EE 435. Lecture 17. Compensation of Feedback Amplifiers

Transcription

1 EE 435 Lecture 17 Compensation of Feedback Amplifiers

2 . Review from last lecture. Can now use these results to calculate poles of Basic Two-stae Miller Compensated Op Amp From small sinal analysis: A s s 2 C p m5 2= C L C C md L s m5 m5 sc C C C oo p= oo od 1 m5 C C A m5 md = oo od od md GB= C C m5 md p m5 md oo od md 1 oo od oo od m5 C C od oo m1 o2 o5 o4 m2 o6

3 . Review from last lecture. V IN V DD M 3 M 4 M 5 M 1 M 2 V IN C C I T V B2 M7 V B3 M 6 V SS Basic Two-Stae Op Amp mdm sc c A FB(s ) 2 s C C C L sc C mo β md β mdmo C L V OUT Pole Q?

4 . Review from last lecture. Basic Two-Stae Op Amp mdm sc c A FB(s ) 2 s C C C L sc C mo β md β mdmo V DD M 3 M 4 M 5 It can be shown that V IN M 1 M 2 V IN C C C L V OUT Q C C L C mo mo md md V B2 V SS I T M 7 V B3 M 6 where oo But what pole Q is desired?.77< Q <.5 o5 C C md m1 o6 CLβ 2 Q and β 2 Riht Half-Plane Zero in OL Gain (from Miller Compensation) Limits Performance mo od mo o2 md md mo m5 4

5 Compensation 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 243) The oal of compensation is to maintain stability when neative feedback is applied around the op amp.

6 Compensation From Gray and Meyer (p634) Thus if this amplifier is to be used in a feedback loop with loop ain larer than a f 1, efforts must be made to increase the phase marin. This process is known as compensation. From Sedra and Smith (p 9) 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.

7 Compensation 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!

8 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, neative feedback, phase marin, or oscillation. Note that acceptable performance is strictly determined by the user in the context of the specific application

9 Approach to Studyin Compensation Will attempt to develop a correct understandin of the concept of compensation rather than plune into a procedure for doin compensation Compensation requires the use of some classical mathematical concepts

10 Compensation 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 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 Compensation 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, )

11 Compensation Compensation 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 Compensation requirements are stonly β dependent Characteristic Polynomial obtained from denominator term of basic feedback equation 1+A sβs Asβs defined to be the loop ain of a feedback amplifier

12 Review of Basic Concepts 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

13 Review of Basic Concepts 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. 45 o pole-pair anle corresponds to 9 o pole anle (on pole pair) corresponds to 1 Q = 2

14 Review of Basic Concepts Nyquist Plots The Nyquist Plot is a plot of the Loop Gain (Aβ) versus j in the complex plane for - < < Theorem: A system is stable iff the Nyquist Plot does not encircle the point -1+j. 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

15 Review of Basic Concepts Nyquist Plots Example Im Aβ(j) = = -1+j Re = - Stable since -1+j is not encircled Useful for determinin stability when few computational tools are available Leacy of enineers and mathematicians of pre-computer era

16 Review of Basic Concepts Nyquist Plots Example 1 s 1 β=1/2 A j As 5 j 1 Im = -1 = - Re -1+j = 25 = =1 In this example, Nyquist plot is circle of radius 25

17 Review of Basic Concepts Nyquist Plots Im DFB s = 1+A sβs Im Re -1+j Re s-plane A(s)β -1+j 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 Important in the 3s - 6 s

18 Review of Basic Concepts Nyquist Plots Conceptually would like to be sure Nyquist Plot does not et too close to -1+j Im Nyquist Plot -1+j Re Unit Circle

19 Review of Basic Concepts Nyquist Plots Conceptually would like to be sure Nyquist Plot does not et too close to -1+j Im Nyquist Plot -1+j Re But identification of a suitable neihborhood is not natural Unit Circle

20 Review of Basic Concepts Nyquist Plots Conceptually would like to be sure Nyquist Plot does not et too close to -1+j Im Nyquist Plot -1+j Re But identification of a suitable neihborhood is not natural Unit Circle

21 Review of Basic Concepts Nyquist Plots Im Phase Marin -1+j Re Unit Circle Phase marin is 18 o anle of Aβ when the manitude of Aβ =1

22 Review of Basic Concepts Nyquist Plots Im Gain Marin -1+j Re Unit Circle Gain marin is 1 manitude of Aβ when the anle of Aβ =18 o

23 Nyquist and Gain-Phase Plots Nyquist and Gain-Phase Plots convey identical information but ain-phase plots often easier to work with Ma 7 6 = Im Ma Phase Re j = - = Phase Note: The two plots do not correspond to the same system in this slide

24 Nyquist and Gain-Phase Plots Nyquist and Gain-Phase Plots convey identical information but ain-phase plots often easier to work with = -1+j = - Im Ma Phase = Re Ma Aβ plots chane with different values of β Often β is independent of frequency Phase 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 Instead of plottin Aβ, often plot A and phase of A and superimpose β and phase of β to et ain and phase marins do not need to replot A and phase of A to assess performance with different β

25 Anle in derees Manitude in db Gain and Phase Marin Examples β 1 1 T(s) s Phase Marin -18 o

26 Anle in derees Manitude in db Gain and Phase Marin Examples Be aware of the multiple values of the arctan function!

27 Anle in derees Manitude in db Gain and Phase Marin Examples β Phase Marin

28 Anle in derees Manitude in db Gain and Phase Marin Examples β Phase Marin

29 Anle in derees Manitude in db Gain and Phase Marin Examples A s 1 s s β 1 Phase Marin

30 Anle in derees Manitude in db Gain and Phase Marin Examples A s 1 s s β 1 Phase Marin Unstable!

31 Anle in derees Manitude in db Gain and Phase Marin Examples β 1 T(s) 1 s o Phase Marin

32 Anle in derees Manitude in db Gain and Phase Marin Examples Gain Marin β 1 T(s) 1 s o -25-3

33 Anle in derees Manitude in db Gain and Phase Marin Examples β 1 T(s) 1 s Phase Marin -18 o -25-3

34 Anle in derees Manitude in db Gain and Phase Marin Examples β 1 Gain Marin T(s) 1 s o -25-3

35 Relationship between pole Q and phase marin In eneral, the relationship between the phase marin 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 lowpass, a closed form analytical relationship between pole Q and phase marin exists and this is independent of A and β.. Q cos(φ sin(φ M M ) ) φ M cos Q 4 1 2Q 2 The reion of interest is invariable only for.5 < Q <.7 larer Q introduces unacceptable rinin and settlin smaller Q slows the amplifier down too much

36 Pole Q Phase Marin vs Q Second-order low-pass Amplifier Phase Marin

37 Phase Marin vs Q Second-order low-pass Amplifier Pole Q Phase Marin

38 Phase Marin vs Q Second-order low-pass Amplifier Pole Q Phase Marin

39 Manitude Response of 2 nd -order Lowpass Function Q MAX for no peakin = Q From Laker-Sansen Text

40 Phase Response of 2 nd -order Lowpass Function 1 2Q From Laker-Sansen Text

41 Step Response of 2 nd -order Lowpass Function 1 2Q Q MAX for no overshoot = 1/2 From Laker-Sansen Text

42 Step Response of 2 nd -order Lowpass Function 1 2Q From Laker-Sansen Text

43 Compensation Summary Gain and phase marin performance often stronly dependent upon architecture Relationship between overshoot and rinin and phase marin were developed only for 2 nd -order lowpass ain characteristics and differ dramatically for hiher-order structures Absolute ain and phase marin criteria are not robust to chanes in architecture or order It is often difficult to correctly break the loop to determine the loop ain Aβ with the correct loadin on the loop (will discuss this more later)

44 End of Lecture 17