PART V. PLL FUNDAMENTALS 1

Transcription

1 all-017 Joe Slva-Martnez PART. PLL UNDAMENTALS 1 The phae locked loop a very popular crcut ued n many dfferent applcaton; e.g. frequency ynthezer, M and phae demodulator, clock and data recovery ytem, clock ynchronzaton, etc. PLL have the ablty to extract gnal n a very noy envronment. It fundamental component are the phae detector, loop flter and voltage controlled ocllator. Although th a non-lnear crcut operatng wth large gnal, t typcally lnearzed and analyzed n the phae doman rather than n the frequency repone. In the followng mple analy, we aume that the loop n teady tate and the frequency of both ncomng gnal and CO the ame. The only potental error n th cae a phae error, hence the loop analyzed for phae gnal. Important parameter uch a noe performance, trackng range and lockng range are decrbed n the followng ecton. Steady tate analy. If the phae detector an deal multpler, and f the loop gan very hgh uch that the error gnal mall, and f the hgh frequency component generated by the multplcaton of the nput gnal, then the followng mplfed 1. M. Gardner, Phae Lock Technque, Wley, New York, nd edton, R. E. Bet, Phae Locked Loop: Theory, Degn and Applcaton, McGraw Hll, New York, M. S. Yuan and C.. Wang, PLL Crcut, chapter 51 n The Crcut and lter Handbook, Wa- a Chen, CRC-Pre, nd edton,

2 all-017 Joe Slva-Martnez equvalent crcut can be ued for the analy of the PLL. Pleae be aware of all thoe mplfcaton n the analy, be cautou wth the nterpretaton of the reult. Phae Detector. Neglectng the hgh order term, we have v v LO LO R co t n LO Rt O or an deal analog multpler wth a multplcaton factor, we have v LO LO R R co n t n t LO t co R O LO R O R LO t O If LO~R, the loop cloe to t teady tate, and f the loop flter reduce further the hgh frequency component, the phae detector output then gven by the followng expreon: v n R LO t O Only f the loop operatng n teady tate, LO~R, and for mall phae error 0 < (mall gnal analy) the output of the phae detector can be approxmated a v n O 0 Th term compoed by a tatc term and the gnal error e=0; actually the mall gnal gan of the phae detector. Loop flter. The purpoe of th flter to provde hgh n-band gan, proper locaton of compenatng pole-zero to guarantee loop tablty and to flter out the hgh - -

3 all-017 Joe Slva-Martnez frequency component reultng of the phae/frequency comparon. If LO~R, the loop cloe to t teady tate, and t can be analyzed for phae repone only. The flter can be repreented n the -doman by ung the regular frequency tranfer functon, H() to be dcued n the followng ecton. oltage controlled Ocllator (CO). The voltage controlled ocllator mut be properly modeled. In a frt approxmaton, the CO a voltage to frequency converter that can be repreented by the followng equaton: LO LP Typcal CO tranfer charactertc: O LO LP t O LO LP t Where LO the free runnng frequency and LP(t) the control voltage. the CO gan gven n rad/ec-. a very mportant parameter that gve u a lot of nformaton; typcal ntegrated R CO have gan value n the range of MHz/ (actually th value multpled by ). Therefore you have an f-varaton of 100 khz per m appled to CO control termnal; evdently the nput referred noe level (embedded n LP) mut be mall enough not to generate large varaton n the frequency of the LO. If LP(t) corrupted wth noe n(t) then - 3 -

4 all-017 Joe Slva-Martnez O LO t nt t nt LP LO LP To make the ytem le entve to noe, very large CO gan factor are typcally avoded unle you can guarantee extremely mall nput referred noe fgure. The CO phae defned a the ntegral of the frequency; then t follow that O LO t LP t dt In th equaton, 0 the free runnng frequency, and theoretcally t not changng n tme, then for phae analy t condered a a tme nvarant quantty; th however not true n real mplementaton but tll frequency drft are uually mall to affect ytem tablty unle the free runnng frequency goe beyond the frequency range of nteret. Although 0 can not be elmnated from 0, when the ytem locked, 0= and thoe term get cancelled at the output of the phae detector; therefore, 0 uually the carrer frequency and doe not play a qute relevant role on PLL dynamc. In the frequency doman, and for ac phae varaton, the CO phae tranfer functon can then be repreented by an ntegrator a follow O LP rom the prevou mplfcaton we can contruct the equvalent teady tate phaerepreentaton crcut for the phae-locked loop ytem a follow: Phae detector e = - 0 Loop flter CO () () e 0 S 0-4 -

5 all-017 Joe Slva-Martnez Notce n th plot that nput and output are phae gnal. Although th a very mple repreentaton, we can clearly dentfy the relevant PLL parameter; we have to ue our flter knowledge for the nterpretaton of the fundamental loop properte. The frt one to be dcued the phae tranfer functon defned a o 1 Loop gan defned a A loop In cae the loop flter an allpa, ()=0, the loop gan follow the hape of the CO tranfer functon and the loop alway table nce the ytem a ngle-pole. Notce that the CO ntroduce pole at the orgn, a a reult of th, the loop gan at DC nfnte, unle () preent a zero at dc whch not the cae for all practcal crcut. Infnte dc open loop gan a very derable feature nce for cloed loop tranfer functon the error nverely proportonal to t open-loop gan. It expected then to have zero error at dc. A very mportant parameter the error tranfer functon; t can be defned n everal way, but the mot popular one defned below: e o o A loop A expected, for a frt order loop the error functon very mall at low-frequence. The nterpretaton of th reult that the output track very well the nput at very low - 5 -

6 all-017 Joe Slva-Martnez frequence, but the loop looe th ablty when the frequency of the phae varaton approache the pole frequency 0. In order to get ome nght let u conder the pecal cae that ()=(0). Then the loop gan, tranfer functon and error tranfer functon can be ealy plotted a hown below. () A loop (0) o / e / (0) Notce that the unty gan frequency of the loop gan gven by U=(0); th an mportant parameter nce th the bandwdth of the cloed loop gnal tranfer functon. The phae tranfer functon unty only at very low frequence; for frequence cloe to U the phae error ncreae dratcally. At hgh frequency, the phae tranfer functon goe to zero, whch ndcate that the output phae not longer followng the nput phae. Notce that the error unacceptable at hgh frequence; th may not be an ue becaue the PLL operate a a lowpa flter for hgh frequency phae noe. A major ue f the flter removed that the -3dB frequency of the gnal tranfer functon equal to the loop tranfer functon. On top of that, the hgh frequency component are not uppreed, and mot probably the PLL performance wll be unacceptable, and the loop wll not be able to lock n frequency; thee effect wll be clear n the followng ubecton. Bottom lne, the loop flter can not be elmnated! If the phae error mut be wthn 1 degree, for ntance, the n-band phae error mut be le than 1/57 (error n rad/ec = error n degree*/180 ~ error n degree/57). Hence loop gan mut be larger than 60 for the dered (entre) frequency range. Notce that the - 6 -

7 all-017 Joe Slva-Martnez noe bandwdth (untl unty gan frequency) around 60 tme greater than the bandwdth where the error wthn the pec. THE LOOP BANDWIDTH MUST BE INCREASED AS MUCH AS POSSIBLE TO INCREASE THE LOCING RANGE, BUT AT THE SAME TIME THE PLL IS MORE SENSITIE TO NOISE. HIGHER ORDER LOOPS ALLEIATE THIS ISSUE, BUT.. Wthout a loop flter there are not enough varable to optmze the PLL performance. To optmze the PLL performance t requred to ncreae the loop gan a much a poble wthn the dered bandwdth, and to lmt the bandwdth a much a you can to uppre both noe and hgh frequency harmonc. SECOND-ORDER PLL. A a econd example, let u conder the cae of ung a frt order lowpa flter wth one pole and one zero. The flter needed for the uppreon of the hgh frequency component reultng from the mxer. If the zero not ncluded we do not need to do any major analy, the PLL POTENTIALLY untable or the flter uele (Why?). Th wll be evdent n the followng analy. If the flter ha the followng tranfer functon 1 1 P Then the loop gan become A loop 1 1 P - 7 -

8 all-017 Joe Slva-Martnez where flter DC gan. The phae tranfer functon lead to o P P P P rt at all, notce n th equaton that the zero at nfnty lead to a typcal lowpa tranfer functon. If zero located at very hgh frequence and p placed below () then the phae margn poor, leadng to large Q-factor. On the other hand, f the pole located at hgh frequence the flter uele! Notce that the zero can not be very cloe to the unty gan frequency otherwe overhoot and rngng wll appear n the repone of the PLL. Notce that the noe performance even wore f the zero located at lower frequency than the pole. On top of that, hgh frequency pole ntroduce negatve exce phae that mght make the ytem untable. A LOOP Untable Stable And o what, what are the beneft of havng a flter wth a pole-zero par? rt at all, we aume that both the pole and zero are located below U. Due to the nherent pole at the orgn due to the CO, the phae tranfer functon become the typcal one of a econd order functon. A aforementoned, t advable to keep the Q factor of the pole below - 8 -

9 all-017 Joe Slva-Martnez under all cae. The beneft of the pole-zero are evdent f the gnal and error tranfer functon are condered: e o 1 P P P The error tranfer functon ha one zero at the orgn and another one at. Snce the unty gan frequency reduce, the noe preent at the nput reduced by the lower loop bandwdth. At very low frequence, the phae error mlar to the cae when the flter removed, but the PLL mght be tablzed. TYPICAL LOOP ILTER. A better oluton hould ncreae the low frequency loop gan and to reduce the bandwdth cloe to the dered bandwdth aurng enough phae margn. I t poble? If o, what hould be the hape of the flter tranfer functon? The frt order tranfer functon that atfe thoe condton preent a pole at very low frequence (preferably at DC) wth a zero located cloe to the unty gan frequency n order to mnmze noe bandwdth. The tranfer functon gven n the followng expreon: 1 Th flter ha a very large low-frequency gan, ncreang even further the loop gan, reultng n very low phae error. Then the loop gan A loop 1-9 -

10 all-017 Joe Slva-Martnez And the phae tranfer functon lead to o Some mportant obervaton are: The loop reonant frequency ndependent of the locaton of the zero. The zero control the loop qualty factor. In mot of the control book the factor 0 (=BW) ued. Notce that Q The zero allow u to ncreae the cloed loop bandwdth nce t not affectng 0 but only Q (qute mportant property). I th, however, property mprovng loop performance? Before you make any concluon, pleae check the error functon!

11 all-017 Joe Slva-Martnez Loop gan a functon of or /0 = 1, the loop bandwdth ncreae further! I th an advantage or not? The advantage of the econd order functon hould be evdent f the error functon condered. The error tranfer functon can be computed a: e Th tranfer functon correpond to a clac econd-order hgh-pa tranfer functon where ) Cut-off frequency (n=( ) 1/ ) ndependent of the locaton of the ) zero lter-q (tablty) controlled by the zero; notce that f very large, the pole became complex conjugate wth larger maller real part and larger

12 all-017 Joe Slva-Martnez ) v) magnary part; e.g. pule repone wth hgher rngng and longer ettlng tme. Hgh frequency zero may lead to peakng n error functon! Thnk about th ue! In-band error functon further attenuated (lope -40 db/dec at low frequence). Error ~ 1% for /0<

13 all-017 Joe Slva-Martnez A LOOP BW Error that can be tolerated -0/dec BW -40/dec Error Gan Notce that loop bandwdth can be greater than noe bandwdth f the frequency f the flter properly located. On top of that, the error tranfer functon decreae wth an lope of -40 db/dec; whch mean that for gan error of 0.01 (-40 db) you looe only 1 decade of bandwdth; compare t wth the prevou cae! It make ene to thnk n a thrd order ytem wth 3 pole at very low frequency (loop flter wth two pole and two zero, n t t?). More properte of th knd of loop flter wll be dcued n the followng ecton. Noe properte of PLL. PLL are uually ued to clean noy nput gnal. In th ecton, we aume that the ncomng gnal noy wth a pectral noe denty gven by N0 /Hz. Snce th noe goe drectly to the mxer, and nce the hgh frequency component are fltered-out, the mot mportant noe located around the ncomng frequency. In ome book t repreented a follow: v n N 0 co t The overall PLL nput then compoed by the ncomng gnal plu the nput referred noe

14 all-017 Joe Slva-Martnez v n t N co t co 0 Remember that th repreentaton ued jut to emphaze that relevant nput noe located around the frequency of the ncomng gnal. It certanly better to repreent the noe component around = a v n N 0. Notce that 1/ No the RMS value of the pectral noe denty preent at the nput of the PLL. Th gnal modulated by the CO output gnal, then the gnal at the output of the phae detector become: v d m m v n o co ot o t Noco t co * co t * o o o After neglectng the hgh frequency term, t can be hown that the remanng low frequency component are: v d m O n 0 N 0 More detal are gven n cla! Notce that for the noe modulaton the phae not condered nce the noe a random varable. The gnal and noe are better compared f we put th equaton n the followng form: v d mo n N 0 0 Th equaton qute mportant nce the nformaton uually embedded n the phae of the nput gnal. WHEN THE SYSTEM IS LOC, AND I THE DC LOOP GAIN IS

15 all-017 Joe Slva-Martnez LARGE ENOUGH, then the phae at the output of the phae detector can be approxmated by the followng qute relevant expreon: v d m O 0 N 0 or th equaton we aumed: PLL locked and phae error mall uch that n(phae)~phae. Neverthele, we would lke to emphaze that the phae detector output a non-lnear m O functon of the phae dfference: v n d The noe that folded-back to low frequence the one that around the frequency of the ncomng gnal. A you can notce, low frequency noe not a major ue here. Let u to defne the nput gnal to noe rato (neglectng all other noe ource), SNRnput, at the output of the loop flter a: 0 N 0 SNR N BW 0 df In th equaton we are uppong that the loop tracked and that the ncomng gnal n-band. Another way of lookng th SNR conderng the cae of two nput gnal: gnal and noe gven by the followng equaton

16 all-017 Joe Slva-Martnez n t No n tco No co tn No co t Non t We aumed that N 0 1. Th a very mportant reult that relate ampltude gnal (noe n th cae) nto phae gnal; magntude to phae converon. or mall noe gnal, the followng approxmaton hold: co or t Non t n t No co t Non t No n t Th equaton wll be frequently ued! Work yourelf the cae when the noe located at +! or thermal noe only, th equaton reult n SNR 4N 0 BW df Input phae varance approxmately gven by 1/( SNRnput). n 0 BW N df

17 all-017 Joe Slva-Martnez We alo aumed that the CO output tme nvarant, whch not the cae nce the noe hould modulate the CO, and t phae wll vare randomly. The effect of the noe at the output of the phae detector can be modeled a follow: N eq e = ( - 0 )+N eq Loop flter CO () () e 0 S 0 Input and nternal noe: General reult. Let conder the PLL wth noe at the nput and addtonal noe beng njected at the nput of the CO. The econd noe component can be generated by the flter or by upply noe reflected nto th node. n +n 1 D () 0 S 0 The cloed-loop noe tranfer functon are gven by the followng expreon: o n 1 If the followng flter ued

18 all-017 Joe Slva-Martnez Then n n n o Where n n ; The unty gan frequency C can be obtaned from the followng condton: 1 1 C n C n C Whch can be approxmated for mot practcal cae a: n C

19 all-017 Joe Slva-Martnez A loop o / z C n =(0) e / A dcued n the prevou ecton, the phae noe gven a n 0 BW N df Hence, ncreae he ampltude of the ncomng gnal and bandwdth hould be reduced a much a poble. or the nternal noe ource, the tranfer functon o n T. Lee, The Degn of CMOS Rado requency Integrated Crcut, Cambrdge Unverty Pre,

20 all-017 Joe Slva-Martnez n D () 0 S 0 or the ame type of flter a before (pole at DC and zero at ) the nternal noe tranfer functon yeld; o n n n If n condered a an tep noe, produced by the wtchng of the reference clock for ntance, then oe t t e nh 1 n n n 1 t the ntantaneou frequency varaton due to the ntantaneou nput tep. The phae noe the ntegral of the frequency varaton. The maxmum phae devaton produced by n can be obtaned by takng the dervatve of the output phae error and equatng t wth zero. The reult of d( oe )/dt =0 rather complex, but t can be hown that for mall dampng factor (Q>1) the maxmum phae error can be approxmated a: oe max C - 0 -

21 all-017 Joe Slva-Martnez THE PHASE DEIATION CAN BE MINIMIED I AND ONLY I: THE GAIN O THE CO IS REDUCED, THEN MINIMIED. IS Le entve to nternal Noe C C0 HE ILTER BANDWIDTHC MUST BE MAXIMIED! But external noe ncreae Degn trade-off and optmzaton requred. It nteretng to undertand better th lat condton. When the noe tep appear at the CO nput, t frequency change ntantaneouly, and the phae follow th varaton (ntegral of frequency varaton). The phae varaton appear at the nput of the loop flter, and t nverted and ntegrated agan there. After th, the correctng gnal tart to compenate the tep nput. The larger the tme contant of the loop, the larger the output phae error, nce the correctng gnal need at leat a loop tme contant to compenate for t. If the correctng gnal not rapdly generated, the phae error ncreae wth tme a a reult of the nput noe. Th effect llutrated n the followng fgure

22 all-017 Joe Slva-Martnez t 0 t 0 n - 0 D () 0 t 0 - -

23 all-017 Joe Slva-Martnez TRACING PROCESS. The trackng proce drectly related wth the tranent repone of any ytem. If the nput gnal change (mpule, tep, ramp, etc.) the queton : can the PLL output track (follow) thee varaton? Th proce mlar to what happen wth an amplfer: the lew rate repreent the maxmum peed of the OPAMP; any gnal whoe varaton are beyond th lmt wll be track by the amplfer degradng dratcally ytem performance. In a frt approxmaton, the trackng proce can be explaned by ung the lnear model of the PLL. The trackng proce explaned baed on both gnal and error tranfer functon. Another mportant apect of trackng the tuaton where the nput gnal mooth but uddenly there a dturbance uch a an pke at the nput; the ytem may lot track and to return back to the tate the ytem requre ome tme, uually 4-5 tme contant n the cae of frt order ytem and 4-5 tme the 1/BW (BW=n when the pole are complex conjugate). The PLL fundamental equaton are repeated here for convenence: o 1 e o rom thee equaton you can fnd f the crcut table under teady tate condton (for a nuodal nput gnal long tme after any tranent). The tablty alo teted by applyng an mpule or pule at the nput and analyzng the ytem repone n tme; th actually more commonly ued than the analy baed on frequency repone and Laplace tranform

24 all-017 Joe Slva-Martnez STEP RESPONSE AND SETTLING (TRACING) ERROR. or the cae of a tep varaton n phae (mpule varaton n frequency), then we have that or ut Th one of the mot relevant cae to tudy nce many communcaton ytem employ Phae ey Modulaton cheme that employ puled phae varaton for the repreentaton of the gnal. A an example, the expreon n=mn(ot+m*/4) correpond a 4-phae PS ytem: M a bt word wth value 1, 3, 5, 7 that change wth every baeband clock perod. Let u conder the cae of a phae tep at the nput; hence o and the error (for th pecfc nput) become e Obvouly the performance of the PLL trongly depend on the flter tranfer functon. If we do not ue any flter (()=1 and gnorng the effect of the hgh frequency - 4 -

25 all-017 Joe Slva-Martnez component generated by the mxer) for ntance, the phae output ha a pole at the orgn and another one that depend on the other PLL parameter. Snce the phae gan nfnte at DC, hence huge output phae varaton are produced. The phae error ha a knd of lowpa lke tranfer functon wth ngle pole, one of them located at DC. Notce that phae error zero at very hgh frequence only, other than that you have error. An nteretng queton that can not be anwered by ung frequency analy : f you apply a tep at the nput, what the error when the loop ettle? The anwer traghtforward f you ue the fnal value theorem of the Laplace Tranform: lm t e t lm 0 e Ung th theorem for the output phae n cae there the loop flter (() contant (no flter), yeld; lm t o t 0 lm Hence the output phae ettle to the proper value after a tranent, and the PLL output able to track the nput. You can alo ealy ee that the error phae eventually goe to zero when the loop ettle. What th expreon do not tell u how long take for the ytem to ettle! rom our fundamental on Control theory, for the loop wth a ngle pole located at P=(), wth () contant, the ettlng error follow the equaton e -p t. The ettlng error under 0.5% for t>5/p. In general, for obtanng th reult you have to do tme doman analy. IMPORTANT (MOST PRACTICAL) CASE: If the loop flter ha the followng tranfer functon: - 5 -

26 all-017 Joe Slva-Martnez Notce that the flter gan at DC nfnte, and at hgh frequence t determned by. The loop tep repone (mpule frequency repone) gven by o Ung the fnal value theorem t can be een that the loop ettle to the fnal value equal to the nput phae tep. Hence, after the tranent, the output wll track the nput whenever a phae tep appled at the PLL nput. The error functon become e rom th expreon t evdent that after ettlng the phae error zero. On the other hand, the error ha a bandpa lke tranfer functon, wth a zero at dc; whch bacally mean that after the tep appled, f the phae unchanged the error vanhe n tme. RAMP RESPONSE AND SETTLING ERROR. The mot practcal applcaton ue phae or frequency modulaton, hence the nput frequency could uddenly change n tme. The PLL hould be able to ettle properly under thee condton a well. Actually very relevant that the ytem ettle to the deal fnal value, but t equally mportant to ettle wthn the allocated tme frame. Tme for ettlng wll be covered n the next ecton

27 all-017 Joe Slva-Martnez A tep n frequency mean that the nput frequency change accordng to the followng expreon: fn f0 f ut Therefore the phae varaton can be repreented by the followng tme-doman equaton: t In the frequency doman th ramp equaton repreented (ung the Laplace tranform of the ramp) a follow: Ung th expreon and a loop flter wth a pole at DC and a zero t can be hown that the output phae can be obtaned a o Ung the fnal value theorem t can be een that the loop output goe to nfnte when t approache nfnte (reaon for that?). I th correct or not? If th n unclear, alway conder the error functon to get better nde on th ue! - 7 -

28 all-017 Joe Slva-Martnez or the ramp phae nput, the error functon become e Notce n th equaton that after ettlng the error tll zero! The fnal value lead to 0 0 t m l t e lm Th reult ugget that the PLL hould tll be able to track the nput gnal. or the cae of a frequency ramp uch a t n 0, the equvalent nput phae 3 or t t n n Hence 0 m l t e lm 0 t Notce n th equaton that after ettlng the error not zero! And o what?

29 all-017 Joe Slva-Martnez ey obervaton: lm e t t l m 0 0 AN IMPORTANT CONCLUSION IS THAT, WITH THIS LOOP, I YOU APPLY A PHASE STEP YOU ALWAYS HAE A PHASE ERROR ATER SETTLING. IN OTHER WORDS, THE SECOND ORDER PLL IS NOT ABLE TO OLLOW THE INPUT: NOTICE THAT THE PHASE ARIATION IS QUADRATIC IN TIME! THE PHASE ERROR IS HOWEER MINIMIED I THE LOOP BANDWIDTH IS INCREASED AS MUCH AS POSSIBLE. IMPLICATIONS ON NOISE PERORMANCE? USUALLY NOISE LEEL INCREASES WITH THE SQUARE ROOT O THE LOOP BANDWIDTH IS THE INPUT NOISE DOMINANT I YOU INCREASE THE LOOP BANDWIDTH? o IS THIS THE REASON WHY MANY ENDORS USE EXTERNAL REERENCES (CLEAN) OR THEIR CLOC AND DATA RECOERY SYSTEMS? o IS THIS THE REASON WHY THE LOOP BANDWIDTH IN MOST O THE PRACTICAL PLLS IS UNDER 1 MH? - 9 -

30 all-017 Joe Slva-Martnez ey obervaton: lm e t t lm 0 Phae error nverely proportonal to the flter gan ; notce that the hgh frequency gan of the flter lm t th mot probably the only parameter that you can optmze for better performance. and are lnked to other crtcal loop parameter and uually you do not have enough flexblty to change them. Therefore, ncreae a much a you can, th condton uually lead to extremely large capactor! Even n the range of n! lmted by loop bandwdth: there a trade-off between error and noe floor! Notce that another DC pole needed, otherwe the phae error ncreae dratcally, epecally for S modulaton cheme!

31 all-017 Joe Slva-Martnez What the beneft f you ue a econd order flter? Can we add the pole at DC or at a frequency P. What the beneft or drawback when the extra pole added? Thnk about t. Let u conder the followng flter: P P Notce that n order to tablze the ytem, the followng condton mut be atfed: P>U> where U the unty gan frequency. An advantage of th topology that the hgh-frequency noe further attenuated by the 3-pole ytem. or th flter and conderng the cae of a frequency ramp, the loop repone can be obtaned a P P P P The phae error functon become: P P P P e 3 3 If the ntal value theorem ued, t can be hown that the fnal value of the phae error zero. We mut double check ytem tablty and ettlng tme. Thee parameter wll force you to elect a very large value for P.

32 all-017 Joe Slva-Martnez OPEN QUESTION: Can you degn a loop flter uch that the phae error zero for frequency ramp and alo gve you better performance for n-band gnal? Try to fnd the flter tranfer functon and flter realzaton for th condton! Sytem performance f the loop flter ha the followng tranfer functon? What about mplementng the followng flter tranfer functon? 1 Potental mplementaton? In th cae t n 0 Therefore the phae varaton can be repreented by the followng equaton: t Th quadratc equaton can be repreented n the frequency doman a follow: 3-3 -

33 all-017 Joe Slva-Martnez Ung th expreon and a loop flter wth a pole at DC and a zero t can be hown that the output phae can be obtaned a o 3 How we can enure that phae error mall? What knd of cloed-loop tranfer functon needed?