EE 435. Lecture 16. Compensation Systematic Two-Stage Op Amp Design
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1 EE 435 Lecture 6 Compensation Systematic Two-Stage Op Amp Design
2 Review from last lecture 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 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. 45 o pole-pair angle corresponds to 90 o pole angle (on pole pair) corresponds to Q= =.707 Q =
3 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 -+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
4 Review from last lecture Review of Basic Concepts Nyquist Plots DFB ( s ) = +A( s) β( s) Im Im Re -+j0 Re s-plane A(s)β -+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
5 Review from last lecture Review of Basic Concepts Nyquist Plots Im Phase Margin -+j0 Re Unit Circle Phase margin is 80 o angle of Aβ when the magnitude of Aβ =
6 Review from last lecture Review of Basic Concepts Nyquist Plots Im Gain Margin -+j0 Re Unit Circle Gain margin is magnitude of Aβ when the angle of Aβ =80 o
7 Review from last lecture Review of Basic Concepts Nyquist Plots 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
8 Review from last lecture Review of Basic Concepts Nyquist Plots 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
9 Nyquist and Gain-Phase Plots Nyquist and Gain-Phase Plots convey identical information but gain-phase plots often easier to work with Mag ω 0-50 ω Phase Note: The two plots do not correspond to the same system in this slide
10 Gain and Phase Margin Examples Magnitude in db β ω T(s) = 000 ( s + ) 3 Angle in degrees o ω Phase Margin
11 Gain and Phase Margin Examples Magnitude in db Gain Margin β ω T(s) = 000 ( s + ) 3 Angle in degrees o ω
12 Gain and Phase Margin Examples Magnitude in db β ω T(s) = 000 ( s + ) ω Angle in degrees Phase Margin -80 o -300
13 Gain and Phase Margin Examples Magnitude in db β Gain Margin ω T(s) = 000 ( s + ) 3 Angle in degrees o ω
14 Gain and Phase Margin Examples Magnitude in db β ω 000 T(s) = ( s + ) Angle in degrees ω Phase Margin -80 o
15 Gain and Phase Margin Examples Magnitude in db T(s) = ( s + ) ( s + 0) ω Angle in degrees Be aware of the multiple values of the arctan function! ω
16 Magnitude in db Gain and Phase Margin Examples β T(s) ω = 58 ( s + ) ( s + 0) Angle in degrees Phase Margin ω -300
17 Gain and Phase Margin Examples T(s) = 58 ( s + ) ( s + 0) Magnitude in db β ω -80 Angle in degrees Phase Margin ω -300
18 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 lowpass, a closed form analytical relationship between pole Q and phase margin exists and this is independent of A 0 and β.. Q cos(φ ) φ = cos M = M 4 sin(φm) 4Q Q + The region of interest is invariable only for < Q < 0.7 larger Q introduces unacceptable ringing and settling smaller Q slows the amplifier down too much
19 Pole Q Phase Margin vs Q Second-order low-pass Amplifier Phase Margin
20 Phase Margin vs Q Second-order low-pass Amplifier Pole Q Phase Margin
21 Phase Margin vs Q Second-order low-pass Amplifier Pole Q Phase Margin
22 Magnitude Response of nd -order Lowpass Function Q MAX for no peaking = 707 =. ξ = Q From Laker-Sansen Text
23 Phase Response of nd -order Lowpass Function ξ = Q From Laker-Sansen Text
24 Step Response of nd -order Lowpass Function ξ = Q Q MAX for no overshoot = / From Laker-Sansen Text
25 Step Response of nd -order Lowpass Function ξ = Q From Laker-Sansen Text
26 Compensation Summary Gain and phase margin performance often strongly dependent upon architecture Relationship between overshoot and ringing and phase margin were developed only for nd -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)
27 Design of Two-Stage Op Amps Compensation is critical on two-stage op amps General approach to designing two-stage op amps is common even though significant differences in performance for different architectures Will consider initially the most basic twostage op amp with internal compensation
28 Natural Parameter Space for the Two-Stage Amplifier Design S NATURAL = {W, L, W 3, L 3, W 5, L 5, W 6, L 6, W 7, L 7, I T, I D6, C c, B, B3 }
29 Design Degrees of Freedom Total independent variables: 5 Degrees of Freedom: 5 If phase margin is considered a constraint, 5 independent ariables, constraint and thus 4 degrees of freedom
30 Observation: W,L appear as W/L ratio in almost all characterizing equations Implication: Degrees of Freedom are Reduced S NATURAL-REDUCED = {(W/L),(W/L) 3,(W/L) 5,(W/L) 6,(W/L) 7,I D6,I T,C C } With phase margin constraint, Degrees of freedom: 7
31 Common Performance Parameters of Operational Amplifiers Parameter Description Ao Open-loop DC Gain GB Gain-Bandwidth Product Φm(or Q) Phase Margin (or pole Q) SR Slew Rate T SETTLE Settling Time A T Total Area A A Total Active Area P Power Dissipation σ OS Standard Deviation of Input Referred Offset oltage (often termed the input offset voltage) CMRR Common Mode Rejection Ratio PSRR Power Supply Rejection Ratio imax Maximum Common Mode Input oltage imin Minimum Common Mode Output oltage omax Maximum Output oltage Swing omin Minimum Output oltage Swing noise Input Referred RMS Noise oltage Sv Input Referred Noise Spectral Density
32 Performance Parameters Total: 7
33 Performance Parameters Total: 7 System is Generally Highly Over Constrained!
34 Typical Parameter Space for a Two-Stage Amplifier Small signal model of the two-stage operational amplifier Augmented set of design parameters: S AUGMENTED = {g oo, g od, g mo, g md, C C, EBQ, EB3Q, EB5Q, EB6Q, EB7Q, I T, g o, g o4, g o5, g o6 }
35 Signal Swing of Two-Stage Op Amp M6: > + OUT SS EB6 M: M5: M: M7: ic ic < ic < OUT DD DD + < + DD > T + EB + EB7 EB5 T T3 EB3 + T T5 EB5 SS
36 Signal Swing of Two-Stage Op Amp OUT DD EB5 + + T EB EB7 max{( ( EB3 EB5 + + T3 T5 T T ), )} SS DD ic OUT < OUT DD SS > + EB5 EB6 EB6 SS + ic > T + EB + EB7 SS ic < DD + T T3 EB3 ic < DD + T T5 EB5
37 Signal Swing of Two-Stage Op Amp OUT DD EB T EB EB7 max{( ( EB3 EB5 + + T3 T5 T T ), )} SS DD ic EB6 SS
38 Common Expressions for the Performance Parameters A O GB SR g g md oo g C I C g g md T C C mo od
39 Common Expressions for the Performance Parameters (cont) OMAX OMIN = DD SS = + EB5 EB6 + inmin = T + EB + EB7 SS = max{( 3 + T 3 T ),( EB 5 + T 5 inmax DD EB T )}
40 Parameter Inter-dependence A O g g md GB oo g g g C mo od md C I T affects SR I C T C W md OX T g μc I L
41 A Set of Independent Design Parameters is Needed Consider the Natural Reduced Parameter Set W W3 W5 W6 W 7,,,,,I T, θ L L3 L L 5 6 L7 A I θ= O D6Q g g I T md oo g g mo od A O = C μ μ OX n p ( + ) W W W L L n p T W 7L 6 λ λ I L
42 md C g GB C n p p n L C L W β L L L W W W W L L L W W W L Q C β C μ μ μ μ = n OX T C μ C W I L GB C = For a given pole Q and a feedback factor β, it can be shown that:
43 + + + inmin = T EB EB7 SS IL IL T T 7 imin = T SS μncoxw μncoxw7 Expressions for signal swings are particularly complicated!
44 Observation Even the most elementary performance parameters require very complicated expressions when the natural design parameter space is used Strong simultaneous dependence on multiple natural design parameters Interdependence and notational complexity obscures insight into performance and optimization
45 Practical Set of Design Parameters S PRACTICAL = {P, θ, EB, EB3, EB5, EB6, EB7 } 7 degrees of freedom! P : total power dissipation θ = ΙDQ5/I T, current split factor EBK=GSQK-TK, excess bias voltage for the k th transistor Phase margin constraint assumed (so C C not shown in DoF)
46 Basic Two-Stage Op Amp 7 Degrees of Freedom {P, θ, EB, EB3, EB5, EB6, EB7 } W W3 W5 W6 W 7,,,,,I T, θ L L3 L L 5 6 L7
47 Relationship Between the Practical Parameters and the Natural Design Parameters {P, θ, EB, EB3, EB5, EB6, EB7 } W W3 W5 W6 W 7,,,,,I T, θ L L3 L L 5 6 L7 P I T θ DD ( ) + W L i I DQi μ i C OX EBi I I I,,θI T DQi T T
48 Relationship Between the Practical Design Parameters and the Performance Parameters (Assuming Q = ) A O = 4 ( ) λ + λ n P Pθ ( EB β EB5 ) GB = = ( + θ ) C 4C θβ ( + θ) p EB EB5 DD EB C L DD EB EB5 Pθ ( EB β EB5 ) SR = EBGB = 4C L θβ DD( + θ ) EB EB5 C c = 4C θβ L EB EB5 ( ) θ β EB EB5
49 Relationship Between the Proposed Design Parameters and the Performance Parameters OMAX OMIN = DD SS = + EB5 EB6 + inmin = T + EB + EB7 SS = max{( 3 + T 3 T ),( EB 5 + T 5 inmax DD EB T )}
50 Characteristics of the Practical Design Parameter Space Minimum set of independent parameters Results in major simplification of the key performance parameters Provides valuable insight which makes performance optimization more practical
51 Example for Design Procedure Given specification: Ao: 66dB GB: 5MHz omin= omax=3. imin=. imax=3 P=0.7mw β= Assume: TN =0.6, TP = 0.7, λn=0.04, λp=0.8 7 constraints (in addition to φ m ) and 7 degrees of freedom
52 Design Assumptions Assume the following system parameters: DD = 3.3 C L = pf Typical 0.35um CMOS process Simulation corner: typ/55 C/3.3
53 Example for Design Procedure. Choose channel length. EB3, EB5, EB6 {P, θ, EB, EB3, EB5, EB6, EB7 } 3. EB imax = DD + EB3 + T + T3 omax = DD + EB5 omin = EB6 A O = 4. EB7 imin=eb + EB7 + T 4 ( ) λ + λ n p EB EB5 {P, θ, EB, EB3, EB5, EB6, EB7 } {P, θ, EB, EB3, EB5, EB6, EB7 } 5. Choose P to satisfy power constraint {P, θ, EB, EB3, EB5, EB6, EB7 } I = T P ( ) +θ DD
54 Example for Design Procedure 6. Choose θ to meet GB constraint {P, θ, EB, EB3, EB5, EB6, EB7 } GB = P ( +θ ) DD EB CC 7. Compensation capacitance C C C c = 4C L θβ EB EB 5 ( θ β ) EB 8. Calculate all transistor sizes I = T P ( ) +θ DD EB 5 W k = L k I μ C Dk OXEBk 9. Implement structure, simulate, and make modifications if necessary guided by where deviations may occur Note: It may be necessary or preferable to make some constraints an inequality Note: Specifications may be over-constrained or have no solution k (Assuming Q = )
55 Example for Design Procedure Simulation result: (W/L), 3/ (W/L) 3,4 4.5/ (W/L) 5 54/ (W/L) 6,7 7.4/ C C 3.7pF φphase 45.4deg. I SS 5uA I uA θ.06 GB 5.MHz P 0.7mw EB7 0. Ao 65dB
56 Spreadsheet Summarizing Design Space Settling Characteristics of Two-Stage Operational Amplifie Process Parameters 0.0 Power 0.0 ln 9E-05 ucoxn E- CT 0. lp 5E-05 ucoxp 4 dd tn tp Device Output Range Input Range Performance Characteristics Design Parameters W/L W/L min max max min ISS(mA) CC GB Ao η EB7 EB6 EB5 EB EB E E E E E E E E E E E E ERR.67 ERR E E E E ERR.67 ERR E E E E E E ERR.67 ERR 39 W/L7 W/L6 W/L Device Sizing
57 Summary. Determination of Design Space and Degrees of Freedom Often Useful for Understanding the Design Problem. Analytical Expressions for Key Performance Parameters give Considerable Insight Into Design Potential 3. Natural Design Parameters Often Not Most Useful for Providing Insight or Facilitating Optimization 4. Concepts Readily Extend to other Widely Used Structures
58 End of Lecture 6
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