Transmission Line Analysis of Beam Deflection in a BPM Stripline Kicker

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1 UCR-JC PREPRINT Tansmission ine Analysis of Beam Defletion in a BPM Stipline Kike Geoge J. Capoaso Yu Ju Chen Bian Poole This pape was pepaed fo submittal to the 1997 Patile Aeleato Confeene Vanouve, B. C. Canada May 12-16, 1997 May 1997 awene ivemoe National aboatoy This is a pepint of a pape intended fo publiation in a jounal o poeedings. Sine hanges may be made befoe publiation, this pepint is made available with the undestanding that it will not be ited o epodued without the pemission of the autho.

2 DISCAIMER This doument was pepaed as an aount of wok sponsoed by an ageny of the United States Govenment. Neithe the United States Govenment no the Univesity of Califonia no any of thei employees, makes any waanty, expess o implied, o assumes any legal liability o esponsibility fo the auay, ompleteness, o usefulness of any infomation, appaatus, podut, o poess dislosed, o epesents that its use would not infinge pivately owned ights. Refeene heein to any speifi ommeial podut, poess, o sevie by tade name, tademak, manufatue, o othewise, does not neessaily onstitute o imply its endosement, eommendation, o favoing by the United States Govenment o the Univesity of Califonia. The views and opinions of authos expessed heein do not neessaily state o eflet those of the United States Govenment o the Univesity of Califonia, and shall not be used fo advetising o podut endosement puposes.

3 TRANSMISSION INE ANAYSIS OF BEAM DEFECTION IN A BPM STRIPINE KICKER Geoge J. Capoaso, Yu Ju Chen and Bian Poole, awene ivemoe National aboatoy, ivemoe, Califonia USA Abstat In the usual teatment of impedanes of beamline stutues the eletomagneti esponse is omputed unde the assumption that the soue hage tajetoy is paallel to the popagation axis and is unaffeted by the wake of the stutue. Fo high enegy beams of elatively low uent this is geneally a valid assumption. Unde etain onditions the assumption of a paallel soue hage tajetoy is no longe valid and the effets of the hanging tajetoy must be inluded in the analysis. Hee the usual tansmission line analysis [1] that has been applied to BPM type tansvese kikes is extended to inlude the self-onsistent motion of the beam in the stutue. 1 INTRODUCTION The desie to use one indution aeleato to povide multiple lines of sight fo advaned adiogaphy [2] has stimulated wok in the use of ylindial stipline kikes to deflet kiloampee eleton beams. We onside a ylindial stipline kike as shown in Fig. 1 onsisting of fou eletodes. Two of the eletodes ae gounded while the emaining two ae diven fom the downsteam end by opposite polaity able signals. Fo simpliity, we will assume that the kike impedane is mathed to that of the dive ables but that the upsteam temination ables have an abitay impedane Zt. able and I is intepeted as the dipole etun uent flowing on that stip (sine the monopole etun uent will not geneate a net defleting foe). To these soues we must add distibuted shunt uent soues to aount fo the fat that the beam is hanging its tansvese position within the stutue. If we follow a given "slie" of the beam as it entes the kike imagine that it entes on axis so that thee is no dipole etun uent at z=0. We now allow the beam slie to deflet due to the ation of, say, an extenal bias oil. That slie will then geneate a dipole etun uent on the stip. Sine the uent on the stip must be onseved, an equal and opposite uent must be indued on the othe side of the stip, i.e., in the tansmission line. This an be epesented by the distibuted shunt uent soue g(z,τ) given by g z,τ [ ]. (1) ( ) = z I ( z,τ) I ( 0,τ) Hee the vaiable τ is the slie label given by τ t / z /. We will assume that the axial veloity is, vauum light speed and is the length of the kike. We will solve the tansmission line equations in the vaiables z and t so that we will need to onvet g to the pope fom. The tansmission line equations beome and V t = i t (2) e-beam etun uent i t V = C t + g( z,t) (3) eletode Z k Fig. 1 Shemati of etun uents in the stipline kike Z t + 2V p - 2 TRANSMISSION INE EQUATIONS The tansmission line model of the kike stutue is shown in Fig. 2. The quantity I epesents the beam etun uent whih is intodued into the tansmission lines fomed by the eletodes and the oute vauum housing at the gaps at eithe end of the stiplines. These ae the usual soues used in the analysis of efeene [1]. Also shown is the voltage soue epesenting the pulse. The shemati is shown only fo one of the diven plates I (0,t) g(z,t) I (,t) Fig. 2 Tansmission line iuit showing distibuted soues whee C is the apaitane pe unit length of the line and in the indutane pe unit length of the line. We take Z k = / C and =1/ C vauum light speed whih is also the popagation speed on the line. We have

4 two bounday onditions fo the poblem. At z=0 we have that and ( ) = Z t ( i( 0,t) I ( 0,t) ) (4) V 0,t ( ) = 2V p + Z k i(,t ) + I (,t) V,t ( ). (5) We will also need to ompute the total oentz foe on an eleton v passing though the stutue. It an be shown that E + v B is popotional to the quantity V* defined as V* = V Z k i. (6) We solve these equations by aplae tansfoming in t to s. By using the method of vaiation of paametes we find that V * z,s ( ) = 2 V p ( s) + I (,s)z k e 2s ( ) +Z k dz' g z', s z e s ( z) e s ( +z' )+ s z z' ( ) (7) Note that the foe does not depend on the upsteam temination impedane. This is due to the fat that waves moving in the positive z dietion have the magneti foe aneling the eleti foe. Only waves moving upsteam will exet a foe on the beam. Sine the downsteam temination is mathed to the line any waves efleting off the upsteam temination exet no foe and leave the poblem when they aive at the downsteam temination. whee xo is the patile offset, Q is it's hage, b is the kike eletode adius and η is a geometi fato. We thus find that V* is given by V * ( z,τ ) = Qηx Z 0 k b δ τ 2 + 2z. (10) We an integate the tansvese foe ove the length of the kike to obtain the wake funtion as θ τ 2b (11) whee α is anothe geometi fato. We an now find the tansvese impedane by taking the Fouie tansfom of the wake funtion. W( τ ) = αqηx 0 ( ) θ τ 2 Z ( ω) = i W τ Qx ( )e iωτ 0 dτ (12) to obtain Z ( ω) = αηz k 1 e 2iω 2b ω (13) the nomalized eal and imaginay pats of whih ae plotted in Fig. 3 and 4 espetively as a funtion of x ω /. These foms math those found peviously [1]. 3 BEAM DYNAMICS In ode to ompute the behavio of a slie of the beam we need V*(z,τ). We an invet equation (7) and use the definition of τ to obtain ( ) = 2V p τ + 2z V * z,t +I,τ 2 + 2z Z k +Z k dz' g z', τ + 2z 2z'. (8) z et us examine the onsequenes of equation (8) fo the usual ase. We set Vp=0 and take a paallel beam tajetoy fo the soue hage so that g vanishes and Fig. 3 Plot of nomalized eal pat of the impedane vs. ω / I = Qηx 0 b δ( τ) (9)

5 whee Zo is the impedane of fee spae (377 ohms) and Io is appoximately 17 ka. Upon integating (16) by pats we obtain 2 x ( z,s) [ x ( z, s) 2 z 2 sz s x ( z', s)e 2sz' dz'] (18) whih an be solved in the asymptoti limit fo lage τ. This limit oesponds to the limit s->0. Theefoe, in the limit τ-> we have Fig. 4 Nomalized plot of the imaginay pat of the impedane vs. ω / 4 ASYMPTOTIC DEFECTION We may use the expession fo V* to find the selfonsistent displaement of the beam inside the kike due to the ation of the wakefields. et us onside the ase of a ontinuous beam with no applied voltage. The dipole etun uent is given by I = 2I τ b( )x ( z, τ ) sin φ 0 (14) πb 2 whee Ib is the beam uent (Ib is <0 fo eletons) and whee φo is the angle subtended by the diven stipline. et us onside the ase of a long eleton beam and put Ib=-I B whee I B is a positive onstant. Inseting the appopiate geometi fatos we may wite the diffeential equation of motion fo a slie of the beam as 2 x( z,τ) [x,τ 2 + 2z z z' x z', τ + 2z 2z' dz' ]. (15) et us now aplae tansfom this equation in τ to s. 2 x ( )e 2 s [ x, s z z' + 2sz 2sz x ( z',s)e 2sz' dz']. (16) The quantity I is the "itial uent" and is given by π 16 Z 0 b 2 γβ 2 I 0 Z k 2 sin 2 φ 0 2 (17) 2 x( z,τ) x( z,τ ) 2 (19) whih an be solved to yield x(z,τ ) = x(0,τ )Cosh x'(0, τ) 2I B 2I B Sinh z + 2I B z. (20) Thus the input position and angle ae both amplified by the fato Cosh 2I B / ( ) at the exit of the kike. 5 CONCUSIONS We have studied the defletion due to beam indued voltages in a stipline tansvese kike. The asymptoti displaement of the beam position at the kike output is pedited as a funtion of the beam uent, kike impedane and dimensions of the stutue. In the limit of infinitely stiff beams the usual esult is eoveed fo the tansvese impedane. 6 ACKNOWEDGMENTS The wok was pefomed unde the auspies of the U.S. Depatment of Enegy by N unde ontat W ENG-48. REFERENCES [1] K-Y. Ng, Patile Aeleatos, 23, (1988). [2] G. Capoaso, "inea Indution Aeleato Appoah fo Advaned Radiogaphy", this onfeene. [3] B. Poole, G. Capoaso and W. Ng, "Wake Popeties of Stipline Beam Kike ", this onfeene.

6 Tehnial Infomation Depatment awene ivemoe National aboatoy Univesity of Califonia ivemoe, Califonia 94551

8.022 (E&M) Lecture 13. What we learned about magnetism so far

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