EE 435 Lecture 15 ompensation of ascaded Amplifier Structures
. Review from last lecture. Basic Two-Stae Op Amp V DD M 3 M 4 M 5 V OUT V IN M 1 M V IN L I T V B M 7 V B3 M 6 By inspection A o m1 o p 1 o1 o4 05 o5 m5 o6 V SS o5 m5 o6 p GB m5 L m1
. Review from last lecture. Small Sinal Analysis of Two-Stae Miller ompensated Op Amp V DD M 3 M 4 M 5 V OUT V IN M 1 M V IN L I T V B M7 V B3 M 6 V SS Differential Small Sinal Equivalent M 3 M 4 M 5 V OUT V d M 1 M V d M 6 L Norton Equivalent One-Port Two-Port
. Review from last lecture. Small Sinal Analysis of Two-Stae Op Amp Differential Small Sinal Equivalent V OUT md V d Od V mo V OO L Solvin we obtain: V OUT This simplifies to: V V V V d OUT OUT s sl oo mov sv s od mdvd s OUT V s V d L s s md mo s mo oo od Lod oood md L s mo mo s oo od
. Review from last lecture. Small Sinal Analysis of Two-Stae Op Amp Differential Small Sinal Equivalent Summary: where A s s md L s m5 m5 s oo od md m1 m od o o4 oo o5 o6
. Review from last lecture. Small Sinal Analysis of Two-Stae Miller- ompensated Op Amp A s s md L s m5 m5 s oo od Note this is of the form: A 1 A s 0 s +1 z s s +1 +1 p p 1 This has two neative real-axis poles and one positive real-axis zero jω p p 1 z 1 Re
How does the Gain of the Two-Stae Miller-ompensated Op Amp ompare with Internal ompensated Op Amp? P 1 P P 1 P V OUT V OUT V IN F 1 F EFF L V IN F 1 F L md m5 -s A s = md As m5 s +s + s +1 z 0 s s +1 +1 p1 p A 1 A s L m5 oo od L oo oo od s +s + A0 A s 1 s s +1 +1 p1 p jω jω Re p p 1 Re p p 1 z 1 must be developed ompensation criteria: 4β A p 0 β A0 p1
Review of Basic oncepts onsider a second-order factor of a denominator polynomial, P(s), P(s)=s +a 1 s+a 0 Then P(s) can be expressed in several alternative but equivalent ways s s ω s Q sξ s p s p 1 and if 0 ω 0 ω complex s α jβs α jβ 0 0 conjuate poles, These are all -paramater characterizations of the second-order factor and it is easy to map from any one characterization to any other
Review of Basic oncepts Im s s ω Q 0 ω 0 ω o sinθ 1 Q Re ω o = manitude of pole Q determines the anle of the pole
Simple pole calculations for -stae op amp Since the poles of the -stae op amp must be widely separated, a simple calculation of the poles from the characteristic polynomial is possible. Assume p 1 and p are the poles and p 1 << p D(s)=s +a 1 s+a 0 but determines p 1 D(s)=(s+p 1 )(s+p )=s +s(p 1 +p )+p 1 p s + p s + p 1 p thus determines p p =-a 1 and p 1 = -a 0 /a 1
an now use these results to calculate poles of Basic Two-stae Miller ompensated Op Amp From small sinal analysis: A s s p m5 = L md L s m5 m5 s oo p= oo od 1 m5 A m5 md 0= oo od od md GB= m5 md p m5 md oo od md 1 oo od oo od m5 od oo m1 o o5 o4 m o6
From Previous Inspection V DD A o m1 o o4 o5 m5 o6 M 3 M 4 M 5 V OUT p 1 o GB 05 o4 m5 m1 o6 p m5 L V IN M 1 M I T V B M 7 V SS V IN V B3 M 6 L Note the simple results obtained from inspection aree with the more time consumin results obtained from a small sinal analysis
Feedback applications of the twostae Op Amp X IN X 1 A X OUT β How does the amplifier perform with feedback? How should the amplifier be compensated?
Feedback applications of the twostae Op Amp Open-loop Gain A FB (s) A(s) Standard Feedback Gain 1 N(s) D(s) A(s) A(s)β(s) N FB D FB (s) (s) N(s) D(s) N(s)β(s) defn N(s) D (s) β(s)n(s) N D FB FB (s) (s) Open-loop and closed-loop zeros identical losed-loop poles different than open-loop poles Often β(s) is not dependent upon frequency
Open-loop Gain A(s) N(s) D(s) Alternate Feedback Gain 1 β1(s) FB(s) 1 1 A(s) β(s) Feedback applications of the twostae Op Amp Standard Feedback Gain (s) A FB A 1 A(s) 1 A(s)β(s) (often FB is not of standard form) β(s) N(s) β (s) D(s) N(s) β(s) 1 1 β(s) 1 A(s) β(s) In either case, denominators are the same and characteristic equation defined by D FB (s) D (s) β(s)n(s) Often β(s) and β 1 (s) are not dependent upon frequency and in this case N FB (s) N(s)
Basic Two-Stae Op Amp with Feedback V DD M 3 M 4 M 5 Open-loop ain -s A(s)= s +s + md mo c L mo oo od V OUT V IN M 1 M V IN L I T V B V SS M 7 A FB A V B3 (s) FB where s (s) s M 6 L s Standard feedback ain with constant β md m1 od L s o mdmo sc mo βmd oood βmdmo md m0 sc mo β md β mdmo 04 mo m5 oo o5 o6
V IN V DD M 3 M 4 M 5 M 1 M V IN I T V B M7 V B3 M 6 V SS Basic Two-Stae Op Amp mdm0 sc A FB(s) V OUT s L s mo β md β mdmo L Pole Q?
Basic Two-Stae Op Amp mdm0 sc A FB(s) s L s mo β md β mdmo V DD M 3 M 4 M 5 It can be shown that V IN M 1 M mo md V B V SS I T M 7 V IN V B3 M 6 L V OUT where oo But what pole Q is desired?.707< Q <0.5 Q o5 md m1 Riht Half-Plane Zero in OL Gain Limits Performance o6 L Lβ Q and od mo o md β mo mo md md mo m5 04
ompensation What is compensation or frequency compensation? From Wikipedia: In electrical enineerin, frequency compensation is a technique used in amplifiers, and especially in amplifiers employin neative feedback. It usually has two primary oals: To avoid the unintentional creation of positive feedback, which will cause the amplifier to oscillate, and to control overshoot and rinin in the amplifier's step response. From Martin and Johns no specific definition but makes comparisons with optimal compensation which also is not defined From Allen and Holber (p 43) The oal of compensation is to maintain stability when neative feedback is applied around the op amp.
ompensation From Gray and Meyer (p634) Thus if this amplifier is to be used in a feedback loop with loop ain larer than a 0 f 1, efforts must be made to increase the phase marin. This process is known as compensation. From Sedra and Smith (p 90) This process of modifyin the open-loop ain is termed frequency compensation, and its purpose is to ensure that op-amp circuits will be stable (as opposed to oscillatory). From Razavi (p355) Typical op amp circuit contain many poles. In a foldedcascode topoloy, for example, both the foldin node and the output node contribute poles For this reason, op amps must usually be compensated, that is, their open-loop transfer function must be modified such that the closed-loop circuit is stable and the time response is well-behaved.
ompensation What is compensation or frequency compensation and what is the oal of compensation? Nobody defines it or defines it correctly but everybody tries to do it!
ompensation ompensation (alt Frequency ompensation) is the manipulation of the poles and/or zeros of the open-loop amplifier so that the closed-loop amplifier will perform acceptably Note this definition does not mention stability, positive feedback, neative feedback, phase marin, or oscillation. Note that acceptable performance is strictly determined by the user in the context of the specific application
ompensation ompensation is the manipulation of the poles and/or zeros of the open-loop amplifier so that the closed-loop amplifier will perform acceptably Acceptable performance is often application dependent and somewhat interpretation dependent Acceptable performance should include affects of process and temperature variations Althouh some think of compensation as a method of maintainin stability with feedback, acceptable performance enerally dictates much more strinent performance than simply stability ompensation criteria are often an indirect indicator of some type of desired (but unstated) performance Varyin approaches and criteria are used for compensation often resultin in similar but not identical performance Over compensation often comes at a considerable expense (increased power, decreased frequency response, increased area, )
ompensation ompensation requirements usually determined by closed-loop pole locations: D FB (s) D(s) β(s)n(s) Often Phase Marin or Gain Marin criteria are used instead of pole Q criteria when compensatin 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 desin of amplifiers Phase Marin and Gain Marin criteria are directly related to the Nyquist Plots ompensation requirements are stonly β dependent haracteristic Polynomial obtained from denominator term of basic feedback equation 1+A sβs Asβs defined to be the loop ain of a feedback amplifier
Review of Basic oncepts Pole Locations and Stability Theorem: A system is stable iff all closed-loop poles lie in the open left half-plane. Im Im Re Re Unstable Im Stable Re Unstable
Review of Basic oncepts 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 havin closed-loop amplifier poles close to the imainary axis to provide reasonable stability marin, to minimize rinin in the time-domain, and to minimize peakin in the frequency domain. Im 45 o Typical Acceptable Reion for Poles Re 45 o 1 pole-pair anle corresponds to Q= =.707 90 o 1 pole anle (on pole pair) corresponds to Q =
Review of Basic oncepts 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 crossins of the real axis by the Nyquist Plot, the term encirclement requires a formal definition that will not be presented here
Review of Basic oncepts Nyquist Plots Example Im ω = ω = 0-1+j0 Re ω = - Stable since -1+j0 is not encircled Useful for determinin stability when few computational tools are available Leacy of enineers and mathematicians of pre-computer era
Review of Basic oncepts Nyquist Plots Example 100 s 1 β=1/ A j As 50 j 1 Im ω = -1 ω = - Re -1+j0 ω = 5 ω=0 ω=1 In this example, Nyquist plot is circle of radius 5
Review of Basic oncepts Nyquist Plots DFB s = 1+A sβs Im Im Re -1+j0 Re s-plane A(s)β -1+j0 is the imae of ALL poles The Nyquist Plot is the imae of the entire imainary axis and separates the imae complex plane into two parts Everythin outside of the Nyquist Plot is the imae of the LHP Nyquist plot can be enerated with pencil and paper
Review of Basic oncepts Nyquist Plots onceptually would like to be sure Nyquist Plot does not et too close to -1+j0 Im Nyquist Plot -1+j0 Re Unit ircle
Review of Basic oncepts Nyquist Plots onceptually would like to be sure Nyquist Plot does not et too close to -1+j0 Im Nyquist Plot -1+j0 Re But identification of a suitable neihborhood is not natural Unit ircle
Review of Basic oncepts Nyquist Plots onceptually would like to be sure Nyquist Plot does not et too close to -1+j0 Im Nyquist Plot -1+j0 Re But identification of a suitable neihborhood is not natural Unit ircle
Review of Basic oncepts Nyquist Plots Im Phase Marin -1+j0 Re Unit ircle Phase marin is 180 o anle of Aβ when the manitude of Aβ =1
Review of Basic oncepts Nyquist Plots Im Gain Marin -1+j0 Re Unit ircle Gain marin is 1 manitude of Aβ when the anle of Aβ =180 o
End of Lecture 15