Frequency-Domain Study of Lock Range of Injection-Locked Non- Harmonic Oscillators

Transcription

1 0 teratial Cferece mage Visi ad Cmputig CVC 0 PCST vl ACST Press Sigapre DO: 0.776/PCST.0.V50.6 Frequecy-Dmai Study f Lck Rage f jecti-lcked N- armic Oscillatrs Yushi Zhu ad Fei Yua Departmet f Electrical ad Cmputer Egieerig Ryers Uiversity Trt Otari Caada Abstract: This paper ivestigates the lck rage f ijecti-lcked -harmic scillatrs usig a frequecy-dmai apprach. By represetig a -harmic scillatr with a set f harmic scillatrs the itrisic relati betwee the lck rage f the harmic scillatrs ad that f the -harmic is btaied. We shw that -harmic scillatrs exhibit a larger lck rage as cmpared with harmic scillatrs. We further shw that -harmic scillatrs with a multi-te ijecti exhibit a larger lck rage as cmpared with that with a sigle-te ijecti. The theretical fidigs are verified usig a sub-threshld relaxati scillatr desiged i BM 0 m. V CMOS techlgy. Keywrds: N-armic Oscillatrs armic Oscillatrs jecti-lckig.. trducti N-harmic scillatrs such as relaxati scillatrs ad rig scillatrs have bee widely used i passive wireless micrsystems such as as RFD radi-frequecy idetificati systems passive wireless sesrs ad bimedical implats []. Arisig frm prcess spread supply vltage fluctuati ad temperature variati the frequecy f these scillatr exhibits a high degree f ucertaity. The ucertaity f the frequecy f the system clck f passive wireless micrsystems is further escalated by the fact that these systems are usually fabricated usig lw-cst CMOS techlgies which typically have a high degree f prcess spread. Sice the perati f the basebad uits f a passive wireless micrsystem ad the uplik frm the micrsystem t its base stati are ctrlled by its system clck a striget requiremet the frequecy f the system clck exists. Fr example EPC radi-frequecy idetity prtcls class- geerati- UF RFD prtcls require that the accuracy f the frequecy f the backscattered data be upper-buded by ±4% []. Remte calibrati f the system clck f passive wireless micrsystems frm their base stati is critical. jecti-lckig has bee demstrated t be a pwer-efficiet meas fr remte frequecy calibrati f passive wireless micrsystems []. The success f ijecti-lckig based remte frequecy calibrati is largely affected by the lck rage f these -harmic scillatrs. Ultra lw-pwer -harmic scillatrs with a large lck rage are highly desirable i these applicatis. Usig a phaser-dmai apprach Razavi shwed that the lck rage f a harmic scillatr i weak ijecti is give by [4] Q sc where ij ad sc are the amplitude f the ijecti curret ad that f the LC tak curret respectively ij ad Q are the resace frequecy ad quality factr f the LC tak respectively. Revealed by is that the lck rage is directly prprtial t the stregth f the ijecti sigal ad iversely prprtial t the Crrespdig authr: Fei Yua. Tel Fax: fyua@ee.ryers.ca

2 quality factr f the LC tak. creasig the amplitude f ijecti sigal ij thugh icreasig the lck rage is at the cst f pwer csumpti. armic scillatrs such as LC-tak scillatrs typically exhibit better phase ise perfrmace as cmpared with -harmic scillatrs because f its relative high quality factr give by [5]. S y S x 4Q The lck rage f harmic scillatrs hwever is smaller maily due t the arrw-bad characteristics f the LC taks f these scillatrs. additi the ijecti-iduced impedace variati f these scillatrs which is e f the key factrs determiig the lck rage is als smaller as cmpared with that f -harmic scillatrs [6]. A phasr dmai mdel f the lck rage f harmic scillatrs was give i [7]. Maffezzi [8] prpsed a mathematical mdel t predict the sychrizati regi f strgly liear scillatrs. A i-depth mathematical treatmet f lck rage f -harmic scillatrs ad their itrisic relati with that f harmic scillatrs i the frequecy dmai hwever are t available. This paper presets a geeral frequecy-dmai apprach t aalyze the lck rage f -harmic scillatrs. Usig the defiiti f the pe-lp quality factr f harmic scillatrs itrduced by Razavi [5] where Ae j A Q is the lp gai f the scillatr we shw that -harmic scillatrs exhibit a larger lck rage as cmpared with their harmic cuterparts. We further shw that the lck time f harmic scillatrs with a multi-te ijecti is smaller as cmpared with that f -harmic scillatrs with a sigle-te ijecti. The theretical fidigs are verified usig a sub-threshld relaxati scillatr desiged i BM 0 m. V CMOS techlgy.. Lck Rage A distict characteristic f -harmic scillatrs is their square-wave-like utput wavefrm. T capture the essece f -harmic scillatrs we assume that the wavefrm f -harmic scillatrs is a purely square wave such that it ca be represeted by a Furier series with fudametal frequecy / T where T is the frequecy f the scillatr. f the wavefrm f the utput f the scillatr is shw i Fig.a its spectrum will ly ctai the fudametal ad dd harmics. Sice the spectrum f the scillatr csists f a trai f impulses at 5 the scillatr ca be csidered as the assembly f a set f ideal harmic scillatrs that scillate at 5 respectively. Each harmic scillatr ca be represeted by a feedback system that satisfies Barkhause scillati criteria. Whe the ise ad lss f the scillatr are csidered the square wavefrm f the utput f the -harmic scillatr is replaced with the skirt-like wavefrm as shw i Fig.b. Figure : Time-dmai wavefrm ad spectrum f harmic scillatrs Figure : Represetati f ijecti-lcked harmic scillatrs with ijecti-lcked harmic scillatrs. The utput f the -harmic scillatr ca be deted as 4

3 where ij / ad is discrete Dirac fucti depictig the multi-frequecy ature f the spectrum f t. Nte that if 0 ad 0 therwise.whe a ijecti sigal ij t is withi the lck rage f a -harmic scillatr the frequecy f the scillatr will be shifted t that f the ijecti sigal i the lck state. f ij t is a sigle-te at ω ij it ca be represeted by ij t = ij δω ω ij. Sice the utput f the scillatr is a multi-te ij t will affect the iput f the harmic scillatrs represetig the -harmic scillatr. The iputs f the harmic scillatrs are deted by ij ij ijm where the sub-idex m idetifies m th harmic scillatr ad the sub-idex sigifies the sigle-te iput at ω ij as illustrated graphically i Fig.. f ij is a multi-te such as a square wave it ca be represeted by ij ij ij ij ij t 5 Bth the fudametal ad harmic cmpets f ij t will affect the iputs f the harmic scillatrs. The ctributi f ij t the iput f the st rd 5 th harmic scillatrs is represeted by ij ij ij5 respectively. Similar tati applies t ther harmics scillatrs as well. Let us assume that the ijecti sigal ij t causes the fudametal frequecy f a -harmic scillatr t shift frm ω t ω + Δω with Δω <<ω i the lck state. The lp gai f th harmic scillatr ca be represeted by. 6 Therefre A 7 where j e A is the lp gai f th harmic scillatr ad detes the cmplex cjugate peratr. Ntig that 0 A at. Exted Razavi s defiiti f the pe-lp quality factr f harmic scillatrs t th harmic scillatr A Q. 8 Eq.7 becmes Q. 9 Csider a sigle-te ijecti sigal at ij. Let the iputs t st rd 5th harmic scillatrs be ij ij ij5 respectively. The utput f the scillatr is btaied frm ij ij ij ij 0 Usig the first-rder Taylr apprximati Eq.0 becmes ij ij ij ij. Because j i if i=j ad 0 therwise it fllws frm ij ij ij ij. Utilizig 9 derived earlier ad matchig the cmpets that have the same frequecy i yield

4 Q ij ij Q. Sice are the fudametal ad harmics f t we defie the ttal utput pwer f the scillatr assume a Ω lad The summati f the expressis i yields Defie Eq.5 ca be writte as. 4 5 ij ij. Q Q 5 ij Q Similarly if the ijecti sigal is a multi-te the utput f the scillatr ca be btaied by csiderig the effect f the harmic cmpets f the ijecti sigal i.e. idividually. Utilizig ad we have ij ij ij ij ijij ij ij ij ijij ij 8 Matchig the cmpets that have the same frequecy i 8 ad summig each f cmpets as 5 yields. ij ij ij ij ij ij ij 9 ij Q Q Defie Eq.9 ca be writte as ij ij ij ij Q 0. 5 We cmmet the precedig results: i Sice the demiatr f 6 is the ttal utput curret f the scillatr Eq.6 quatifies the ctributi f ij t the verall lck rage f the scillatr. ii Althugh the ijecti sigal is a sigle-te it affects all harmic scillatrs. This is evidet i 7. iii f the scillatr uder ijecti is a ideal LC scillatr as 0 fr = 5 revealig that -harmic scillatrs have a larger lck rage as cmpared with harmic scillatrs. iv Frm 7 ad fr sigle-te is evidetly smaller tha fr multi-te ijectis. v A cmparis f ad reveals that the lck rage f -harmic scillatrs with a multi-te ijecti is larger as cmpared with that with a sigle-te ijecti. The larger lck rage is due t the ctributi f the harmics f the ijecti sigal t each f the harmic scillatrs represetig the -harmic scillatr. vi Althugh plays a critical rle i aalysis f -harmic scillatrs the fial expressi f the lck rage f -harmic scillatrs d t ivlve. Oly Q is eeded.. Example

5 this secti we use the relaxati scillatr shw i Fig t verify the theretical fidigs preseted earlier. The scillatr is desiged i BM 0 m.v CMOS techlgy ad scillates at Mz. All devices are i sub-threshld t miimize pwer csumpti. t is aalyzed usig Spectre frm Cadece Desig Systems with BSM4 device mdels. The scillatr with i a siusid ijecti sigal: iij t ij si ijt ii a dual-siusid ijecti sigal: iij t ij si ijt ij / si ijt ad iii a square-wave ijecti sigal multi-te is aalyzed. Fig.4 shws the spectrum f the scillatr i bth ulcked ad lcked states. Whe the scillatr is ulcked its spectrum ctais multiple frequecy cmpets. Whe it is lcked ly a sigle frequecy cmpet exists. The spectrum f the scillatr is therefre used t determie whether the scillatr is i lck r t. Figure : Schematic f relaxati scillatr with ijectilckig. Circuit parameters: W 670 = 0.5μmW 4 =.μm W 89 =.5 μm W 5 = μm V ref = 0 mv b =5 A V b = 60 mv. Figure 4: Simulated spectrum f the scillatr i lck state. The spectrum f the scillatr i the lck state is left-shifted by Mz fr a better view. ij =.65 A ad = 6.5 A i.e. ijecti rati=0.. Fig.5 ivestigates the depedece f the lck rage the stregth f the ijecti sigal. t is see that icreasig ijecti sigal stregth yields a larger lck rage. Als bserved is that sigle-te ijecti gives the smallest lck rage while square-wave ijecti multi-te yields the largest lck rage. Figure 5: Simulated depedece f lck rage LR f relaxati scillatr ijecti stregth ad the umber f frequecy cmpets f the ijecti sigal. Figure 6: Asymmetry f lck rage f relaxati scillatr. Whe ijecti stregth rises the asymmetry f the lck rage will becme severe. This is evidet i Fig.6 where the psitive lck rage ij ad egative lck rage ij are cmpared. The asymmetry f lck rage is due t the liear characteristics f -harmic scillatrs [9]. the theretical results give i the paper ly first-rder apprximati that gives a symmetrical lck rage is csidered. Whe high-rder terms are csidered the lck rage asymmetry ca be captured. 4. Cclusis

6 We have preseted a geeral frequecy-dmai treatmet f the lck rage f -harmic scillatrs. By represetig -harmic scillatrs with a set f harmic scillatrs the itrisic relati betwee the lck rage f harmic scillatrs ad that f -harmic has bee btaied. We have shw that harmic scillatrs with a multi-te ijecti exhibit a larger lck rage as cmpared with that with a sigle-te ijecti. The fidigs have bee validated usig a sub-threshld relaxati scillatr desiged i BM 0 m. V CMOS techlgy. 5. Ackwledgemet Fiacial supprt frm Natural Sciece ad Egieerig Research Cucil NSERC f Caada ad cmputer-aided tls frm CMC Micrsystems Kigst Otari Caada are gratefully appreciated by the authrs. 6. Refereces [] F. Yua CMOS Circuits fr Passive Wireless Micrsystems Spriger New Yrk 00. [] EPC radi-frequecy idetity prtcls class- geerati- UF RFD prtcl fr cmmuicatis at 860 Mz Mz versi.0.9 EPCglbal c [] N. Sltai ad F. Yua N-harmic ijecti-lcked phase-lcked lps with applicatis i remte frequecy calibrati f passive wireless traspders EEE Tras. Circuits ad Systems Regular Papers Vl. 57 N. pp.8-9 Sept. 00. [4] B. Razavi A study f ijecti lckig ad pullig i scillatrs EEE J. Slid-State Circuits Vl. 9 N. 9 pp Sept [5] B. Razavi A study f phase ise i CMOS scillatrs EEE J. Slid- State Circuits Vl. N. pp. - 4 Mar [6] Y. Zhu ad F. Yua A study f lck rage f ijecti-lcked CMOS active-iductr scillatrs usig a liear ctrl system apprach EEE Tras. Circuits ad Systems Express Briefs Vl. 58 N. 0 pp Oct. 0. [7] X. Lai ad J. Rychwdhury Capturig scillatr ijecti lckig via liear phase-dmai macrmdels EEE Tras. Micrwave Thery ad Techiques Vl.5 N. 9 pp. 5-6 Sept [8] P. Maffezzi Cmputig the sychrizati regis f ijecti-lcked strgly liear scillatrs fr frequecy divisi applicatis EEE Tras. Cmputer-Aided Desig Vl.9 N. pp Dec. 00. [9] S. Shekhar M. Masuri F. O Mahy G. Balamuruga J.E. Jaussi J.Keedy D. J. Allstt R. Mey ad B. Casper Strg ijecti lckig i lw-q LC scillatrs: mdelig ad applicati i a frwarded-clck /O receiver EEE Tras. Circuits ad Systems Regular Papers Vl. 56 N. 8 pp Aug. 009.