UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences

Transcription

1 UNIVEITY OF CAIFONIA Collee of Enineerin Departent of Electrical Enineerin and Coputer cience E. Alon Hoework # olution EEC 40 P. Nuzzo Ue the EEC40 90n CMO proce in all hoework and project unle noted otherwie. In thi hoework you ay ue jut the typical (tt) deice paraeter.. The coon-ource aplifier hown in the fiure below drie an external load C with a DC ain A V and a bandwidth ω bw. et C p be the total paraitic capacitance at the drain node of the NMO tranitor, C the total all-inal equialent capacitance at the ate of the tranitor when the other terinal are (all-inal) rounded, and γ the ratio between C p and C (i.e. γ C p /C ). Uin thee definition, derie an expreion for the power conuption P DC of thi aplifier a a function of V DD, C, A V, ω bw, the tranitor V*, the unity current-ain bandwidth ω T, and γ. Note that you can aue that the current ource i ideal. (Hint: Derie an expreion for C p a a function of the tranitor.) V DD C p V o C V i olution: By uin the definition of V*, the aplifier DC power conuption can be expreed a follow * V PDC IDDVDD VDD, () where I DD i the tranitor bia DC current, which need to be proided by the power upply. Therefore, all that we need to do i find an expreion for a a function of the paraeter in the proble tateent. To do thi, we firt obere that ince the current ource i ideal, the ain-bandwidth product for thi aplifier can be eaily written a AV ω bw ro r C C C ( + ) o p + C p. ()

2 In fact, ince the all-inal equialent input capacitance of the tranitor i drien by the ideal oltae ource at the input of the aplifier, and C p include by definition the total paraitic capacitance at the drain node of the NMO tranitor, the aplifier pole (and hence bandwidth) will be ien by the total output capacitance (C p + C ) and the tranitor output ipedance r o. Uin the definition of γ and ω T we can alo obtain C p γc γ. (3) ω T By ubtitutin (3) into () and olin for we finally et AV ωbwc AV ωbwγ ω T, (4) which finally lead to the deired expreion P DC * AV ωbwc V V AV ωbwγ ω T DD. (5) For a ien aplifier unity-ain bandwidth, load, and power upply, the aboe expreion offer a ueful way of etiatin power a a function of a all nuber of the tranitor bia point paraeter. In particular, it proide iniht into which alue to chooe for V* in order to iniize the power conuption. ince ω T i alo an increain function of V*, (5) uet that there i an optial bia point for iniu power-conuption. Decreain V* i adantaeou in ter of power conuption a lon a ω T reain uch larer than γa V ω bw. On the other hand, when ω T et too cloer to γa V ω bw power conuption will bein to increae draatically due to the tranitor own elf-loadin.. Dein an NMO coon-ource tae driin a.5pf load with 800-MHz unity-ain bandwidth. Chooe a ufficiently lon to achiee an open-loop ain of at leat 35 for output oltae between 0. V and V and iniize the power diipation of the circuit. Deterine the deice width and bia current, and erify the ain and bandwidth with PICE. Alo, plot the all-inal ain a 0 dv o /dv i a a function of the output DC leel. olution: Firt, we need to pick the channel lenth of our deice. A quick iulation how that with ufficiently low V*, a channel lenth of 30 n i ufficient to achiee the ain pecification oer the entire output win rane. Moreoer, knowin that 800 MHz i till lower than the f T of the 0.-μ tranitor at a low V*, we opt for a V* of 0 V, which ie f T 9 GHz. Once we hae picked the channel lenth and V*, equation (4) in proble directly proide u with the deired, which alo account for the tranitor own elf-loadin. Howeer, to apply (4) we need oe ore data fro PICE. In particular, uin the ae iulation

3 et-up a in proble 3 of Hoework, it i eay to extract: C.73 ff/μ of width, C p 0.94 ff/μ, γ0.54. The deired can then be coputed a follow: 8 AV ωbwc 8 0 π AV ωbwγ ω 9 0 T Thu, the drain current will be I D V*/ 463 μa, and we can turn back to PICE to actually ize the deice. At a V* of 0V, the tranitor ha a drain current denity of.4 μa/μ. Therefore, to achiee the required I D (and hence the required ), we need a tranitor whoe width i 463/.4 37 μ. The dein i baically now coplete all that left i to erify it with HPICE. An AC iulation ie a unity ain bandwidth of 80. MHz riht on our taret. A hown below, all our pecification are atified. NMO coon-ource (W37 µ, 30 n, I 463 D µa, C.5 pf) 40 NMO coon-ource (W37 µ, 30 n, I D 463 µa, C.5 pf) Gain [db] Frequency [Hz] a V O [V] Note that the plot of Voltae Gain eru Output Voltae i unrealitic. Becaue we iulated with an ideal current ource the ain doe not drop off at hih output oltae. Howeer, if we replace the current ource with a PMO tranitor the win would be liited on the hih end a well. The dein procedure decribed aboe i iple and effectie, and any olution followin thi ethodoloy (een with a different choice for V*) would receie full credit. Notice that, if you initially inored the capacitie loadin at the output fro the tranitor, you would then etiate a required πf u C π 800MHz.5pF Thi would ie a unity ain bandwidth of 78.3 MHz within.5% of our taret. In order to actually hit 800 MHz, you would then increae the width/bia current to copenate for the paraitic capacitance of the deice. ince thi capacitance increae linearly with the tranitor W, you would need a few iteration to conere to the propoed reult. Finally, if we truly wanted to iniize the power diipation of the circuit, we would not jut chooe V* baed on intuition. The key tep to truly iniizin the power would till be to take the paraitic capacitance of the tranitor into account when calculatin the required, a we hae done aboe, and then ue the equation (5) fro proble to weep oer V* and find the optiu alue. A it turn out, for the et of paraeter ien in thi proble, the optiu V* i approxiately 00V (I D 406μA and W38μ). 3

4 3. Interate the input referred flicker noie of a /0.09 NMO tranitor in a coon-ource confiuration fro 0. Hz to GHz. pecify the reult in Volt r. Ue V * 00 V and your reult fro hoework # to calculate, C ox 7 ff/µ, and the K f paraeter fro the lecture note. How doe the reult chane if you reduce the lower liit of interation to year -? Find the factor M by which W and I D ut be increaed (keepin /I D contant) to accoodate the lower interation liit without an increae of the total noie. olution: The flicker noie pectral denity at the output of the tranitor and the total interated flicker noie are: i / f / f f K I hi i K I f D f f D / f df f Cox f f Cox f hi, i ln flo lo ince V * 00 V, fro Hoework, proble 3, part d) we obtain I D 76.µA (/0.09 TT NMO) and 76µ. To refer the noie to the input a a oltae, all we hae to do i diide by. o: 8 f D hi 9 i, eq,/ f ln ln.5 0 V Cox flo (76 µ ) (0.09 µ ) (7 ff/ µ ) 0.Hz i, eq,/ f K I f. 0 AF 76.µ A GHz 50µ V If we decreae the lower interation liit to year -, the flicker noie power increae by GHz GHz ln ln.63 /( ) Hz 0.Hz.63 50µ V 63.7µ V i, eq,/ f Thu, the new input-referred flicker noie oltae i ien aboe. To aintain the ae flicker noie perforance a before, we need to increae, I D, and W by a factor of M Derie an expreion for the input referred oltae noie of an NMO coon-ource aplifier M with PMO load M. Aue M and M are both in aturation and inore all capacitie paraitic except for an explicit load capacitor C. pecify the reult a the noie fro M ultiplied by a factor that i a function of the V * of the two tranitor. What doe thi iply about the V * you hould chooe for M. M? You can nelect flicker noie for thi analyi. olution: By auin that the alue of γ i the ae for both the PMO and the NMO tranitor, the total noie current pectral denity at the output i 4γ( + ), o the input referred noie oltae denity i: 4

5 γ i, eq, tot i, eq,m f f. eeberin that I D /V* and that the drain current for the two deice are the ae, we can rewrite the total noie a: V* + V* i, eq, tot i, eq,m. Thu, to iniize the total noie, the V* of M hould be ade uch larer than the V* of M. In other word, we would want to ake the of M uch larer than that of M. 5. In thi proble we will be workin with the circuit hown below. You can aue that M i in aturation and it all inal output reitance (r o ) i infinite. Moreoer, you can inore flicker noie for thi proble, and you hould aue that all capacitor except for C are neliible. V DD D V o M C V i a) Auin that the witch i alway on, calculate the oltae noie denity a a function of frequency at the output (V o ) due to the drain current noie fro M. olution: et firt draw the all inal odel of the circuit: 5

6 o i M D C To copute the tranfer function H() o ()/i M (), all we need to do i calculate what percentae of the noie current flow into the ipedance at the output. The noie current fro M et plit into two part: one that flow into the parallel cobination of and, and one that flow into. Any of the current that flow into the parallel cobination of and will eentually flow into the output ipedance, and therefore: + i i i ( ) o M M and finally i i + C D D o o M + C D ( ) D () Therefore, the noie oltae denity at o due to M i: + ω ( ω) om, D 4 γ H ( jω) 4γ ω + ( C D ω) + ( + ) ω b) epeat part a), but now find the oltae noie denity a a function of frequency at the output due to the current noie fro D and fro. olution: et once aain draw the all inal odel but includin only the noie current fro and D thi tie: 6

7 o i D D C i Notice that no current will flow throuh and due to the noie current fro D, and therefore thi noie current iply ee the parallel cobination of D and C. We then obtain: o D id C D + () 4 D 4D o, ( ω) ω D + Cω + Cω ( ) ( ) D D D When coputin the contribution of, we obere aain that thi noie current alo et plit between and the parallel cobination of and. Howeer, thi tie it i the portion of the current that flow into that will alo flow to the output ipedance: + io i i ( ) And hence o D i + C D + ( + ) ( ω) o, 4 ω D ω + ( C D ω) + ( + ) ω (3) ince and are independent ource of noie, the total noie power pectral denity i jut the u of the two contribution () and (3) coputed aboe. c) Now calculate the noie apled on C when witch i abruptly turned off after bein on for a lon period of tie. olution: 7

8 ince the different contribution in the frequency doain hae been coputed in (), () and (3) fro part a) and b), all that we need to do i interate the different noie oltae denitie oer frequency and u up all the contribution. The contribution fro D i the eaiet to be coputed. The tranfer function i in (). After interatin and pluin in the noie current denity fro D we obtain (not too urpriinly): i o D C+ D D 4 4 o, D D D 4C D C iilarly, fro () we obtain + o D im ( ) D + C D + C D + ( + ) C D ( ) + + and therefore om, 4γ D + 4 C ( ) C D ( ) D (4) D ( ) C + + D γ C ( + ) C D + + Finally, fro (3) we obtain 8

9 o D i + C D + ( + ) C D ( ) + + o, ( ) 4 D C D ( + ) 4 C D ( ) + + D C ( + ) DC + + (5) We can now u up all the contribution to et + o D C + γ DC( + ) +. (6) ( + ) DC + + d) Baed on your reult fro part c, coent on how the choice of (with all other circuit paraeter held fixed) affect the noie at the output due to itelf, M, and D. olution: The output noie due to D i obiouly not affected by the choice of. To tudy the effect on the output noie due to, it i conenient to define D C / and iolate thoe ter in (5) that do not depend on a follow:, n( ) o D D C ( + ) ( + ( + ) ) C It i traihtforward to ee that n( ) 0 for all, n( ) 0 for 0 and. By takin the deriatie we can alo conclude that the output noie power increae when <* and decreae for >*. Therefore it reache it axiu for + *. Thi reult totally confir our intuition. In fact, when i uch aller than / the current noie power denity increae, but only a all portion of it et to the output throuh the ource of M, ince the parallel of and offer uch aller ipedance at all frequencie. The ipedance een by the noie current ource i practically and the aount of noie current power that et to the output i directly proportional to it, i.e. 9

10 i o, +. On the other hand, when becoe larer than /, the ipedance een at the tranitor ource practically coincide with / (in parallel with ) and the output noie becoe inerely proportional to. et u dicu now the effect of the choice of on the output noie due to M. Fro (4) we et ( + ) + o, M γd γd( ) C + ( + ) ( + ) C It i now eay to ee that ( ) 0 for all and ( ) for 0 and ( ) /(+ ) for. By takin the deriatie we can thi tie conclude that the output noie power decreae when < and increae when >. Therefore, the output noie ha now a iniu at + + '. The intuition behind thi reult i that when i uch aller than /, / i alo uch aller than /C D by definition of. Therefore, the ource deeneration ipedance i otly reitie in the bandwidth of interet, and noie will decreae with increain the alue, ince there i le tranitor noie that can ake it to the output. On the other hand, increain reduce noie le when the inductor doinate the ource ipedance oer the bandwidth of interet. ince ha a different ipact on the noie due to itelf and M, it i worth coniderin it effect on the joint noie ien by the u of the aboe contribution. Fro (6) we obtain q( ) γ o γ D γ D C ( + ) + + C ( ( ) ) (7) We will aain hae q( ) 0 for all, q( ) for 0 and q( ) /(+ ) for. Howeer, after takin the deriatie we obere that thi tie q( ) i alway decreain. Thi happen for any alue of the paraeter in (7) a far a γ>0.5, a i true for typical 0

11 tranitor. We conclude that to iniize the oerall noie and et cloer to the iniu alue o,in γ D C + we hould chooe a hih a poible. Notice, howeer, that hih deeneration reitor would alo iply aller ain for the inal path. The optiu alue hould then be deterined baed on the inal propertie and pectru in order to axiize the inal-tonoie ratio.