbelow.The longitudinal impedance of a kicker magnet system can be characterized in termsof an equivalent circuit as the mutual coupling between a beam and a kicker magnetsystem. The beam would be the primary winding and the magnet the secondarywinding. The speed—up network, saturating inductor, and transmission cables of themagnet system, all of which are connected to the magnet, are also coupled to thebeam. Thus the mutual coupling is determined by the whole magnet system consistingof the magnet and other components. Consequently, the longitudinal impedance of themagnet system depends on all the components of the system not just on the magnet.As part of the KAON Factory Project Definition Study, a prototype 30 S) magnethad been designed and built at TRIUMF based on those of the CERN PS Division.The magnet has 10 LC cells. PSpice modelling [4, 5] was used to determine the optimalvalues of circuit elements for a speed-up network and a saturating inductor to improvethe performance of the prototype magnet system. The system was set up for thetwo configurations [3] in which it would operate in the KAON Factory to determineits longitudinal impedance. The effects of transmission cable, speed-up network andsaturating inductor on the longitudinal impedance was determined.142 Principles of the Coaxial Wire Method2.1 Principles of the MethodThe set—up of the coaxial wire method to measure the longitudinal impedance in thefrequency domain or the effects of a wakefield in the time domain is different fromthe actual environment in which a bunch and an accelerator component interact witheach other. We reproduce the arguments put forward by M. Sands and J. Rees [8] toshow that a central conductor can reproduce the wakefield in an accelerator componentinduced by a passing bunch. Thus, a measurement of the wakefield in terms of energyloss can be made with this method in the time domain. Although the arguments arepresented for the time domain measurement, they are equally valid for the frequencydomain measurement, since the set—ups are identical in both cases.Suppose we wish to measure the longitudinal wakefield in a section of beam pipe.Inserting a central conductor along the axis transforms the beam pipe into a coaxialstructure with the pipe as the ground conductor and the wire as the inner positiveconductor. A short pulse can now be sent down this coaxial structure from one endto the other. To emulate a charged particle bunch passing through the beam pipe, thecurrent pulse should be made to have the same shape as that of the bunch. The currentpulse will travel along the central conductor with approximately the same speed as thatof the bunch, namely close to the speed of light, and will retain an instantaneous chargedensity that corresponds closely to that of the travelling bunch.Figure 2 shows a wakefield in the beam pipe due to a point—like bunch. The longi-tudinal wakefield at the bunch subtracts energy from it but does not significantly alterthe charge distribution. The energy lost by the bunch due to the wakefield in this wayis the integral of the field along its path. Under the same circumstances the energylost by the current pulse due to its image currents is the same as that of the bunch. Ifwe can measure the energy lost by the current pulse, then we will have determined theenergy lost by the bunch as it traverses the same beam pipe. In general, the energylost by the bunch due to the longitudinal wakefield of any accelerator component can15be determined in this way by transforming the component into a coaxial structure withthe insertion of a central conductor.The most striking difference between the coaxial structure and the beam pipe is thepresence of the central conductor in the coaxial structure. In the absence of the centralconductor, wakefield energy is stored in a large number of normal-mode oscillations ofthe beam pipe and is dissipated as heat energy in the resistive walls. With the centralconductor, the characteristics of the normal-mode oscillations are changed, and theoscillations, to a certain extent, are coupled to the central wire and travel along itto be dissipated in the terminations at the two ends. On the surface it may appearthat the difference between the two cases is so great as to make them irreconcilable.However, if the central conductor is sufficiently thin, as is usually the case, then thenormal-mode oscillations are only slightly modified. In addition, if the transit time ofthe current pulse or bunch is short compared to the decay times of the oscillations, thenthe energy removed from the pulse and left behind in the oscillations will be similar tothe energy removed from the bunch.In Appendix A, the error due to a central conductor is roughly estimated. We usedtwo conductor sizes of 3 mm and 41 mm for the measurements. The error due tothe large conductor (50 Si line) is estimated to be comparable with the measurementuncertainty of the data, which is at most 5%. However, the error due to the smallerconductor (180 12 line) is estimated to be negligible. Also in the appendix, the transittime is roughly estimated to be 1.5 ns which is small compared with the decay timesof the normal-mode oscillations with a typical time of 15 ns.2.2 Energy Loss by a Beam BunchWe will start with a description of a coaxial wire set-up for sending and recording acurrent pulse. In the time domain, the energy loss can be measured by the coaxialwire method. Equations of energy loss will be derived in the time domain for use inthe following section where they are Fourier transformed into their frequency domaincounterparts to identify the longitudinal impedance. It is much simpler to derive these16Matching sectionCurrent pulseDevice under testpulsetest cablePulse generatorandTermination 0 0 0OscilloscopeandTerminationFigure 8: Experimental set-up for sending and recording a current pulse.equations in the time domain and then Fourier transform as needed.A schematic diagram of an experimental set-up for sending and recording a currentpulse is shown in Figure 8. The beam pipe segment or accelerator component is trans-formed into a coaxial structure by inserting a wire along the axis. Tapered sections,matching the coaxial structure to the test cables, are attached to both ends and are inturn connected to a pulse generator and an oscilloscope by test cables. It is necessarythat the entire coaxial line is matched to avoid reflections. The pulse generator mustproduce a current pulse with the same time shape as that of a bunch because timedomain measurement is bunch-shape dependent.To determine the energy difference or loss, it is necessary to record the passage ofa current pulse through a reference pipe. The length and cross-sectional dimensionsof the reference pipe must be the same as that of the accelerator component to bemeasured because the same matching sections are used. It should be made of a goodconducting material. A current pulse passing through the metallic reference pipe isonly slightly attenuated and hence it should retain its original time shape. An over-17all check on the performance of the pulse generator, tapered matching sections, andterminations can be obtained by comparing the time shape of the current pulse beforeand after its passage through the reference pipe.The basic measurement of consists of recording the time shape of the current pulse4(0 after its passage through the reference pipe and 4,(t) then again when the refer-ence pipe is replaced by the accelerator component. The raw data will be the two timefunctions recorded by the oscilloscope. The secondary current A/(t) is defined as thedifference between the reference pulse and the modified one asA/(t) = /,.(t) — Im (t).^ (8)The energy Ur contained in the reference pulse is^ r = Zo I 1, 2 dt,^ (9 )where Zo is the impedance of the matched termination. Similarly, the energy containedin the modified pulse is^U  = Zo f im 2 dt.^ (10)The energy loss AU by the pulse as it passes through the accelerator component isgiven byAU = Ur — Um,^ (11)^= 2Z0 I ImAI dt + Zo^i AI 2 dt.^(12)The last equation is obtained by rewriting it in terms of A/(t). This energy lossis attributed to the work done against the electric wakefield 4, as the pulse passesthrough it. The energy loss in terms of the electric wakefield iv, is^AU = q i few • di,^ (13)where q is the total charge of the pulse.After the reference pulse I,. and modified pulse I, have been recorded by the os-cilloscope, the above equations can be used to calculate the energy loss by the pulse18as it traverses the accelerator component. The energy loss by a bunch depends onits length and shape. Hence many measurements must be made for the different timeshapes and lengths that might be encountered. This dependence makes the directmeasurement of the energy loss in the time domain inefficient. If the response of thewhole coaxial structure is linear at all the relevant frequencies and amplitudes, thenan equivalent measurement can be made in the frequency domain. The energy lossby a particular bunch can then be calculated using Fourier analysis. The frequencydomain measurement is also known as the coaxial wire method and is explained in thefollowing sections.2.3 Method of Transmission MeasurementTime domain measurements of the type described in the last section, are limited tomeasuring the energy loss and phase difference of a bunch, whereas frequency domainmeasurements can completely characterize the longitudinal interaction of a bunch withan accelerator component in the measured frequency range. Frequency domain mea-surements are relatively easy compared to time domain measurements and can providemuch more useful information. We will start with a description of the experimentalset—up. Fourier transformation of the energy loss (Equation 12) into the frequencydomain will be used to identify the longitudinal impedance.Figure 9 shows a schematic experimental set—up for measuring the transmissioncoefficient S21 of a coaxial structure. In the frequency domain, the transmission coef-ficient of a coaxial structure is measured instead of the energy loss. The transmissioncoefficient is the ratio of the amplitudes of the transmitted voltage over the incidentvoltage. In transmission measurements, a coaxial structure is known as a two—portdevice with input and output ports. The definition of the transmission coefficient of atwo port device is given in Appendix C.The set—up in Figure 9 is made up of the transformed coaxial structure of a referencepipe or an accelerator component, tapered matching sections, transmission lines anda network analyzer. The method consists of measuring the transmission coefficients of19Test cableadaptorS210 0Port^Port1 2Device under test50 0 test cables0 0 DTest cableConnectorMatching sectionsetupMating FlangesNetwork analyzerFigure 9: Experimental set-up for measuring S21 •the reference pipe and the accelerator component in some frequency range. The rawdata will be two transmission coefficients at each frequency. One is the transmissioncoefficient ,S21 of the reference pipe and the other is the transmission coefficient S21of the accelerator component in place of the reference pipe.The energy loss AU by a current pulse is given (Equation 12) byAU .zs_ 2Z0 flmAI dt,^ (14)if the second order term 01 2 is neglected. The magnitude of AI is small comparedwith /, so neglecting the second order term is a good approximation. The same energyloss can also be written in terms of the work done against the wake potential WII(t) asAU = q^dt,^ (15)where q is the total charge. Equating Equations 14 and 15, we obtainqWji(t) = 2441(0.^ (16)20Fourier transforming the above equation into the frequency domain yieldsZi(w)i(w) = —2Z0 [4.(w) — i(w)],^(17)where Zo is frequency independent. Using the definition of transmission coefficient, theabove equation can be written asS;ff — S21 ZII = 2Z0^,..,021(1 8)Equation 18 gives the longitudinal impedance directly in terms of the complex trans-mission coefficients involved.2.4 Longitudinal Impedance and Transmission CoefficientThe naive analysis leading to Equation 18 glosses over the question of whether Z1 is alocalized impedance, i.e. almost zero length, or an extended impedance, such thatfL,ZII = jo Z(s) ds,^ (19)where the integration is performed over the length L. The analysis of Hahn and Peder-sen [9] makes this distinction more explicit and indicates that Equation 18 is appropri-ate to a localized impedance. They derive a new expression for an extended impedancebased on repeated application of Equation 18 to each differential element. This newexpression is appropriate to kicker magnets whose impedance is distributed. In thissection, we summarize their derivation of the expression that relates the transmissioncoefficient to the longitudinal impedance of an extended and uniform source.For accelerator components which exhibit uniform structure in the axial direction,the longitudinal impedance can be assumed to be uniformly distributed. Beam pipes,bellows, and kicker magnets are examples of accelerator components that have approx-imately uniform axial structure and so for these components, a longitudinal impedanceper unit length can be defined. In Section 3.2, we derived an expression (Equation 18)which relates a localized longitudinal impedance between the ports to the transmissioncoefficients. For an extended and uniform component for which an impedance per unit21'rapedZna^ ZIlbZo^Zo^ ZoS 21^ Sb21Figure 10: Two localized impedances.length can be defined, a new expression can be derived by the repeated application ofEquation 18 to each differential element of the extended impedance.We will start with a combination of two localized impedances and then generalizeto a distributed source of impedance. Figure 10 shows a schematic diagram of twolocalized impedances Z IT and Zti along the beam pipe of characteristic impedance Zoseparated by a distance 1. The transmission coefficients S2 1 and ,S/ 1 of the two localizedimpedances can be obtained from Equation 18S11^ZaS;7f^2Z011 Z1blS42.fZI  2Zo .The combined transmission coefficient Sg of the two cascaded impedances can beshown [9] to be^cref^1 — Sli ShSab^Syr ys1 1^(22)where S I``i and Sit are the reflection coefficients of the two impedances, respectively.Substituting expressions for S3 .1 and SI (Equation 18) into the above equation yieldski21^are./^2Z0^Qab^ZIT + ZII ^ (23)(20)(21)22after simplification, since IZ I11 < Zo and Rill < Zo as required by Equation 18. Thusthe two contributions simply add up independent of their location provided that eachcontribution is small compared to Z o .This result can easily be extended to a distributed impedance [9] Zii in Ohms (S2)S21 ,,,, ( 1^RH'52. 1 - 1 - 2Z0) exP(—P(11/2Z0),(24)where the total impedance Z11 = RN + jX11 is separated into the resistive part RH andthe reactive part X11. One can see that the real part affects the magnitude while theimaginary part affects the phase of a transmitted signal. This equation will be used toextract the longitudinal impedance of the prototype kicker magnet from the measuredtransmission coefficients.233 Experimental Set-up and Experiment3.1 OverviewIn order to determine the longitudinal impedance of the prototype kicker magnet sys-tem, the magnet aperture was transformed into a coaxial structure by inserting acentral wire. The magnet was enclosed in an aluminum tank to simulate a vacuumtank coupled to a section of beam pipe. To minimize unwanted reflections in thecoaxial line, tapered matching sections were used. The transmission coefficient of thecoaxial line was measured with an HP network analyzer. The longitudinal impedanceof the magnet was calculated from the transmission coefficient. A reference coaxialline was built which has the same length and cross-sectional dimensions as the magnetaperture. This was necessary to fix the phase of the transmission coefficient. It alsoprovided a check on the overall accuracy of the measurements.The longitudinal impedance of our prototype kicker system was determined in thefrequency range of 0.3 to 200 MHz. The transmission coefficient from 0.3 to 200MHz was measured in two steps: from 0.3 to 50 MHz we transformed the magnetaperture into a 50 it coaxial line and from 45 to 200 MHz into a 180 coaxial St line.A 50 ft line was chosen to match the impedance of an HP network analyzer andstandard test cables. This also permitted the use of existing matching sections andcable adaptors from an other experiment [21]. A line with a larger impedance wasnecessary in the higher frequency range to improve the accuracy of the approximateformula used to calculate impedance. One of the conditions of the formula is thatlongitudinal impedance must be smaller than line impedance. A 180 f2 was chosen toutilize existing 180 It apparatus from another experiment [21].For both lines, calibration was necessary to eliminate systematic errors of the HPnetwork analyzer and test cables. Semi-rigid 50 ft test cables were used to connect thecoaxial line to the HP network analyzer. For the 50 ft line, HP calibration proceduresand HP standard terminations were used to eliminate the systematic errors of thetest cables and the HP network analyzer. However, for the 180 it lines, we used the24TSD (Through, Short, Delay) calibration method, which calibrated the measurementassembly from the network analyzer up to the transformed coaxial line. With the TSDcalibration method, the impedance of the line did not need to be matched to the 50 CItest cables. The raw data of a calibration and measurement were processed by an HPcomputer. Error—correction factors were calculated from the calibration and then usedto extract corrected transmission coefficients.3.2 Experimental Set—up3.2.1 Magnet TankThe prototype kicker magnet was placed in an aluminum tank which has two openingsaligned with the aperture of the magnet. The opening dimensions of the tank arethe same as the cross—section of the aperture. Figure 11 shows the magnet in thetank The dimensions of the tank are rectangular with the bottom 27.5 cm of thetank 4 cm wider in the axial direction of the magnet to accomodate the magnet stand.The tank is approximately 90 cm tall and 55 cm by 42 cm in cross—section. Thealuminum tank simulated the condition of a vacuum tank in a beam line and ensuredthe presence of a ground conductor around the central conductor of the line. Twogaps between the matching sections and the magnet were bridged by rf finger stocks(Figure 11). The rf finger stocks simulated the beam pipe connection to the magnet.The ground connection from the input matching section to the output matching sectionwas continuous through the ground conductor of the magnet when rf finger stocks wereinstalled.3.2.2 Reference PipeThe reference pipe for the prototype kicker magnet was made of brass for mechanicalstiffness. It has the same rectangular cross—section as the magnet aperture. Thedimensions are 15.5 cm by 7.8 cm and 45.1 cm long. Its length is the same as thatof the tank. It ends are welded to flanges for mating with other sections as shownin Figures 12 and 13. Brass is a very good conductor in the frequency range of 0.325Ground tankGapMatching sectionINI111111111111111H111111rICITITITIlt1411111-14-11t,Central conductorRF fingersFigure 11: The prototype kicker magnet in the ground tank.26Mating FlangesMatching sectionCentral conductorReference pipeor Magnet tank50 f2 Test cableadaptorTest cableConnectori501ineFigure 12: The 50 SI coaxial line.to 200 MHz and so the brass reference pipe has a very low longitudinal impedance(< 1.612/m).3.2.3 Matching SectionsThe purpose of the matching sections is to match segments of the coaxial line withdifferent cross—sectional dimensions. Two matching sections, one at each end of themagnet tank or reference pipe, were used to connect the tank or pipe to the end sectionswhich were in turn connected to test cables of the HP network analyzer. See Figures12 and 13 for more details. The cross—sectional dimensions of the matching sectionsare 15.5 cm by 7.8 cm at one end and taper to 10.3 cm by 7.8 cm. They are 18.4 cmin length and also made of brass. For the 50 ft line, the central conductor was alsotapered at the end sections, but for the 180 n line, the central conductor did not needto be tapered.3.2.4 Central ConductorsCentral conductors were required to transform the reference pipe and magnet apertureinto coaxial lines. For the 50 5 .1 transformed line, the central conductor was a circular27 180fineTest cable^ Central conductorConnectorMating FlangesEnd capReference pipeor Magnet tank^Matching sectionFigure 13: The 180 CZ coaxial line.copper pipe with a diameter of 41.35 mm. The pipe was held in the center by plasticspacers in the end sections. Because the pipe was rigid, no additional spacers wererequired in the middle section. In the 180 it transformed line, a copper wire with adiameter of 3.175 mm was strung between the end sections. It was held in place undertension to reduce sagging at the middle.3.3 Calibrations3.3.1 Full Two—Port CalibrationIn any measurement there are errors associated with the measurement system thatcontribute to the uncertainty of the results. Over the years, transmission measurementerrors have been studied and methods developed to correct them. The standard methodof correcting transmission measurements is to use an error model which characterizesthe errors of the measurement ports. Both the HP calibration and TSD calibration arebased on the principles of a full two—port calibration, the principles and procedures ofwhich are outlined in this section.Our measurement system consisted of an HP 8753B network analyzer and 50 fl semi-rigid test cables. Sources of systematic errors that can be modelled are slight impedance28mismatch and leakage in the test cables, isolation between the reference and test signalpaths, and system frequency response. All these errors introduce uncertainty in themagnitude and phase of a measured transmission coefficient. The two—port calibrationmeasures the reflection and transmission coefficients of test cables, a standard short—circuit termination and a matched resistive load at each of the two calibration planes.Error parameters of the sources being modelled are calculated from the calibrationusing equations from Speciale and Franzen [12]. The correction factor of the two—port transmission measurement can then be calculated from the error parameters ofthe calibration. When the correction factor is applied to a measured transmissioncoefficient the sources of error being modelled are effectively eliminated.A check on the overall accuracy of a calibration was provided by measuring thelongitudinal impedance of the reference pipe and comparing the measured values to theexpected values. If an excessively large impedance of the reference pipe was measuredthen the calibration was faulty and was repeated.3.3.2 HP Calibration for the 50 ft LineIn this section we will describe the method of HP calibration and steps taken to assurethe accuracy of the calibration.For the 50 it line, the calibration planes were at the ends of the semi—rigid 50 SI testcables coming from the two ports of the network analyzer. See Figure 14 for details.Full two—port calibration was performed following the menu and using the standardterminations of the HP 8753B network analyzer. The frequency range was from 0.3 to45 MHz. The HP standard terminations, which consist of a short—circuit terminationand a resistive 50 S2 load, were connected one by one to the ends of the test cables. Thetransmission and reflection coefficients of the terminations could then be measured. Ameasurement of the transmission and reflection coefficients of the test cables was alsoincluded in the calibration procedures. The test cables were connected together forthis measurement.Sources of error during calibration and measurement were insecure connections and29any movement of the test cables that could loosen connections and cause slight vari-ations in the lengths of the test cables. Small movements of the test cables wereunavoidable during calibration procedures when standards were being mounted anddismounted. However, large movements could be avoided by securing the test cablesat various points along their length. For best results, the calibration was done as closeas possible to the transformed line being measured. This eliminated large movementsof the test cables after calibration when they were reconnected to the transformed linefor test measurements.The success of the calibration was confirmed by measuring the transmission coef-ficient of the reference line to determine its longitudinal impedance. It was expectedthat the brass reference pipe should have a distributed longitudinal impedance lessthan 1 ft/m [71 in the 0.3 to 45 MHz frequency range. For our reference pipe of about0.5 m, the measured longitudinal impedance was 0.3 Q, satisfying the expectation.3.3.3 TSD Calibration for the 180 S/ lineThe TSD calibration is based on the same principles as those of the HP calibration.A full two-port calibration requires the use of a resistive load as one of its standardterminations. Because of the difficulty of building accurate 180 It resistive load forpipe-size dimensions, a delay pipe, of the same dimensions as the reference pipe, wasused to simulate a resistive load. In this section we will describe the the method ofTSD calibration.For the 180 n line, the TSD (Through, Short, Delay) method of calibration was usedto calibrate the measurement assembly. The calibration planes were at the end flangesof the end caps and are shown in Figure 14. The frequency range was from 45 to 200MHz. The TSD standard terminations consisted of a short-circuit plate, and a 180 itload in the form of a pipe segment. The pipe segment, used to simulate a matched load,was a short section of pipe less than one-quarter wavelength at the highest frequency.For a pipe segment of known length, the input impedance can be calculated for use asa matched load. The reference pipe was a segment less than one-half wavelength at the30OneThe 50 0 LineMineFigure 14: Calibration planes.31highest frequency. The terminations were mounted on the end caps and held in placeby nuts and bolts to ensure good electrical contact. The transmission and reflectioncoefficients of the terminations were measured. The reference pipe was also used in thecalibration to measure the transmission of the matching sections, end sections, and testcables. The measured transmission and reflection coefficients were used to calculatethe error parameters of the two—port error model of the measurement assembly.The calibration procedures were adopted from Walling [7]. A computer was neededto control the calibration procedures. For best results, movements of the test cable werekept to a minimum and the same precautions outlined in Section 4.3.2 were followed.Unlike the small standard HP terminations, which were mounted at the ends of thetest cables using standard cable connectors, TSD terminations were mounted at theend flanges of the end caps using nuts and bolts. These terminations were large andheavy and required a great deal of care to ensure good electrical contact. A flash lightwas used each time to search for air gaps between the interfaces. If any gaps werefound, the bolts were further tightened until the gaps closed. The presence of an airgap would mismatch different sections of the transformed coaxial line leading to faultyerror parameters.The success of the calibration was confirmed by the measured longitudinal impedanceof the reference pipe which was ill for a length of 0.451 m. Comparing this to a mea-sured value of 1 Slim for a similar pipe [7] indicated the calibration was not faulty.3.4 Digital Noise ReductionThe 8753B HP network analyzer is a highly sophisticated instrument which offersmany features enhancing accuracy. For instance, greater accuracy in the test datacan be obtained with little effort by utilizing its digital—data processing capability.An average sampling size of 8 was used for both calibrations and test measurements.Averaging reduces the noise level of the signal. The value of each data point was basedon a weighted average [19] of 8 consecutive samplings. An averaging rate of 8 wasmoderate and hence did not slow down the update time significantly. Another noise32reduction feature IF (Input Frequency) Bandwidth Reduction was also used to lowerthe noise floor. This reduction was accomplished by digitally reducing the receiverinput bandwidth. It is more reliable than sample averaging in filtering out unwantedresponses such as spurs, high—frequency spectral noise, and line—related noise. An IFbandwidth of 10 Hz was used, the smallest bandwidth available.3.5 Computer Calibration ProgramFor the 180 11 line, which used the TSD method of calibration, a computer program,originally written by Walling [7], was adapted to take the measurements and performthe calibration. First the calibration was performed following the calibration menu ofthe program and then the error correction factor of the measurement assembly wascalculated. To check the accuracy of the calibration, the transmission coefficient ofthe reference pipe was measured and the corrected transmission coefficient calculatedusing the error correction factor. The longitudinal impedance of the reference pipewas calculated from the corrected transmission coefficient and compared with that ofa pipe of similar dimensions. If the measured value was within an acceptable range ofother measured values, the calibration was used for subsequent measurements of thekicker magnet.For the 50 1/ line, which used the HP method of calibration, a computer was notneeded to take the measurements or perform the calibration. A calibration menu forthe 50 ft line was available on the HP network analyzer. The measurement steps werethe same as those for the 180 f2 line.3.6 Longitudinal Impedance CalculationsThe basic transmission data consisted of the transmission coefficients S;If and S21 ofthe reference pipe and the kicker magnet, respectively. These transmission coefficientswere measured only after a successful calibration of the test instrument had beenaccomplished. The longitudinal impedance was then calculated from the transmissioncoefficients using Equation 24, from which the resistance R11 and the reactance X11 can33[magnitude -S21 I ^Rils;;/ = 1 - 2z0 ' (26)be extracted separately as followsphase[  S21 S72-11 I^'''' — XII2Z0 '(25)where Zo is the characteristic impedance of the reference line.A slightly different method of determining longitudinal impedance was required forthe reference pipe. In order to determine the longitudinal impedance of the referencepipe, for use as an overall check on the accuracy of the measurement assembly, its trans-mission coefficient must be compared to that of another pipe as required by Equations25 and 26 to fix the phase and background of the test signal. A transmission—coefficientequation for an ideal lossless reference pipe was substituted into Equations 25 and 26,S21 = exp(—jkl), (27)where k is the wave number and 1 is the total length of the line between the calibrationplanes. The magnitude of S21 is unity because an ideal line has no losses. The measuredlongitudinal impedance of the reference pipe would then include any errors in the lineand the calibration.The length of a line is the distance between the calibration planes. The length ofthe 180 Il ideal line was given by the length of the reference pipe whose ends wereat the calibration planes. The pipe was measured with calipers to be 0.820 m. Forthe 50 E2 ideal line, the calibration planes were at the ends of the test cables, whichwere further back from the ends of the reference pipe. Its length could not be preciselymeasured with calipers. A different method, consisted of fitting the measured S21 ofthe reference pipe to Equation 27, was used to extract the length from the equation.Its length was found to be 1.62 m.344 Measurement Results and Discussions4.1 Longitudinal Impedance of the Reference pipeIn order to extract the impedance of the magnet using Equations 25 and 26, one mustobtain a transmission measurement of a perfectly matched reference coaxial line. Sucha reference line must include all the transition sections of the magnet measurement line,but with a reference pipe in place of the magnet (Figure 11). In practice, a coaxial linewith different segments cannot be perfectly matched at all frequencies. A well matchedline can be used instead, provided any loss and mismatch are negligible. The referenceline for the magnet was formed with a brass pipe.In a transmission measurement of the magnet, the mismatch between different seg-ments of the line causes reflections which reduce the transmission. If this reductionin transmission is attributed to attenuation rather than reflection, then the measuredimpedance is not the true impedance but also includes a contribution from the mis-match. We shall call this contribution the mismatch impedance. The mismatch isapproximately the same for the reference line because the components are the same forboth lines except for the interchange of the magnet with the (brass) reference pipe.Under the assumption that the measured impedance can be separated into the trueimpedance and a mismatch impedance, and that the mismatch impedance is additive,we may writegmagmesszrefmeas= Zimruaf Zmismatchl= Z;retf,e Zmismatch •(28)(29)If we now form the quotient of transmission parameters we findsmogStVas= exp[ZPr: — Zirreufe (30)In principle, the systematic error Zr,?.f e could be removed if the impedance of the refer-ence line was precisely known. However, no greater accuracy could be obtained.if themeasured reference pipe impedance were used, because of the presence of a mismatch35impedance which could be of comparable size. A direct measurement of the mismatchimpedance requires an ideal lossless line, which is not available. Consequently, no at-tempt was made to remove the systematic error, Z trreufe . However, its upper limit canbe determined from the inequality1 Zrrele I < I Zmr ela s I • (31)Consequently, the systematic error will be small if the measured reference pipe impedanceis small compared with the measured magnet impedance.The impedance of the brass reference line was determined as outlined in Section 3.6 .Its value indicates the upper limit on the systematic error of the measured impedanceof the magnet. The longitudinal impedance of the brass reference pipe was measuredfrom 0.3 to 50 MHz by transforming it into a 50 ft coaxial line. The results of themeasurements are shown in Figure 15. The absolute maximum resistance is less than0.3 CI for a pipe length of 0.451 m and the absolute maximum reactance is less thanj0.1 Si.From 45 to 200 MHz, the reference pipe was transformed into a 180 S2 coaxial line,and the longitudinal impedance was measured. The absolute maximum resistance is0.8 f2 and the absolute maximum reactance is j3.8 ft. The results of the measurementsare shown in Figure 16. Note in Figure 16 there is a rising trend in the longitudinalreactance. This was likely due to the phase instability of the test cables caused by cablemovements during the calibration procedures. Hence a large longitudinal reactance wasmeasured.With the exception of the longitudinal reactance in the 45-200 MHz range, thevalues obtained are approximately the same as those measured by Walling [7] for sim-ilar pipe dimensions and length. Direct comparison of the measured values with themeasurements of Reference [7] is not possible because those values are for slightly dif-ferent pipe dimensions. However, comparison between pipes of similar dimensions canprovide a strong indication of the accuracy of the measurements.36ototoOviev*40^500.75 —0.50 —a)2, 0.25 —000.00 —0.E —0.25 —-o51 -0.50 —c—J—0.75 ——1.00 I^I^I10 20 30Frequency (MHz)0 40 501.000.75 —ref500.50 —a)- 0.25cr)cr)0.00 —0.c —0.25 —-oEr —0.50 —0—J—0.75 ——1.00 I^I^i0^10 20 30Frequency (MHz)ref5Or1.00Figure 15: Longitudinal Impedance of the Reference Pipe from 0.3-50 MHz.37160 180I^I^I^I^I^i^160 80 100 120^140Frequency (MHz)I40 2001 ref 1802.01.5 —C:1.0 —oco 0.5 ---1--'cncna)(X 0.0 —0.c —0.5 —-0DET) —1.0 —C0_J—1.5 ——2.0 i^i^i^1^i^1^140^60 80 100^120^140^160 180^200Frequency (MHz)Figure 16: Longitudinal Impedance of the Reference Pipe from 45-200 MHz.384.2 Electrical Resonant ModesThe magnet system behaves like a transmission line up to its network cut—off frequency(30 MHz). In this frequency range, the resonances in the impedance of the magnetsystem can be explained in terms of the propagation of voltage waves in a transmissionline of the same length. Using this model, resonant frequencies are determined by thoseof the standing waves. These frequencies can easily be calculated from the terminationand length of the line. The widths and heights of the resonances in the longitudinalimpedance, however, cannot be obtained from this simplistic model. The agreementbetween the calculated and measured resonant frequencies was confirmed by the data.Above the cut—off frequency, there is strong attenuation of travelling waves and hencedifficulty in obtaining standing waves. This was also confirmed by the lack of resonancesin the longitudinal impedance from the data. Consequently, below 30 MHz, we expectthe transmission line model to work well. We show data that support this model.In an equivalent circuit diagram (Figure 17), the prototype kicker magnet is mod-elled as a section of transmission line. The one—way delay time of the magnet is 30ns and its characteristic impedance is 30 n. The ends of the transmission line cor-respond to the input and the output of the magnet. They can be short—circuited,open—circuited, or connected to input or output 30 ft cables. The magnet and thecables form an extended transmission line which can support standing waves.39InputOpen—circuitedOutputOpen—circuitedquarterkickKicker magnet (30ns)InputOpen—circuitedOutputShort—circuited(2n+1)X/4 lineKicker magnet (3Ons)n(X/2) lineFigure 17: Resonant structures of the prototype kicker magnet.40When the magnet is excited by a sinusoidal current travelling along the centralconductor, image currents are set up in the magnet and propagate as travelling wavestowards the ends of the magnet. These travelling waves will be either absorbed orreflected at the ends depending on the terminations. If an end is terminated resistivelywith an impedance matching the characteristic impedance of the magnet, no waveswill be reflected. For voltage waves reflected from a short—circuit termination, thereis a 180 degree phase change. For an open—circuit termination, however, there is nophase change in the reflected waves. If the line is not terminated resistively, thenat certain frequencies the right conditions exist for standing waves to resonate alongthe line. With the assumption that the wakefield excited by a current pulse on thecentral conductor is the same as the wakefield excited by a charged particle beam, thesestanding waves are identical to the local modes of the oscillating wakefield, which inturn interacts with the beam. In a longitudinal impedance graph the local modes showup as resonances.41Resonantmode (n)quarter–wavelength resonator half–wavelength resonatorCalculated Measured Calculated Measured1238.3 MHz25.041.68.1 MHz21.641.016.6 MHz33.350.015.6 MHz30.2—Table 1: Resonant frequencies of the 30 ns quarter–wavelength resonator and the 30ns half–wavelength resonator.In Figure 18, the measured longitudinal impedance of the prototype kicker magnetis shown. 30 S/ feed cables were not connected to either the input or the output of themagnet. The solid line is the longitudinal impedance of the magnet with the input (oroutput) open–circuited and the output (or input) short–circuited. Note that with nocables connected, the input and the output of the magnet are interchangeable. Theresonances appearing in the solid line correspond to those of a quarter–wavelengthresonator: a resonance appears for the first three (2n + 1)A/4 modes where n = 1, 2, 3.The dashed line in Figure 18 is the longitudinal impedance of the magnet with boththe input and the output open–circuited. The resonances in the dashed line correspondto those of a half–wavelength resonator: a resonance appears for the first two nA/2modes where n = 1, 2.Table 1 lists the frequencies of the first three resonances of the magnet from mea-surements and calculations. The measured resonant frequencies are very close to thecalculated frequencies. Above the network cut–off frequency (30 MHz), the magnetceases to behave like a transmission line and no more standing waves can exist: hence,the lack of resonances.42120C 100 —a)Uc 80-0-4—J. cf)(11Crr 60 —vc 40—=--6C0__J 20—IMagnet outputShort Circuit- - - Open Circuit open1I^I^I10 20 30Frequency (MHz)0i40^5040I^I^I0^10 20 30Frequency (MHz)1 openlr1Figure 18: Longitudinal impedance of the prototype kicker magnet4350n(X/2) lineFigure 19: Resonant structure of the kicker magnet system with a feed cable.44Figure 20 shows the longitudinal impedances of the magnet with a feed cable, itsoutput being short—circuited or open—circuited (Figure 19). The one—way delay of thecable is 30 ns and it has a characteristic impedance of 30 li, matching that of themagnet and thus forming a combined line of 60 ns. The resonance distribution issimilar to that of Figure 18 but corresponds to that of resonators that are twice aslong but with lower resonant frequencies. Table 2 shows the calculated and measuredresonant frequencies of the two resonators. The agreement is very close, within 4%.The longitudinal impedance shown in Figure 21 is that measured for the magnetwith a longer feed cable connected to its input. The one—way delay of the long feed ca-ble is 193 ns and it has a characteristic impedance (30 fi) matching those of the magnet.The combined length of the cable and the magnet is 223 ns. The resonance distribu-tions correspond to those of quarter—wavelength and half—wavelength resonators. Theresonant frequencies can be calculated assuming an ideal transmission line as in thelast two cases. An envelope on the overall magnitude of the longitudinal impedancecan be observed.45Magnet InputShort Circuit—^— Open CircuitI%\^ / x/1^/1 /^'1 /^1.■ s. ....^.... •/ s'•1iI^I^I10 20 30Frequency (MHz)0openr5040open10080 —a)0c0-4--,^6cn. c7i^0 —a)2 40 —'5=alc0 20 —_1I^I^I10 20 30Frequency (MHz)I40^50Figure 20: Longitudinal impedance of the prototype kicker magnet with a 30 ns feedcable.461i■IIiiI iii .IV^.•1.1 '1/4 / --IciII^1II^4I^li001 111IIII1I00IIIIII11I1 11^ 1^ I^ 1 open3r601 i^fIII / 1 II^11 ^/ t/^►I^I/^1 1 I^II^■1^I/^I i I / /1I/ t / 11^.-^1 ■Magnet Output^ Open Circuit- - - Short CircuitI^I^I^110 20 30 40Frequency (MHz)11—400I/ II^II^I^i lI I^I / I,11^I50411a 40 —a)UCo-1-6 20oa)IY6.-c 0—•-6=-,-,•ifncc) —20 —Jopen3Magnet OuputOpen Circuit- Short CircuitI^I^I^10^10 20 30 40^50Frequency (MHz)Figure 21: Longitudinal impedance of the prototype kicker magnet with a 193 ns feedcable.70 -C0 —6a)UC° 50 —In' c7)C( u 40 ——61.c 30 —-om-,-,.520 —cO_J10 —8047Resonantmode (n)quarter-wavelength resonator half-wavelength resonatorCalculated Measured Calculated Measured1234.1 MHz12.520.85.3 MHz12.121.08.3 MHz16.625.08.4 MHz17.224.0Table 2: Resonant frequencies of the 60 ns quarter-wavelength resonator and the 60ns half-wavelength resonator.The heights and the widths of resonances do not exhibit any simple dependence onthe parameters of the magnet and the cable that can easily be deduced. In general,the attenuation of travelling waves by the magnet is responsible for broadening theresonance widths. Above the cut-off frequency, the travelling waves become evanescentand hence standing waves cannot be maintained. Consequently, the conditions forresonances do not exist. The cut-off frequency of the prototype kicker magnet wasmeasured to be 30 MHz (Appendix B). Losses in the C-core ferrite largely determinethe degree of attenuation and hence the cut-off frequency.Figure 23 shows the effect of terminating the kicker magnet system with a 30 Siresistor. The 30 S/ resistor was connected at the end of the output cable. The equivalentcircuit diagram of the system is shown in Figure 22. For a magnet system terminatedwith a matched resistor, standing waves cannot exist because reflections do not occurfrom a matched resistor. Without standing waves, resonances do not occur.Nevertheless, some reflections are possible as indicated by the figure. The reflectionsoccurred since the magnet could not be really terminated at all frequencies with a 30 CZresistor. This is due to the fact that the input impedance of the magnet slightly variesfrom 30 I/ below the cut-off frequency (Appendix D). Above the cut-off frequency(30 MHz), the input impedance varies considerably and can no longer be consideredconstant nor be terminated with a resistor.48ri(1 0^J C0Output\O )Input30matKicker magnet (30ns) Output cable (30ns)0°^n^No f)^v^\ 1input Output30 0n(A/2) lineKicker magnet (30ns) Output cable (30ns)30 0Terminated lineFigure 22: Equivalent diagram of the prototype kicker magnet with a 30 ns outputcable49100C 80 —a)C060 —in•tr)a)cc0 40 —0 20 ——J 11 p1/^1Output CableOpen Circuit- -- Matched Terminati nopen2^I ^I^I10 20 30Frequency (MHz)40^5080—80open2rOutput CableOpen CircuitMatched Termina:ion^I^1^I10 20 30^40^50Frequency (MHz)60 —5C40 —Ca)3 20 —000 —IT).E-20 —•5-40 —C0_J—60 —Figure 23: Longitudinal impedance of the prototype kicker magnet with a 30 ns outputcable resistively terminated.504.3 Magnet ConfigurationsblockmatchPulse–formingnetworkFeed cable kicker magnet Output cableO D^ 0—0NO^ThyratronswitchMatchedresistorFigure 24: Schematic diagram of matched-termination configuration.In the KAON Factory, kicker magnets will be operated in two electrical configurations.In the short-circuit configuration the magnet output is short-circuited, and in thematched-termination configuration the magnet output is connected to a cable which isresistively terminated (Figures 25 and 24, respectively). As shown in Section 4.2, thelongitudinal impedance of a magnet system depends not only on the magnet itself butalso on other components of the system. The longitudinal impedance of the prototypemagnet for both configurations is described so that it can be used for comparison inlater sections.Figure 26 and Figure 27 show the longitudinal impedance of the two configura-tions from 0.3-50 MHz and 45-200 MHz, respectively. For the short-circuit configura-tion, the longitudinal impedance contains a resonance distribution that is typical of aquarter-wavelength resonator. Resonances are not present in the matched-terminationconfiguration, as discussed in Section 4.2 .51blockshortFeed cable^kicker magnet Short-circuitPulse-formingnetworkC—oNoThyratronswitchFigure 25: Schematic diagram of short-circuit configuration.The longitudinal impedance in the 30-200 MHz range for both configurations isdominated by losses in the C-core ferrite of the LC cells. The increase in loss as afunction of frequency is characteristic of hysteresis loss in the ferrite. In general, thelongitudinal impedance from 30-200 MHz does not depend on the termination of themagnet nor on other components of the magnet system. The maximum real impedanceof both configurations is approximately 32 li.5240^50config40Magnet Configuration-Short Circuit- - -Matched Termination35 —0 —3a)° 2 5 —.cn(7)a)= 20 —155)10 —CO5—0 110^20^30Frequency (MHz)•I//40'Es: 30a)C 20 —IDa)10—0C0—CPC0—10 —configrMagnet Configuration^ Short Circuit- - - -Matched Termination—20 I^I^110 20 30Frequency (MHz)40^50Figure 26: Longitudinal impedance of short-circuit and matched-termination config-urations (0.3-50 MHz).531 config lr140.--. 120 -C.---0) 100 -c.)c0g 80 -a)IY0 60 -c.-E)=-I., 40 -c0_1 20 -0^i^I^I^I^I^I^i40^60 80 100 120^140Frequency (MHz)160 180 200Magnet Configuration^ Short Circuit3530 -C8 25 -co-,--•• -(2 20 -cnaux75 15 -.c-o=10 -Enco-J 5 -1 configlIIMagnet Configuration^ Short Circuit- - - - Matched Termination040I^I^I^I^I^i60 80 100^120^140^160Frequency (MHz)i180^200Figure  27: Longitudinal impedance of short-circuit and matched-termination config-urations (45-200 MHz).54Losses due to the ferrite can be reduced by shielding them with a conducting ma-terial. Aluminum sheets 1.5 mm thick were placed over the walls of the magnet aper-ture. Figure 28 shows the effectiveness of this shielding in reducing the longitudinalimpedance in the 50 to 200 MHz range. It appears there is some degradation in thelongitudinal impedance below about 100 MHz, attributable to undamped resonancesof some unidentified modes. The reactance is reduced almost at all frequencies and theresistance is increased at frequencies below 100 MHz.In actual operation, such a thickness is not practical due to its adverse effect on therise—time of the magnetic field. The thickness of the shielding material must be lessthan the skin depth of the high—frequency components of the main current pulse thatdrives the magnet, in order not to increase the rise—time of the magnet drastically. Dueto the cut—off frequency of the magnet, the high—frequency components of the currentpulse are restricted to below 30 MHz. For aluminum, the skin depth (15 pm) at 30MHz is much less than the 1.5 mm aluminum sheets used. The improvement in thelongitudinal impedance above about 100 MHz shown in Figure 28 is therefore an upperlimit. A feasible solution would be to insert a ceramic chamber in the magnet aperture,with its inner surface lined with a metallic film, thin relative to the skin depth (15 pm)[13].55layer50C40 —a)UC0-rf) 30 —cr)a)2200 1 ——10Shield— — — No Shieldp/1///040^60^80^100^120^140^160^180^200Frequency (MHz)120 —Ccl.) 100 —C01.50 80—a)75 60 —c*-640 —00.120 —Shield--- No ShieldIayerr4vi1400 I^I^I^I^i^140^60^80 100 120 140 160 180^200Frequency (MHz)Figure 28: Effect of shielding the ferrite of the prototype kicker magnet.56C1Open—circuit(*-No^LiFeed cable (193 ns) Saturating inductor Kicker magnet Short—circuitFigure 29: Schematic diagram of magnet system with saturating inductor.4.4 Effect of a Speed—up Network and a Saturating InductorIt has been proposed [5] that a saturating inductor be connected in series betweenthe input of the magnet and the feed cable (Figure 29) to absorb small displacementcurrents before the main power pulse so as to improve the rise—time of the magneticfield. In addition, travelling waves would be reflected by such an inductor because ithas a high impedance for small current magnitudes. As a result, the length of themagnet system is reduced and so there are fewer resonances.Figure 30 shows the longitudinal impedance of the prototype magnet in the short—circuit configuration with and without the ferrite saturating inductor. An equivalentcircuit diagram for the magnet system is shown (Figure 29). The saturating inductorhas a very high impedance for small currents below saturation. As a result, it effectivelyterminated the feed cable at the magnet input as an open—circuit. Consequently theeffective length of the magnet system did not include the length of the feed cable whenthe saturating inductor was connected. For a shorter line, the number of resonances ina given frequency interval is reduced. Therefore, the saturating inductor has the effectof reducing the number of resonances.57C 40 —a)0030 —• —cna)cr220 —on0 10 ——J1 0WithoutWith Saturating Inductor —20-0Ca)0010 —1770—=C:710—10 —30 —5020^30Frequency (MHz)40^50WithoutWith Saturating Inductorsat r40—200I^I^I10 20 30Frequency (MHz)40^50Figure 30: Effect of saturating inductor.58A speed—up network was connected at the input of the prototype magnet to improvethe performance of the magnet system [4, 5]. It consisted of a capacitor and resistorin series. Figure 33 shows the effect of the speed—up network, with different values ofcapacitance and resistance, on the longitudinal impedance of the prototype magnet,in the matched—termination configuration (Figure 31). Below the cut—off frequency,the resistor of the network slightly modifies the impedance. Most importantly, thespeed—up network did not give rise to any resonances if the resistor of the networkwas present. Hence connecting the speed—up network did not adversely affect thelongitudinal impedance.Figure 34 shows the damping effect of a 30 SI network resistor on the resonancesof the prototype magnet in the short—circuit configuration (Figure 32). The networkresistor damped the last two resonances but did not influence the first (5 MHz) reso-nance. At low frequencies, the impedance of the capacitor is high, thus preventing theresistor from absorbing energy and thereby providing some additional damping for thefirst resonance.59MatchedresistorSpeed—upnetworkThyratronswitchMatched termination Configuration^blockFeed cableKicker magnet Output cable Figure 31: Schematic diagram of magnet system with speed-up network (matched-termination configuration.Short—Circuit ConfigurationblockconFeed cableKicker magnet■., 0 Short—CircuitThyratronswitchSpeed—upnetworkFigure 32: Schematic diagram of magnet system with speed-up network (Short-circuitconfiguration).60pora3530 -25 -c0• D-▪ 20 -cn15 -c:17)10 -En-0__I 5-Speed-up Network Parameters^ Not Connected— — - (140pF,30C2)- - - (140pF,00)...■■•••••a 30 ---...a)C0-r.5 20-vT;) 10. _0-1010^20^30Frequency (MHz)40^5040Speed-up Network Paramenters^ Not Connected(140pF,300)- - - - (140pF, on)I^I0^10^20 30Frequency (MHz)40^50parerFigure 33: Effect of speed-up network (matched-termination configuration).6145 -30 -a)02 15 -0(1)0--•S -15 -o-4-,.51-30 -0-J- 45 -Not connected- - - - 140pF, 30 060shortparaC:"50 —a)0c 40 -0-U)•cric2 30 -a5 20 -0__I 10 -I^ I13 25^38^50Frequency (MHz)60- 600shortpararNot connected- - - 140pF, 30 0^-I^I^I^I10 20 30 40^50Frequency (MHz)Figure 34: Effect of speed-up network (short-circuit configuration).624.5 Gaps Between the Beam Pipe and Kicker MagnetsIn the proposed KAON Factory, the design of the kicker magnet system follows thatof CERN where there is a gap between the kicker magnet, which is in a vacuum tank,and the beam pipe which is connected to the vacuum tank. This is the easiest designchoice because there is no direct coupling between the beam pipe and the kicker magnet.Such a gap can contribute additional strong resonances compared to the longitudinalimpedance of the magnet system.See Figure 11 for the location of the gaps. The gaps, approximately 1 cm wide, wereat both ends of the magnet aperture facing the beam pipe. The gaps were bridged byrf finger stock when the longitudinal impedance of the kicker magnet was measured.Finger stock is usually used for electrical connections in rf devices and consists of springloaded wires for making pressure contact.The resonances due to the gap were measured when the rf finger stock was removedfrom the gaps. Figures 35 and 36 show the gap resonances for the case of the matched—termination configuration. The longitudinal impedances of the gaps are of the localizedtype because the gaps are discrete sources. The measured peak values of the resonanceswere approximately 100-200 SI, which are large compared to the longitudinal impedanceof the magnet system. In general, these gaps should be avoided because they give riseto strong resonances.63air80C 60—a)Ca(,) 40 —•F)a)tYC 200 0-—20— Gap— — - No Gap10^20^30Frequency (MHz)40^50140I^I^I10 20 30Frequency (MHz)—2040^50airrFigure 35: Resonances due to gaps (0.3-50 MHz).64200 airl^ Gap- - - No Gap0. _cr) 50 -c0040raw"'" — ••• .....^•-•I^I^I^I^I^—1^I60 80 100^120^140^160 180Frequency (MHz)200airlr150C: 100a)500a)tY0-*-6folo -50 —J.**-100 I^I^I40^60 80 100^120Frequency (MHz)140^160^180^200Figure 36: Resonances due to gaps (45-200 MHz).654.6 Kicker Magnet Contribution to the Total LongitudinalImpedanceThe collective instabilities of a beam in a synchrotron ring are determined by the totallongitudinal and transverse impedance of all the components which make up the ring.The total impedance can be broken down into major contributions from rf cavities,resistive beam pipe, beam instrumentation devices, and kicker magnets. In all ringsof the KAON Factory, the largest contribution to the total impedance is from the rfcavities. The longitudinal impedance of a cavity consists of many sharp peaks at itsresonant frequencies. An upper limit of 1 MI has been set for each peak[14].For the purpose of instability calculations in the KAON Accelerator Design Re-port [14], it was assumed that a kicker magnet module has maximum real longitudinalimpedance equal to half its characteristic impedance. Thus a maximum real longitu-dinal impedance of 12.5 12 was assumed for each 25 S) magnet module.The total impedance of the kicker magnets in a ring is the sum of the impedancesof the magnet modules that make up the kicker magnets. Each ring has a few kickermagnets and several magnet modules make up a kicker magnet. The number of magnetmodules required in a ring depends on the total kick angle and the beam momentum.Because the number of cells that make up a magnet module varies from ring to ring, thetotal magnetic length of the magnet modules in a ring is used instead as the measure ofkick strength. The magnetic length of the prototype kicker magnet is 34.5 cm. Becausethe maximum longitudinal impedance of the prototype kicker magnet does not dependon other components that make up the magnet system, the total maximum longitudinalimpedance of the kicker magnets in a ring can be scaled by the total magnetic length.The maximum real longitudinal impedance of the prototype kicker magnet wasmeasured to be approximately 32 f up to 200 MHz in both the short-circuit and thematched-termination configurations (Section 4.3). The total magnetic length in theBooster ring is 4.58 m and so the total longitudinal impedance is 425 11, by magneticlength scaling. In the Driver ring, the total magnetic length is 12.77 m and so the totallongitudinal impedance is 1184 a The total kicker magnet impedance in each ring is66small compared with the rf cavity impedance, 12 kfi in the Booster ring and 18 kf/ inthe Driver ring, respectively. Due to their small contribution to the total longitudinalimpedance of each ring, the kicker magnets are not expected to be a significant sourceof instabilities.For short bunch (1 ns) operations of the rings, it is required that the total impedanceZ11 of a ring divided by the mode number n must be smaller than 2 u. The impedanceof the magnet does not satisfy such a low 141/n I nor do other components of the ringswithout major efforts. Nothing less than a redesign of the magnet system can such alow impedance be obtained.675 ConclusionsThe longitudinal impedance of a prototype kicker magnet system for the KAON Fac-tory has been determined in both the short-circuit and the matched-termination con-figurations. The effects on the longitudinal impedance of a saturating inductor and aspeed-up network, which were installed to improve the kick performance of the sys-tem, have also been assessed. When a saturating inductor was installed, the numberof resonances in the impedance was reduced. The resistor of of a speed-up network,which was connected to the input of the magnet, damped the high-frequency reso-nances. Most importantly, the network did not give rise to any additional resonancesprovided the network resistor was present. In the frequency range from 30 to 200MHz, the longitudinal impedance did not depend much on components external to themagnet (such as the saturating inductor, the speed-up network, and the feed cables),because of the onset of strong attenuation of travelling waves by the magnet. Gapsbetween the kicker magnet and the beam pipe contributed very strong resonances tothe longitudinal impedance of the magnet system. Therefore gaps must be eliminated.The maximum real longitudinal impedance of the prototype kicker magnet wasmeasured to be approximately 32 C2 up to 200 MHz in both the short-circuit andthe matched-termination configurations. The total real longitudinal impedance of thekicker magnets in the Booster ring would then be 425 SI and in the Driver ring 1184 Si,respectively. The total kicker magnet impedance in each ring is small compared withthe total rf cavity impedance with typical values of 10 kf2. Hence the contribution ofkicker magnets to the total longitudinal impedance of each ring is negligible and anylongitudinal instabilities due to kicker magnets can be damped with existing dampingsystems for parasitic resonances of rf cavities. However, the magnet impedance doesnot satisfy the strigent requirements for short bunch (1 ns) operations of the rings.68References[1] E. Keil and W. Schnell, CERN Report TH-RF/69-48 (1969)[2] G. D. Wait, M. J. Barnes, D. Bishop, G. Waters, "Interleaved Wide and Nar-row Pulses for the KAON Factory 1 MHz Beam Chopper", Proc. 15 th ParticleAccelerator Conference, Washington, D.C., May 1993, (IEEE, in press).[3] M. J. Barnes, G. D. Wait, "Kickers for the KAON Factory", Proc. 15 th Inter-national Conference of High Energy Accelerators , Hamburg, July 1992, (Int. J.Mod. Phys. A (Proc. Suppl.) 2A, 194-196, 1993).[4] M. J. Barnes, G. D. Wait. "Optimization of Speed-Up Network Component Valuesfor the 30 SI Resistively Terminated Prototype Kicker Magnet", Proc. 15 th ParticleAccelerator Conference, Washington, D.C., May 1993, (IEEE, in press).[5] M. J. Barnes, G. D. Wait, "Improving the Performance of Kicker Magnet Sys-tems", Proc. 15 th International Conference of High Energy Accelerators , Ham-burg, July 1992, (Int. J. Mod. Phys. A (Proc. Suppl.) 2A, 191-193, 1993).[6] M. J. Barnes and G. D. Wait, "A Mathematical Model of a Three-Gap ThyratronSimulating Turn-on" Proc. 9 th IEEE Pulsed Power Conference, Albuquerque, June1993, (IEEE, in press).[7] L.S. Walling et al, "Transmission Line Impedance Measurements for an AdvancedHadron Facility", MS-H817, (LANL, 1988).[8] M. Sands, J. Rees, " A Bench Measurement of the Energy Loss of a Stored Beamto a Cavity", PEP-95, (SLAC, 1974).[9] H. Hahn, F. Pedersen, "On Coaxial Wire Measurement of the Longitudinal Cou-pling Impedance", BNL-50870, (Brookhaven, 1974).[10] S.A. Heifets and S.A. Kheifets, "Coupling Impedance in Modern Accelerators",PUB-5297, (SLAC, 1990).69[11] K-.Y. Ng, "Fields, Impedances and Structures", FN-443, (Fermi Lab., 1987)[12] R. Speciale, "A Generalization of the TSD Network Analyzer Calibration Proce-dure, Covering n-Port Scattering Parameters, Affected by Leakage Errors," IEEETrans. Microwave Theory and Techniques, MTT-25, 12, 1101-1115, December1977.[13] S. Kurennoy, "Using a Ceramic Chamber in Kicker Magnets," SSCL-Preprint-311, (SSC, 1993).[14] TRIUMF, "KAON Factory Study: Accelerator Design Report", (TRIUMF, 1990).[15] F.F. Kuo, "Network Analysis and Synthesis," John Wiley & Sons, New York,1962.[16] M.J. Barnes, G.D. Wait, "Suppression of the Effect of Thyratron DisplacementCurrent Upon the Field in the 30 CI Prototype Kicker Magnet," TRI-DN-91-K170, (TRIUMF, 1991).[17] P.E. Faugeras, "The Full Aperture Kicker Magnet for the CPS," CERN/MPS/SR71-6, (CERN, 1972).[18] D.A. Edwards and M.J. Syphers, "An Introduction to the Physics of High EnergyAccelerators," John Wiley Si Sons, New York, 1993.[19] HP 8753B Network Analyzer Reference Manual.[20] P.B. Wilson, J.B. Styles and K.L.F. Bane, "Comparison of Measured and Com-puted Loss to Parasitic Modes in Cylindrical Cavities With Beams Ports,"SLAC-PUB-1908, (SLAC, 1977).[21] Y. Yin et al, "Measurement of Longitudinal Impedance for a KAON Factory TestPipe Model with TSD-calibration method," Proc. 13 th Particle Accelerator Conf.,San Francisco 1991, 1722-1724, (IEEE, 1991).70Appendix AEffects of a Central Conductor and Transit TimeA central conductor perturbs the wakefield of an accelerator component. The largerthe radius of the conductor, the more it modifies the wakefield. However, if the centralconductor is sufficiently thin, then the wakefield is only slightly modified and more ac-curate measurements can be made. For mechanical support and impedance matching,the radius cannot be as small as one wishes. A chosen radius size is a compromise be-tween practicality and measurement accuracy. In this appendix, we make an estimateof the error caused by using a central conductor [8]. We also show by an example thatthe transit time of a bunch is small compared with the decay time of normal—modeoscillations.The energy loss factor K can be obtained from the reference pulse 4.(t) and themodified pulse 1m (t) of the test component by the equation2ZoK . —q-2 i IdIr (t) dt + 8K,where Id = Im — Ir , Zo is the characteristic impedance of the line, and q the totalcharge in the pulse. The correction term 6K acknowledges the effect of the centralconductor. A first—order—of—magnitude estimate for the size of 6K is given bySK , Idwhere Id and L. are the maximum magnitudes of the currents. For the case of zeroenergy loss 6K -4 0 since Id -4 0. Let us assume that 6K is small compared to K fora case where the conductor is sufficiently thin, and that it is approximately equal tothe measurement uncertainty AK, that is,SK , AKK -- K • (34)(32)(33)K  1^,. '71For this conductor, where the effect of size is minimal, it follows from Equation 33thatId ,.., AidIr ^IrAid<^, ,Idwhere /a < Ir and AL is the measurement uncertainty. The above inequality showsthat, for a sufficiently thin conductor, the effect of the conductor size is not as importantas the measurement uncertainty.For short pulses (10 ns), the integral of Equation 32 can be approximately evaluatedusing rectangular pulses. The reference current Ir (t) can be approximated by a rect-angular pulse of magnitude Ir and duration 7 and, likewise, Ia (t) by /a and T. Withthis approximation the integral is roughly equal to /J a r, and so Equation 32 can berearranged asId T KI r^2Z0 'where q = IrT. K and 7 are constant, so the only free parameter is Z0 , which isdetermined by the size of the central conductor. This equation justifies the statement,made in Section 2.1, that the effect of the central conductor can be made small bychoosing a central conductor of a sufficiently small radius.Now, we have all the equations to estimate the measurement error due to a centralconductor. For a typical bunch length in the KAON Factory, we take 7.10 ns. Weestimate the loss factor K of the prototype kicker magnet to be 100 times less thanthat of a typical cavity. This estimate is very conservative since cavities are known tohave very large loss factors compared with those of kicker magnets. We take a cavity[20] with a loss factor of 4.8 x 10 10 V/C and so the loss factor K of the kicker magnetis =4.8 x 108 V/C. We used two conductor sizes (41 mm and 3 mm). The smallconductor size gives the coaxial line a characteristic impedance Z 0 of 180 5/ and thebigger one 50 52.(35)(36)(37)72Equation 37 givesId ,,, 4.8 SI/, -- 2 x 50 St ^0.05,Id^4.8 ft^ '-- 0.01 ./,. N2 x 180 StWe assume that the uncertainty in the measured data is at most 5%. According tothe arguments that lead to Equation 36, the error due to the presence of the centralconductor of the 50 It line is comparable to the measurement uncertainty. For the 180St line, this kind of error is negligible compared with the measurement uncertainty.To estimate the effect of transit time, we can roughly calculate the time it takesa pulse to pass through the prototype kicker magnet and the decay time constant ofthe wakefield. For the prototype kicker magnet, the transit time is approximately thespeed of light divided by its length, which is about 1.5 ns. The decay time constantis given by 2Q/wo , where Q is the quality factor and coo the resonant frequency. Forthe kicker magnet, there are no resonances above 30 MHz. We assume a poor qualityfactor of 1. The decay time constant is about 11 ns, which is much larger than thetransit time.(38)(39)73LC)-One Cellcell^C15-irnSM^Figure 37: An improved model of the prototype kicker magnetAppendix BThe Cut-off Frequency of the Prototype Kicker MagnetThe low-pass behavior of a kicker magnet is characterized by the cut-off frequency L.The cut-off frequency of a low-pass network is the frequency at the half power pointon the roll-off curve of frequency versus power transmitted. Because of the mutualinductance L, between adjacent cells, the cut-off frequency of the prototype kickermagnet is further reduced. The cut-off frequency is given approximately by [16, 17]1 fc^^ (40)71- \AL + 4/,,,)Cwhere L and C are the inductance and capacitance of each cell respectively. For theprototype kicker magnet, L is 139 nH for an end cell , L, is 119 nH, and C is 74 pF.The cut-off frequency given by Equation 40 is approximately 47 MHz. The measured74cut—off frequency of the prototype magnet at the 3 dB point is 30 MHz ( Figure 38 ).The discrepancy between the calculated and measured values is due to the fact thatEquation 40 does not take into account the shunt capacitance between adjacent cellsor resistive losses in the cells.75Figure 38: Magnitude of transmission coefficient of the prototype kicker magnet as alow pass filter.76Ref .=R01 Ref.=R02al =a2 —bi =1b2 = 2,,(41)(42)(43)(44)scatterFigure 39: Parameters defining the scattering matrix.Appendix CThe Scattering Matrix DefinitionsThe scattering matrix is a convenient and standard way to describe the transmission ofa two—port device in a transmission line environment [15]. The incident and reflectedwave amplitudes a and b are defined by21 (;,roi +21 (,_; _2,2 +21 (14 (A%Iii—ol 1.1) ,j-101 12) ,Xi li)1/Roi 12)77(47)2V2 VRol—a2=0 = 717v gl^1 1,02b2S21 —a l RI =R01 ,R2 =R02(49)2VI /R02ai=0 = 7—v g2 1101S12 = —a2R I=R0 1,R2=R02where Rol and Ro2 are arbitrary, positive, real reference impedances. The scatteringparameters Si; are defined bybl ^{ Sll S12^{ al•b2^S21 S22^a2(45)The input reflection coefficient Si' is defined bybl—a lZ1 — Rol a2=0 — Z1 + RO1 R2=R02(46)where Zl is the input impedance with the output port terminated in the referenceimpedance R02.The forward transmission coefficient S21 is given bywhere R1 = R92 gives a simple relationship between Vgi and a l , and R2 = R02 impliesa2 = O.By reversing input and output ports we get the output reflection coefficientZ2 RO2 IS22 ^b2l ai=° =a2^Z2 + RO2 Ri=Roi(48)and the reverse transmission coefficientIf the two-port is a lossless transmission line it is convenient to choose the referenceimpedance equal to the characteristic impedance: R ol = R02 = Z0 . The wave param-eters can then be related to incident and reflected voltage. The scattering matrix isthenEs) ={0 e-jkle—iki 0(50)where k is the propagation constant and / the line length.78Appendix DInput Impedance of the Prototype Kicker MagnetFigure 40 shows the input impedance of the prototype kicker magnet when the outputof the magnet is short—circuited. Even though the characteristic impedance of theprototype kicker magnet is 30 ft, the input impedance of the short—circuited magnetdepends on the frequency in a predictable way until the cut—off frequency (30 MHz).Figure 41 shows the input impedance of the kicker magnet with the output ter-minated with a resistor whose impedance matches the characteristic impedance of themagnet (30 9). The input impedance of the terminated magnet would be a constant 309 if the magnet behaved like an ideal transmission line. In fact the magnet behaves likea low—pass filter with a cut—off frequency of 30 MHz. As shown in Figure 41, the inputimpedance is approximately 30 11 below the cut—off frequency and drops considerablyabove the cut—off frequency.791Figure 40: Magnitude of input impedance of the short—circuited prototype kicker mag-net.80Figure 41: Magnitude of input impedance of the terminated prototype kicker magnet81

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