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EE 435 Lect 16 Spring 2010

# EE 435 Lect 16 Spring 2010 - EE 435 Lecture 16 Compensation...

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EE 435 Lecture 16 Compensation Systematic Two-Stage Op Amp Design

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Pole Locations and Stability Theorem: A system is stable iff all closed-loop poles lie in the open left half-plane. Note: Practically want to avoid having closed-loop amplifier poles close to the imaginary axis to provide reasonable stability margin, to minimize ringing in the time-domain, and to minimize peaking in the frequency domain. Review of Basic Concepts 45 o pole-pair angle corresponds to 1 Q= =.707 2 90 o pole angle (on pole pair) corresponds to 2 1 Q = Review from last lecture
Nyquist Plots The Nyquist Plot is a plot of the Loop Gain (A β ) versus j ω 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

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Nyquist Plots Review of Basic Concepts -1+j0 Re Im -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 Re Im s-plane A(s) β ( ) ( ) ( ) FB Ds = 1 + A s β s Review from last lecture
Nyquist Plots Review of Basic Concepts -1+j0 Re Im Unit Circle Phase Margin Phase margin is 180 o – angle of A β when the magnitude of A β =1 Review from last lecture

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Nyquist Plots Review of Basic Concepts -1+j0 Re Im Unit Circle Gain Margin Gain margin is 1 – magnitude of A β when the angle of A β =180 o Review from last lecture
Nyquist Plots Review of Basic Concepts Theorem: A system is stable iff the phase margin is positive Theorem: A system is stable iff the gain margin is positive The phase margin is often the parameter that is specified when compensating operational amplifiers Phase margins of 45 o to 60 o or sometimes even 75 o are often used The definition of phase margin does not depend upon the order of the system and is affected by the location of the zeros of the system The phase margin is a function of β Review from last lecture

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Nyquist Plots Review of Basic Concepts Engineers have some comfort in how far an amplifier is from becoming stable when specifying phase margin criteria (but this is often not mathematically justifiable) Pole Q criteria are generally much better to use than phase margin criteria but industry is heavily “phase-margin” entrenched ! Separate magnitude and phase plots are often used rather than Nyquist Plots when assessing phase margins or gain margins The magnitude and phase plots convey exactly the same information as Nyquist Plots but have a linear (or logarithmic) axis rather than the highly skewed imaginary axis of the Nyquist Plot Review from last lecture
Nyquist and Gain-Phase Plots -40 -30 -20 -10 0 10 20 30 40 50 60 70 Nyquist and Gain-Phase Plots convey identical information but gain-phase plots often easier to work with -300 -250 -200 -150 -100 -50 0 Note: The two plots do not correspond to the same system in this slide Mag Phase ω ω

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EE 435 Lect 16 Spring 2010 - EE 435 Lecture 16 Compensation...

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