Numerical optimization of hybrid dielectric/HTS resonators for surface impedance evaluation of HTS films

1956 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 9, NO. 2, JUNE 1999 Nume ical Op imiza ion o Hyb id Dielec ic/HTS Resona o s o Su ace Impedance E alua ion o HTS Films C. Collado, D. Gonzalo, E. Rozan, J.M. O’Callaghan. Uni e si a Poli ecnica de Ca alunya. Campus No d UPC-D3. Ba celona 08034, Spain. C. Sans Uni e si a de Vic. C/ Mi ama ges 4. Vic 08500, Spain Abs ac - This wo k desc ibes an al e na i e o he adi ional dielec ic esona o opology used o measu ing su ace impedance in High Tempe a u e Supe conduc ing (HTS) ilms. A gap is in oduced abo e he dielec ic so ha only he lowe ilm is in di ec con ac wi h i . This a angemen has been used ex ensi ely o mechanical uning o dielec ic esona o s and, when used o su ace impedance measu emen , i can be designed o make he losses in he uppe ilm small ela i e o he o e all esona o losses. Then, measu ed esul s a e mos ly due o one o he ilms and no he a e age o wo. The speci ics o a esona o design o measu ing 2-inch wa e s a e p esen ed. An analysis and op imiza ion o he esona o is done using a nume ically e icien mode-ma ching algo i hm. I. INTRODUCTION Commonly used echniques o e alua ing su ace impedance in HTS hin ilms a e based on measu emen s o esona o s on ei he pa e ned, plana s uc u es o in a dielec ic esona o ca i y in which he uppe and lowe walls a e made o HTS ilms and a e in con ac wi h he dielec ic [l]. Only he la e app oach is non-des uc i e, bu i yields he a e age su ace esis ance o wo ilms. Thus, o measu e a sample wi h unknown p ope ies, i has o be pai ed wi h a e e ence sample o de e mine i s su ace impedance. The deg ee o accu acy o which he su ace esis ance o he e e ence is known can be a signi ican sou ce o e o in hese measu emen s. The e o e, o he non-des uc i e me hods need o be explo ed u he . 11. HYBRID DIELECTRICIHTS RESONATOR The dielec ic esona o opology unde s udy is a modi ica ion o he adi ional dielec ic-HTS one [ 11, whe e a gap is in oduced be ween he uppe endpla e and he dielec ic cylinde (Fig. 1). The pu pose o in oducing his gap is o lowe he RF losses in he uppe ilm and Manusc ip ecei ed Sep embe 15, 1998 This wo k has been unded by he Spanish Minis y o Educa ion and Cul u e h ough CICYT g an MAT95-1038-C02-02 and by he Ca alan ClRlT h ough g an SGROOO43. Coupling p Dba1 Sapphi e/ c ys ol .G- I , U‘ U Fig. 1. Dielec ic esona o s uc u e. De ini ions o dielec ic heigh (hd), dielec ic adius ( d) and gap size (h,) concen a e hem in he lowe one, so ha i s R, is he la ges con ibu ing ac o o he o al losses o he ca i y. Expe imen al use o his ype o ca i y o measu emen o R, is men ioned in [2], bu no design de ails a e gi en. Am analy ical s udy o his opology, based on an a.xia1 mode- ma ching me hod wi h a single mode is p esen ed in [3]. No desc ip ion is gi en in any o hese wo ks o he p ocedu e o op imize he ca i y o R, de e mina ion pu poses. In [3] i is poin ed ou ha , as he uppe gap is inc eased, he elec omagne ic ields on he no mal me al side walls also inc ease, so he con ibu ion o he RF losses o he uppe ilm a e educed a he expense o inc easing hose o he side walls. Fo R, de e mina ion, an op imal heigh exis s o .a gi en esona o size and a gi en diame e o he supe conduc ing endpla es, o which he a io o he RI3 losses in he lowe pla e o he o e all losses in he ca i y is maximum. This op imum may be ound using an e icien. mode-ma ching code [4], which calcula es he elec omagne ic ield dis ibu ion in s uc u es wi h azimu all symme y. The algo i hm is no es ic ed o small gap sizes and can be used o op imize he whole s uc u e (gap size:, dielec ic heigh and dielec ic diame e ) o a gi en diame e o he supeconduc ing wa e s o be measu ed. Re e ence [2] p oposes a su ace esis ance de e mina ion p ocedu e based in wo consecu i e measu emen s o quali y ac o s in which he op and bo om samples a e ‘exchangecl. The op imiza ion p oposed he e migh make his unnecessa y, so ha a single measu emen and a ough es ima e o he su ace esis ance o he op ilm and side walls migh lead o an accu a e de e mina ion o he su ace esis ance o he lowe ilm. 1051-8223/99$10.00 0 1999 IEEE 1957 Fig. 2. Radial egions (A,B,C) and axial laye s (1,2,3) conside ed by he mode-ma ching algo i hm. Dielec ic pe mi i i y can be de ined independen ly o each o he 9 cells (AI..C3). A. Full-wa e adial mode-ma ching me hod The so wa e de eloped [4] can analyze up o h ee di e en adial egions, each one di ided in o h ee di e en laye s (Fig.2). The dielec ic cons an in each egion and laye can be se independen ly, and he whole olume analyzed is assumed o be enclosed by conduc ing walls. Fields in e e y adial egion a e exp essed as an in ini e se ies o pa icula solu ions o he wa e equa ion in cylind ical coo dina es. Azimu al and axial eigen unc ions a e ha monic unc ions while adial eigen unc ions a e Bessel unc ions, which mus be chosen app op ia ely o a oid singula i y in he axis ( egion A), ul ill bounda y condi ions on he ex e nal walls ( egion C), o p o ide con inui y be ween adjacen egions ( egion B). The eigen alues o each adial egion a e ound by sol ing a pai o anscenden al equa ions ela ed o he h ee-laye p oblem. A sys em o equa ions is hen ound by o cing con inui y condi ions o he angen ial ields be ween adial egions A and B, and B and C, om which he coe icien s o he mode expansion a e ound. In ou case 12 modes a e su icien . Fig. 3 illus a es he ield and cu en dis ibu ions o a sapphi e dielec ic wi h a 12 mm diame e and a 6 mm heigh . B. Op imiza ion. Basics and al e na i es Fo R, de e mina ion, he op imal dimensions o he ca i y in Fig. 1 a e hose ha make he losses o he lowe pla e dominan wi h espec o he o he ype o losses. This is equi alen o maximizing Qo/Qlc,w, whe e: U is he o al s o ed ene gy in he ca i y, Pd he o al dissipa ed powe , Pd-/ow he powe dissipa ed in he lowe wall, and Qo he unloaded quali y ac o . Bo h he o al ene gy U and he dissipa ed powe s Pd and P l-lOw can be calcula ed om he ield dis ibu ion inside he ca i y wi h p ope ly de ined su ace and olume in eg als o he elec omagne ic ields [5]. De ini ions analogous o ha o Qlclw in (1) can be used o Qu/,, Q.,d and Qd which accoun o he losses in he uppe wall, side wall, and dielec ic espec i ely. Thus, he unloaded quali y ac o can be w i en as: ...._.......... .... , I,,................ I#,,.-......... . ... Il . ............. I ,.- ]I!..-::::::::' :::: 'id, ........ .. -),,,.......... /j/i ,,,........ . $:j:::::;:;;:;;: hzj::: : :: : : : : : : : I -.%.-A " .. . . . . . . . . . . . . -+c+ 4- - . . . . . . . . . . . Fig. 3. Field and cu en dis ibu ions in he ca i y wi h a TEOI mode and a sapphi e dielec ic. Bo om pla e diame e is 48mm, dielec ic diame e is 12mm and dielec ic heigh is 6mm. The dashed line ep esen s he sapphi e ou line. (Top) Radial cu o he esona o showing magne ic ield lines. (Bo om) Cu en s a he su ace o he lowe HTS ilm. The ele ance o maximizing Q0/Qluw is due o he ac ha his de ines he sensi i i y o he sys em: (3) which can. be p o en om (1) and (2), and om he p opo ionali y be ween and R,. Also, as Qo/Qlc,w is inc eased, he accu acy in es ima ing Q,,, QSd and Qd has a lesse e ec on he R,y p edic ed om he measu ed Qo and (2). A so wa e code has been de eloped, which includes he elec omagne ic mode-ma ching algo i hm desc ibed p e iously, and is capable o calcula ing Q clw , Q,, QSd and Qd as a unc ion o dielec ic heigh (hJ, dielec ic adius ( d) and gap size (h,) (Fig. 1). To es i , he ca i y dimensions gi en in [2] we e used, and he code esul s we e compa ed wi h hose in Table I in ha e e ence. The ag eemen is excellen , as shown in Table I o his wo k. The so wa e de eloped can use se e al s a egies o a y hd, d and h, in he sea ch o a maximum o Qo/QlcIw : 1) a ia ion o he gap size o a gi en ( ixed) dimension o he dielec ic; 2) exhaus i e sea ch a ying hd, d and h, o e a speci ied ange, wi h a speci ied inc emen al s ep size; and 3) 1958 -D- Q-Side wall (Qsd) ~ "i j_ + .... 4 - Q-Uppe wall (Qup) ' .i.. .... ~. I. :. i.. ........ : - . Q-Dielec ic (Qd) ~ - .. . --+-.Q~/QIow~ . ,. F ,< j .TABLE I COMPARISONS BETWEEN 121 AND THIS WORK Glow is de ined as he p oduc o he su ace esis ance o he lowe pla e and Plow; Cup and Gsd a e de ined simila ly. (GHz) Glow GUP Gsd PI 9.5 546 4.90e3 1.15e6 This wo k 9.510 541 4.81e3 l.lle6 E o % 0.8% 1.7% 3.3% ,. ,. -' I .. -. ....... + .....; ........................................... I +, L.- -- ............... & .................................. .> +-I--- __ x- -. 1 *. - -&--*----+-; ;A' I - /I y; .. ,, . ,- ~, . I- ; : ....... ~ ................................ g adien sea ch using Ma lab [6]. In his p ocess i is assumed ha he uppe and lowe walls a e made o YBa2Cu3O7.8 (YBCO), and he side walls o a no mal me al (usually OFHC coppe , bu i can be changed by he use ). Thei R,T is scaled as he calcula ed esonan equency o he ca i y changes due o a ia ions in hd, d and h,. The scaling law is aken as * o YBCO and o he me al, and he alues o R,y a lOGHz and 77K a e aken as 300@ (YBCO) and 9mQ (Cu) [7] espec i ely. The equency dependence o he loss angen o sapphi e is neglec ed, aking ank4 [8] h oughou he op imiza ion. C. Op imum gap o ixed dielec ic and wa e sizes 10 i ; [1 *i- xi.- A -/ Y U. 11'1,,,,111,,,,,111,1,, Fig. 4 shows he esul s o he gap heigh op imiza, ion o a dielec ic wi h hp6mm, p6mm and a 2 inch wa e whe e only an a ea o 48mm in diame e is exposed o he elec omagne ic ields in he ca i y. This igu e indica es ha , while he losses in he side walls a e s ongly dependen on h,, his dependence is mode a e in he uppe wall, and weak in he dielec ic and lowe wall. The e o e, he op imum heigh ep esen s a ade-o be ween he losses in he uppe wall and hose in he side wall. This esul s in an op imum gap o h,=6mm and Qo/Q1,,,,=O.85, which essen ially coincides wi h he esul s epo ed in [2]. As shown la e , subs an ial imp o emen s can be made by adjus ing h,l, d and h, join ly. Finally, Fig. 5 shows ha hese esul s deg ade !s ongly i he diame e o me al side wall is educed. D. Subs i u ion o he uppe HTSpla e by a coope one Unsuccess ul a emp s ha e been made a ep lacing he YBCO in he uppe wall by coppe . E en when he gap heigh op imiza ion p ocess was pe o med aking in o accoun he new R,, o he uppe pla e, he sensi i i y (Qo/Ql(,,,) alues we e oo low o allow an (accep able accu acy in he de e mina ion o he R,, o he bo oin pla e. Inl o he wo ds, he gap can ne e be high enough o eplace he: uppe ilm by a no mal me al wi hou making i a dominani. sou ce o loss in he ca i y and ende ing i useless €o su ace: esis ance de e mina ion. Fig. 6 shows he e ec s o his subs i u ion in he sensi i i y (Qo/Q,,J. When dimensions o he wa e and he dielec ic a e known and ixed, only h, needs o be a ied o ind a maximum o Qo/Q o,,,. This is done o wo easons: o explo e a possible imp o ed design o he ca i y in [2] (while main aining he sapphi e used); and o ind a good ini ial se o alues o hd, d and h, om which a second op imiza ion can be done using a g adien sea ch in which all h ee a iables a e allowed o change. E. Join op imiza ion o hd, d and h, The alues o hd, d and h, ha e been op imized using wo pa allel app oaches: a g adien op imiza ion and an IO" 1 O'O 1 0' 1 o6 1 0.8 1 o8 0.6 Q . gi 4 0.4 0.2 0 Fig. 4. Dependence o Qw, elow, Qsd wi h gap heigh (h,). The op imum heigh (max. QO /Q& ep esen s a ade-o be ween he losses in he uppe and side walls (elq, Qsd), whose dependence on h, is much s onge han ha o he lowe wall (elow) and dielec ic (Qd). The maximum o Qo/QlOw is a h,=6mm. 0.8 0.6 8 . 0.4 0.2 Ca i y adius ( n n) Fig. 5. Dependence o Qw, elow, Q,Td wi h ca i y adius. As he adius is dec eased, losses in he side walls inc ease, become dominan , and deg ade QO /el(,,,,. Gap heigh is hg=6mm. As in Fig. 4. dielec ic heigh is h~6mm 1959 Fig. 6. De e io a ion o he sensi i i y (Qo/QlOw) as he alue o R, o he uppe sample is inc eased. A s ong deg ada ion occu s o he ange o R,T alues co esponding o coope a 77K a he esonan equencies o he ca i y (abou 10 mn). Fo hese alues o Rs in he uppe wall, he Qo/Qim is oo low o de e mina ion o he R,y o he lowe HTS ilm. exhaus i e sea ch o e a b oad ange o hd, d and h, o ensu e ha he g adien op imiza ion did no s op a a local maximum o Qo/Qbw. The exhaus i e sea ch was done om 2mm< hdlOmm, 3mm< 614mm and 1mm<hg<15mm. A maximum was ound a hd=2mm, ~5mm and h,=6mm wi h Qo/Ql,,,~0.94. The sea ch showed ha his op imum alue would deg ade sha ply when hd was dec eased beyond i s op imum alue. Howe e , he dependence wi h hd and d was much weake , G adien op imiza ions we e un s a ing om se e al alues, including he ones om Sec . 11-C. They all eached simila alues o Qo/Q o,,,. The bes one was he one s a ed om hs2mm, d=5mm and h,=6mm ( he op imum o he exhaus i e sea ch), which was used o ine- une he p e ious sea ch esul ing in hd=l.7mm, &.Omm and hg=4.6mm wi h Qo/Qlow =0.95. In o he wo ds, i no o he losses a e conside ed in a ca i y wi h hese dimensions, only a 5% o e es ima ion o he su ace esis ance o he lowe sample will be made, and his may be g ea ly wi h es ima es o he su ace esis ance o he side walls, uppe ilm and dielec ic loss angen . 111. CONCLUSION An op imiza ion o he image dielec ic esona o ca i y has been done by nume ical echniques. The so wa e de eloped o he op imiza ion has been c oss-checked wi h p e iously published esul s. The op imized ca i y should be allow he de e mina ion o su ace esis ance o YBCO wi h an e o lowe , han 5% i no o he sou ces o RF loss in he ca i y a e aken in o accoun . REFERENCES Z-Y Shen, C. Wilke , P. Pang, W. L. Hols ein, D. Face, D.J. 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