# Stability and Frequency Compensation

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1 類比電路設計 (3349) Stability and Frequency ompensation hing-yuan Yang National hung-hsing University Department of Electrical Engineering Overview Reading B Razavi hapter 0 Introduction In this lecture, we deal with the stability and frequency compensation of linear feedback systems to the extent necessary to understand design issues of analog feedback circuits Beginning with a review of stability criteria and the concept of phase margin, we study frequency compensation, introducing various techniques suited to different op amp topologies We also analyze the impact of frequency compensation on the slew rate of twostage op amps 0- hing-yuan Yang / EE, NHU

2 Basic negative-feedback system General considerations Y H ( s) lose-loop transfer function: () s X + βh ( s) if βh(s jω), the gain goes to infinity, and the circuit can amplify is own noise until it eventually begins to oscillate at frequency ω Barkhausen s riteria: βh(s jω) βh(s jω) 80 o The total phase shift around the loop at ω is 360 o because negative feedback itself introduces 80 o of phase shift The 360 o phase shift is necessary for oscillation since the feedback must add in phase to the original noise to allow oscillation buildup By the same token, a loop gain of unity (or greater) is also required to enable growth of the oscillation amplitude 0-2 hing-yuan Yang / EE, NHU Bode diagram of loop gain A negative feedback system may oscillate at ω if () the phase shift around the loop at this frequency is so much that the feedback becomes positive (2) the loop gain is still enough to allow signal buildup Unstable system Stable system 0-3 hing-yuan Yang / EE, NHU

3 Time-domain response Root locus: Pole frequency s P jω P + σ P (a) σ P > 0, unstable with growing amplitude (b) σ P 0, unstable with constantamplitude oscillation σ P < 0, stable (c) 0-4 hing-yuan Yang / EE, NHU Bode plots of one-pole system Assuming H(s) A 0 /( + s/ω 0 ), β is less than or equal to unity and does not depend on the frequency, we have A0 Y H ( s) + βa0 () s X + βh ( s) s + ω ( + βa ) Bode plots of loop gain: Root locus: s P ω 0 ( + βa 0 ) dB/dec 0ω 0 0ω 0 A single pole cannot contribute a phase shift greater than 90o and the system is unconditionally stable for all non-negative valuesof β 0-5 hing-yuan Yang / EE, NHU

4 Bodes plots of two-pole system Assuming the open-loop transfer function Bode plots of loop gain: H ( s) s + ω p A0 s + ω p 2 The system is stable because βh drops to below unity at a frequency for which βh < 80 o To reduce the amount of feedback, we decrease β, obtaining the gray magnitude plot in the figure For a logarithmic vertical axis, a change in β translates the magnitude plot vertically Note that the phase plot does not change The stability is obtained at the cost of weaker feedback 0-6 hing-yuan Yang / EE, NHU Bodes plots of three-pole system Assuming the open-loop transfer function A0 H( s) s s s ω p ω p2 ω p3 Bode plots of loop gain: If the feedback factor β decreases, the circuit becomes more stable because the gain crossover moves toward the origin while the phase crossover remains constant 0-7 hing-yuan Yang / EE, NHU

5 Phase margin lose-loop frequency and time response Small margin Large margin 0-8 hing-yuan Yang / EE, NHU Phase margin (cont d) Phase margin (PM) is defined as PM 80 o + βh(ω ω ) where ω is the gain crossover frequency Example A two-pole feedback system is designed such that βh(ω ω P2 ) and ω P << ω P2 Since βh reaches 35 o at ω ω P2, the phase margin is equal to 45 o 0-9 hing-yuan Yang / EE, NHU

6 How much phase margin is adequate? For PM 45 o, at the gain crossover frequency βh(ω ) 35 o and βh(ω ), yielding Y H ( jω) H ( jω) X + exp( j35 ) j It follows that Y X 3 β j β The frequency response of the feedback system suffers from a 30% peak at ω ω lose-loop frequency response for 45 o phase margin: 0-0 hing-yuan Yang / EE, NHU How much phase margin is adequate? (cont d) lose-loop time response for 45 o, 60 o, and 90 o phase margin: For PM 60 o, Y(jω )/X(jω ) /β, suggesting a negligible frequency peaking This typically means that the step response of the feedback system exhibits little ringing, providing a fast settling For greater phase margins, the system is more stable but the time response slows down Thus, PM 60 o is typically considered the optimum value The concept of phase margin is well-suited to the design of circuits that process small signals In practice, the large-signal step response of feedback amplifiers does not follow the illustration of the above figure For large-signal applications, time-domain simulations of the close-loop system prove more relevant and useful than small-signal ac computations of the open-loop amplifier 0- hing-yuan Yang / EE, NHU

7 How much phase margin is adequate? (cont d) Example: Unity-gain buffer: PM 65 o, unity-gain frequency 50 MHz However, the large-signal step response suffers from significant ringing The large-signal step response of feedback amplifiers is not only due to slewing but also because of the nonlinear behavior resulting from large excursions in the bias voltages and currents of the amplifier Such excursions in fact cause the pole and zero frequencies to vary during the transient, leading to a complicated time response Thus, for large-signal applications, time-domain simulations of the close-loop system prove more relevant and useful than small-signal ac computations of the open-loop amplifier 0-2 hing-yuan Yang / EE, NHU Frequency compensation Typical op amp circuits contain many poles For this reason, op amps must usually be compensated, that is, the open-loop transfer function must be modified such that the closed-loop circuit is stable and the time response is well-behaved Stability can be achieved by minimizing the overall phase shift, thus pushing the phase crossover out Moving PX out Discussion: This approach requires that we attempt to minimize the number of poles in the signal path by proper design Since each additional stage contributes at least one pole, this means the number of stages must be minimized, a remedy that yields low voltage gain and/or limited output swings 0-3 hing-yuan Yang / EE, NHU

8 Frequency compensation (cont d) Stability can be achieved by dropping Moving GX in the gain thereby pushing the gain crossover in Discussion: This approach retains the low frequency gain and the output swings but it reduces the bandwidth by forcing the gain to fall at lower frequencies 0-4 hing-yuan Yang / EE, NHU Telescopic op amp with single-ended output Determine the poles of the circuit: We identify a number of poles in the signal paths: path contains a high-frequency pole at the source of M 3, a mirror pole at node A, and another high-frequency pole at the source of M 7, whereas path 2 contains a high-frequency pole at the source of M 4 The two paths share a pole at the output Dominant pole: the closest to the origin ω p,out /(R out L ), usually sets the open-loop 3-dB bandwidth Nondominant poles: ω p,a g m5 / A, the closest pole to the origin after ω p,out (mirror pole) where A GS5 + GS6 + DB5 + 2 GD6 + DB3 + GD3 Pole locations: ω p,n g m7 / N, ω p,x g m3 / X g m4 / Y ω p,y Since g m 2I D / V GS V TH, if M 4 and M 7 are designed to have the same overdrive, they exhibit the same transconductance From square-law characteristics, we have W 4 /W 7 µ p /µ n /3 Thus, nodes N and X(Y) see roughly equal small-signal resistances but node N suffers from much more capacitance 0-5 hing-yuan Yang / EE, NHU

9 Telescopic op amp with single-ended output (cont d) Bode plots of loop gain for op amp: using β for the worst case The mirror pole ω p,a typically limits the phase margin because its phase contribution occurs at lower frequencies than that other nondominant poles 0-6 hing-yuan Yang / EE, NHU Telescopic op amp with single-ended output (cont d) Translating the dominant pole toward origin to compensate the op amp: Assuming ω p,a > 0ω p,out, we must force the loop gain crossover point moves toward the origin We can simply lower the frequency of the dominant pole (ω p,out ) by increasing the load capacitance The key point is that the phase contribution of the dominant pole in the vicinity of the gain or phase crossover points is close to 90 o and relatively independent of the location of the pole That is, translating the dominant pole toward the origin affects the magnitude plot but not the critical part of the phase plot 0-7 hing-yuan Yang / EE, NHU

10 Telescopic op amp with single-ended output (cont d) How much the dominant pole must be shift down? Assume () the 2 nd nondominant pole (ω p,n ) is quite higher than the mirror pole so that the phase shift at ω ω p,a is equal to 35 o, and (2) PM 45 o L (ω p,out /ω p,out ) L The load capacitance must be increased by a factor of ω p,out /ω p,out The unity-gain bandwidth of the compensated op amp is equal to the frequency of the nondominant pole To achieve a wideband in a feedback system employing an op amp, the first nondominant pole must be as far as possible Although ω p,out (R out L ), increasing R out does not compensate the op amp A higher R out results in a greater gain, only affecting the low-frequency portion of the characteristics 0-8 hing-yuan Yang / EE, NHU Fully differential telescopic op amp N GS5 + SB5 + + GD7 + DB7 Z N r o7 ( N s), where body effect is neglected We have r ( + g o7 m5ro 5 ) ro 7 Z ( + g 5r 5 ) Z + r 5 ( + g 5r ) Z out m o N o m o5 out r 7 s +,and o N Ls [( + gm5ro 5 ) ro 7L + ro 7N ] s + A pole with τ ( + g m5 r o5 )r o7 L + r o7 N, where ( + g m5 r o5 )r o7 L is simply due to the low-frequency output resistance of the cascode The pole in the PMOS cascode is merged with the output pole, thus creating no addition pole It merely lowers the dominant pole by a slight amount For g m r o >> and L > N, then τ g m5 r o5 r o7 L 0-9 hing-yuan Yang / EE, NHU

11 ompensation of two-stage op amps We identify three poles at X(orY), E(orF) and A(orB) A pole at X(orY) lies at relatively high frequencies Since the small-signal resistance seen at E is quite high, even the capacitances of M 3, M 5 and M 9 can create a pole relatively close to the origin At node A, the small-signal resistance is lower but the value of L may be quite high onsequently, the circuit exhibits two dominant poles One of the dominant poles must be moved toward the origin so as to place the gain crossover well below the phase crossover If the magnitude of ω p,e is to be reduced, the available bandwidth is limited to approximately ω p,a, a low value Furthermore, this required dominant pole translates to a very large compensation capacitor 0-20 hing-yuan Yang / EE, NHU Miller compensation of a two-stage op amp In a two-stage amp as shown in Fig(a), the first stage exhibits a high output impedance and the second stage provides a moderate gain, thereby providing a suitable environment for Miller multiplication of capacitors In Fig(b), we create a large capacitance at E, the pole is ω p, E Rout[ E + ( + Av 2 ) ] As a result, a low-frequency pole can be established with a moderate capacitor value, saving considerable chip area In addition to lowering the required capacitor value, Miller compensation entails a very important property: it moves the output pole away from the origin (pole splitting) 0-2 hing-yuan Yang / EE, NHU

12 Miller compensation of a two-stage op amp (cont d) Pole splitting as a result of Miller compensation Discussion Two poles: (based on the assumption ω p, << ω p,2 ) Simplified circuit of a two-stage op amp: ω p R S [( + gm9rl )( + GD9 ) + E ] + RL ( + GD9 + L ) R S the output resistance of st stage RS [( + gm9rl )( + GD9 ) + E ] + RL ( + GD9 + L ) ω R L o9 r o r p2 R R [( + ) + ( + ) + ] S L GD9 E GD9 L E For 0 and relatively large L, ω p,2 /(R L L ) For 0 and + GD9 >> E, ω p,2 g m9 /( E + L ) Typically E << L, we conclude that Miller compensation increases the magnitude of the output pole (ω p,2 ) by a factor of g m9 R L, a relatively large value Miller compensation moves the interstage pole toward the origin and the output pole away from the origin, allowing a much greater bandwidth than that obtained by merely connect the compensation capacitor from one node to ground L 0-22 hing-yuan Yang / EE, NHU Miller compensation of a two-stage op amp (cont d) Effect of right half plane zero The circuit contains a right-half-plane zero at ω z g m9 /( + GD9 ) because + GD9 forms a parasitic signal path from the input to the output A zero in the right hand plane contributes more phase shift, thus moving the phase crossover toward the origin Furthermore, from Bode approximations, the zero slows down the drop of the magnitude, thereby pushing the gain crossover away from the origin As a result, the stability degrades considerably For two-stage op amps, typically ω p < ω z < ω p2, the zero introduces significant phase shift while preventing the gain from falling sufficiently The right-half-plane zero is a serious issue because g m is relatively small and is chosen large enough to position the dominant pole properly 0-23 hing-yuan Yang / EE, NHU

13 Miller compensation of a two-stage op amp (cont d) Addition of R z to move the right hand plane zero The zero is given by If R z g, then ω z 0 m 9 We may move the zero well into the left plane so as to cancel the first nondominant pole That is gm9 L + E + L + R, because E << L + z ( gm9 Rz ) L + E gm9 gm9 Drawbacks: It is difficult to guarantee the relationship of the above equation, especially if L is unknown or variable 2 The actual implementation of R z is variable R z is typically realized by a MOS transistor in the triode region 0-24 hing-yuan Yang / EE, NHU Miller compensation of a two-stage op amp (cont d) Generation of V b for proper temperature and process tracking If I is chosen with respect to I D9 such that V GS3 V GS9, then V GS5 V GS4 Since g m4 µ p ox (W/L) 4 (V GS4 V TH4 ) and R on5 [µ p ox (W/L) 5 (V GS5 V TH5 )], we have R on 5 ( W / L) 4 g ( W / L) m4 5 ( W / L) 4 L + For pole-zero cancellation to occur, g ( W / L) g m4 5 m9 and hence ( W / L) ( W / L) ( W / L 5 4 ) 9 I I D9 D4 + L If L is constant, it can be established with reasonable accuracy because it contains only the ratio of quantities 0-25 hing-yuan Yang / EE, NHU

14 Miller compensation of a two-stage op amp (cont d) Method of defining g m9 with respect to R S Goal: R z (A) The technique incorporates M b -M b4 along with R S to generate Thus, g L + g m9 m9 I D9 D I b R R S 2 S Proper ratioing of R Z and R S therefore ensures (A) is valid even with temperature and process variations I 0-26 hing-yuan Yang / EE, NHU Effect of increased load capacitance on step response One-stage op amps: Two-stage op amps: In one-stage op amps, a higher load capacitance brings the dominant pole closer to the origin, improving the phase margin (albeit making the feedback system more overdamped) In two-stage op amps, since Miller compensation establishes the dominant pole at the output of the first stage, a higher load capacitance presented to the second stage moves the second pole toward the origin, degrading the phase margin Illustrated in the figure is the step response of a unity-gain feedback amplifier, suggesting that the response approaches an oscillatory behavior if the load capacitance seen by the two-stage op amp increases 0-27 hing-yuan Yang / EE, NHU

15 Slewing in two-stage op amps Positive slewing: negative slewing: The positive slew rate equals I SS / During slewing, M 5 must provide two currents: I SS and I If M 5 is not wide enough to sustain I SS + I in saturation, then V X drops significantly, possibly driving M into the triode region During negative slew rate, I must support both I SS and I D5 For example, if I I SS, then V X rises so as to turn off M 5 If I < I SS, then M 3 enters the triode region and the slew rate is given by I D3 / 0-28 hing-yuan Yang / EE, NHU ompensation technique using a source follower Two-stage op amp with right half plane zero due to : Addition of a source follower to remove zero: Equivalent circuit: Source follower Since GS of M 2 is typically much less than, we expect the right frequencies Vout gm V Vout ( RL + Ls) V ( + RLLs) gmr L V and out V V + I, then in + RS Zero in the left hand plane gm2 s Vout gm R LRS ( gm2 + s) 2 Vin RLL ( + gm2r S ) s + [( + gm gm 2RLR S ) + gm2r LL ] s + gm2 Assume ω p << ω p2, since typically + g m2 R S >>, ( + g m g m2 R L R S ) >> g m2 R L L, we have ω gm p gm and ω p2 (Note ω gm RLRS p2 : ) RLL L L 0-29 hing-yuan Yang / EE, NHU

16 ompensation technique using a G stage Equivalent circuit: The primary issue is that the source follower limits the lower end of the output voltage to V GS2 + V I2 In the G topology, and the G stage M 2 convert the output voltage swing to a current, returning the result to the gate of M g 2V V m 2 V V, and, we obtain out + g 2 m V + Vout + Ls gm2v I 2 in + gm2v2 s RL RS Vout gmrs RL( gm2 + Ss) 2 Iin RLLs + [( + gm RS ) gm2rl + + gm2rll ] s + gm2 gm2rs gm Using approximations, ω p and ω p2 g R R m L S L 0-30 hing-yuan Yang / EE, NHU ompensation technique using a G stage (cont d) Positive slewing: Negative slewing: For positive slewing, M 2 and I must support I SS, requiring I I SS + I D If I is less, then V P drops, turning M off, and if I < I SS, M 0 and its tail current source must enter the triode region, yielding a slew rate equal to I / For negative slewing, I 2 must support both I SS and I D2 As I SS flows into node P, V P tends to rise, increasing I D Thus, M absorbs the current produced by I 3 through, tuning off M 2 and opposing the increase in V P We can therefore consider P a virtual ground node For equal positive and negative slew rates, I 3 (and hence I 2 ) must be as large as I SS, raising the power dissipation 0-3 hing-yuan Yang / EE, NHU

17 Alternative method of compensation two-stage op amps 0-32 hing-yuan Yang / EE, NHU

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