Cooper, R K

Neil, V K

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Particle Accelerators, 1972, Vol. 4, pp. 75-79 © GORDON AND BREACH, SCIENCE PUBLISHERS LTD. Printed in Glasgow, ScotlandTHE EFFECT OF RANDOM RF VOLTAGE FLUCTUATIONS ON ABUNCHED BEAMtV. KELVIN NEIL AND RICHARD K. COOPERtLawrence Livermore Laboratory, University of California, Livermore, California 94550, USAThe noise present in the driving current of the rf system for a storage ring will cause an increase in the average phaseangle of protons stored in a bunched configuration. The purpose of this work is to calculate the rate of increasein the phase angle. We consider only small amplitude oscillations about the synchronous phase angle. Results aregiven in terms of the signal-to-noise ratio (in power terms) of the cavity voltage, and indicate that the noise may be astringent limitation on storage times unless the rf system is designed with care.1. INTRODUCTIONIf protons are kept bunched by an rf system in astorage ring, random fluctuation in the rf voltagemay cause a diffusion of particles out of stablephase. The purpose of this work is to calculate arate of increase of the phase angle of a particlesubjected to random fluctuations of the rf voltage.Results are expressed in terms of the signal-to-noise ratio (in power terms) of the rf voltage.Alternatively, this ratio may be fo'und from thespectral density of the current driving the rf cavity.This is a preliminary report, and the variouspossible sources of noise are not investigated.Future work (if performed) will include an investi-gation of these sources as well as a solution of theFokker-Planck equation for the distribution ofparticles in stable phase.The methods used in this work are conventionalmethods for the mathematical treatment of randomphenomena. 1 A reader unfamiliar with thesemethods may at times feel he is being hoodwinked.Let him be assured that these methods, which mayappear to be black magic to the uninitiated, have beenproven in countless applications.2. SPECTRAL DENSITY AND AUTO-CORRELATION FUNCTION OF CAVITYVOLTAGEWe consider only one mode of the rf cavity tobe excited. The cavity has a very high qualityt Work performed jointly under the auspices of the U.S.Atomic Energy Commission and the Advanced ResearchProjects Agency, Order No. 1377.t Permanent address: California State University-HaywardHayward, Calif.factor Q, and operates on a harmonic number h.That is, the angular frequency of the cavity is W,.and the angular circulation frequency of theparticles is W o with W r = hwc• The particularcavity mode in question can be represented by aseries L-C-R circuit. We take the external drivingmechanism to be such that the cavity wall currentI satisfies the equationd2[ dI 1 d2IdL dt2 +R dt +"(71 = 1\11- dt 2 ' (2.1)in which L, C, and R are respectively the inductance,capacitance, and resistance of the mode; M isthe coupling factor with units of inductance; andI d is the driving current. This coupling mechanismis convenient, but is not fundamental to the follow-ing treatment. A driving current varying ase- iQt will produce a voltage Va = IIdtjC given byV _ Mw/Qldd-[(QR/L)+i(w/_Q2)J' (2.2)in which w,.2 = (LC) -1 . We need not assume inthis work that the Q = W r • It is usually advan-tageous to drive a cavity slightly off resonance tocompensate for the beam-induced voltage.The source of noise we wish to consider is arandom fluctuation of the driving current. Weaugment our driving current with an additionalcurrent I(t)( ~ Ia), which is considered to be astationary random quantity with a Fourier trans-form jew). (Actually, jew) is the Fourier trans-form of a function which is identical to I(t) within along time interval T and zero outside this interval.The Fourier transform of a random variable doesnot exist.) Referring to Eq. (2.2) we see that the76 V. KELVIN NEIL AND RICHARD K. COOPER(2.3)Fourier transform Y(co) of the cavity voltageproduced by I(t) is of the formV(w) = Mw/wl(w)[(coR/L) + i(COr 2 _C02 )]'Noting t~at R/L may be identified with cor/Q, wesee that V(co) is sharply peaked at co = COr'We now introduce the spectral density G(co) ofthe random voltage Vet) by the standard definition(2.4)where K is a constant with dimensions of inverseseconds. It can be shown by means of the Wiener-Khintchine theorem that G(co) is also given by lG(w) = f~oo <V(t)V(t+u) coswudu, (2.5a)integral in Eq. (2.5b) is then readily evaluated bycontour integration with the result<l/(t)V(t+u) _ G/(wr ) QVd2 - 2Wr 3 Idix [(cor!!/Q)2+(wr2_Q2)2Je-w/,/2Q COSWrU, (2.8)in which we have neglected terms of order Q-lcompared to unity. The correlation time of therandom variable V is equal to 2Q/wr• Physicallythis time is a measure of how long the voltageproduced by a fluctuation in I will persist in thecavity.It is important to note that a particle goesaround the machine in a time 2n/wc = 2nhfcor •If Q~ h, then the random changes of energyexperienced by the particle on successive turns arecorrelated.(2.6)(3.3)(3.2)in which y is the energy of the synchronous particleand Yt is the transition energy of the accelerator.We go to a rotating coordinate system and intro-duce the phase angle ¢ by the definition¢ = (Wr/wc) (f)-coc t) == h((J-coc t). (3.4)dO (_ dP)-=Wc 1- rl--dt Pcin which Pc is the momentum of the synchronousparticle (that particle with () = w c) and dp is thedeviation in momentum. Only extreme relativisticenergies are considered. The quantity 1] is given by3. INCREASE IN PHASE ANGLE ARISINGFROM RANDOM VOLTAGE FLUCTUA-TIONSWe consider the orbit of the particle in theaccelerator to be a circle. The azimuthal positionof the particle is (J(t). The cavity is located at(J = 0 and has a negligible extent along the orbit ofthe particle. The total voltage on the cavityVt(t) is of the formV t (t) = - Vd sin Wr t + J~,7(t). (3.1)We assume throughout this work that Vet) ~ Vd•The angular velocity of the particle is given by<V(t)V(t+u) = ~foo G(w) coswudw.2n -00(2.5b)with the inversion relationThe quantity (V(t)V(t+u) is the autocorrelationfunction of the random variable Vet). Thisquantity is independent of t for a stationary randomvariable, and is an even function of the variable u.We use Eq. (2.2) and write Eq. (2.3) in the formYew) = ~j(w) [(Qcor/Q) + i(cor2-n2)]Vd Q Id [(cowr /Q)+i(w r 2 _w2)J'We thus obtain from Eq. (2.4)G(co) = Gl(W)(~)2[(QWr/Q)2+(Wr2-Q2)2JV d2 Id2 Q (wwr/Q)2+(wr2_w2)2'(2.7)in which G](w) is the spectral density of the randomcurrent I(t).We can use Eq. (2.7) in Eq. (2.5b) to calculate theautocorrelation function of the random voltage V.Provided G](w) is not a wildly varying function,the main contribution to the integral arises fromvalues of co near ±cor. We assume that G](w) isconstant in the neighborhood of w = wandreplace co2G](w) by wr2G/(wr) in Eq. (2.7).' Wefurther replace the factor (COr/Q)2 by unity. TheEFFECT OF RF NOISE ON A BUNCHED BEAM 77bPn = (elc) [Vd sin 4>n + V(tn)]. (3.6)From Eqs. (3.5) and (3.6) one may derive the phaseequation,(3.13)(3.15)x sin ,u(n -1+j) sin ,u(n -1).x sin fl(n -I) sin ,u(n - m).in which the constants a and et are determined fromthe initial conditions,and fl == 2 sin -1 k12.We now square both sides of Eq. (3.12) and takean ensemble average, which we indicate with ().Making use of the property of random variablesthat (v) = 0 at any time, we havek4 n n(4)n2) = a2 sin2(,un+et)+-'.-2- L L (VlVm)SIn ,u 1= 1 m = 1If nh/Q ~ 1, the quantities <vlvm ) for m i= I may beignored and there is no correlation in the values ofV( t) experienced by the particle on successivepassages through the cavity. In this limit Eq. (3.14)is a good approximation to Eq. (3.13).We next consider terms for which m = I±j.There are 2(n - j) such terms. Denoting thecontribution from these terms as (cPn 2 ) j' we have14 n-j(¢n2>.i =~ L (VlVl+ j ) sinfl(n-I-j)SIn fll= 1 'Clearly the first term on the right does not con-tribute to an increase of cP with increasing n, andwill be ignored. The quantity (v l vm) is the auto-correlation function of the random variableV(t)IVd introduced in the last section. It is afunction only of u == It l - 1m I and is given by Eq.(2.8).Let us first examine terms with m = I in thedouble summation in Eq. (3.13). The quantity(Vl2 ) is the expectation value of v2 , and is inde-pendent of time. Denoting the contribution fromthe m =/ terms as (cPn2)0 we havek4( 2) n<¢ n 2 )0 = --- ~-~_._- L si112 fl(n -I). (3.14)SIll fl 1= 1But the autocorrelation function depends only onthe absolute value of the time difference, thus(Vi V2) = (V2 Vi), etc. so that we may combine thetwo sums and obtain2k4 n-j(cPn2)j = -'-2-- L <VlVl + j ) sinfl(n-l)SIn fll= 1(3.5)(3.8)(3.9)(3.10)In the absence of the random quantity Vm thefinite difference Eq. (3.7) is traditionally approxi-mated by the differential equation2d2¢ 2',h 0-2- + COs slnljJ = ,dtAt e= 0, 4> = - COr t. The synchronous particlehas 4> = 0 and therefore passes through the cavitywhen the driven voltage is zero. From Eq. (3.2)we have (for any particle)with the so-called synchrotron frequency COsdefined bycos2 = (kwc/2n)2 == h'1e Vdcoc2/2npc c. (3.11)We shall not use this approximation at this stage,but we shall limit our treatment to small values of ¢so that we may approximate sin <Pn by <Pn in Eq. (3.7).In practice the quantity k 2 is quite generallyvery much less than unity. With the approximationsin <Pn = <Pm the solution of Eq. (3.6) is3k2 n¢n = a sin (fln +et)- -.- L Vl sinfl(n-l),SIll fll= 1The rate of change of 4> is constant during oneturn around the machine. A particle with a valueof <P = <PII as it passes through the cavity for thenth time will have a value <Pn+ 1 as it passes throughthe cavity the next time. On the nth passagethrough the cavity (at time t = tn) the particleexperiences a change in momentum bPn == dPn-dPn-l given by4>n+ 1 + 4>n-l - 24>n = - k2(sin cPn +vn), (3.7)in which we have introduced the quantities k 2 andVn by the definitions(3.12) x sin fl(n - l- j). (3.16)78 V. KELVIN NEIL AND RICHARD K. COOPERThe quantity (V1V1+j ) is only approximatelyindependent of I. It is a function of f1+j-f1,which is given byt1+j - t1= jC:-(cPl+j -cPl)/OJr , (3.17)in which 1: == 2n/OJc • The only dependence of(¢n2) on the amplitude of ¢ is contained in thistime difference. As long as we are restricted tosmall enough values of ¢ such that sin ¢ ~ ¢, thisdependence is a weak one indeed. We shall simplyset f1+j-f1 =jc: in the following. This approxi-mation is equivalent to neglecting the change in ¢ .over the number of periods for which the randomvoltage is correlated.Introducing the notation (vvj ) == (v(t1)V(t1+jc:),)we rewrite Eq. (3.16) in the form2k4(vv.) n- j(¢n2)j = . 2 J L [sin2 Jl(n-I) cosjJlSIll Jl 1= 1- sinjJl sin Jl(n -I) cos Jl(n -I)J.We take n sufficiently large so that the sum extendsover at least a few periods of Jl(n -I). The portionof the sum that increases with increasing n may befound by replacing sin2 Jl(n-/) by its averagevalue of 1/2 and neglecting the second term whichhas an average value of zero. Thusk4(¢n2)j = (n-j)-'-2-<vvj ) cosjJl. (3.18)SIn JlReplacing sin2 Jl(n-/) by 1/2 in Eq. (3.14), we maynow express (4)n 2 ) as2 nk4 <v2 ) k4n,. . .<cPn ) = -2-'-2-+-'-2- L (n-)) cosJJl(VVj ).SIn Jl SIn Jl j= 1(3.19)Inspection of Eq. (2.8) reveals that<VVj ) = <v2) e- j ror t / 2Q cos(jOJ ,1:),but OJrr = 2nh, thus Eq. (3.19) becomes<4J/) = .k: <v2)[~+I (n-j)cosj/le-j1th/Q].SIll Jl 2 j= 1(3.20)Writing cosjJl as Re eijll gives the summationthe form of a finite geometric series and the deriv-ative of such a series, with the result that the sumhas the approximate value (neglecting 1 withrespect to n and taking e -n7th/Q ~ 0)n n (cos Jl- e- 7th/Q)j~l ~ 2. (cosh nhjQ-cos/l)'Combining this result with the first term on theright hand side of Eq. (3.20) we have-1,2 ne <v2 ) sinh (nhjQ) (<'I'n ) = sin2 /l-2 [cosh(nhIQ)-cos/lJ · 3.21)This expression has a maximum value at cosh(nh/Q) = sec Jl. To second order in k and nh/Qthis maximum occurs at nh/Q = k .and is given by<¢n2 )max = nk<v2 )/2. (3.22)We note that for very small Q the summation isnegligible (from Eq. (3.20) or (3.21)). For verysmall Q the random voltage encountered by theparticle on successive turns is uncorrelated. Onthe other hand, in the limit Q~ 00, the right handside of Eq. (3.21) vanishes. In this limit the voltagevet) is not a random variable. It is simply a constanterror in the amplitude (or phase) of the drivingvoltage.The quantity (v 2 ) may be known or measurablefor a given rf system. Alternatively, we may useEq. (2.8), which to a good approximation may besimplified to(V2 ) = G[(OJr)OJr/2Qld2 • (3.23/This equation allows a calculation of (v 2 ) if thespectral density G[(OJr) of the driving current isknown.As a numerical example, we consider the para-meters of a proton storage ring as presented inRef. 4. From these parameters we obtain k = 0.1and a circulation frequency Ie of 2 x 105 cps. Ifonly the first term on the right hand side of Eq.(3.20) is retained (turn-to-turn correlation com-pletely neglected) we find that 4>,,2 has an expectationvalue of unity after a number of turns n = 200/<v2 ). This corresponds to a time t = nile = 10- 3I<v2 ) sec. Little information is available regardingthe value of <v2) for rf systems in operation atpresent. A value of 10- 6 could reasonably beexpected in a conventionally designed system. 5Using this value we find t = 103 sec. But now ifnh/Q = k, Eq. (3.22) applies and the time becomes102 sec. These numbers indicate that rf systemsEFFECT OF RF NOISE ON A BUNCHED BEAM 79for bunched-proton storage rings must be designedvery carefully if long storage times are to beachieved.6Finally, we remark that particles with largeamplitude of phase oscillations will experience agrowth rate of cPn that is slower than that given byEq. (3.21). This fact can be discovered by a moreaccurate treatment of the quantities <vzvz+ j ),taking into account the change in phase over acorrelation time.ACKNOWLEDGEMENTSThe authors- wish to thank William A. Newcomband Warren Heckrotte for enlightening discussionsand guidance, particularly in the theory of randomvariables.REFERENCES1. F. Reif, Fundamentals of Statistical and Thermal Physics(McGraw-Hill Book Co., New York, 1965), Chap. 15.2. D. Bohm and L. Foldy, Phys. Rev. 70, 249 (1946).3. The authors are indebted to the referee for pointing outthis general form of the solution.4. C. Pellegrini et al., in Proc. VIII International Conferenceon High Energy Accelerators CERN, Geneva, 1971,p.153.5. Bob H. Smith, private communication.6. Admittedly, our treatment does not include the presenceof a phase lock or other feedback control of the rfsystem. Such a control may have both good and badinfluence on the storage time. Preliminary investiga-tion shows that the feedback suppresses the growth ofcoherent phase motion, but has little effect on theincoherent motion within the rf bucketReceived 24 July 1972;and in final form 1 September 1972

The Effect Of Random Rf Voltage Fluctuations On A Bunched beam

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