EE 508 Lecture 29. Integrator Design. Metrics for comparing integrators Current-Mode Integrators

Transcription

1 EE 508 Lecture 29 Integrator Design Metrics for comparing integrators urrent-mode Integrators

2 eview from last time nti-aliasing filter often required to limit frequency content at input to S filters ontinuous-time Discrete-Time Filter VIN t Filter VINFILT t VOUT kt Signal Band nti-aliasing Filter Switched- apacitor Filter f LK 2 lias Frequencies Why not just make the clock frequency >> signal band edge? ecall in the continuous-time -S counterparts f POLES flk Since f POLES will be in the signal band (that is why we are building a filter) large f LK will require large capacitor ratios if f LK >>f POLES Large capacitor ratios not attractive on silicon (area and matching issues) High f LK creates need for high GB in the op amps (area,power, and noise increase) Often f LK /f POLES in the 0: range proves useful (20: to 5: typical) f LK f

3 eview from last time Elimination of edundant Switches edundant Switches φ φ 2 φ φ 2 φ φ 2 Noninverting Input Noninverting Input φ φ 2 φ φ φ 2 φ φ 2 φ Noninverting Input Noninverting Input Switched-apacitor Input with edundant Switches Switched-apacitor Input with edundant Switches emoved lthough developed from the concept of S-resistor equivalence, S circuits often have no esistor-apacitor equivalents

4 eview from last time Switched apacitor mplifiers 2 φ φ 2 φ 2 φ 2 φ 2 φ φ 2 Summing, Differencing, Inverting, and Noninverting S mplifiers Widely Used Significant reduction in switches from what we started with by eliminating in S integrator Must be stray insensitive in most applications

5 eview from last time Switched-esistor Voltage Mode Integrators Precharge X M Filter FET P F There are some modest nonlinearities in this MOSFET when operating in the triode region Precharge X M M 2 X Filter FET P F Significant improvement in linearity by cross-coupling a pair of triode region resistors Perfectly cancels nonlinearities if square law model is valid for M and M 2 Only modest additional complexity in the Precharge circuit

6 eview from last time Switched-esistor Voltage Mode Integrators Integrator Output with Perfect omplimentary locks Integrator Output with Nonoverlapping locks Integrator Output with Overlapping locks F P F P berrations are very small, occur very infrequently, and are further filtered Play almost no role on performance of integrator or filter

7 eview from last time Switched-esistor Voltage Mode Integrators EF f EF FET FET Pretune ircuit Switched-resistor integrator ccurate FET products is possible rea reduced compared to ctive structure because FET small Single pretune circuit can be used to calibrate large number of resistors lock frequency not fast and not critical (but accuracy of f EF is important) Since resistors are memoryless elements, no transients associated with switching Since filter is a feedback structure, speed limited by BW of op amp

8 Voltage Mode Integrators ctive (Feedback-based) MOSFET- (Feedback-based) OT- T- Switched apacitor Switched esistor Other Structures Sometimes termed current mode Will discuss later Have introduced a basic voltage-mode integrators in each of these approaches ll of these structures have applications where they are useful Performance of all are limited by variability or Op mp BW

9 How can integrator performance be improved? Better op amps Better Integrator rchitectures How can the performance of integrator structures be compared? Need metric for comparing integrator performance

10 re there other integrators in the basic classes that have been considered? Miller Inverting V s = - V s = - s s High-Q Inverting B Zero Second Derivative Inverting s = - s V ascaded Inverting V s = - s V Zero Second Derivative Inverting s = - s Zero Sensitivity Inverting V s = - s

11 re there other integrators in the basic classes that have been considered? Miller Noninverting V s = s V s = s Modified High-Q Noninverting V Zero Sensitivity Noninverting s = s s = s V Phase Lead Integrator V High-Q NonInverting Modified Miller Noninverting V s = s s = s

12 re there other integrators in the basic classes that have been considered? Miller Noninverting V s = s V s = s Balanced Time onstant Noninverting V 2 s = s Zero Sensitivity Noninverting If = 2 and 3 = 4

13 De Boo Integrator onsider the Howland urrent Source Howland urrent Source V X 3 4 GG 2 3 IOUT VING VX -G G4 2 I OUT If resistors sized so that I V G OUT IN G GG G G L If sizing constraints are satisfied, behaves as a grounded constant-current source

14 DeBoo Integrator I Howland urrent Source V G OUT IN I OUT V X 2 2 G L Observe that if a current source drives a grounded capacitor, then the nodal voltage on the capacitor is given by V =I X X s I X V X Thus, if we could make I X proportional to, the voltage on the capacitor would be a weighted Integral of

15 2 2 De Boo Integrator Howland urrent Source V X 2 Howland urrent Source V X 22 I OUT I OUT I V G OUT IN G L If = and 2 = 22 OUT V s V IN X= V V =V + = + 22 IN 22 X s V s = + s 22

16 De Boo Integrator Howland urrent Source V X 2 Howland urrent Source 2 V X 2 2 I OUT De Boo Integrator V s = + s 22 I OUT IOUT VIN G G L Howland urrent Source V X 2 B 2 I OUT De Boo Integrator V s = - + s B 22

17 Many different integrator architectures that ideally provide the same gain Similar observations can be made for other classes of integrators How can the performance of an integrator be characterized and how can integrators be compared?

18 How can the performance of an integrator be characterized and how can integrators be compared? onsider Ideal Integrator Gain Function I 0 V s = s I jω = jω V 0 Ideal Integrator Im e onsider a nonideal integrator Gain Function I 0 V s = OO s s+ Nonideal Integrator Im e Key characteristics of an ideal integrator: Magnitude of the gain at I 0 = Phase of integrator always 90 o Gain decreases with /ω re any of these properties more critical than others? In many applications: Key property of ideal integrator is a phase shift of 90 o at frequencies around I 0!

19 How can the performance of an integrator be characterized and how can integrators be compared? Is stability of an integrator of concern? Ideal Integrator Im e Ideal integrator is not stable Integrator function is inherently ill-conditioned Integrator is almost never used open-loop Stability of integrator not of concern, stability of filter using integrator is of concern Some integrators may cause unstable filters, others may result in stable filters

20 How can the performance of an integrator be characterized and how can integrators be compared? Express V (jω) as V jω= ω +jx ω where (ω) and X(ω) are real and represent the real and imaginary parts of the denominator respectively Xω Phase tan ω Ideally (ω)=0 Definition: The Integrator Q factor is the ratio of the imaginary part of the denominator to the real part of the denominator Q INT X ω ω Typically most interested in Q INT at the nominal unity gain frequency of the integrator

21 How can the performance of an integrator be characterized and how can integrators be compared? Express V (jω) as I V jω= ω +jx ω Lead/Lag haracteristics for Inverting Integrators Im Im Im Phase Lag Integrator I(jω 0 ) e Ideal Phase Integrator I(jω 0 ) e I(jω 0 ) Phase Lead Integrator e For Phase Lag Integrators, (ω) is negative For Phase Lead integrators, (ω) is positive

22 How can the performance of an integrator be characterized and how can integrators be compared? Lead/Lag haracteristics for Inverting Integrators I V - jω= ω +jx ω Im Im Im Phase Lag Integrator I(jω 0 ) e Ideal Phase Integrator I(jω 0 ) e I(jω 0 ) Phase Lead Integrator e For Phase Lag Integrators, (ω) and X(ω) have opposite signs. For Phase Lead integrators, (ω) and X(ω) have the same sign. Phase shift ideally 90 o Lead/Lag haracteristics for Noninverting Integrators Im Im Im e e e I V jω= ω +jx ω Phase Lag Integrator I(jω 0 ) Ideal Phase Integrator I(jω 0 ) Phase Lead Integrator I(jω 0 ) For Phase Lag Integrators, (ω) and X(ω) have opposite signs. For Phase Lead integrators, (ω) and X(ω) have the same sign. Phase shift ideally 270 o

23 Integrator Q Factor onsider Miller Inverting Integrator V - s = s+ s +s s s V - s = jω+ jω +jω - s = -ω +j ω + V 2 Normalizing by ω n = ω and τ n = τ/ = I 0n /GB - s = - ω +j ω + V 2 n n n n Observe this integrator has excess phase shift (more than 90 o in the denominator) at all frequencies

24 V Integrator Q Factor jω= ω +jx ω onsider Miller Inverting Integrator s s Q INT - s = - ω +j ω + V 2 n n n n INT 2 nωn X ω ω n n n n ω + GB Q = - - ω ω Since the phase is less than 90 o, the Miller Inverting Integrator is a Phase Lag Integrator and Q INT is negative Im ω 0 I(jω 0 ) Phase Lag Integrator e

25 Integrator Pole Locations onsider Miller Inverting Integrator V - s = s+ s +s I 0 =/() Poles at s=0 and s = -I 0 (+GB/I 0 ) ; -GB Im -GB e

26 Is the integrator Q factors simply a metric or does it have some other significance? Two-Integrator Loop Q V OLP V OBP INT INT 2 B F Summer QP Tow Thomas Biquad It can be shown that the pole Q for the TT Biquad can be approximated by + + QPN QINT QINT2 where Q INT and Q INT2 are evaluated at ω=ω o

27 Is the integrator Q factors simply a metric or does it have some other significance? Two-Integrator Loop Q V OLP V OBP INT INT 2 B F Summer Tow Thomas Biquad It can be shown that the pole Q for the TT Biquad can be approximated by QP + + QPN QINT ω0 QINT2 ω0 Similar expressions for other second-order biquads Observe that the integrator Q factors adversely affect the pole Q of the filter Observe that if Q INT and Q INT2 are of opposite signs and equal magnitudes, nonideal effects of integrator can cancel

28 What can be done to correct the phase problems of an integrator? V - s = s+ s +s One thing that can help the Miller Integrator is phase-lead compensation X X V - +xs s = s+ s +[+ x ]s - +xs V s = s+ s +[+ x ]s x and x will add phase-lead by introduction of a zero X and X will be small components

29 Integrator Q Factor onsider Miller Noninverting Integrator - - V s = s+ s +s s 2 V jω= -3ω 2 +j ω(+2ω) V jω -3ω 2 +jω Observe this integrator has excess phase shift (more than 90 o in the denominator) at all frequencies ω QINT 2-3ω 3ω QINT -GB jω 3 ω 3 Note: The Miller Noninverting Integrator has a modestly poorer Q INT than the Miller Inverting Integrator

30 Example: If f 0 =0KHz, GB=MHz, Q NOM =0, estimate the pole Q for the Tow-Thomas Biquad if the Miller Integrator and the Miller Noninverting Integrators are used. lso determine the relative degradation in performance due to each of the integrators. Two-Integrator Loop Q V OLP V OBP INT INT 2 B F Summer Tow Thomas Biquad QP + + QPN QINT QINT2

31 Example: If f 0 =0KHz, GB=MHz, Q NOM =0, estimate the pole Q for the Tow-Thomas Biquad if the Miller Integrator and the Miller Noninverting Integrators are used. lso determine the relative degradation in performance due to each of the integrators. Q QP INT + + QPN QINT QINT2 GB MHz -= 00 ω 0KHz -GB MHz QINT2 jω - = ω 3 0KHz QP Note the nonideal integrators cause about a 75% shift in Q P Note that 3 times as much of the shift is due to the noninverting integrator as is due to the inverting integrator! Similar effects of the integrators will be seen on other filter structures

32 Example: If f 0 =0KHz, GB=MHz, Q NOM =0, estimate the pole Q for the Tow-Thomas Biquad if the Miller Integrator and the Miller Noninverting Integrators are used. lso determine the relative degradation in performance due to each of the integrators. Q QP INT How can the problem be solved? + + QPN QINT QINT2 GB MHz -= 00 ω 0KHz -GB MHz QINT2 jω - = ω 3 0KHz QP ompensate Integrator 2. Use better integrators 3. Use phase-lead and phase/lag pairs

33 Example: If f 0 =0KHz, GB=MHz, Q NOM =0, estimate the pole Q for the Tow-Thomas Biquad if the Miller Integrator and the Miller Noninverting Integrators are used. lso determine the relative degradation in performance due to each of the integrators. QP + + QPN QINT QINT2 How can the problem be solved? QINT 00 QINT2-33 Phase ompensation of INT X Pick x so that Q INT =33 at ω=/() Solving, obtain x =4/GB V - +xs s = s+ s +[+ x ]s - GB Q ω INT -GB X Useful for hand calibration but not practical for volume production because of variability in components

34 What are the integrator Q factors for other integrators that have been considered? Miller Inverting V s = - s Q INT = - High-Q Inverting V s = - s Q INT = - 2 B Zero Second Derivative Inverting V s = - s (stability problems) ascaded Inverting V s = - s Q INT = - 2 V Zero Second Derivative Inverting s = - s Zero Sensitivity Inverting V s = - s

35 What are the integrator Q factors for other integrators that have been considered? Miller Inverting I OUT V V X s = - s Howland urrent Source 2 2 X Q INT = - De Boo Integrator + 2 V s = 2 s+ s + + +s 2 QINT If = 2 = Q + 2 INT 2 V 2 s = s

36 What are the integrator Q factors for other integrators that have been considered? Miller Noninverting V s = s Modified High-Q Noninverting QINT Zero Sensitivity Noninverting V Q jω s = s V INT 3 s = s jω 3 Modified Miller Noninverting Phase Lead Integrator V V High-Q NonInverting QINT QINT V jω s = s s = s jω s = s

37 Improving Integrator Performance:. ompensate Integrator 2. Use better integrators 3. Use phase-lead and phase/lag pairs These methods all provide some improvements in integrator performance But both magnitude and phase of an integrator are important so focusing only on integrator Q factor only may only improve performance to a certain level In higher-order integrator-based filters, the linearity in /ω of the integrator gain is also important. The integrator magnitude and Q factor at ω 0 ignore the frequency nonlinearity that may occur in the /ω dependence There is little in the literature on improving the performance of integrated integrators within a basic class. t high frequencies where the active device performance degrades, particularly in finer-feature processes, there may be some benefits that can be derived from architectural modifications along the line of those discussed in this lecture

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