SERVO CONTROL OF RF CAVITIES UNDER BEAM LOADING

SERVO CONTROL OF RF CAVITIES UNDER BEAM LOADING Alexande Gamp DESY, Hambug, Gemany Abstact I begin by giving a desciption of the RF geneato cavity beam coupled system in tems of basic quantities. Taking beam loading and cavity detuning into account, expessions fo the cavity impedance as seen by the geneato and as seen by the beam ae deived. Subsequently methods of beam-loading compensation by cavity detuning, RF feedback and feedfowad ae descibed. Finally, a dedicated phase loop fo damping synchoton oscillations is discussed. 1. INTRODUCTION In moden paticle acceleatos RF voltages with an extemely lage amplitude and fequency ange, fom a few hunded volts to hundeds of megavolts and fom seveal kilohetz to many gigahetz, ae equied fo paticle acceleation and stoage. The RF powe needed to satisfy these demands can be geneated, fo example, by tiodes, tetodes, o klystons. The Continuous Wave (CW) output powe available fom some tetodes used at DESY is 60 kw at 08 MHz and 500 800 kw fo the 500 MHz klystons. Such RF powe geneatos geneally delive RF voltages of only a few kilovolts because thei souce impedance is small compaed with the cavity shunt impedance. Fo the TeV Enegy Supeconducting Linea Acceleato (TESLA) poject a pototype pulsed L-band Multibeamklyston has deliveed up to 10 MW peak powe fo 0.5 ms long pulses, and at 3 GHz 150 MW wee achieved fo pulses of 3 µs in length. Typically, a tetode has its highest efficiency fo a load esistance of less than 1 k,wheeas the cavity shunt impedance usually is of the ode of seveal megaohms. This is the eal impedance, which the cavity epesents to a geneato at the esonant fequency. It must not be confused with ohmic esistances. Optimum fixed impedance matching between geneato and cavity can be easily achieved with a coupling loop in the cavity. Thee is, howeve, the complication that the tansfomed cavity impedance as seen by the geneato depends also on the synchonous phase angle and the beam cuent and is theefoe not constant (as we shall show quantitatively). The beam cuent induces a voltage in the cavity that may become even lage than the one induced by the geneato. Owing to the vecto addition of these two voltages the geneato now sees a cavity that appeas to be detuned and unmatched except fo the paticula value of beam cuent fo which the coupling has been optimized. The eflected powe occuing at all othe beam cuents has to be handled. In addition, the beam-induced cavity voltage may cause single- o multi-bunch instabilities, since any bunch in the machine may see an impotant faction of the cavity voltage induced by itself o fom pevious bunches. This voltage is given by the poduct of beam cuent and cavity impedance as seen by the beam. Minimizing this latte quantity is theefoe essential. It is also called beamloading compensation, and some sevo contol mechanisms, which can be used to achieve this goal, will be discussed in this lectue. 73

. THE COUPLING BETWEEN THE RF GENERATOR, THE CAVITY, AND THE BEAM Fo fequencies in the neighbouhood of the fundamental esonance, an RF cavity can be descibed [1] by an equivalent cicuit consisting of an inductance L, a capacito C, and a shunt impedance R S, as shown in Fig. 1. In pactice, L is made up by the cavity walls, wheeas the coupling loop L 1 is usually small compaed with the cavity dimensions. In this example a tiode with maximum efficiency fo a eal load impedance R A has been taken as an RF powe geneato. Fo simplicity we conside a shot and lossless tansmission line between the geneato and L 1. Then thee is optimum coupling between the geneato and the empty (i.e. without beam) cavity fo N = R S /R A = L /L 1, (1) whee R A equals the dynamic souce impedance R 1. The tem N is the tansfomation o step-up atio. Fig. 1: Equivalent cicuit of a esonant cavity nea its fundamental esonance. In pactice, the inductance L is made up by the cavity walls, wheeas L 1 is usually a small coupling loop. Since, in geneal, thee may be powe tansmitted fom the geneato to the cavity and also, in the case of impefect matching, vice vesa, the voltage U 1 is expessed as the sum of two voltages U 1 = U fowad + U eflected, () wheeas the coesponding cuents flow in the opposite diections, hence I 1 = I fowad I eflected. (3) The minus sign in Eq. (3) indicates the counteflowing cuents, while voltages of fowad and backwad waves just add up. So, in the simplest case, whee the beam cuent I B = 0 and whee the geneato fequency f GEN = f CAV, thee is no eflected powe fom the cavity to the geneato, and U 1 and I 1 ae identical to the geneato voltage and cuent, espectively. One has = NU 1. (4) 74

Now we can deive an expession fo the complex cavity voltage as a function of geneato and beam cuent and of the cavity and geneato fequency. Accoding to Fig. 1 the cavity voltage U CAV can be witten as I = L + I 1 N (5) I = I B + R S +C. (6) All voltages and cuents have the time dependence ˆ U = U e it. (7) I B = I B ( ) is the hamonic content at the fequency of the total beam cuent. Thoughout this lectue we conside only a bunched beam with a small bunch spacing compaed to the cavity filling time. In this case I B () is quasi sinusoidal. We also estict the discussion to the inteaction of the beam with the fundamental cavity esonance. Dedicated damping antennas built into the cavity can minimize the inteaction with highe-ode cavity modes. Inseting Eq. (6) in Eq. (5) and using f CAV = CAV = 1 L C (8) one finds CAV = 1 1 I 1 I B 1 C N R S. (9) We define = 1 = CAV CR S Q (10) whee the quality facto of the cavity can be expessed as times the atio of total electomagnetic enegy stoed in the cavity to the enegy loss pe cycle. Hee we would like to mention that the atio R S Q = L C (11) is a chaacteistic quantity of a cavity depending only on its geomety. We can ewite Eq. (9) as + U 1 CAV + CAV = R S I 1 I B N. (1) This equation descibes a esonant cicuit excited by the cuent I = ( I 1 ( N I B )). The minus sign occus because the geneato-induced cavity voltage has opposite sign to the beam-induced voltage, which would deceleate the beam. It can be shown that the beam actually sees only 50% of its own induced voltage. This is called the fundamental theoem of beam loading [, 3]. 75

In ode to find the cavity impedance as seen by the beam we make use of Eqs. (), (3), and (4) to expess the geneato cuent tem of Eq. (1) in the fom 1 I 1 = 1 U fowad U 1 N NR A [ ] = 1 N I fowad NR A. (13) This leads to a modification of the damping tem in Eq. (1) + ( 1 + ) U CAV + CAV = L R SL I f I B N. (14) With the coupling atio we can intoduce the loaded damping tem = R S /N ( R I ) (15) L = ( 1+ ) (16) and consequently, in accodance with Eq. (10), the loaded cavity Q and loaded shunt impedance ae Q L = Q/1+ ( ) and R SL = R S /( 1+ ). (17) In the case of pefect matching in the absence of beam, i.e. = 1, the damping tem simply doubles and Q and R S take half thei oiginal values. This is because the beam would see the cavity shunt impedance R S in paallel o loaded with the tansfomed geneato impedance N R I = R S. Theefoe we find in Eq. (14) that the tansfomed geneato cuent I G = I f /N (18) gives ise to twice as much cavity voltage as a simila beam cuent would do. Hee and in Eq. (15) we assume that the tansfomed dynamic souce impedance N R I is identical to the geneato impedance seen by the cavity. This is stictly tue only if a ciculato is placed between the RF powe geneato and the cavity. Without a ciculato it may be appoximately tue if the powe souce is a tiode. Owing to its almost constant anode-voltage-to-cuent chaacteistic the impedance of a tetode as seen fom the cavity is, howeve, much bigge than the coesponding R I and theefoe R SL R S hee, whee a shot tansmission line (o of length n/, n intege) is consideed. Following Ref. [4] we wite the solution of Eq. (14) in the Fouie Laplace epesentation ˆ = CAV i ˆ L R SL I G ˆ I B +i. (19) L Fo CAV this can be appoximated by ˆ R ˆ SL I G ˆ I B 1 +iq L CAV, (0) whee = CAV +. Fo a esonant cavity the beam-induced voltage U B, o the beam loading, is thus given by the poduct of loaded shunt impedance and beam cuent: 76

The ideal beam loading compensation would, theefoe, minimize geneato powe necessay to maintain the cavity voltage. U B = R SL I B (1) R SL without inceasing the Having just discussed the impedance that the combined system geneato and cavity epesents to the beam, we would like to discuss the impedance Z, o athe admittance Y = 1/Z, which the combined cavity and beam system epesents to the geneato. Fom Eqs. (1), (5), and (6) one sees [5] that I Y = 1 = N I + B N + N 1 U 1 R S il CAV. () which educes to Y = 1/R A fo a tuned cavity without beam cuent in the case of = 1. As we ae now going to show, a non-vanishing eal pat of the quotient I B will necessitate a change in to maintain optimum matching, wheeas the imaginay pat can be compensated by detuning the cavity. In ode to wok out Re and Im( I B ), we define the angle s as the phase angle between the synchonous paticle and the zeo cossing of the RF cavity voltage. The acceleating voltage is theefoe given by U ACC = U CAV sin s (3) and the nomalized cavity voltage and beam cuent ae elated by I B U = CAV e i s. (4) I B Consequently and Re I B Im I B = = I B sin s (5) I B cos s. (6) The eal pat of the admittance seen by the geneato then becomes Re(Y) = N 1+ R S I B sin s R S U. (7) CAV We see that the tem in the backet descibes the change of admittance caused by the beam. In ode to maintain optimum coupling the coupling atio must now take the value = 1+ R S I B sin s U. (8) CAV This esult tells us that the change in the eal pat of the admittance is popotional to the atio of RF powe deliveed to the beam to RF powe dissipated in the cavity walls. Fo cicula electon 77

machines, whee the consideable amount of enegy lost by synchoton adiation has to be compensated continuously by RF powe, values of s 30 and 1. ae typical fo high beam cuent and nomally conducting cavities. A typical set of paametes fo this case would be R S = 6M, U CAV = 1 MV and I B = 30 ma. This implies, of couse, that fo a that has been optimized fo the maximum beam cuent, thee will be eflected geneato powe fo lowe beam intensities. If the powe souce is a klyston, this can be handled by inseting a ciculato in the path between geneato and cavity o, in the case of a tube, by a sufficiently high plate dissipation powe capability. Fom the imaginay pat of Eq. () and fom Eq. (6) we find that the appaent cavity detuning caused by the beam cuent can be compensated by a eal cavity detuning (fo example, by means of a mechanical plunge cavity tune) of the amount CAV = 1 R S I B QU cos s. (9) CAV Expanding the squae oot to fist ode we find a cavity detuning angle tan R SL I B cos s Q L. (30) CAV This is essentially the atio between beam-induced and total cavity voltage. In ode to calculate the maximum amount of eflected powe seen by the geneato as a consequence of beam loading we conside, fo = 1, a tuned cavity, i.e. = CAV. Then, with Eqs. (5) and (6), Eq. () eads Y = 1 1+ R S I B sin s +i R S I B cos s R A. (31) Solving fo U efl. by means of Eqs. () and (3) the eflected powe P efl. = ˆ U efl. /R I becomes ˆ P efl. = R S I B /8. (3) This coesponds to half of the powe given by the beam to the coupled system cavity and geneato. The second half of this powe is dissipated in the cavity walls. All we found is that two equal esistos in paallel dissipate equal amounts of powe. As pointed out above, this is stictly tue only if a ciculato is placed between the RF powe souce and the cavity. Nevetheless, the amount of eflected powe can be quite impessive. Fo an aveage DC beam cuent of, say 0.1 A, the hamonic ˆ cuent I B () may become up to twice as lage. Then, taking R S = 8 M, fo example, we find 40 kw of eflected powe have to be dissipated. Fo a cavity whee only the eactive pat of the beam loading has been compensated by detuning accoding to Eq. (30), but = 1, the eflected powe is given by ˆ P efl. = R S I B sin s /8. (33) Summaizing the esults of this section we state that the beam sees the cavity shunt impedance in paallel with the tansfomed geneato impedance. The esulting loaded impedance is educed by the facto 1/(1 + ). The optimum coupling atio between geneato and cavity depends on the amount of enegy taken by the beam out of the RF field. The coupling is usually fixed and optimized fo maximum beam cuent. The amount of cavity detuning necessay fo optimum matching, on the othe hand, depends on the atio of beam-induced to total cavity voltage. 78

3. BEAM-LOADING COMPENSATION BY DETUNING As we have shown in the pevious Section, stationay beam loading can be entiely compensated by detuning the cavity, povided that the synchonous phase angle is small o zeo. This is usually the case in poton synchotons duing stoage, whee the enegy loss due to the emission of synchoton adiation is negligible. Hee, the RF voltage is needed only to keep the bunch length shot. Enegy amping also takes place at vey small s. In the following we estict ouselves, fo simplicity, to hadon machines. Consequently = 1, s 0, and the geneato- and beam-induced voltages ae in quadatue. Thee ae, howeve, also limitations to detuning as the only means of beam-loading compensation. One is known as Robinson s stability citeion [6]. If the amount of detuning calculated by Eq. (30) becomes compaable to the evolution fequency of the paticles in a synchoton, the beam will become unstable. Anothe one is the finite time of, say, a second, which is needed fo the tune to eact. Actually, the time scale of the cavity voltage tansients, which may cause beam instabilities, is much shote. The cavity filling time CAV is given by CAV = Q L / CAV. (34) The cavity voltage ise afte injection of a bunched beam with a cuent I B ( CAV ) can be appoximated by U B R SL I B ( 1 e t/ ). (35) This voltage will add to the cavity voltage poduced by the geneato, and afte a time t 3 the total cavity voltage becomes R SL I g + I B with a phase shift given by Eq. (30). Since typical values of ae below 100 µs and theefoe much smalle than the poton synchoton fequency in a stoage ing (T S is usually seveal ms), these tansients will, in geneal, excite synchoton oscillations of the beam with the consequence of emittance blow-up and paticle loss. Additional compensation of tansient beam loading is theefoe necessay. This will be discussed in the following paagaphs. In Fig. a diagam of a tune egulation cicuit is shown. The phase detecto measues the elative phase between geneato cuent and cavity voltage which depends, accoding to Eq. (0), on the fequency, by which the cavity is detuned. The phase detecto output signal acts on a moto that dives a plunge tune into the cavity volume until thee is esonance. An altenative tune could be a esonant cicuit loaded with feites. The magnetic pemeability µ of the feites and hence the esonance fequency of the cicuit can be contolled by a magnetic field. This latte method is especially useful when a lage tuning ange in combination with a low cavity Q is equied. If pope tune action is necessay in a lage dynamic ange of cavity voltages, limites with a minimum phase shift pe db compession have to be installed at the phase detecto input. Since this phase shift is deceasing with fequency, all signals should be mixed down to a sufficiently low intemediate fequency. The signal popotional to the geneato cuent I fow. can be obtained fom a diectional couple. In case the RF amplifie is so closely coupled to the cavity that no diectional couple can be installed, the elative phase between RF amplifie input and output signal can also be used to deive a tune signal [7]. (36) 79

4. REDUCTION OF TRANSIENT BEAM LOADING BY FAST FEEDBACK The pinciple of a fast feedback cicuit is illustated in Fig.. A small faction of the cavity RF signal is fed back to the RF peamplifie input and combined with the geneato signal. The total delay in the feedback path is such that both signals have opposite phase at the cavity esonance fequency CAV. Fo othe fequencies thee is a phase shift =. (37) Theefoe the voltage at the amplifie input is now given by U in = U in e i U CAV. (38) With the voltage gain K of the amplifie we can ewite Eq. (0) and obtain fo the cavity voltage with feedback K [ U in e i U CAV ]U B 1+ iq L (39) CAV o K U in U B 1 +iq L CAV + e i K. (40) Fig. : Schematic of sevo loops fo phase and amplitude contol of the HERA 08 MHz poton RF system Fo = 0 and A F»1 this educes to U in U B K. (41) 80

The open loop feedback gain A F is defined as A F =K. (4) One sees that thee is a eduction of the beam-induced cavity voltage by the facto 1/A F due to the feedback. This is equivalent to a simila eduction of the cavity shunt impedance as seen by the beam. Z L R SL 1 +iq L CAV R SL 1+ iq L CAV + A F e i. (43) The pice fo this fast eduction of beam loading is the additional amount of geneato cuent I B N that is needed almost to compensate the beam cuent in the cavity. In tems of additional tansmitte powe P this eads ˆ P = R S I B /8. (44) It is the powe aleady calculated by Eq. (3). As thee is no change in cavity voltage due to P this powe will be eflected back to the geneato, which has to have a sufficiently lage plate dissipation powe capability. Othewise a ciculato is needed. This citical situation of additional RF powe consumption and eflection lasts, howeve, only until the tune has eacted, and it may be minimized by pe-detuning. The geneato-induced voltage is, of couse, also educed by the amount 1/A F, but this can be easily compensated on the low powe level by inceasing U in by the facto 1/ as Eq. (41) suggests. The pactical implications of this will be illustated by the following example. Let the powe gain of the amplifie be 80 db. Fo a cavity powe of 50 kw an input powe P IN of 0.5 mw is thus equied. This coesponds to a voltage gain of 10 4 so, fo a design value of A F = 100, becomes 10. Hence the powe that is fed back to the amplifie input, is 5 W. In ode to maintain the same cavity voltage as without feedback, P IN has to be inceased fom 0.5 mw to 5.0005 W. This value can, of couse, be educed by deceasing. But then the amplifie gain has to be inceased to keep A F constant. This leads to powe levels in the 100 µw ange at the amplifie input. All this is still pactical, but some pecautions, such as extemely good shielding and suppession of geneato and cavity hamonics, have to be taken. The maximum feedback gain that can be obtained is limited by the afoementioned delay time of a signal popagating aound the loop. Accoding to Nyquist's citeion, the system will stat to oscillate if the phase shift between U in and U CAV exceeds 135. A cavity with high Q can poduce a ±90 phase shift even fo vey small. Theefoe, once the additional phase shift given by Eq. (37) has eached ± /4, the loop gain must have become 1, i.e. whee K A F ( max ) 1+ iq L max CAV 1 (45) max =± 4. (46) Hee we assume that all othe fequency-dependent phase shifts, like the ones poduced by the amplifies, can be neglected. Taking U B = 0 and inseting Eq. (45) we can solve Eq. (39) fo A F : A F = Q L 4 f CAV. (47) 81

This is the maximum possible feedback gain fo a given. A fast feedback loop of gain 100 has been ealized at the HERA 08 MHz poton RF system. With a loaded cavity Q L of 7000 the maximum toleable delay, including all amplifie stages and cables, is = 330 ns. Theefoe all RF amplifies have been installed vey close to the cavities in the HERA tunnel. In addition, thee ae independent slow phase and amplitude egulation units fo each cavity with still highe gain in the egion of the synchoton fequencies, i.e. below 300 Hz. Without fast feedback these units might become unstable at heavy beam loading [8, 9] since changes in cavity voltage and phase ae then coelated, as shown by Eqs. (30) and (36). The effect of a fast feedback loop is evealed in Fig. 3, whee the tansient behaviou of the imaginay (uppe cuve) and eal (middle cuve) pat of a HERA 08 MHz cavity voltage vecto is displayed. The lowe cuve is the signal of a beam cuent monito, which shows nicely the bunch stuctue of the beam and a µs gap between batches of 6 10 bunches each. A detailed desciption of this measuement and of the IQ detecto used is given in Ref. [10]. In this paticula case the uppe cuve is essentially equivalent to the phase change of the cavity voltage due to tansient beam loading, and the middle cuve coesponds to the change in amplitude. Fig. 3: Tansient behaviou of the cavity voltage unde the influence of fast feedback. This figue is taken fom Ref. [10]. The appaent time shift between the bunch signals and the cavity signals is due to the time of flight of the potons between the location of the cavity and the beam monito in HERA. The tansients esulting fom the fist two o thee bunches afte the gap cause step-like tansients, which accumulate without significant coection. Late the fast feedback delives a coection signal, which causes the subsequent tansients to look moe and moe saw-tooth-like. Fom this one can estimate the time delay in the feedback loop to be of the ode of 50 ns. Afte about 1 µs equilibium with the beam is eached. Similaly, one obseves in the left pat of the pictue that the feedback coection is still pesent fo 50 ns afte the last bunch befoe the gap has left the cavity. The equilibium without beam is also eached afte about 1 µs. Without fast feedback the time taken to each equilibium is about 100 times longe, as one would expect fo a feedback gain of 100. To summaize this Section we state that fast feedback educes the esonant cavity impedance as seen by an extenal obseve (usually the beam) by the facto 1/A F. It is impotant to ealize that any 8

noise oiginating fom souces othe than the geneato, especially amplitude and phase noise fom the amplifies, will be educed by the facto 1/A F because the cavity signal is diectly compaed to the geneato signal at the amplifie input stage. Cae has to be taken that no noise be ceated, by diode limites o othe non-linea elements, in the path whee the cavity signal is fed back to the amplifie input. This noise would be added to the cavity signal by the feedback cicuit. 5. FEEDBACK AND FEEDFORWARD APPLIED TO SUPERCONDUCTING CAVITIES So fa, we have only consideed nomally conducting cavities in a poton stoage ing, whee the potons aive in the cavities at the zeo cossing of the RF signal, i.e. at s = 0 o o a few degees. In the following I would like to pesent an example of the othe exteme: supeconducting cavities in a linea electon acceleato whee the electons coss the cavities nea the moment of maximum RF voltage, i.e. at s 90 o. (Note that fo linea collides a diffeent definition of s is usually used, namely s = 0 o when the paticle is on cest. In this aticle we do not adopt this definition.) A test facility fo TESLA is cuently being built at DESY. We efe to the special example of the TESLA Test Facility cavities, which ae 9-cell cavities made of pue niobium. The opeating fequency is 1.3 GHz. Fig. 4: Schematic of the low-level RF system fo contol of the TESLA Test Facility 1.3 GHz cavities. This figue is taken fom Ref. [11]. The unloaded Q-value of these cavities is in the ange 10 9 10 10, o even highe. Hence the bandwidth is only of the ode of 1 Hz, and also the shunt impedance of these cavities exceeds that of nomally conducting cavities by many odes of magnitude. Theefoe, we have a coupling facto 1000 in this case, which also eflects the fact that the atio of the powe taken away by the beam to the powe dissipated in the cavity walls is much lage fo supeconducting cavities than fo nomally conducting ones. Owing to the coupling, the nominal loaded Q-value is only 3 10 6, and the loaded cavity bandwidth seen by the beam is 433 Hz. Since in this case thee is a ciculato with a load to potect the klyston fom eflected powe, this loaded bandwidth is also seen by the RF geneato, which is a klyston. Since the paticles ae (almost) on cest, only the eal pat of the admittance [Eq. (7)] seen by the geneato is changed due to beam loading. This means that in this example 83

beam loading causes a change only in the cavity impedance seen by the geneato and detuning plays no ole as a means of beam-loading compensation. Theefoe thee is only pefect matching fo the nominal beam cuent to which the cavity powe input couple has been adjusted. Fom the cicuit diagam in Fig. 4 we see that one RF geneato is planned to supply up to 3 cavities with RF powe. The RF powe that is needed pe cavity to acceleate an electon beam of 8.5 ma to 5 MeV, is close to 15 kw, hence a klyston powe close to 7 MW is needed. This powe is entiely caied away by the beam. In contast to the pevious example, whee all the RF powe was essentially dissipated in the nomally conducting cavity walls, the powe needed to build up the RF cavity voltage in the supeconducting cavities is only a few hunded watts. A high-efficiency, 10 MW, multibeam klyston has been developed fo this poject. Fo completeness we mention that this is pulsed powe, with a pulse length of 1.5 ms and a maximum epetition ate of 10 Hz. So the maximum aveage klyston powe is 150 kw. The RF seen by the beam coesponds to the vecto sum of all cavity signals. Theefoe fo the RF contol system such a vecto sum needs to be geneated. This is done by down convesion of the cavity signals to 50 khz intemediate fequency signals that ae sampled in time steps of 1 µs. Each set of two subsequent samples coesponds to the eal and imaginay pat of the cavity voltage vecto. The vecto sum is geneated in a compute and is compaed to a table of set point values. The diffeence signal, which coesponds to the cavity voltage eo, acts on a vecto modulato at the lowlevel klyston input signal. In addition to this feedback a feedfowad coection can be added. The advantage of feedfowad is that, in pinciple, thee is no gain limitation as in the case of feedback. If the eo is known in advance, one can pogam a counteaction in the feedfowad table. Examples of such eos could be a systematic decease in beam cuent duing the pulse due to some popety of the electon souce, o a systematic change in the cavity esonance fequency duing the pulse. This effect does indeed exist. The mechanical foces esulting fom the stong pulsed RF field in the supeconducting cavities cause a detuning of the ode of a few hunded hetz at 5 MV/m. This effect is called Loentz foce detuning. Fom Eq. (47) one might infe that, because of the lage value of Q L, the maximum possible feedback gain in this case could become significantly lage than fo nomally conducting cavities. Howeve, one has to check whethe thee ae poles in the system at othe fequencies, and, at least in this case, thee is a faily lage loop delay of about 4 µs caused by the 1 m length of the cyogenic modules in which the cavities ae placed and by the time delay in the compute. This esults in a ealistic maximum loop gain of 140. So fa, we have demonstated fo up to 16 cavities that all this eally does wok in pactice. The impessive phase and amplitude stability of 0.1 degee and 0.5% that has been eached with this feedback system is shown in Fig. 5. Thee one also sees that the addition of feedfowad impoves the amplitude and phase stability to 0.05% and 0.03 degees espectively. 6. DAMPING OF SYNCHROTRON OSCILLATIONS OF PROTONS IN THE PETRA II MACHINE In the peceding sections phase and amplitude contol of the cavity voltage was discussed. In this last Section we would like to give an example of beam contol by looking at the dedicated RF system fo the damping of synchoton oscillations of potons in the PETRA II synchoton at DESY. Pio to injection into HERA, potons ae pe-acceleated to 7.5 and 40 GeV/c in the DESY III and PETRA II synchotons, espectively [13]. Timing impefections duing the tansfe of potons fom one machine to the next and RF noise duing amping wee obseved to cause synchoton oscillations that, if not damped popely, may lead to an incease in beam emittance and to significant beam losses. Theefoe a phase loop acting on the RF phase to damp these oscillations of the poton bunches is a necessay component of the low-level RF system. The PETRA II poton RF system, which consists of two 5 MHz cavities, each with a closely coupled RF amplifie chain and a fast 84

feedback loop of gain 50, is simila to the one shown in Fig.. The block diagam of the PETRA II phase loop, on which I shall now concentate, is shown in Fig. 6. Fig. 5: Phase and amplitude stability achieved by digital feedback and feedfowad in a 1.3 GHz cavity of the TESLA Test Facility. The obtained phase and amplitude stability with feedback alone is 0.5% and 0.1 degees. With feedfowad an impovement to 0.05% and 0.03 degees was eached. This figue is taken fom Ref. [1]. 85

Fig. 6: Block diagam of the PETRA II phase loop. In the phase detecto, synchoton oscillations of the bunches ae detected by compaing the filteed 5 MHz component of the beam with the 5 MHz RF efeence souce. An aveage phase signal fo each of the eight batches of ten bunches is phase shifted by 90 with espect to the synchoton fequency, stoed in its egiste, and popely multiplexed to the phase modulato acting on the RF dive signal. 6.1 Loop bandwidth The maximum numbe of bunches is 11 in DESY III and 80 in PETRA II, so eight DESY III cycles ae needed to fill PETRA II. If synchoton oscillations aise due to injection timing eos, all bunches of the coesponding batch ae expected to oscillate coheently. Theefoe one single coection signal can damp the bunch oscillations in that batch and in total up to eight such signals ae needed, one fo each batch. This phase loop is a batch-to-batch athe than a bunch-to-bunch feedback. Ideally, the coection of expected eos of about two degees in the injection phase has to be switched within the 96 ns sepaating the last bunch of batch n fom the fist bunch of batch n + 1. Owing to the fast feedback of gain 50, the RF system has an effective bandwidth of about 1 MHz. Howeve, it is capable of pefoming small phase changes of the ode of 1 pe 100 ns, which should be sufficient fo damping synchoton oscillations also in the multibatch mode of opeation. 6. The phase detecto Each bunch passage geneates a signal in the inductive beam monito, also shown in Fig. 6. A passive LC filte of 8 MHz bandwidth filtes out the 5 MHz component. The inging time is compaable to the bunch spacing time as shown in Fig. 7. Amplitude fluctuations of this signal ae educed to ±0.5 db in a limite of 40 db dynamic ange. So the amplitude dependence of the synchoton phase measuement between the bunch signal and the 5 MHz RF souce signal is minimized. The phase detecto has a sensitivity of 10 mv pe degee. By inseting a low pass filte one can diectly obseve the synchoton motion of the bunches at the phase detecto output. This is shown in Fig. 8(a) fo one batch of nine poton bunches ciculating in PETRA II with a momentum of 7.5 GeV/c a few milliseconds afte injection. The obseved synchoton peiod T S = 5 ms agees with the expected value fo the actual RF voltage of 50 kv. 86

Fig. 7: Filteed signal of a batch of nine poton bunches ciculating in PETRA. The bunch spacing time is 96 ns. Fig. 8(a): The synchoton oscillation measued at the phase detecto output a few milliseconds afte injection of a batch of nine poton bunches into PETRA II. It is smeaed out by Landau damping afte seveal peiods. The damping loop is not active. Fig. 8(b): Same as Fig. 8(a) but with the phase loop active. The synchoton oscillation is completely damped within half a synchoton peiod of 5 ms. 6.3 The FIR filte as a digital phase shifte A feedback loop can damp the synchoton motion if, as is indicated in Fig. 6, the synchoton phase signal is shifted by 90 elative to the synchoton fequency f S, delayed popely, and fed into a phase modulato acting on the 5 MHz dive signal. The necessity of the 90 phase shift elative to f S can be seen fom the equation of damped hamonic motion x + ax +bx = 0 with the solution x = Asin ( S t )e at. The damping tem a x is popotional to the time deivative of the solution x, i.e. a phase shift of 90. The coection signal will coincide with the coesponding batch in the cavity if the total delay = t f + nt ev, whee t f is the tansit time fom the beam monito to the cavity, n an intege, and T ev = 7.7 µs is the paticle evolution time in PETRA. Since T S» T ev, a delay even of moe than one tun (n > 1) would not be citical. 87

Rathe than using a simple RC integato of diffeentiato netwok as a 90 phase shifte, which is not without poblems [14], a moe complex digital solution with a softwae contolled phase shift has been adopted. This is vey attactive since duing injection, acceleation, and compession of the bunches the synchoton fequency vaies in the ange fom 00 Hz to 350 Hz. In addition, stoing and multiplexing the eight coection signals fo each of the eight possible batches in PETRA II can also be ealized most comfotably on the digital side. The phase shifte has been built up as a theecoefficient digital Finite-length Impulse Response (FIR) filte accoding to with an amplitude esponse g = h k f µ k k = 0 µ (5) H() = h k e ikt S, (53) k= 0 whee f and g ae input and output data espectively. Using the coefficients h 0 = sin, h 1 = cos, h = sin one obtains a phase shift that, in the fequency ange of inteest 00 Hz f S 359 Hz, deviates by less than ±0.4 fom the nominal value = / in accodance with Eqs. (5) and (53). The fequency dependence of the phase shift is mainly due to the delay in the filte, which is of the ode of 1 ms, i.e. two sampling peiods. It can always be coected by softwae, if necessay. The amplitude esponse is constant within a few pe cent fo all fequencies. A block diagam of the filte is shown in Fig. 9. The synchoton phase infomation of the eight batches is sampled at intevals T S = 0.5 ms and passed though eight times thee shift egistes. The thee coefficients ae stoed in ROMs and ae appopiately combined with the phase infomation. So the fist filte output is available afte thee sampling peiods and is then enewed evey 0.5 ms. Fig. 9: Block diagam of the FIR filte. Fom thee successive sampling peiods the aveaged phase signals fo the eight poton batches in PETRA II ae stoed in shift egistes and combined with the thee coefficients, which ae stoed in ROMs. The fist phase-shifted output is available afte thee sampling peiods of 0.5 ms and is enewed evey sampling peiod. 6.4 Pefomance of the phase loop The pefomance of the loop is demonstated in Fig. 8, whee the phase detecto output ecoded by a stoage scope is displayed. Complete damping of the synchoton oscillation is achieved within less than one peiod. This coesponds to a damping time of less than 4 ms. If the loop is opeated in the antidamping mode, the beam is lost within some milliseconds. With the loop, losses of the poton beam in PETRA II duing enegy amping could be significantly educed. 88

REFERENCES [1] R.E. Collin, Foundations fo Micowave Engineeing (McGaw-Hill, New Yok, 1966). [] P.B. Wilson, CERN ISR TH/783 (1978). [3] D. Boussad, CERN, SPS/86 10 (ARF) (1986). [4] R.D. Kohaupt, Dynamik intensive Teilchenstahlen in Speicheingen, Lectue Notes, DESY (1987). [5] A. Piwinski, DESY H 70/1 (1970). [6] K.W. Robinson, CEA Repot CEAL-1010 (1964). [7] F. Pedesen, IEEE Tans. Nucl. Sci. NS 3 (1985) 138. [8] D. Boussad, CERN SPS/85 31 (ARF) (1985). [9] F. Pedesen, IEEE Tans. Nucl. Sci. NS (1975) 1906. [10] E. Vogel, Ingedients fo an RF Feedfowad at HERA, DESY HERA 99 04 (1999) p. 398. [11] S. Simock, pivate communication, DESY (1999). [1] M. Liepe, Diploma Thesis, DESY (000). [13] A. Gamp, W. Ebeling, W. Funk, J.R. Maidment, G.H. Rees, C.W. Planne, in Poc. 1st Euopean Paticle Acceleato Confeence (EPAC 1), Rome, 1988, Ed. S. Tazzai (Wold Scientific, Singapoe, 1989). [14] A. Gamp, in Poc. nd Euopean Paticle Acceleato Confeence (EPAC), Nice, 1990, Eds. P. Main and P. Mandillong (Ed. Fontièes, Gif-su-Yvette, 1990). 89