ecture 7: Frequency epone of Aplifier Gu-Yeon Wei Diiion of Engineering and Applied Science Harard Unierity guyeon@eec.harard.edu Wei
Oeriew eading S&S: Chapter 7 Ski ection ince otly decribed uing BJT circuit. ecture note focu on MOS circuit. Suppleental eading azai, Deign of Analog CMOS Integrated Circuit: Chapter 6 Background So far, our treatent of all-ignal analyi of aplifier ha been for low frequencie where internal capacitance do not affect operation. Howeer, we did ee that internal capacitance do exit and we deried the f T of tranitor. Moreoer, we pent oe tie looking at aplifier odeled with a ingle pole. Now, we will ee how thee capacitance affect the frequency repone of aplifier. To fully undertand and odel the frequency repone of aplifier, we utilize Bode plot again. We will ue a technique called open-circuit tie contant (OCT) to approxiate frequency repone calculation in the preence of eeral capacitor and and Miller theore to deal with bridging capacitor. Wei ES54 - ecture 7
Aplifier Tranfer Function A (db) A (db) A 0-3dB A M -3dB H H Voltage-gain frequency repone of aplifier een o far take one of two for Direct-Coupled (DC) aplifier exhibit low-pa characteritic flat gain fro DC to H Capacitiely coupled aplifier exhibit band-pa characteritic attenuation at low frequency due to ipedance fro coupling capacitance increaing for low frequencie We will focu on the high-frequency portion of the repone ( H ) Gain drop due to effect of internal capacitance of the deice Bandwidth i the frequency range oer which gain i flat BW H or H - H ( H >> ) Gain-Bandwidth Product (GB) Aplifier figure of erit GB A M H where A M i the idband gain We will ee later that it i poible to trade off gain for bandwidth Wei ES54 - ecture 7 3
Gain Function A() We can repreent the frequency dependence of gain with the following expreion: A ( ) A F ( ) F ( ) M Where F () and F H () are the function that account for the frequency dependence of gain on frequency at the lower and upper frequency range We can ole for A M by auing that large coupling capacitor are hort circuit and internal deice capacitance are open circuit (what we hae done o far for low-frequency all-ignal analyi) H Wei ES54 - ecture 7 4
High-Frequency epone We can expre function FH() with the general for: F H () ( + Z)( + Z ) ( + ZnH ) ( + )( + ) ( + ) P Where P and Z repreent the frequencie of high-frequency pole and zero The zero are uually at infinity or ufficiently high frequency uch that the nuerator and auing there i one doinant pole (other pole at uch higher frequencie), we can approxiate the function a F H () Thi iplifie the deterination of the BW or H If a doinant pole doe not exit, the upper 3-dB frequency H can be found fro a plot of F H (j). Alternatiely, we can approxiate with following forula (ee S&S p593 for deriation). + + H P ( + ) P P P Z Z PnH H P Note: if P i a doinant pole, then reduce to H P Wei ES54 - ecture 7 5
Open-Circuit Tie Contant Method It ay be difficult to find the pole and zero of the yte (which i uually the cae). We can find approxiate alue of H uing the following ethod. We can ultiply out factor and repreent F H () in an alternatie for: nh + a + a + + anh FH () nh + b + b + + b Where a and b are coefficient related to the zero and pole frequencie We can how that b + + + P P and b can be obtained by conidering the ariou capacitance in the highfrequency equialent circuit one at a tie while reducing all other capacitor to zero (or open circuit); and calculating and uing the C tie contant due to the circuit aociated with each capacitor. Thi i called the open-circuit tie contant ethod (OCT) nh PnH Wei ES54 - ecture 7 6
Calculating OCT The approach: For each capacitor: et input ignal to zero replace all other capacitor with open circuit find the effectie reitance ( io ) een by the capacitor C i Su the indiidual tie contant (C or alo called the open-circuit tie contant) nh C i io Thi ethod for deterining b i exact. The approxiation coe fro uing thi reult to deterine H. H nh C Thi equation yield good reult een if there i no ingle doinant pole but when all pole are real We will ee an exaple of thi ethod when we analyze the high-frequency repone of different aplifier topologie b i i i io Wei ES54 - ecture 7 7
Miller Theore Before we begin analyzing the high-frequency repone of aplifier, there i an iportant phenoenon that we hould firt inetigate called Miller Effect Conider the circuit network below on the right with two node, and. An adittance Y (Y/Z) i connected between the two node and thee node are alo connected to other node in the network. Miller theore proide a way for replacing the bridging adittance Y with two adittance Y and Y between node and gnd, and node and gnd. I Y I I I V V V Y Y V The relationhip between V and V i gien by KV /V To find Y and Y I I I Y Y ( V V ) YV ( V V ) I Y ( V V ) YV ( V V ) YV ( K ) I YV ( K ) YV Y ( K ) I Y Caeat: Y V Y ( K ) The Miller equialent circuit i alid only a long a the condition that exited in the network when K wa deterined are not changed. Wei ES54 - ecture 7 8
High-Frequency epone of CS Ap Take the following circuit and inetigate it high-frequency repone Firt, redraw uing a high-frequency all-ignal odel for the nmos There are two way to find the upper 3-dB frequency H Ue open-circuit tie contant ethod Ue Miller theore Brute force calculation to find out / in et inetigate the all Wei ES54 - ecture 7 9
Uing OCT on CS Aplifier Find the C tie contant aociated with C and C in the following circuit C i g C r o o ( r ) o A M g i o eplace C with an open-ckt and find the reitance een by C tt i tt I tt g r o o eplace C with an open-ckt and find the reitance een by C i tt tt g r o o τ τ tt tt C C i tt tt i tt g + + r ( ro ) + g ( ro ) i + [( r o ) + g ( r o ) ] C C + tt o Wei ES54 - ecture 7 0
Uing OCT Cont d Suing to two tie contant yield H H H τ + τ C + + Fro the aboe equation, it i not difficult to iagine that C ha a ore ignificant effect on reducing BW The reulting frequency dependence of gain i [( r o ) + g ( r o ) ] C A () A M H + et copare thi reult with what we get uing Miller theore Wei ES54 - ecture 7
Uing Miller Theore on CS Aplifier edraw the high-frequency all-ignal odel uing Miller theore C (+g ') g i C ' o C [+/(g')] ~ C C T Auing a doinant pole introduced by C in parallel with C H [ C + C ( + g ')] CT Miller ultiplication of C reult in a large input capacitance Notice that thi approxiation for H i cloe to the approxiation found uing OCT auing that C (+g ) doinate et erify our auption by deriing the exact high-frequency tranfer function of the CS aplifier Wei ES54 - ecture 7
High-Frequency epone of CS Aplifier eplace the input ource and erie reitance with a Norton equialent C ( - o ) C i C () + C + C ( ) o o ( ) () g + i o () g A0 () + C + C ( + g ') ' i / g [ + C '] + C C ' o C C ' o The exact olution gie a zero (at a high frequency) and two pole Notice that the ter i the ae a the olution uing the OCT ethod Unfortunately, the denoinator i too coplicated to extract any ueful info So, auing the two pole are widely eparated (greater than an order of agnitude), we can rewrite the expreion for the denoinator a Wei ES54 - ecture 7 3
HF epone of CS Aplifier ewrite the denoinator a: D + + + () ( P)( P ) ( P + P ) () + P + P P D And fro the olution on the preiou lide we can write P P C C + C + C ( + g ') ( + g ') + C C C ' + C ' ' g C + P P So the econd pole i uually at a uch higher frequency and we can aue a doinant pole Uing either Miller theore or OCT enable a way to quickly find approxiation of the aplifier high-frequency repone Wei ES54 - ecture 7 4
Frequency epone of CG Aplifier One way to aoid the frequency liitation of Miller ultiplication of C i to utilize a CG aplifier configuration C -(g +g b ) x -(g +g b ) x C C db D D C D C +C db x C b x C S C +C b Uing OCT ethod, we find two tie contant At the input (ource node) CS τ S + g + gb At the output (drain node) τ C D D The output uually drie additional load capacitance uch that the output pole i doinant The frequency repone of CG aplifier i when cobined with a CS tage to build a cacode circuit D Wei ES54 - ecture 7 5
Cacode Stage Cacoding enable high bandwidth by uppreing Miller ultiplication of C. et inetigate how with the following high-frequency odel of a cacode tage. C OUT C db +C -g +g b x C db +C out V b C C C C db +C b in C x g C db +C b +C IN C Ue OCT ethod to find the tie contant aociated with each capacitor. The tie contant aociated with C i τ + g + ( g + g ) b C g + g b + Wei ES54 - ecture 7 6 g g + g b C NOTICE: Miller ultiplication i ~
Frequency epone of Source Follower Start with a high-frequency all-ignal odel of the ource follower circuit IN C OUT in C C g out C Directly oling for out / in yield: in out in C () + g [ C + ( + ) C ] in out in C g + g C + C + out ( CC + CC + CC ) + ( gc + C + C ) + g + C out P g C g + C + C The zero i due to C that directly couple the ignal fro the input to the output If pole are far apart, then the ter repreent the doinant pole Wei ES54 - ecture 7 7
More on Source Follower Other iportant apect of a ource follower are it input and output ipedance (ince they are often ued a buffer) et calculate the input ipedance uing the high-freq all-ignal odel g Z in + + C C gb + C Zin C /g b g out C Now calculate the output ipedance (ignoring g b for iplicity) C + Zout g + C At low frequency, Z out /g At high frequency, Z out Shape of the repone depend on the relatie ize of and /g C g out Z out Z out /g Z out /g Z out can look inductie or capacitie depending on and /g Wei ES54 - ecture 7 8
Differential Pair We hae een that a yetric differential aplifier can be analyzed with a differential half circuit. Thi till hold true for high-frequency all-ignal analyi. D D + d / - d / out I d / C C g C db D out The repone i identical to that of a coon-ource tage Wei ES54 - ecture 7 9
High-Frequency CM The CM of a differential pair degrade at high frequency due to a nuber of factor. The ot iportant i the increae in CM gain with frequency due to capacitance on the tail node. Ue the coon-ode equialent half circuit to undertand how CM gain increae with frequency in,cm D out,cm Draw the all-ignal equialent odel and ee the effect of C TAI on the out / in tranfer function out out in g () + g D in ( r C ) ( r o CTAI ) in + g( r o CTAI ) ( + r C ) g D + g and r o o o in TAI + r C TAI - x and out x g g D o TAI in,cm r o C TAI I/ g out,cm D Zero at Z /r o C TAI (ince r o i big, Z occur at a low frequency) There are additional pole at higher frequencie due to C TAI and other internal capacitance (that we hae ignored) The zero caue the CM gain to increae with frequency until the higher frequency pole kick in CM degrade due to the zero r o C TAI Wei ES54 - ecture 7 0
HF CM plot The ipact of the zero in the CM gain on CM can be illutrated a hown A c (db) eeber CM A d /A c Z (log cale) There i a trade off between CM and oltage headroo A d (db) Wider current ource deice enable lower d Wider current ource deice ean larger C TAI P (log cale) CM (db) -0dB/dec -40dB/dec Z P (log cale) Wei ES54 - ecture 7
Next Tie eading S&S Chapter 8 Suppleental eading azai: Chapter 8 What to look forward to Negatie feedback for aplifier wa inented in 97 by Harold Black to tabilize the gain and correct the ditortion of aplifier ued in lonitance telephone network. Negatie feedback (a well a poitie feedback) i widely ued in analog circuit today. In fact, we ued negatie feedback when we contructed op ap with gain et uing reitor. Throughout the next lecture, we will inetigate the general theory of feedback and look at four baic feedback topologie. We will alo learn how to undertand and analyze the tability of aplifier. Wei ES54 - ecture 7