DRIVE LEVEL DEPENDENCE IN QUARTZ CRYSTAL RESONATORS AT LOW DRIVE LEVELS: A REVIEW

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1 DRIVE LEVEL DEPENDENCE IN QUARTZ CRYSTAL RESONATORS AT LOW DRIVE LEVELS: A REVIEW R. Bendel a, M. Addouche b, P. Salzentein a, E. Rubiola c, and Y. S. Shmaliy d a FEMTO ST Laboatoie de Phyique et Métologie de Ocillateu 3, avenue de l'obevatoie F544 Beançon cedex Fance bendel@lmo.edu Fax +33 () b Laboatoie d Atohyique de l Obevatoie de Beançon 41, avenue de l'obevatoie F53 Beançon cedex Fance c ESSTIN LPMI Univeité H. Poincaé, avenue Jean Lamou F54519 Vandœuve lè Nancy Fance d FIMEE Guanajuato Univeity.o. Box 15-A, Pol. Tamico S/N 3679, Salamanca Gto Mexico Keywod: Quatz, Reonato, Dive Level Senitivity Abtact Uually, tating of ocillation in a quatz cytal ocillato equie a eonato inut owe in the ange of dbm, but unde toage a henomenon known a Dive Level Deendency (DLD) o Dive Level Senitivity (DLS) may aea that event the tating of ocillation. Seveal tudie efomed in the at have hown that at low dive level ome quatz eonato may exhibit a lage inceae of thei eie eitance eventing the tating of ocillation. Thi wok eview the tudie and eult obtained fo nealy fifty yea on vey low dive level enitivity of quatz. The vaiou mechanim and model baed on the hyothei of moving aticle and uface defect in the eonato inducing eitance inceae and it elation with noie mechanim ae eviewed a well. Alo, the ae decibe eveal exeimental et-u, and meauement ocedue ued to obtain vey low dive level motional aamete. Thi wok i a contibution to undetand the oblem of tating quatz afte a long toage eiod. Some eliminay eult of the eie eitance meaued at vey low dive level ae alo eented. 1 Intoduction Fo about fifty yea, enginee and manufactue have been faced with the aticula behaviou of cetain ocillato that do not tat deite the fact that the eonato aamete meet the ecification at nomal dive level [3]. It wa found that the eaon why thi haen come fom a lage inceae of the eonato eie eitance at low dive level that cannot be comenated by the negative eitance of the feedback amlifie [1]. The numbe of name thi henomenon ha been given how that it i vey common: autho often efe to Stating Reitance [3], Had Stating Chaacteitic, Sleeing Sickne [4], Cuent Deendency of Cytal Reitance [1], Low Level Dive Senitivity [8], Second Level of Dive [15]. Although the name Dive Level Deendency [9] i widely ued in thi context, it i often ued to mean the aniochonim effect occuing at high dive level, to avoid miundetanding with thi non-linea henomenon that ha a quite diffeent oigin, it eem wie to ue a diffeent exeion uch a Dive Level Senitivity (DLS) [1] ued hee 1. Vey ealy, DLS ha been attibuted to uface defect coming fom micocoic ca of vaiou oigin often aociated with a ticky uface coating [3] o uface catche [8,1]. The inceae of the eonato eie eitance i one of the mot obviou effect of thee uface defect, but othe aaitic effect hould have the ame oigin uch a intemodulation in monolithic filte [8,14], ageing [4,6], fequency jum [1], o noie [,1,3]. On the othe hand, the exteme enitivity of the eonato chaacteitic to a tiny uface modification ha been tuned to account by uing it a a eno fo the chaacteization of bonding foce between a uface and gold aticle [9], tainle teel [31], o olytyene hee [16]. DLS behaviou and chaacteitic Many exeiment have been eoted duing eveal decade, ome of the mot demontative will be ummaized hee. Fo examle, a monotonic deceae of the eie eitance a the dive level inceae (Fig. 1) [1], ometime, eie eitance tat low, inceae u to a maximum then deceae (Fig. ) [1], mot of the time, the henomenon i not eveible, and exhibit an hyteei with unedictable thehold a hown in Fig. 3 [1]. Note that the eitance change i accomanied with a change of the eonance fequency eithe negative o oitive a in thi figue. Alo, aymmetical behaviou a hown in Fig. 4 ha been eoted [8]. 1 A moe igoou and unambiguou definition of the effect would equie to call it Low dive level DLD o DLD at low dive level. Hee the tem Dive Level Senitivity o DLS hould be conideed only a a hot cut.

2 Cytal Cuent (µa) Figue 1: Monotonic deceae [1] Cytal Cuent (µa) Figue : Maximum eitance fo a aticula dive level [1]. f/f (m) Cytal Cuent (µa) Figue 3: Hyteei of the eonance fequency and eie eitance [1]. A lage numbe of exeiment caied out fo decade have led to the following obevation: Inceae of the eie eitance i alway aociated with a oitive o negative fequency hift. The DLS ignatue tongly deend on the temeatue [1]. The DLS behaviou can be modified o ueed emanently o temoaily by ovediving the eonato [3,1], by olihing, etching o cleaning the cytal, thi latte oce being often conideed a the mot efficient [7,15]. The mot futating aect of thi henomenon i it lack of eoducibility, hence cytal eemingly identical may be dive enitive o not, and DLS of cytal aaently cued may eaea afte a long time of inactivity. 3 Oigin of the DLS A lot of wok and effot have been ut into undetanding the oigin of the henomenon, and vey ealy the attention ha been focued on the uface imefection a a oible caue of the DLS. Among the mot often eoted uface defect involved one can cite: Paticle of metal o quatz o abaive, Thin coat of ein o oil, Suface catche, Flaking of quatz uface o metal electode, Pooly adheive electode o blite, Suface te. Vaiou exeiment have oved the elationhi of caue and effect between the uface ollution and DLS. Fo examle, talc blown in the vicinity of an unealed quatz eonato may induce DLS [5]. Anothe inteeting and damatic demontation of the coelation between uface contamination and DLS ha been eoted a few yea ago [6]: mall quae in Figue 6 i a ecod of the elative eitance v. dive level of a 1 MHz 5 th ovetone AT-cut cytal that doen t exhibit a noticeable DLS. The ame eonato ha been oened and the uface ha been inkled with alumina aticle a hown in Fig. 5, afte the eonato ha been eealed, it eent an imotant DLS (mall cicle in Fig. 6). Once the eonato ha been eoened, cleaned and eealed, it aoximately ecoveed it oiginal tate (mall tiangle in Fig. 6). 6 Motional Reitance (Ohm) Dive Level (dbm) Figue 4: Peak and hyteei of the eie eitance [8]. Figue 5: Paticle of alumina on the uface of a eonato [6].

3 Dive Level (dbm) Figue 6: Effect of a uface contamination [6]. 4 Mechanim of the DLS No aticle AlO3 aticle Paticle emoved Many attemt have been made to exlain the mechanim eonible fo the DLS. It hould be noted fit that a aticle by oneelf, that i without any bound with the uface, cannot induce the obeved henomena. On the othe hand, if the aticle wee tightly bound to the uface, it would act a a ma loading and hould induce a negative fequency hift which i often efuted by exeiment. So, obeved henomena can be exlained only if the aticle i bound to the uface by an elatic foce that can be due to a thin ticky coating of oil o ein, o any othe attactive foce uch a Van de Waal, electotatic, caillaity foce fo examle [31] (Fig. 7). In thi cae, a the uface move back and foth unde the hea motion, the bounded aticle act a a mall ocillating ytem (Fig. 7c). (a) (b) (c) Paticle Quatz late Electode Figue 7: Paticle bound to the uface by an elatic foce. K u M k F = F e j ω t Figue 8: Simle mechanical model [9]. m x 4.1 Simle mechanical model A imle model of couled ocillato ha been ooed on the bai of a mechanical analogy eeented in Fig. 8 [9]. The lage ing ma ytem at left mimic the eonato motion, while the mall one at ight eeent the aticle motion. The hea motion of the uface act a an extenal diving foce F. Uing Newton law with the ytem aamete defined hee, the motion of the two mae can be obtain unde the fom of a diffeential ytem [Equation (1)]. jω t Mu&& + Ku + k ( u x) = Fe (1) mx && + k ( x u) = Seaching a hamonic olution of the ytem, the eonant fequency of the couled ytem [Equation ()] can be exeed a a function of the eonant fequencie of the iolated ing ma ytem and of the tiffne atio α [Equation (3)]. ω + (1 + α) ω 4ω ω = ω 1 ± 1 () c ω + (1 + α) ω m k ω = ω = M K α = K k (3) By lotting the couled ytem eonant fequency a a function of the tiffne atio, it tun out in Fig. 9 that fo a weak aticle binding the eonant fequency hould inceae while fo a tong couling, the fequency hould deceae a fo the ma loading effect. f/f (m) Weak aticle binding Couling Contant Ratio k / K (m) Stong aticle binding (ma loading) Figue 9: Reonant fequency of the couled ytem. 4. Suface taed aticle model Although the imle couling ytem we jut eented doe exlain that the eonant fequency can eithe deceae o inceae, it doen t give infomation on the daming mechanim. Even if linea daming tem wee intoduced in the eonato and aticle ing ma equivalent ytem, they would induce a linea equivalent daming tem indeendent of the dive level. Thu, a dive level enitive daming tem mut invoke ome non-linea mechanim uch a ooed by Dwoky [8] auming that the aticle taed in ome uface imefection hould exeience inelatic colliion with catch wall a illutated in Fig. 1 fo examle, thu inducing the equied non-linea daming tem. Figue 11 how the mechanical equivalent couled ocillato that look like the eviou one with daming tem added to the ing ma ytem.

4 g Figue 1: Suface taed aticle [8]. K R u M k F = F e j ω t Figue 11: Mechanical equivalent of a taed aticle [8]. Motion Equation (4) ae deived fom the Newton law whee the aticle daming facto ha been given the fom of Equation (5) uggeted by ome tatitical and hyical conideation. jω t Mu&& + Ru& + Ku + k ( u x) + ( u& x& ) = Fe (4) mx && + k ( x u) + ( x& u& ) = + n = 1 x g u (5) The mechanical couled ytem eviouly eented ha an electical equivalent cicuit eeented in Fig. 1 whee the non-linea daming tem of the aticle i modelled by a nonlinea eito the eitance of which deend on the diving cuent [Equation (6)]. n i( ) = 1 + (6) g + V = F Z m L = M C = 1/ K R x C d = 1 / k L d = m Figue 1: Electical equivalent of a taed aticle [8] =.1 =.1 = Dive Level (dbm) M = 1 K = 1 R =.5 m =.1 k =. Figue 13: Modelling of a dive level enitive eitance [8]. The equivalent eitance of the couled ytem can be obtained fom the eal at of the equivalent imedance Z calculated by uing numeical method. Accoding to the value given to the linea aticle daming tem, Fig. 13 how that the model atifactoily deict vaiou behaviou obeved exeimentally excet the hyteei henomenon. 4.3 Phyical model U to now, the model ued give only qualitative exlanation on the obeved henomenon. By uing a hyical ditibuted model fo the eonato [1] it i oible to exe it electical admittance unde the fom given by Equation (7) whee the fit tem of the ight hand ide eeent the eie banch admittance. The aamete K q deend on the geomety, on the hyical contant, on the oientation, and on the aticula vibation mode of the cytal late. The econd tem i the tatic o aallel caacitance of the eonato. In thi model, the cytal loe ae intoduced unde the fom of an imaginay at in the exeion of the eie eonant fequency [Equation (8)]. jωkq Y = + jωc (7) ω ω ω = ω + jω (8) The influence of a aticle located on the eonato can be calculated auming it act a a mall ing ma dahot ytem inducing an equivalent uface foce F a hown in Fig. 14. F Quatz late i Paticle: ma = m, couling contant = k, daming facto = Figue 14: Action of a lumed ing ma dahot aticle on a ditibuted eonato model [1]. In thi cae, it can be hown that the aticle intoduce a etubation tem in the denominato of the eie banch admittance [Equation (9)], that ha the fom given by Equation (1). The etubation tem can be exeed a a function of the aticle eonant fequency, and the ma atio of the aticle to the ma of the actively vibating quatz egion [Equation (11)]. jωkq Y = j C mu F + ω (9) ω ω j m mu F ω ( ω + ω ) = α ω ω + jω m (1) = M m α ω = m k (11) Figue 15 how the elative eie eitance change of the eonato a a function of the aticle daming to ma atio fo two value of the aticle to eonato ma atio. Note that a ma atio of only 1 m would induce a 1 % inceae of the eie eitance and thi value would damatically inceae if the aticle eonant fequency i cloe fom the eonato one.

5 5 4 α = 5 m L 1 C 1 R α = 1 m Paticle Daming / Ma Ratio ( / m) Figue 15: Phyical modelling of a dive level enitive eitance [1]. Beide of it influence on the ue hea mode equivalent admittance eviouly decibed, a localized iegulaity can alo be eonible fo eonant mode ditotion due to mall aaitic flexual motion that induce additional vibation enegy loe [11]. Futhemoe, the tong deendence of the aaitic effect with the aticula location of the iegulaity hould exlain the hyteei o othe ieoducible behaviou of the DLS a well a the high dive level cuing mechanim [1]. 5 Exeimental method and eult Fom the exeimental oint of view, eveal method can be ued to meaue DLS. In the ealiet exeiment, the cytal imedance mete wa the mot common method ued [3]. Then, an eaie way wa to ue an ocillato with vaiable gain a chematically deicted in Fig. 16. At low gain, the negative eitance of the amlifie cannot comenate fo the eonato lo o that the ocillato doen t tat. By ogeively inceaing the amlifie gain u to each the ocillation level it i oible to know if the cytal i dive level enitive o not. Nevethele, thi fat and inexenive method doen t give deendency with dive level and i mainly ued fo ceening of cytal to ecified accetance limit. [17,9]. R q Figue 18: Cytal equivalent cicuit. The aamete of the eonato equivalent cicuit (Fig. 18) can be calculated fom the amlitude and hae tanfe function of the meauement netwok the cytal i ineted in [8]. Fo high fequency eonato, thoe aamete can be obtained fom the catteing aamete meauement by uing automatic netwok analye [1,3,4,5]. With the develoment of mat intumentation and comute aided meauement technique it i now vey convenient to ue netwok/imedance analye that enable the ue to vay eveal aamete uch a the inut ignal magnitude and to calculate the equivalent cicuit aamete fom the imedance magnitude and hae ecod [18,19]. In the eent cae, the exeiment have been efomed by uing an Agilent 4395A Netwok/Sectum/Imedance Analye and the 43961A Imedance Kit [,33]. To imove the accuacy of the meauement, the eonato unde tet ha to be iolated fom the envionmental etubation mainly due to temeatue fluctuation. To thi end, the imedance tet kit ha been comlemented with a eciion oven keeing the cytal at it tunove oint (Fig. 19, ). C Netwok/Sectum/Imedance Analye Agilent 4395A RF out R in A in 5 DB Attenuato Imedance Tet Kit Agilent 43961A Ovenized cytal +5 db dual tage low noie amlifie Figue 19: Low dive level cytal aamete meauement. R Figue 14: Active method fo DLS meauement [9]. Moe accuate meauement of DLS can be achieved by uing aive tanmiion method uch a the oula IEC-444 Pinetwok (Fig. 17) widely ued to meaue the eonato motional aamete [13,,3]. Ovenized cytal Syntheize Attenuato Z q Vecto Voltmete Figue 17: Simlified IEC-444 Pi-netwok cytal aamete meauement method [3]. Imedance analye Figue : Exeimental et-u. Imedance tet kit

6 Beide, o a to lowe the dive level ange initially et between 5 to + 15 dbm, the exeimental et-u ha been modified a hown in Fig. 19: A wide band 5 db attenuato i ineted in the RF inut ignal and two dual-tage low noie amlifie ae ineted in the meauing ot of the tet kit allowing meauement fom aoximately 1 dbm to - 35 dbm. It hould be noted that a caeful attention ha to be aid to the calibation ocedue to guaantee the meauement accuacy. Figue 1 eeent two tyical ecod of the imedance the cytal aamete ae calculated fom. Of coue, a the dive level deceae, the ecod become moe and moe noiy and eveal meauement ae equied to inceae by aveaging the ignal to noie atio. Figue eent the dive level enitivity of a 1 MHz SC-cut 3 d ovetone quatz eonato. Nevethele, the exeimental et-u i not yet accuate enough to obtain atifactoy meauement of the cytal eactive aamete L 1 and C 1, and thi lot i only a eliminay eult that i till to be confimed by additional exeiment. Seie eitance R1 (Ohm) Dive level (dbm) Figue : Low dive level enitivity of a 1 MHz SC-cut 3 d ovetone quatz eonato. 6 Noie and DLS The main eaon why thee exeiment ae cuently caied out i to check fo a oible coelation between dive level enitivity and noie of the eonato that hould have the ame oigin. One of the oible mechanim elating thee two henomena ha been uggeted in the at [3]. It aumed that ome contaminant ecie ae andomly taed to and eleaed fom N oible uface ite at ate and 1 eectively, each taed aticle cauing an aveage elative fequency hift f/f (Fig. 3). Adotion ate Deotion ate 1 Numbe of adotion ite: N Quatz late Actual owe: 6 dbm Actual owe: 9 dbm Figue 1: Low dive level cytal imedance ecod. Figue 3: Noie induced by a contaminant ecie [3]. In thi cae it ha been hown that the ectal denity of hae fluctuation ha the fom given by Equation (1) eeented in Fig f S ( f ) = 1 φ N( ) f (1) + ( ) ( ) πf 1 Exeimental veification of the exected coelation between dive level enitivity and eonato noie ha alo been invetigated [,6]. The eult eented in Fig. 5 eem demontate that thee i a oitive coelation between DLS meaued by the loe of the eitance change and the noie meaued by the eidual hae noie at 1 Hz fom the caie []. Nevethele, thee omiing eult have not been confimed by anothe et of exeiment efomed on a lage amle of eonato a hown in Fig. 6 [6].

7 Sφ (dbad /Hz) White fequency noie Random walk fequency noie Fequency (Hz) Figue 4: Sectal denity of hae fluctuation fo a 55 MHz eonato [3]. Reidual Phae 1 Hz (dbc/hz) R 1 Change Between 1 dbm and dbm (Ohm) Figue 5: Noie v. DLS of a 1 MHz AT-cut quatz []. Reidual Phae 1 Hz (dbc/hz) Vaiance of R max / R min Between 4 dbm and 15 dbm Figue 6: Noie v. DLS of a 1 MHz AT-cut quatz [6]. Thu, the quetion of a oible coelation between noie and uface defect that could be dicloed by meauing DLS i till oen. A theoetical noie model baed on a non-linea eie eitance R 1 (Fig. 18) whoe value i a function o the cytal cuent a hown in Equation (13) ha been intoduced in the at [7] and i till unde conideation [6]. γ R = R 1 + n (13) 1 1 I In Equation (13): R 1 i the nomal dive level eitance, γ i the DLS coefficient, I i the eonato dive cuent amlitude, n i an intege numbe (1 n 4). 7 Concluion In ummay, the low level dive enitivity of quatz eonato i mainly exeed by a change of the eie eonant fequency and eie eitance that may exhibit vaiou and often ieoducible behaviou. The coelation between DLS and uface defect i now well etablihed and eveal model have been develoed to decibe the hyical mechanim. Nevethele, the quetion of the coelation between DLS and eonato noie i till oen. Many othe quetion deeve caeful attention uch a: What haen at vey low dive level, nea the themal noie floo? How behave the othe motional aamete at thee low dive level? Ae the mechanim eonible fo the DLS the ame a fo the noie? How the aumed coelation can be demontated? The wok in oge will ty to give an anwe to thee quetion. Refeence [1] J. P. Auby, E. Gead, S. Lechoie. S. Y. Paamete Method fo Accuate Meauement of Bulk Wave Cytal Reonato at Fequencie u to GHz, Poc. 37 th AFCS, , (1983). [] P. Bate. Dive Level Deendence Veu Reidual Phae Noie of Fifth Ovetone AT Cut Quatz Cytal, Poc. 54 th IEEE IFCS, , (). [3] M. Bentein. Inceaed Cytal Unit Reitance at Ocillato Noie Level, Poc. 1 t AFCS, , (1967). [4] V. E. Bottom. Intoduction to Quatz Cytal Unit Deign, Van Notand Reinhold Co.,. 8 19, (198). [5] V. E. Bottom. The Behavio of Quatz Reonato at Low Dive Level, Poc. 5 th Cytal Confeence, , (1983). [6] E. E. Calon, T. E. Wickad. A Study on the Meaued Coelation of Dive Level Deendency and Phae Noie of Quatz Cytal Reonato, Poc. 55 th IEEE IFCS, , (1). [7] R. C. Chittick, D. M. Hyland. A Peliminay Invetigation into a Novel Aoach fo the Removal of Dive Level Deendence in Quatz Cytal Reonato, Poc. 11 th EFTF, , (1997).

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