Lectures on STABILITY

University of California Berkeley College of Engineering Department of Electrical Engineering and Computer Science νin ( ) Effect of Feedback on Frequency Response a SB Robert W. Brodersen EECS40 Analog Circuit Design f Lectures on STABILITY macro block of a ao R C Effect of Feedback on Frequency Response (Cont.) SB Effect of Feedback on Frequency Response (Cont.) SB3 j RHP Let a( ) be a single pole response, X p σ exp( p t) a( s) a s p j p Impulse p p j LHP exp ( t) p UT ( s) a( s) T( s) νin( s ) a( s) f f T( s) X p σ Impulse s p a O f f s s p ( f) p

Effect of Feedback on Frequency Response (Cont.) SB4 Effect of Feedback on Frequency Response (Cont.) SB5 Let T O f T O s p ( T O ) Pole is at p ( T O ) p ( T O ) T o X p ( T o ) T o 0 X p Root Locus motion of poles as loop Gain is increased. 0log ( a o) a o 0 log T o open loop a( ) p 5HZ a closed loop A( ) T o 0 4 p ( T o ) 50kHZ 0log ( T o ) Gain reduction by negative feedback reduces Gain by and increases bandwidth by ( T o ) T o Effect of Feedback on Frequency Response (Cont.) Why not let T O? ( a o ) db a Problems if we have more than one pole. s s s p p p 3 SB6 Effect of Feedback on Frequency Response (Cont.) Lets look at the motion of a single pole with positive feedback : νin ( ) a SB7 0. p 0. p 0. p3 f 90 0 80 0 70 0 p 0 p p 0 p corresponds to a sign inversion p3 0 p3 At the frequency ( 0 p ) the pahse shift is 80 0 or negative feedback at DC is now positive feedback. a ao p p s p ao T O s p ( f)

Effect of Feedback on Frequency Response (Cont.) SB8 Effect of Feedback on Frequency Response (Cont.) SB9 Since, The condition for stability of a multipole response is the Nyquist Criteria. T o a o f Stable a o f Unstable Simple Version : a( s) a( s) f a( s) T( s ) If Tj > at the frequency where the phase of T( j) 80 0, then the circuit is unstable. p a o f < a o f > T( j) T( s) s j pole is at p ( ao f) θ Tj tan ( ) arc Im{ T( j) } Re{ T( j) } If T < or (T) < 0 the circuit is unstable Effect of Feedback on Frequency Response (Cont.) SB0 Effect of Feedback on Frequency Response (Cont.) SB Complex Nyquist Criteria : Plot T( j) on complex plane. As increases count number of times ν IN ( ) a is circled even number means unstable (I Think). f p3 < 0 3 Poles T( j) < 80 0 p p > 0 Worst Case Stability Condition

Effect of Feedback on Frequency Response (Cont.) SB Effect of Feedback on Frequency Response (Cont.) SB3 aodb T( j) db 0dB 0dB 90 0 80 0 70 0 f 0. T ( ) θ 80 0 0. p p 0p p f T a 0.p3 0. p 0 p p3 Frequency where θ 80 0 0 p3 PHASE MARGIN : Difference between the actual phase shift and 80 0 when i.e. θ m T ( ) Phase Margin θ[ T ] ( 80 0 ) if θ m > 0 then the amplifier is stable typically 45 0 60 0 A R OUT more gain more stable f r o higher R OUT with more gain T Effect of Feedback on Frequency Response (Cont.) SB4 Effect of Feedback on Frequency Response (Cont.) SB5 ν IN a Always Stable As θ m approaches 0 the amplifier is becoming unstable. ν IN a( ) A db closed loop gain θ m 30 0 θ m 45 0 f ( A O ) db θ m 60 0 θ m 90 0 ν IN a( ) Worst Case for Stability p ( T)

Effect of Feedback on Frequency Response (Cont.) SB6 Compensation SB7 A a( s) a a f N( s) D( s) N( s) D( s ) N( s) zeros of a(s) N( s) D( s) N( s) f f D( s ) poles of a(s) if the feedback factor is frequency dependent, then, N f ( s ) f( s) D f( s) N( s)d( s) D( s)d f ( s ) N( s)n f( s) Compensation is the method in which an amplifier is modified so that it is stable. One way is to decrease f (less feedback). If 80 is the frequency where, then if, then, θ( a( 80 0) ) 80 0 f < a ( ) 80 T ( ) f a( ) 80 0 800 and stability is ensured. Narrowbanding for Compensation This entails the addition of a dominant pole p p p3 db T( j) db SB8 For, θ m 45 Narrowbanding for Compensation (Cont.) add a compensation pole, C at the fequency, SB9 0dB 90 0 80 0 70 0 p C f Compensation pole θ m 45 0 80 0 35 0 or θm 45 0 90 45 P a o f C P MHz a o f 0 4 00Hz C of phase shift from the new compensation pole. from the ond pole.

Pole Splitting SB0 It is better to use an existing pole rather than add another. Lets say p and p4 are given with, SB p4» p C GD R L C GD «C GS, C D then, ν in ν in Diff. Pair C GB C GS C D Output Stage νout p p3 RDIFFC GS R LC D Z C GD a j p R L j j p p3 a j p4 X X X X X p4 p p3 p after X p with O gm C GD SB SB3 A db closed loop gain ( A O ) db < 3dB If we add a compensation capacitor, p R DIFF ( R L ) in parallel with C GD : p3 C GS C D t RISE t FALL tsettle t RISE t FALL Lets put numbers in : R DIFF R L C GS C D 0MegΩ 5MegΩ 0.pF 0.pF 0 3 Mhos p 0 0 6rad p4 00 0 6rad a 0 3 a

SB4 SB5 Before compensation, and with, C GD 0 p 0 6rad 0 7 0 3 p3 0 6rad 5 0 6 0 3 Compensate this amplifier for the worst case, f with, θ m 45 0 a 0 3 0 3 5 0 6 j j j j 4 0 6 0 6 0 6 0 8 ν in UT Worst Case Stability Condition ( 5 0 6 ) 3.4dB 0 6 0 6 0 0 6 SB6 P P 0 7 rad 5 0 6 5 0 6 SB7 0 5 θ 35 0 θ 5 0 Formula for» C GS, C D P & P 3 with : Somewhere between MHz and 0MHz, θ 80 0 but the loop gain, T» T 0 5 So how to compensate it? Add C c so that the gain at the first nondominant pole (w p ). since w p3 will move to a higer frequency and w p will move lower P R DIFF ( R L ) P3 Z gm C GS C D

0pF SB8 ν in νout SB9 P 0 7 5 0 3 P 3 0 3 0. 0 5 0 9 rad 5000Mrad Z 0 3 0 8 rad 0 So by adding a 0pF capacitor this circuit is made stable. Slew Rate Limited SB30 Slew Rate & Compensation Miller Op Amp SB3 small θ m Output Stage large θ m ν id ν id M M 0 GAIN ν o ISS A ν G m ν cd

Slew Rate & Compensation Miller Op Amp (Cont.) SB3 Slew Rate & Compensation Miller Op Amp (Cont.) SB33 ν in UT Circuit situation with large ν id basically V B few 00mV A ν ν o Slewrate Slew Rate Volts/µ 0 low 0 50 med 00 high νo dν C o dν C or o ISS slew rate dt dt ISS dt C t linear with time ISS C Slew Rate & Compensation Miller Op Amp (Cont.) SB34 Slew Rate & Compensation Miller Op Amp (Cont.) SB35 p p T ( ) ν id adiff a g ν o R DDIFF, a DIFF R DDIFF, θ m 45 C P R Ddiff, ag ao P a DIFF ag f a o a DIFF a g P a DIFF R DDIFF, P R DDIFF, Location of compensation pole : C P ao The size of the compensation depends only on & P P

Slew Rate & Compensation Miller Op Amp (Cont.) SB36 g I DS m V DSAT V I DS DSAT Slew Rate & Compensation Miller Op Amp (Cont.) How to increase slew rate : Increase V DSAT More current, smaller Increase P W L R S SB37 dν o dt IDS I SS P P Slew rate limits max change : A A sin S t dν Slewrate o V DSAT P dt V DSAT 0.V P 0MHz π SR 6.3V µ A A V S 0 6 π SR 3V µ Max rate of change here dv SIG dt S Acos S Max Value S A t MOS Miller Amp Right Half Plane Zero i ν diff C ν C R odiff, ignore for now ν o R OUTGAIN, SB38 SB39 MOS Miller Amp Right Half Plane Zero (Cont.) Z R Z R Z R L ν o also R t gm O O i diff R L removes zero entirely