R sh /Q 0 Measurements in Klystron Cavities

Transcription

1 R sh /Q Masurmnts in Klystron Cavitis Robson K. B. Silva Navy Tchnology Cntr at Sao Paulo CTMSP Av. Prof. Linu Prsts, 468 Danil T.Lops Nuclar and Enrgy Rsarch Institut IPEN Av. Prof. Linu Prsts, 4 daniltl@usp.br Cláudio C. Motta Univrsity of Sao Paulo USP Av. Prof. Linu Prsts, 468 ccmotta@usp.br Abstract This papr prsnts th xprimntal rsults of masurmnts in klystron cavitis using th prturbation tchniqu. Th thory involving this tchniqu rsults in an xprssion which dpnds on th natural frquncy f and th prturbd frquncy shift Δ f, both masurd using a cylindrical rntrant cavity with offst gap built in th laboratory. In addition, it is usd a cylindrical cavity (pill-box) to calculat a constant that dpnds on th gomtry of th prturbd objct, as wll as analytical xprssions to calculat an intgral factor that rlats th squar of th voltag on axis and th lctric nrgy originally stord in th small volum of th prturbd objct. Th valus masurd of f, Δ f and ar, rspctivly,.86 GHz, MHz and 79.8 Ω. Thy ar also compard with th rsults simulatd by a 3D ignsolvr obtaining a good agrmnt. Kywords Klystron cavity, Slatr s prturbation tchniqu, cylindrical rntrant cavity, prturbd frquncy shift. I. INTRODUCTION Th cavity is on of th most important paramtrs to b dtrmind in th dsign of a klystron amplifir. In this problm th gain of a klystron is proportional to th cavity shunt rsistanc R sh, whil bandwidth proportional to th cavity 1/Q, so that is a usful figur of mrit in dscribing th ffct of th ach cavity on th gain-bandwidth product of ovrall amplifir. Thrfor, it is rlvant whn, for xampl, on of th dsign rquirmnts is high product gain-bandwidth. Morovr, sinc th quantity is indpndnt of cavity losss it is a vry rlvant figur of mrit. Furthrmor, it is dpndnt on th gomtric shap of th cavity and frquncy. Although thr ar som lctromagntic fild cods availabl now to calculat of th cavity, th xprimntal masurmnts provid considrabl insight on th rol of a cavity and its gap [1]-[3] ovr th amplifir. Accordingly to Slatr s prturbation thorm [4], whn som paramtrs such as th configuration of th boundary, th matrial in th volum or th matrial of th boundary chang slightly, th lctromagntic systm is said to b prturbd. If th solution of an unprturbd problm is known, thn th solution of th prturbd problm, which is slightly diffrnt from th unprturbd on, can b obtaind by mans of th principl of th prturbation. Thr ar two possibl ways to do this: th cavity wall prturbation or th conductor prturbation of th cavity. Th first on mans to introduc a small dformation in th wall. Th scond on mans to introduc a small conductiv prturbing objct into th cavity. In this work th lattr tchniqu is usd, although th dformation of th wall may also b considrd as a conductiv prturbing objct stuck on th wall. Givn a cavity at rsonanc, it is known that avrag stord magntic and lctric nrgis ar quals. If a small prturbation is mad by introducing a small conductiv objct into th cavity, this changs on typ of nrgy mor than th othr, and rsonant frquncy would thn shift by an amount ncssary to qualiz th nrgis again. Th prturbation mthod assums that th actual filds of a cavity with a small shap or matrial prturbation ar not diffrnt from thos of th unprturbd cavity. So, if th cavity frquncy shift du to th small objct can b masurd as wll its gomtry shap, th Slatr s prturbation thorm is usful to dtrmin th cavity figur of mrit. This papr is organizd as follows. Sction II prsnts th mathmatical formulation of th problm. Th dscription of th xprimnt stup and rsults ar shown and discussd in Sction III. Finally, th conclusion is prsntd in Sction I. II. MATHEMATICAL FORMULATION Th physical intrst problm is constitutd of a ral rsonant cavity considring small losss whr th surfac currnts ar ssntially thos associatd with th loss-fr fild solutions. This cavity is formd by a surfac S inclosing a volum. It is considrd, as an hypothsis, th volum of th prturbing objct as bing and th surfac nclosing th prturbing objct as bing Δ S. Th positiv dirction of Δ S is th outward dirction of th volum Δ. Hr, th volum of th prturbd cavity is and th surfac nclosing it is S. Considr that th positiv dirction of S and S is th outward dirction of th cavity volum and. Bsids, it is considrd S S Δ S and Δ. Lt ω, E and H rprsnt th natural angular frquncy, th lctric and magntic filds of th unprturbd cavity, rspctivly, and ω, E and H rprsnt th corrsponding quantitis of th prturbd cavity. In both cass Maxwll s quations must b satisfid, that is E ω H jμ, (1) E ω H This work was supportd, in part, by FINEP (Rsarch and Projcts Financing) undr contract /11/$6. 11 IEEE 76

2 H ω E jε, and () H ω E whr ε is th lctric prmittivity of fr spac and μ is th magntic prmability of th fr spac. Th Maxwll s quations (1)-(), convnintly manipulatd [5] [6], rsult in th xact quation for th chang in natural frquncy du to th introduction of a conductiv prturbing objct into th cavity. Initially, on must multiply th complx conjugat of th scond quation of (1) by H, th first quation of () by E, th complx conjugat of th scond quation of () by E and th first quation of (1) by H, obtaining H E jωμ H H, (3) E H jωε E E, (4) E H jωε E E, and (5) H E jωμh H. (6) Thn, subtracting (4) from (3) and (6) from (5) and using th vctor idntity givn by ( A B) ( A) B ( B) A, (7) it is possibl to writ ( E H ) jωμ H H jωεe E, and (8) ( E H ) jωε E E jωμh H. (9) Th nxt stp is to add (8) and (9), intgrat ovr th volum, and us th divrgnc thorm to obtain ( E H + E H) ds j( ω ω) ( εe E + μh H ). S (1) Now, sinc n ˆ E on S, xprssion (1) mans that E H ds j( ω ω) ( εe E + μh H ). (11) S In addition, sinc S S Δ S and nˆ E on S, it is possibl to writ that E H ds E H ds E H ds E H ds, (1) S S ΔS ΔS Finally, using (14) in (13) on gts to j E H ds ΔS Δ ω ω ω. (13) ( ε E E + μ H H ) For practical application of (13) and onc that E, H, or ω ar not gnrally known, it must rplac ths unknown prturbd quantitis by th unprturbd filds E and H, bsids ω. For small prturbations, whn Δ S is small, this is crtainly rasonabl and th dnominator rsults ( ε E E + μ H H ) ( ε E + μ H ). (14) In th numrator, th tangntial componnt of th prturbd magntic fild is, approximatly, qual to th unprturbd valu whn th conductiv prturbing objct is small. Thn, it is possibl to writ that E Hds E H ds ( E H). (15) ΔS ΔS Equation (15) can b writtn any othr way. First, on must multiply th complx conjugat of th scond quation of (1) by H, and th scond quation of () by E, obtaining H E jωμ H, and (16) E H jωε E. (17) Thn, subtracting (17) from (16) and using th vctor idntity (7) it is possibl to writ that ( E H ) j ωμ H j E ωε. (18) Thrfor, (15) can b rwrittn as E H jω ( μ H ε E ). (19) ( ) Substituting (14) and (19) into (13), it obtains ( μ H ε E ). () ω ε E + μ H ( ) This is th prturbation quation for conductor prturbation of a cavity. It is vrifid that th dnominator is proportional to th total nrgy stord in th cavity, whras th trms in th numrator ar proportional to th lctric and magntic nrgis rmovd by th prturbation. Thn, () can b rwrittn as ΔWm ΔW, (1) ω W whr W dnots th total nrgy stord in th original cavity, and Δ Wm and Δ W dnot th tim avrag magntic nrgy and lctric nrgy, rspctivly, originally stord in th small volum Δ. Th prturbation quation shows that th introduction of a conductiv prturbing objct into th cavity will rais th natural frquncy if it is mad at a point with larg magntic fild (high W m ) and small lctric fild (low W ), and will lowr th natural frquncy if it is mad at a point with larg lctric fild (high W ) and small magntic fild (low W m ). Th prturbation formula () or (1) ar valid only whn th introduction of th prturbing objct dos not influnc th filds outsid th prturbing body. Othrwis, th prturbation formula must b modifid as follows K ω ( μh ε E ) ΔWm K ( ε E + μ H ) W ΔW, () whr K is a constant and dpnds upon th shap of th prturbing objct and th orintation of it in th filds. This prturbation constant K may b obtaind by mans of xprimnt. In trms of th rsonant frquncy f and th prturbd frquncy shift Δ f du to application of th mthod of prturbation it is possibl, from (), to writ that Kf Δ f ( μh εe) 4W, (3) for small prturbations and whr 4W is du to th fact that 1 W ( ε E + μh ). (4) 763

3 In TM 1 klystron cavitis, only th lctric fild is prsnt along th cavity axis. Bsids, xprimntally, it is usd th concpt of th Slatr s prturbation thorm, which stats that, considring a small prturbing objct along th axis, th magntic fild can b ngligibl [7] and th prturbd frquncy shift, from (4), can b writtn as Δ f Kε E ( ) z f, r, z f 4W. (5) Th cavity, a figur of mrit rlating th rsonant structur with its ability in doing work on an lctron bam, is dfind as th ratio of th shunt impdanc R sh of th cavity, in cas of klystron cavitis, givn by + (,, E dl Ez f dz axis, (6) PL PL and th unloadd quality factor Q, givn by ωw Q, (7) PL whr P L is th powr loss on th cavity walls [8]. Combining (6) and (7), on has + Ez ( f,, dz. (8) Q ω W Now, multiplying and dividing (8) by (,, ) z rsults + E f r z E f z dz (,, ) z (,, ) Isolating th trm ( ) E f r z z. (9) Q ωw E z ( f, r, E z f, r, z ωw in (5) and substituting in (9), it is possibl to writ that Ez ( f, b, dz axis. (3) Q Kπε f E z ( f, r, Th calculation of K is prsntd as th first rsult of th xprimnt in th nxt sction. Th intgral ratio is obtaind, xprimntally, using th Slatr s prturbation thorm. Fig. 1. Gomtry and dimnsions of th cylindrical cavity (pill-box) and prturbing objct usd in th prturbation mthod. Th dimnsions ar in mm. TABLE I. DIMENSIONS OF THE CYLINDRICAL CAITY (PILL-BOX) AND THE METALLIC PERTURBING CYLINDER. alu Cylindrical cavity lngth L, mm 8 Cylindrical cavity radius r c, mm 3 Mtallic prturbing cylindr lngth L p, mm 5. Mtallic prturbing cylindr radius r p, mm olum of th prturbing objct Δ, mm 4.1 If th prturbing objct is placd along th axis of pillbox in th TM 1 mod, can b thortically dtrmind [8] [9] by L L 185, (31) Q rc Kπε f whr th gap factor rducs to 1. bcaus th lctric fild in a TM 1 cylindrical cavity is constant across L. Thus, K bcoms rl c K. (3) 185πεf A mtallic prturbing cylindr, shown in Fig., was calibratd in this way using th cylindrical cavity shown in Fig. 1. III. EXPERIMENT AND RESULTS First of all, it is ncssary to calculat K. For this, it is usd a cylindrical cavity (pill-box) and a mtallic prturbing cylindr as shown in th Fig. 1. Tabl I prsnts thir dimnsions. Fig.. Mtallic prturbing cylindr usd in th prturbation mthod. Th prturbation masurmnts starting point is th calibration of a port of th Agilnt PNA N53C Ntwork Analysr usd in th xprimnt. Nxt, th calibratd port of th ntwork is connctd to th coupling prob in th cavity, on should slct th paramtr S 11 or S to find out th 764

4 frquncy f without th prturbing objct. Thraftr, th xprimnt is rpatd with th mtallic prturbing cylindr placd along th axis of a cavity and th prturbd frquncy shift Δ f is rcordd. Substituting th xprimntal valus of f and Δ f in (3), bsids th dimnsions shown in Tabl I, on can find th valu of K, prsntd in Tabl II. Morovr, a 3D ignsolvr was usd to simulat th cavity for comparison purposs. dscribd bfor, th first dtrmination is th frquncy f without th prturbing objct. Actually, th prturbing objct is kpt all th tim insid th cavity axis but, for this first masurmnt, its position is far nough from th cntr and, sinc th lctric fild is vanscnt, it is considrd that th fild is ngligibl and th prturbing objct dos not affct th masurmnt. TABLE II. RESULTS OF THE PERTURBATION METHOD APPLIED TO CALCULATE THE CAITY AND COMPARISON WITH THE QUANTITIES CALCULATED BY THE 3D EIGENSOLER. IT WAS USED THE CYLINDRICAL CAITY ( PILL-BOX) SHOWN IN FIG.. alus Exprimnt 3D ignsolvr Frquncy f, GHz Prturbd frquncy shift Δ f, MHz Prturbation constant K Th nxt stp is, using th klystron cavity built in th laboratory with dimnsions as in Fig. 3 and th mtallic prturbing cylindr, showd in th Fig., to dtrmin th using th prturbation mthod. Th rntrant cavity gomtry is shown in Fig. 4. Fig. 5. Exprimntal apparatus usd in th calculation of th cavity R Q using th prturbation mthod. / sh Aftrwards, th xprimnt is rpatd with th mtallic prturbing cylindr placd along th axis at th position whr th prturbd frquncy shift Δ f is a maximum. This iation can b sn in Fig. 6, which is th PNA display. Fig. 3. Gomtry and dimnsions of th rntrant cavity usd in th prturbation mthod. Th dimnsions ar in mm. Fig. 6. Th prturbd frquncy shift mthod. calculatd by th prturbation Fig. 4. Rntrant cavity built in laboratory which gomtry and dimnsions ar prsntd in Fig.3. Th xprimntal stup is shown in Fig. 5. Initially, two coppr foils of. mm thicknss ach wr placd btwn th cavity body and th opnings of th cavity, to act as lctromagntic choks, in ordr to stop th lctromagntic lakag from th cavity. Thn, following th sam stps Furthrmor, it is possibl, using th concpt of th Slatr s prturbation thorm and th xprimnt, to plot th graph of th axial lctric fild along th cavity axis. Th rsult is compard with th profil of th axial lctric fild in a cavity with offst gap considring a hyprbolic profil, according to th lopmnt prsntd in [9] and whr th gap of intraction is symmtric with rspct to th position z g and th lctric fild in th gap rgion is considrd as a combination of two functions: for z < zg th lctric fild in th intraction gap has a hyprbolic profil givn by EM cosh( qz 1 ), othrwis it has a hyprbolic profil givn by EM cosh( qz ), whr E M is th lctric fild amplitud, q 1 and q ar constants. Considring b as th drift tub radius, Fig. 7 shows th distribution of lctric fild in th axial coordinat of th gap ( r b) and for som valus insid th drift tub. It is possibl to obsrv that it is an vanscnt fild. 765

5 Fig. 7. Distribution of lctric fild in th axial coordinat of th gap ( r b) and for som valus insid th drift tub, considring th hyprbolic profil. It is obsrvd that this fild is vanscnt [9]. Th sam rsult can also b obtaind by simulating a klystron cavity using a 3D ignsolvr. Fig. 8 shows th thr rsults for r. Fig. 8. Normalizd axial lctric fild calculatd xprimntally and using, for comparison purposs, th 3D ignsolvr and th analytic xprssion prsntd in [9]. Substituting th xprimntal valus of f, Δ f and K in (3), bsids th valu of th intgral ration obtaind from th xprimntal rsults of th lctric fild shown in Fig. 8, on can find th valu of cavity, which is prsntd in Tabl III, togthr with th valu calculatd using th analytic xprssions and th simulatd rsult from th 3D ignsolvr, both usd for comparison purposs. TABLE III. RESULTS OF THE PERTURBATION METHOD APPLIED TO CALCULATE THE CAITY AND COMPARISON WITH THE QUANTITIES CALCULATED USING THE ANALYTIC EXPRESSIONS AND THE 3D EIGENSOLER. IT WAS USED THE REENTRANT CAITY SHOWN IN FIG. 4. alus 3D Exprimnt Analytic ignsolvr Frquncy f, GHz Prturbd frquncy shift Δ f, MHz Prturbation constant K Cavity, Ω Th iation btwn th xprimntal and th analytic valu is around 6 % whil th xprimntal and simulatd valu is lss than.5 %, which dmonstrats that th xprimntal procdur prsntd is vry ffctiv in masuring. It is intrsting to obsrv th ngativ signal of th prturbd frquncy shift Δ f, which indicats that th prsnc of th prturbing objct dcrasd th natural frquncy. This phnomnon occurs whn th prturbing objct is put in a position with larg lctric fild (high W ) and small magntic fild (low W m ). This is xactly th cas, confirmd by calculating th intgral ratio using th analytic xprssions without nglcting th magntic fild (3). It was obtaind for cavity th valu of 75. Ω, lss than.14 %. This rsult is intrsting bcaus it indicats that th calculation using only th lctric fild (magntic fild is ngligibl) is sufficint to hav a good accuracy. I. CONCLUSION In this work th rsults of a klystron cavity using th prturbation masurmnts was prsntd. Th xprssion involvs th natural frquncy f, th prturbd frquncy shift Δ f, th prturbing constant K and th axis factor which rlats th lctric fild along th cavity axis and th lctric fild in th volum of th prturbd objct. Th two first paramtrs wr masurd using a cylindrical rntrant cavity with offst gap built in th laboratory: f using an unprturbd cavity and using th sam cavity prturbd by a mtallic prturbing cylindr. Th prturbing constant K was calculatd by masuring th sam paramtrs mntiond abov using a cylindrical cavity (pill-box). Th intgral ratio was obtaind xprimntally using th Slatr s prturbation thorm. It was vrifid a good agrmnt btwn th cavity obtaind by th xprimnt and th valus found with th analytic xprssions and th 3D ignsolvr as wll. REFERENCES [1] G. Gopln, L. Ludking, D. Smith, and G. warrn, Usr-configurabl MAGIC for lctromagntic PIC simulations, Comput. Phys. Commun., vol. 87, pp , [] K. Halbach and R. F. Holsingr, "SUPERFISH -- A Computr Program for Evaluation of RF Cavitis with Cylindrical Symmtry," Particl Acclrators 7 (4), 13- (1976). [3] CST, Computr Simulation Tcnology. CST STUDIO SUITE TM 8 gtting startd. 8. [4] J. C. Slatr, Microwav Elctronics. Nw York, N.Y.: Dovr Publications, 195. [5] D. M. Pozar, Microwav Enginnring. Nw York, USA: Addison- Wsly Company Inc, [6] K. Zhang, D. L, Elctromagntic Thory for Microwavs and Optolctronics. nd d. Lipzig, Grmany: Springr, 8. [7] D. J. Liska, Elctric fild masurmnts in klystron cavitis, IEEE Transactions on Elctron Dvics., vol. 18, no. 7, pp , jul [8] R. J. Barkr, J. H. Boosk, N. C. Luhmann Jr., G. S. Nusinovich, Modrn Microwav and Millimtr-wav Powr Elctronics. nd d. Nw Jrsy, N.J.: John Wily & Sons, 5. [9] R. K. Silva, D. T. Lops and C. C. Motta, Analytical dtrmination of lumpd paramtrs of rntrant klystron cavitis, IEEE Transactions on Elctron Dvics, to b publishd. 766