EECE 311 Lecture 7, Spring 2012

EECE 311: Electronic Circuits Part 7 2  Stability in Feedback Amplifiers         1 f A s A s  s A s                             180 1 0 . : 180 j s A s L s A s s L j A j j A j j e L j at                            Characteristic equation: Loop gain: When does loop gain become equal to –1? check |L| at w180 3  Effect of Feedback on Amplifier Poles  Single Pole Amplifier           1 Assuming is independent of frequency: 1 1 1 1 o p o o f p o Pf p o A A s s A A A s s A A                  Bandwidth Extended  Gain Decreased  Feedback did not affect stability 4  Two-Pole Amplifier               1 2 2 1 2 1 2 2 1 2 1 2 1 2 1 1 characteristicequation: 1 0 1 0 1 1 4 1 2 2 o p p p p o p p p p p p o p p A A s s s A s s s A s A                                          Amplifier unconditionally stable 5  Three-Pole Amplifier  Amplifier conditionally stable  Frequency compensation to be used to stabilize the amplifier  Poles move to the right-half plane  Amplifier becomes unstable 6  Stability Study Using Bode Plots of Loop Gain Bode plot of : A  For stability, make sure that w1 < w180  Typically, we want: GM = 30 dB and PM = 60º 7  Example: 5 5 6 7 5 10 ( ) 1 1 1 2 10 2 10 2 10 1 20log 85 dB ( = 5.62 10 ) A s s s s                                 20log 20log 20log 1 20log 20log A A A              1 plot 20log and 20log on same bode plot A         5 5 6 7 1 1 1 5 6 7 10 ( ) 1 1 1 10 10 10 tan tan tan 10 10 10 A jf jf jf jf f f f A                                         8 20log(1/b) = 85 dB |Ab | = 1 -108o Phase Margin = 72o f180 Gain Margin = 25 dB Zero Margin Limit 20log(1/b) = 50 dB UNSTABLE f1 9 10  Rule of Thumb:  General Rule: if the 20log 1 line crosses the 20log where dB its slope is 20 , the system is stable. dec A   1 at the intersection of 20log and 20log ( ) , ( ) the difference of slopes should not exceed 20 dB . dec A j j    Example: Stabilize amplifier for b = 0.01 11 New pole frequency Example: Stabilize amplifier for b = 0.01 12 Shift first pole to here 13  Frequency Compensation Gain curve A intersects the 20 log (1/β) line on the -40 dB/dec line.  Option 1: add a low frequency pole (A’)  Option 2: shift dominant pole to a lower frequency (A’’) 14  Pole Shifting Pole shifting is done by introducing a capacitor between B and B’. Therefore shifting the pole from   1 1 1 to 2 2 p x x x x C f R C R C C     One problem with this scheme is that the CC capacitor is often too large to be implemented on integrated circuits. 15  Miller Compensation and Pole Splitting Introduce a compensating capacitor, Cf, in the feedback path of the amplifier stage       1 2 1 1 2 2 1 2 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 without : 1 1 2 2 with ,transfer function becomes : 1 f p p f f m o i f m f C f f R C R C C sC g R R V I s C R C R C g R R R R s C C C C C R R                       16       1 2 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 2 ' ' ' ' ' ' 1 2 1 2 1 2 ' 1 ' 1 1 1 1 ( ) 1 1 1 normally, one of the poles, say , is dominant ( ) 1 f m o i f m f p p p p p p p p sC g R R V I s C R C R C g R R R R s C C C C C R R s s s D s s s D s                                                             2 ' ' 1 2 1 1 1 2 2 1 2 1 2 2 1 2 1 2 1 2 2 p p p f m m f p f s f C R C R C g R R R R g C f C C C C C                       Miller Compensation therefore shifts the dominant pole to the left while at the same time shifting the second pole further to the right (Pole Splitting).  Also, the compensating capacitor is multiplied by gmR2 and allows us to use much smaller on-chip capacitors. 17  Example 1 2 1 2 1 pF 0.3 pF 0.2 pF 50 K 1 mA/V C m C C C R R g        1 2 without : 3.2 MHz 10.6 MHz C p p C f f   ' 1 ' 2 with : 0.3 MHz 57 MHz 800 MHz 2 C p p m z C C f f g f C      18  The Two-Stage CMOS OP-AMP 19  High Frequency Model 1 1 2 2 2 2 2 2 1 2 2 2 p m C m p m Z C f R G R C G f C G f C       1 1 1 2 2 2 1 2 2 should be less than and less than m t p C t z m m C p m m G f A f C f f G G C C f G G         to let the gain curve intersect with the 0 dB-line (feedback with b = 1) with a slope of -20 dB/dec. Considering feedback with b = 1 20  Positive Feedback  For positive feedback: Af = A / (1 – b A)  Barkhausen criterion for oscillation  b A = 1  (j) A(j) = 1  magnitude(b A) = 1  phase angle(b A) = 0 Sinusoidal Oscillators  Wien-Bridge Oscillator  Amplifier gain  Feedback factor 21 1 f r R A R           0 0 2 0 0 // 1 1 // 1/ 1 (1 ) (1 ) / 1 / 3 / 1 P S P R R sC Z sRC R Z Z R sC R sC R sRC sC sRC sRC sRC sRC sRC s RC s s                         Sinusoidal Oscillators  Wien-Bridge Oscillator  Oscillation condition:  Pure real:  At oscillation frequency: 22               0 2 0 0 0 2 0 0 ( ) ( ) 1 1 / / 1 / 3 / 1 1 / / 1 3 / 1 / f r f r A j j R R j j j j R R j                           2 0 0 1 1 / 0 RC            ( ) ( ) 1 / 1 2 3 f r f r A j j R R R R         Sinusoidal Oscillators  Phase-shift oscillator  Amplifier phase shift = 180 degrees  Inverting OP-AMP amplifier  The other 180 degrees come from the three RC sections  Feedback block 23 Sinusoidal LC Oscillators  Feedback block is a tuned LC circuit 24 C2 C1 L L2 L1 C LC Oscillators  Colpitts Oscillator  Amplifier assumed to have infinite input resistance 25 Colpitts Oscillator  KCL at output node  LC2 voltage divider  Therefore 26 1 2 1 ( ) ( ) ( ) 0 m i o f G V s sC V s sC V s R           2 2 1 ( ) ( ) 1 f o V s s LC V s     2 2 1 2 1 1 ( ) 1 ( ) i f m V s sC sC s LC V s G R                 Colpitts Oscillator  Oscillation condition 27           2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 m m eq eq j C j C LC G R j C C LC LC G R C C LC C C C C C C C LC L C C                                              Colpitts Oscillator  Amplifier gain 28 2 1 2 1 1 m m C G R C C G R C               LC Oscillators  Hartley Oscillator  Oscillation frequency 29 L2 L1 C -K   1 2 1 L L C   