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Compensation of Feedback Amplifiers Two-Stage Op Amp Design Strategies .• • • • • Review from Last Lecture .• • • • •

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Compensation of Feedback Amplifiers Two-Stage Op Amp Design Strategies .• • • • • Review from Last Lecture .• • • • • EE 435 Lecture 18 Compensation of Feedback Amplifiers Two-Stage Op Amp Design Strategies .• • • • • Review from last lecture .• • • • • • Simple pole calculations for 2-stage op amp • • • • Since the poles of the 2-stage op amp must be widely separated, a simple calculation of the poles from the characteristic polynomial is possible. Assume p1 and p2 are the poles and p1 << p2 2 D(s)=s +a1s+a0 determines p1 but Review from last lecture .• • .• Review from last lecture 2 2 D(s)=(s+p1)(s+p2)=s +s(p1+p2)+p1p2 s + p2s + p1p2 • • • • • thus • determines p2 • . p2=-a1 and p1 = -a0/a1 .• • • • • Review from last lecture .• • • • • Compensation Compensation (alt Frequency Compensation) is the manipulation of the poles and/or zeros of the open-loop amplifier so that when feedback is applied, the closed-loop amplifier will perform acceptably Note this definition does not mention stability, positive feedback, negative feedback, phase margin, or oscillation. Note that acceptable performance is strictly determined by the user in the context of the specific application .• • • • • Review from last lecture .• • • • • Compensation Compensation requirements usually determined by closed-loop pole locations: D FB (s ) D (s ) β (s )N (s ) • Often Phase Margin or Gain Margin criteria are used instead of pole Q criteria when compensating amplifiers (for historical reasons but must still be conversant with this approach) • Nyquist plots are an alternative stability criteria that is used some in the design of amplifiers • Phase Margin and Gain Margin criteria are directly related to the Nyquist Plots • Compensation requirements are stongly β dependent Characteristic Polynomial obtained from denominator term of basic feedback equation 1+A sβs As βs defined to be the “loop gain” of a feedback amplifier .• • • • • Review from last lecture .• • • • • Review of Basic Concepts Nyquist Plots The Nyquist Plot is a plot of the Loop Gain (Aβ) versus jω in the complex plane for - ∞ < ω < ∞ Theorem: A system is stable iff the Nyquist Plot does not encircle the point -1+j0. Note: If there are multiple crossings of the real axis by the Nyquist Plot, the term encirclement requires a formal definition that will not be presented here Review of Basic.• • •Concepts • • Review from last lecture .• • • • • Nyquist Plots Example Im Aβ(jω) ω = ∞ ω = 0 -1+j0 Re ω = - ∞ • Stable since -1+j0 is not encircled • Useful for determining stability when few computational tools are available • Legacy of engineers and mathematicians of pre-computer era .• • • • • Review from last lecture .• • • • • Review of Basic Concepts Nyquist Plots DFB s = 1+A sβs Im Im Re -1+j0 Re s-plane A(s)β -1+j0 is the image of ALL poles The Nyquist Plot is the image of the entire imaginary axis and separates the image complex plane into two parts Everything outside of the Nyquist Plot is the image of the LHP Nyquist plot can be generated with pencil and paper Important in the ‘30s - ‘60’s Review of Basic Concepts Nyquist Plots Im 1+j0 Re Conceptually would like to be sure Nyquist Plot does not get too close to -1+j0 Review of Basic Concepts Nyquist Plots Im But identification of a suitable neighborhood is not natural 1+j0 Re Conceptually would like to be sure Nyquist Plot does not get too close to -1+j0 Review of Basic Concepts Nyquist Plots Might be useful to be sure image of 45o lines do not encircle -1+j0 β A(s) Im Im Re 1+j0 Re Conceptually would like to be sure Nyquist Plot does not get too close to -1+j0 Review of Basic Concepts Nyquist Plots At least one pole would make an angle What if this happened ? of less than 45o wrt Im axis β A(s) Im Im Re 1+j0 Re Conceptually would like to be sure Nyquist Plot does not get too close to -1+j0 Review of Basic Concepts Nyquist Plots Im Phase Margin -1+j0 Re Unit Circle Phase margin is 180o – angle of Aβ when the magnitude of Aβ =1 Review of Basic Concepts Nyquist Plots Im Gain Margin -1+j0 Re Unit Circle Gain margin is 1 – magnitude of Aβ when the angle of Aβ =180o Nyquist and Gain-Phase Plots Nyquist and Gain-Phase Plots convey identical information but gain-phase plots often easier to work with Mag 70 60 50 40 30 Im 20 ω 10 0 ω -10 ω = ∞ ag P M h -20 a s e Re -30 -40 -1+j0 0 ω = 0 ω -50 ω = -∞ -100 -150 -200 -250 -300 Phase Note: The two plots do not correspond to the same system in this slide Nyquist and Gain-Phase Plots Nyquist and Gain-Phase Plots convey identical information but gain-phase plots often easier to work with Mag 70 60 50 40 30 Im 20 ω 10 ω 0 -10 ω = ∞ ag P M h -20 a s e Re -30 ω -40 -1+j0 0 ω = 0 -50 ω = -∞ Phase-100 Aβ plots change with different values of β Often β is independent of frequency -150 -200 in this case Aβ plot is just a shifted version of A in this case phase of Aβ is equal to the phase of A -250 Instead of plotting Aβ, often plot |A| and phase of A and-300 superimpose |β| and phase of β to get gain and phase margins do not need to replot |A| and phase of A to assess performance with different β Gain and Phase Margin Examples 70 1000 60 T(s) 50 s 1 40 30 20 1 10 β 0 Magnitude in dB Magnitude -10 -20 ω -30 -40 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 Phase Margin Angle in degrees Angle -180o Gain and Phase Margin Examples 80 60 40 20 0 -20 Magnitude in dB Magnitude -40 ω -60 -80 100 Be aware of the multiple 80 values of the arctan function ! 60 40 20 0 -20 -40 -60 Angle in degrees Angle -80 -100 Discontinuities do not exist in magnitude or phase plots Gain and Phase Margin Examples 80 60 40 β1 20 0 -20 -40 ω Magnitude in dB Magnitude -60 -80 0 -50 -100 -150 Phase Margin -200 -250 Angle in degrees Angle -300 Gain and Phase Margin Examples 80 60 40 20 β1 0 -20 -40 ω Magnitude in dB Magnitude -60 -80 0 -50 -100 Phase Margin -150 -200 -250 Angle in degrees Angle -300 1000 Gain and Phase Margin Examples As s s+1 +1 200 β=.031 β1 ω Magnitude in dB Magnitude Phase Margin Angle in degrees Angle 1000 Gain and Phase Margin Examples As s s+1 +1 200 β=.31 β1 ω Magnitude in dB Magnitude Phase Margin Unstable ! Angle in degrees Angle Gain and Phase Margin Examples 70 1000 60 T(s) 3 50 s 1 40 30 1 20 β 10 0 -10 Magnitude in dB Magnitude -20 -30 ω -40 0 -50 -180o -100 -150 Phase Margin -200 -250 Angle in degrees Angle -300 Gain and Phase Margin Examples 70 1000 60 T(s) 3 50 s 1 40 30 Gain Margin 1 20 β 10 0 -10 Magnitude in dB Magnitude -20 -30 ω -40 0 -50 -180o -100 -150 -200 -250 Angle in degrees Angle -300 Gain and Phase Margin Examples 70 1000 60 1 T(s) 3 50 β s 1 40 30 20 10 0 -10 Magnitude in dB Magnitude -20 -30 ω -40 0 -50 Phase Margin -180o -100 -150 -200 -250 Angle in degrees Angle -300 Gain and Phase Margin Examples 70 1000 60 1 β T(s) 3 50 s 1 40 30 Gain Margin 20 10 0 -10 Magnitude in dB Magnitude -20 -30 ω -40 0 -50 -180o -100 -150 -200 -250 Angle in degrees Angle -300 Relationship between pole Q and phase margin In general, the relationship between the phase margin and the pole Q is dependent upon the order of the transfer function and on the location of the zeros In the special case that the open loop amplifier is second-order low- pass, a closed form analytical relationship between pole Q and phase margin exists and this is independent of A0 and β.. 1 1 cos(φM ) φ cos1 1 Q M 4Q4 2Q2 sin(φM ) The region of interest is invariable only for 0.5 < Q < 0.7 larger Q introduces unacceptable ringing and settling smaller Q slows the amplifier down too much Phase Margin vs Q Second-order low-pass Amplifier 7 6 5 4 3 Pole Q 2 1 0 0 20 40 60 80 100 Phase Margin Phase Margin vs Q Second-order low-pass Amplifier 1.6 1.4 1.2 1 0.8 Pole Q 0.6 0.4 0.2 0 40 50 60 70 80 Phase Margin Phase Margin vs Q Second-order low-pass Amplifier 1.6 1.4 1.2 φ ≈65o φ ≈77o 1 M M 0.8 Q=.707 Pole Q 0.6 0.4 Q=0.5 0.2 0 40 50 60 70 80 Phase Margin Magnitude Response of 2nd-order Lowpass Function 1 QMAX for no peaking = .707 2 1 2Q From Laker-Sansen Text Phase Response of 2nd-order Lowpass Function 1 2Q From Laker-Sansen Text Step Response of 2nd-order Lowpass Function 1 2Q QMAX for no overshoot = 1/2 From Laker-Sansen Text Step Response of 2nd-order Lowpass Function 1 2Q From Laker-Sansen Text Compensation Summary • Gain and phase margin performance often strongly dependent upon architecture • Relationship between overshoot and ringing and phase margin were developed only for 2nd-order lowpass gain characteristics and differ dramatically for higher-order structures • Absolute gain and phase margin criteria are not robust to changes in architecture or order • It is often difficult to correctly “break the loop” to determine the loop gain Aβ with the correct loading on the loop (will discuss this more later) End of Lecture 18.
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