Lecture 30 Compensation of Op AmpsII (/26/04) Page 30 LECTURE 30 COMPENSATION OF OP AMPSII (READING: GHLM 638652, AH 260269) INTRODUCTION The objective of this presentation is to continue the ideas of the last lecture on compensation of op amps. Outline Compensation of Op Amps General principles Miller, Nulling Miller Selfcompensation Feedforward Summary Lecture 30 Compensation of Op AmpsII (/26/04) Page 302 Conditions for Stability of the TwoStage Op Amp (Assuming p 3 GB) Unitygainbandwith is given as: gmigmiiririi GB = Av(0) p = gmiiririicc = g mi Cc = gmgm2rr2 gm2rr2cc = g m Cc The requirement for 45 phase margin is: ω ±80 Arg[AF] = ±80 tan p tan ω p2 tan ω z = 45 Let ω = GB and assume that z 0GB, therefore we get, GB ±80 tan p tan GB p2 tan GB z = 45 GB 35 tan(av(0)) tan p2 tan GB (0.) = 90 tan p2 5.7 GB 39.3 tan p2 GB p 2 = 0.88 p 2.22GB The requirement for 60 phase margin: p2 2.2GB if z 0GB If 60 phase margin is required, then the following relationships apply: gm6 Cc > 0g m Cc gm6 > 0gm and g m6 C2 > 2.2g m Cc Cc > 0.22C2
Lecture 30 Compensation of Op AmpsII (/26/04) Page 303 Controlling the RightHalf Plane Zero Why is the RHP zero a problem? Because it boosts the magnitude but lags the phase the worst possible combination for stability. jω 3 jω jω 2 80 > θ > θ 2 > θ 3 jω θ θ θ3 2 z σ Fig. 4300 Solution of the problem: If zeros are caused by two paths to the output, then eliminate one of the paths. Lecture 30 Compensation of Op AmpsII (/26/04) Page 304 Use of Buffer to Eliminate the Feedforward Path through the Miller Capacitor Model: The transfer function is given by the following equation, V o (s) V in (s) = (g mi )(g mii )(R I )( ) s[r I C I R I g mii R I ] s2[r I (C I )] Using the technique as before to approximate p and p 2 results in the following p R I C I R I g mii R I g mii R I and HighGain Stage C V c I v OUT V in gmi v in C I R I V g mii V R I II C ou II g mii p 2 (C I ) Comments: Poles are approximately what they were before with the zero removed. For 45 phase margin, p 2 must be greater than GB For 60 phase margin, p 2 must be greater than.73gb Fig. 43002
Lecture 30 Compensation of Op AmpsII (/26/04) Page 305 Use of Buffer with Finite Output Resistance to Eliminate the RHP Zero Assume that the unitygain buffer has an output resistance of R o. Model: R o HighGain Stage C V c I v OUT V in gmi v in C I R I R R o o g mii V R I II C II Fig. 43003 It can be shown that if the output resistance of the buffer amplifier, R o, is not neglected that another pole occurs at, p 4 R o [C I /(C I )] and a LHP zero at z 2 R o Closer examination shows that if a resistor, called a nulling resistor, is placed in series with that the RHP zero can be eliminated or moved to the LHP. Lecture 30 Compensation of Op AmpsII (/26/04) Page 306 Use of Nulling Resistor to Eliminate the RHP Zero (or turn it into a LHP zero) R z HighGain Stage v OUT V I R z V in g mi v in C I R I g mii V I Nodal equations: g mi V in V I sc c R I sc I V I s R z (V I ) = 0 g mii V I V o sc c s s R z ( V I ) = 0 Solution: (s) V in (s) = a{ s[(/g mii ) R z ]} bs cs2 ds3 where a = gmi g mii R I b = ( ) (C I )R I g mii R I R z c = [R I (C I C I ) R z (R I C I )] d = R I R z C I Fig. 43004 W.J. Parrish, "An Ion Implanted CMOS Amplifier for High Performance Active Filters", Ph.D. Dissertation, 976, Univ. of CA., Santa Barbara.
Lecture 30 Compensation of Op AmpsII (/26/04) Page 307 Use of Nulling Resistor to Eliminate the RHP Continued If R z is assumed to be less than R I or and the poles widely spaced, then the roots of the above transfer function can be approximated as p ( g mii )R I g mii R I g mii p 2 C I C I g mii p 4 = R z C I (p 3 has been used previously for the mirror pole so we choose p 4 for the nulling resistor pole) and z = (/g mii R z ) Note that the zero can be placed anywhere on the real axis. Lecture 30 Compensation of Op AmpsII (/26/04) Page 308 Conceptual Illustration of the Nulling Resistor Approach V DD R z The output voltage,, can be written as V'' V' M6 Fig. Fig. 43005 g m6 R z s R II g m6 R z g m6 s = R z V s R z V = s R z s when V = V = V. Setting the numerator equal to zero and assuming g m6 = g mii gives, z = (/g mii R z ) V
Lecture 30 Compensation of Op AmpsII (/26/04) Page 309 A Design Procedure that Allows the RHP Zero to Cancel the Output Pole, p 2 We desire that z = p 2 in terms of the previous notation. Therefore, (/g mii R z ) = g mii jω The value of R z can be found as C c R z = (/g mii ) p 4 p 2 p z With p 2 canceled, the remaining roots are p and p 4 (the pole due to R z ). For unitygain stability, all that is required is that A v (0) p 4 > A v (0) p = g mii R I = g mi and (/R z C I ) > (g mi / ) = GB Substituting R z into the above inequality and assuming >> results in g mi > g mii C I σ Fig. 43006 This procedure gives excellent stability for a fixed value of ( C L ). Unfortunately, as C L changes, p 2 changes and the zero must be readjusted to cancel p 2. Lecture 30 Compensation of Op AmpsII (/26/04) Page 300 Increasing the Magnitude of the Output Pole The magnitude of the output pole, p 2, can be increased by introducing gain in the Miller capacitor feedback path. For example, M0 M M8 M4 M9 V Bias The resistors R and R 2 are defined as R = g ds2 g ds4 g ds9 and R 2 = g ds6 g ds7 V DD r ds8 M7 v OUT Iin R V Vs8 R 2 C 2 gm8vs8 g m6 V Ignore r ds8 M6 Iin R V V s8 R 2 C 2 gm8 g m8 V s8 V g m6 V SS Fig. 43007 where transistors M2 and M4 are the output transistors of the first stage. Nodal equations: g m8 s I in = G V g m8 V s8 = G V g m8 sc V c out and 0 = g m6 V G 2 sc 2 g m8s g m8 sc V c out B.K. Ahuja, An Improved Frequency Compensation Technique for CMOS Operational Amplifiers, IEEE J. of SolidState Circuits, Vol. SC8, No. 6 (Dec. 983) pp. 629633.
Lecture 30 Compensation of Op AmpsII (/26/04) Page 30 Increasing the Magnitude of the Output Pole Continued Solving for the transfer function /I in gives, I in = g m6 G G 2 C c s g m8 C 2 G 2 s g m8 G 2 g m6 G G 2 s 2 C 2 g m8 G 2 Using the approximate method of solving for the roots of the denominator gives 6 p = g m8 G 2 C 2 G 2 g m6 g m6 r ds 2 G G 2 and p 2 g m6r ds 2 6 g m8 r ds 2G 2 C 2 = g m8 G 2 6 g m6 C 2 = g m8 r ds 3 p 2 where all the various channel resistance have been assumed to equal r ds and p 2 is the output pole for normal Miller compensation. Result: Dominant pole is approximately the same and the output pole is increased by g m r ds. Lecture 30 Compensation of Op AmpsII (/26/04) Page 302 Concept Behind the Increasing of the Magnitude of the Output Pole V DD V DD r ds7 v out GB C 0 M8 c M6 g m8 r ds8 3 r ds7 M6 v out Fig. Fig. 43008 3 3 R out = r ds7 g m6 g m8 r ds8 g m6 g m8 r ds8 Therefore, the output pole is approximately, p 2 g m6g m8 r ds8 3 Besides, the common gate amplifier stops the feedforward path preventing the RHP zero.
Lecture 30 Compensation of Op AmpsII (/26/04) Page 303 FEEDFORWARD COMPENSATION Use two parallel paths to achieve a LHP zero for lead compensation purposes. RHP Zero A LHP Zero A LHP Zero using Follower V i V i Vi High Gain Amplifier High Gain Amplifier (s) V in (s) = V i A s g mii /A s /[ ( )] A g mii V i Vout Fig.43009 To use the LHP zero for compensation, a compromise must be observed. Placing the zero below GB will lead to boosting of the loop gain that could deteriorate the phase margin. Placing the zero above GB will have less influence on the leading phase caused by the zero. Note that a source follower is a good candidate for the use of feedforward compensation. Lecture 30 Compensation of Op AmpsII (/26/04) Page 304 SELFCOMPENSATED OP AMPS Self compensation occurs when the load capacitor is the compensation capacitor (can never be unstable for resistive feedback) db v in R out(must be large) G m v out R out C L A v (0) db Increasing C L 20dB/dec. Fig. 4300 Voltage gain: v out v in = A v (0) = G m R out Dominant pole: p = R out C L Unitygainbandwidth: GB = A v (0) p = G m C L Stability: Large load capacitors simply reduce GB but the phase is still 90 at GB. 0dB ω
Lecture 30 Compensation of Op AmpsII (/26/04) Page 305 SUMMARY Compensation Designed so that the op amp with unity gain feedback (buffer) is stable Types Miller Miller with nulling resistors Self Compensating Feedforward