PRESENTED AT THE 2004 AMERICAN CONTROL …

1 PRESENTED AT THE 2004 AMERICAN CONTROL CONFERENCE1 internal and External Op-Amp compensation :A CONTROL - centric TutorialKent H. LundbergDepartment of Electrical Engineering and Computer ScienceMassachusetts Institute of Technology, Cambridge, MA Frequency compensation of two-stage integrated-circuit operational amplifiers is normally accomplished with acapacitor around the second stage. This compensation capaci-tance creates the desired dominant-pole behavior in the open-loop transfer function of the op amp. Circuit analysis of thiscompensation leads to a mathematical observation of polesplitting: that as the compensation capacitance is increased, theparasitic poles of the amplifier separate in of op-amp compensation as minor-loop feedback,instead of pole splitting, greatly simplifies and generalizesthe analysis and design of op-amp frequency response. Usingclassical- CONTROL techniques instead of direct circuit analysis,insight and intuition into the behavior and flexibility of the systemare INTRODUCTIONO perational amplifiers have been used by CONTROL engineersfor many decades as key components in compensators [1],sensor circuitry [2], and analog computers [3], [4].

2 They arestill one of the most ubiquitous electronic elements in theworld. However, despite the required use of feedback in all op-amp applications, and the presence of feedback in the internalcircuitry, the design of operational amplifiers is often presentedand completed without a useful CONTROL amps require a deliberately designed frequency responseto ensure stability and satisfactory transient performance inend-user applications. Standard frequency compensation isdesigned for general-purpose op-amp applications such as am-plifiers, buffers, and integrators. Sophisticated compensationtechniques can be employed in specific applications in whichstandard compensation methods perform compensated op amps have a fixed transfer func-tion set by the manufacturer. In the design of the circuit, theop-amp designer must choose a compensation network thatis appropriate for the intended applications of the op compensated op amps [5] allow the end user toselect the compensation network that determines the transferfunction of the op amp.

3 The determination and implementationof appropriate op-amp transfer functions in various applica-tions is easily understood with the tools of classical textbooks in analog circuit design [6], [7], [8] treatop-amp compensation in a network-theory context, writing outmany node equations and discussing the concept of pole split-ting [9]. This approach is unnecessarily abstruse. Treatment +A(s)vIvOR2R1 Fig. for a simple non-inverting amplifier general-purpose use (and commercial success) this circuit must be stable for anyresistor op-amp compensation as minor-loop feedback, instead ofpole splitting, greatly simplifies and generalizes the analysisand design of op-amp frequency- compensation paper demonstrates the use of classical- CONTROL tech-niques instead of direct circuit analysis in the design of com-pensation for general-purpose and special-purpose operationalamplifiers. Intuition and insight into the solution are gainedby using these feedback THEGENERAL-PURPOSETRANSFERFUNCTIONThe frequency response of general-purpose op amps isdesigned to be stable in the largest number of schematic for a simple non-inverting amplifier circuit isshown in Figure 1.

4 This amplifier circuit is implemented witha negative-feedback loop around the op amp, and the closed-loop gain isVoVi=R1+ general-purpose use the op amp must be designed suchthat this circuit is stable for any resistor block diagram of this circuit is shown in Figure 2. Thecircuit loop transfer function isL(s) =A(s)R1R1+R2=A(s) stability in this application, this loop transfer function mustcreate a stable feedback system for any value ofFless thanPRESENTED AT THE 2004 AMERICAN CONTROL CONFERENCE2+_replacementsViVoA(s)FFig. 2. Block diagram for the non-inverting amplifier circuitin Figure 1. Thefeedback path isF=R1/(R1+R2). The op-amp transfer functionA(s)must be designed to guarantee stability for any such attenuative uFig. 3. Frequency response of the desired op-amp transfer functionA(s). Thesingle-pole roll-off (slope of 1) behavior over a wide frequency range givesthe desired transfer function for a general-purpose op amp.

5 The frequency uis the unity-gain frequency of the op The ideal transfer function that meets this requirement isA(s) =A0 s+ 1.(1)With this op-amp transfer function, the closed-loop circuit willbe stable for any choice of resistive feedback. The frequencyresponse of this desired op-amp transfer functionA(s)rollsoff with a slope of 1over a wide frequency range, as shownin Figure 3. In the ideal case, this transfer function gives90 of phase margin, regardless of the real op amp will have additional high-frequency polesbeyond its unity-gain frequency u. Including the effect ofan additional pole at2 u, the frequency response of theloop transfer function of the op-amp circuit with a variety offeedback terms is shown in Figure 4. Even with this additionalhigh-frequency pole, the loop transfer function always crossesover with60 (or more) of phase margin for any attenuativefeedback. Thus, stability is guaranteed for any set of implementation of this desired op-amp transfer functionis easier said than done.

6 Even a simple op-amp circuit modelgives an unacceptable op-amp transfer example, a simplified schematic of the Fairchild A741[10] op amp is shown in Figure 5. This circuit can be modeledby the equivalent-circuit block diagram shown in Figure frequency response of this circuit, when uncompensated,is shown in Figure 7. The two low-frequency poles severely10 2100102104106108 40 20020406080100120140 Magnitude (dB)F = 1F = = = DiagramFrequency (rad/sec)Fig. 4. Frequency response of the op-amp-circuit loop transfer functionL(s)with a variety of feedback terms. Since the loop transfer function alwayscrosses over with60 or more of phase margin for any attenuative feedback,stability is guaranteed.+-Q1Q2Q5Q6Q17Q16Q3Q4 VCCVEEQ20Q14Q22 Fig. 5. Simplified schematic of the uncompensated Fairchild A741 op amp,showing the signal-path transistors. The full schematic is shown and explainedin Appendix the phase margin at crossover.

7 Additional high-frequency poles in the circuit make matters stability in amplifier applications, the op amp must becompensated to achieve a frequency response similar to theideal transfer function in equation (1) and shown in Figure general-purpose compensation is usually accomplishedwith a capacitor [5]. (This technique is often called Millercompensation. See Appendix I.) The simplified schematic ofthe A741 op amp with a compensation capacitor is shown inFigure 8. The compensation capacitor goes around the high-gain stage as shown in the equivalent-circuit block diagraminFigure two-port circuit models for each stage, the equivalent-circuit schematic in Figure 10 can be drawn. Each gain stageis represented by a Norton-equivalent two-port model withinput resistance, output resistance, output capacitance,anda transconductance generator. The output buffer is ignoredin this equivalent circuit since the output voltage of thePRESENTED AT THE 2004 AMERICAN CONTROL CONFERENCE3 +A2A11 VoVinFig.

8 Block diagram of a two-stage op amp. The inputstageA1converts the input signal from differential to single-ended. Thesecond stageA2is the high-gain stage. The output buffer provides currentgain and protection at the 2100102104106108 40 20020406080100120140 Magnitude (dB)Bode DiagramFrequency (rad/sec)Fig. response of an uncompensated op amp. The two low-frequency poles in the uncompensated transfer function severely degrade thephase margin at crossover.+-Q1Q2Q5Q6Q17Q16Q3Q4 VCCVEEQ20Q14Q2230 pFFig. 8. Simplified schematic of the Fairchild A741 op amp with compensa-tion capacitor. The compensation capacitor goes around the high-gain secondstage created byQ16andQ17. +A2A11 CVoVinFig. block diagram of a two-stage op amp withcompensation capacitor. The compensation capacitor goes around the high-gain second +-V1GM1 VinCC1C2 Fig. schematic for the two-stage op amp with com-pensation capacitor of Figure 9, whereA1=GM1R1andA2= stage is equal to the buffer output voltageVo.

9 Thetransfer function of this equivalent circuit will be derived inthe following sections, using the pole-splitting approachinSection III and using a feedback approach in Section POLE-SPLITTINGAPPROACHTo investigate the effects of the compensation capacitor, thetransfer function of the op-amp equivalent-circuit schematic inFigure 10 is calculated to findA(s) =VoVin(s).The pole-splitting approach [9] uses brute-force circuit analy-sis to determine this transfer function. The approach starts withthe constitutive current equations at the two circuit nodesV1andVoGM1 Vin V1R1 sC1V1 sC(V1 Vo) = 0(2)sC(V1 Vo) GM2V1 VoR2 sC2Vo= 0.(3)After a page of algebra (as shown in detail in Appendix III)the transfer function is foundA(s) =VoVin(s) =GM1R1GM2R2(Cs/GM2 1)a2s2+a1s+ 1where the coefficients of the denominator area2=R1R2(C1C2+C1C+C2C)a1=R1C1+R1C+R2C2 +R2C+ that the gain of the second stage is large(GM2R2 1), the final term in the first-order coefficienta1dominates the sum, and the transfer function can be simplifiedasA(s) GM1R1GM2R2(Cs/GM2 1)R1R2(C1C2+CC1+CC2)s2+GM2R2R1Cs+ locations of the transfer-function poles can be found byassuming that the pole locations are widely separatedA(s) A0( 1s+ 1)( 2s+ 1)=A0 1 2s2+ ( 1+ 2)s+ the two poles are widely separated ( 1 2), thenA(s) A0 1 2s2+ 1s+ AT THE 2004 AMERICAN CONTROL CONFERENCE4increasingC 1 1 2 2 Fig.

10 Mathematical observation of pole splitting. Asthe size ofthe compensation capacitorCis increased, the frequency of the first pole 1decreases and the frequency of the second pole 2increases. The polesapparently split in , the approximate pole locations of the op-amptransfer function are 1=1 1=1a1=1GM2R2R1C(4) 2= 1 1 2=a1a2=GM2CC1C2+CC1+CC2.(5)Figure 11 shows the resulting pole-splitting behavior inthe frequency response of this transfer function. It is observedthat as the size of the compensation capacitor is increased,the low-frequency pole location 1decreases in frequency,and the high-frequency pole 2increases in frequency. Thepoles appear to split in frequency. For a large enoughcompensation capacitor, a single-pole roll off over a widerange of frequency results, as shown in Figure 11, whichmatches the desired transfer function in Figure MINOR-LOOPFEEDBACKW hile the above results are correct and useful, they arean impediment to intuition [11].