ECEN 326 Electronic Circuits
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1 ECEN 326 Electronic Circuits Stability Dr. Aydın İlker Karşılayan Texas A&M University Department of Electrical and Computer Engineering
2 Ideal Configuration V i Σ V ε a(s) V o V fb f a(s) = V o V ε (s) = a o 1 s p 1 A(s) = V o V i (s) = a(s) 1 + a(s)f = 1 a o 1 + a o f s (1 + a o f)p 1 ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 1
3 Gain-Bandwidth 1-pole amplifier Gain magnitude, db 20 log a o a(j ω) 20 log a o 1+a o f A(j ω) 20 db/dec 0 p 1 (1+a o f) p 1 ω t ω log scale ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 2
4 Instability and the Nyquist Criterion Transfer function of a 3-pole amplifier: a(s) = 1 s p 1 a o 1 s p 2 1 s p 3 Nyquist criterion for stability of the amplifier: Consider a feedback amplifier with a stable T(s). If the Nyquist plot of T(jω) encircles the point (-1,0), the feedback amplifier is unstable. ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 3
5 Magnitude & Phase 3-pole amplifier a(jω), db 20 log a o 20 log a db/dec 40 db/dec a(jω) p 1 p 2 ω 180 p 3 60 db/dec ω log scale ω ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 4
6 Magnitude & Phase T(s) = a(s)f 1 T(jω), db 20 log a o f 1 20 log a 180 f db/dec 40 db/dec ω p1 p 2 ω180 p 3 60 db/dec log scale ω T(jω) ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 5
7 Nyquist Plot T(s) = a(s)f 1 Im ω negative ( 1,0) a 180 f 1 0 ω= a o f 1 ω=0 Re ω=ω 180 ω positive ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 6
8 Magnitude & Phase T(s) = a(s)f 2 T(jω), db 20 log a o f log a 180 f 2 20 db/dec ω180 3 p p1 p 2 40 db/dec ω log scale 60 db/dec log scale ω T(jω) ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 7
9 Nyquist Plot T(s) = a(s)f 2 Im ω negative ω=ω 180 a 180 f 2 ( 1,0) 0 ω= a o f 2 ω=0 Re ω positive ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 8
10 Gain & Phase Margin 20 log a 20 log a a(jω), db o 180 T(j ω) =0 db ω axis for T(j ω) 20 log 1 f Gain margin a(jω), T(j ω) p 1 ω o p 2 ω 180 p 3 Phase margin ω log scale ω ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 9
11 Stability Criteria Nyquist: T(jω 180 ) = a 180 f < 1 Stable Gain Margin (GM): GM = 20 log 1 T(jω 180 ) = 20 log T(jω 180) GM > 0 Stable Phase Margin (PM): PM = T(jω o ) PM > 0 Stable ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 10
12 Phase Margin T(jω o ) = 1 a(jω o ) f = 1 a(jω o ) = 1 f PM = 45 T(jω o ) = 135, A(jω o ) = a(jω o) A(jω o ) = a(jω o) 1 + e j135 = A(jω o ) = 1 + T(jω o ) a(jω o ) j a(jω o) j = f = 1.3 f PM = 30 T(jω o ) = 150, A(jω o ) = 1.92/f PM = 60 T(jω o ) = 120, A(jω o ) = 1/f PM = 90 T(jω o ) = 90, A(jω o ) = 0.7/f ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 11
13 Closed-Loop Frequency Response 10 5 PM=30 PM=45 PM=60 PM=90 0 Relative Gain, db Relative Frequency ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 12
14 Closed-Loop Step Response PM = 30 PM = 45 PM = 60 PM = 90 ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 13
15 Compensation Dominant pole (p D ) added, f = 1 ω a(j ), db 20 log a o Original After compensation p D Original p 1 p 2 p 3 ω log scale After compensation ω a(j ) p D p 1 /a o ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 14
16 Compensation Dominant pole (p D ) added, f < 1 ω a(j ), db 20 log a o Original After compensation 20 log 1 f p D Original p 1 p 2 p 3 ω log scale After compensation ω a(j ) p D p 1 /(a o f) ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 15
17 Compensation p 1 shifted to p 1, f = 1 ω a(j ), db 20 log a o Original After compensation p 1 Original p 1 p 2 p 3 ω log scale After compensation ω a(j ) p 1 p 2 /a o ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 16
18 Compensation p 1 shifted to p 1, f < 1 ω a(j ), db 20 log a o Original After compensation 20 log 1 f p 1 p 1 p 2 p 3 Original After compensation ω log scale ω a(j ) p 1 p 2 /(a o f) ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 17
19 2-Stage Amplifier Dominant-pole compensation V CC R L1 R L1 R L2 R L2 V i1 Vo1 Vo2 Q C 1 Q2 Q 3 Q 4 V i2 V EE C : Compensation capacitor ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 18
20 2-Stage Amplifier DM half circuit R L1 v o v s R S Q 1 2C Q 3 R L2 p D = 1 2RC R = R L1 (r b3 + r π3 ) The value of C required is usually very large (typically > 1nF) ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 19
21 Compensation of Opamps Simplified BJT V DD Q 8 Q 5 Q 7 v i I bias Q 1 Q 2 C v o Q 3 Q 4 Q 6 V SS C : Compensation capacitor ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 20
22 Compensation of Opamps Simplified MOS V DD M 8 M 5 M 7 v i I bias M 1 M 2 C v o M 3 M 4 M 6 V SS C : Compensation capacitor ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 21
23 Compensation of Opamps Small-signal equivalent C v o C in R g m6 v 1 C 2 R 2 v i R in g m1 v i 1 v1 C 1 v o v i = g m1 R 1 R 2 (g m6 sc) 1+s[R 1 (C 1 +C)+R 2 (C 2 +C)+g m6 R 1 R 2 C]+s 2 [R 1 R 2 (C 1 C 2 +CC 1 +CC 2 ] If p 1 and p 2 are real, p 1 p 2, C is large, g m6 R 1, g m6 R 2 1 : 1 p 1 R 1 (C 1 +C) + R 2 (C 2 +C) + g m6 R 1 R 2 C 1 g m6 R 1 R 2 C g m6 C p 2 C 1 C 2 + C(C 1 +C 2 ) z = g m6 C ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 22
24 Compensation of Opamps Pole splitting C = 0 p 1 = 1 R 1 C 1, p 2 = 1 R 2 C 2 1 C C 1, C 2 p 1 g m6 R 1 R 2 C, p 2 g m6 C 1 + C 2 1 R 2 C 2 poles split 1 R 1 C 1 jω s plane σ As C increases, p 1 decreases and p 2 increases ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 23
25 Effect of RHP Zero z = g m6 /C v o v i (j ω), db a z v o v i ω p1 z p 2 log scale ω (j ω) If a z > 0 db and z < p 2, PM becomes negative ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 24
26 Elimination of RHP Zero Effect Solution #1 C C 1 Buffer V DD M 8 M 5 M 7 v i M 1 M 2 C M 9 I bias v o I D M 3 M 4 M 6 V SS ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 25
27 Elimination of RHP Zero Effect Solution #1 Assuming g m6 R 1, g m6 R 2 1 and C is large, 1 p 1 g m6 R 1 R 2 C p 2 g m6c (C 1 + C)C 2 g m6 C 2 The zero has been eliminated p 1 is unchanged p 2 is approximately the same as before Extra devices and bias current are required Output swing is affected ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 26
28 Elimination of RHP Zero Effect Solution #2 V DD M 8 v i M 5 I D C M 7 M 1 M 2 VB M 9 I bias v o M 3 M 4 I D M 6 V SS ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 27
29 Elimination of RHP Zero Effect Solution #2 Assuming g m6 R 1, g m6 R 2 1 and C is large, 1 p 1 g m6 R 1 R 2 C g m6 p 2 C + C 2 C C 1 The zero has been eliminated p 1 is unchanged p 2 is at a higher frequency Extra devices and bias current are required Input offset voltage is affected if I D s mismatch ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 28
30 Elimination of RHP Zero Effect Solution #2 - alt. V DD M 8 M 5 M 7 v i I bias V B M 1 M 2 M 9 M 10 V B v o M 3 M 4 C M 6 V SS ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 29
31 Elimination of RHP Zero Effect Solution #3 V DD M 8 M 5 M 7 v i I bias M 1 M 2 C R Z v o M 3 M 4 M 6 V SS ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 30
32 Elimination of RHP Zero Effect Solution #3 Assuming g m6 R 1, g m6 R 2 1 and C is large, 1 p 1 g m6 R 1 R 2 C g m6 g m6 C p 2 C 1 C 2 + C(C 1 +C 2 ) C 1 +C 2 p 3 1 R Z C 1 z = 1 1 g m6 R Z C R Z = 1 g m6 z = R Z > 1 g m6 LHP zero (z < 0) ECEN 326 Electronic Circuits - Aydın İ. Karşılayan - Stability 31
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