EE 58 Lecture Sensitivity Functions - Comparison of Circuits - Predistortion and Calibration
Review from last time Sensitivity Comparisons Consider 5 second-order lowpass filters (all can realize same T(s) within a gain factor) R L V OUT C V IN C V IN R R K V OUT C Passive RLC +KRC C R 3 R R C V IN R C R R R 4 R C R 3 V IN V OUT V OUT Bridged-T Feedback Two-Integrator Loop
Review from last time Sensitivity Comparisons Consider 5 second-order lowpass filters (all can realize same T(s) within a gain factor) R 3 R R R 4 R 5 V OUT V IN C C R -K = - R 5 4 -KRC Lowpass
How do these five circuits compare? a) From a passive sensitivity viewpoint? - If is small - If is large b) From an active sensitivity viewpoint? - If is small - If is large - If τω is large
Comparison: Calculate all ω and sensitivities Consider passive sensitivities first a) Passive RLC R L V OUT S S S S S S R L C R C L V IN L = R C ω = LC C
Case b : +KRC Equal R, Equal C ω = R R C C = R C R C R C R C + + K R C R C R C R C SR S R S C S C S K SR = 3 K SR ω = RC SC SC S 3 K
Case b : +KRC Equal R, K= ω = R R C C = R C R C R C R C + + K R C R C R C R C SR S S S S R C C K S S S S S R R C C K ω = = RC C C
c) Bridged T Feedback ω = R R C C For R =R =R 3 =R = C R R RR + + C R3 R R3 SR S R S S S C C R 3 SR 6 S S S S R R 3 C C 3 6 ω = = 3 RC 3 C C
d) integrator loop For: = R4 R R R C C 3 = R RR R = R = R = R C = C = C R 3 = R4 C C SR S R S R S 3 C S C S R 4 SR S S S R R3 C SR S 4 C S S R R = = RC R R
ω = d) -KRC passive sensitivities + R R K + R R R R R R 3 4 3 R R C C = + R R K + R R R R R R 3 4 3 R R C C R C + + R3 RC C RC R4C For R =R =R 3 =R 4 =R, C =C =C 5+K = 5 5 K RC 3 S S + S + 3 5 5 5 R R R 3 S S S S + 4 5 C C R K 3 3 S - S + S + 3 5 5 5 5 R R R 3 S - S S S - 4 5 5 R C C K ω =
Passive Sensitivity Comparisons ω S x S x Passive RLC,/ +KRC Equal R, Equal C (K=3-/) Equal R, K= (C =4 C ),/,, 3,/,/, Bridged-T Feedback,/ /3,/, /6 Two-Integrator Loop -KRC,/,/, less than or equal to / less than or equal to / Substantial Differences Between (or in) Architectures
Where we are at with sensitivity analysis: Considered a group of five second-order filters Passive Sensitivity Analysis Closed form expressions were obtained for ω and Tedious but straightforward calculations provided passive sensitivities directly from the closed form expressions Active Sensitivity Analysis Closed form expressions for ω and are very difficult or impossible to obtain If we consider higher-order filters Passive Sensitivity Analysis Closed form expressions for ω and are very difficult or impossible to obtain for many useful structures Active Sensitivity Analysis Closed form expressions for ω and are very difficult or impossible to obtain Need some better method for obtaining sensitivities when closed-form expressions are difficult or impractical to obtain or manipulate!!???
Relationship between pole sensitivities and and sensitivities p Im Re p = -α+jβ D (s)=(s-p)(s-p*) D (s)=(s+α-jβ)(s+α+jβ) D (s)=s +s(α)+(α +β )
Relationship between active pole sensitivities and and sensitivities Define D(s)=D (s)+t D (s) Recall: Theorem: p s p Theorem: s Theorem: s p Ds -D p s s=p,τ= Re p s p Im Δω Δα Δβ ω ω ω + - 4 (from bilinear form of T(s)) Δ Δα Δβ + - 4 ω 4 ω Claim: These theorems, with straightforward modification, also apply to other parameters (R, C, L, K, ) where, D (s) and D (s) will change since the parameter is different
c) Bridged-T structure
d) Two integrator loop architecture
d) Two integrator loop architecture
e) -KRC
Passive RLC Active Sensitivity Comparisons Δω ω Δ +KRC Equal R, Equal C (K=3-/) - 3- ω - 3- ω Equal R, K= (C =4 C ) Bridged-T Feedback Two-Integrator Loop - ω ω 3 - ω ω - ω 4ω -KRC 5 ω 5 3 ω Substantial Differences Between Architectures
Are these passive sensitivities acceptable? ω S x S x Passive RLC,/ +KRC Equal R, Equal C (K=3-/) Equal R, K= (C =4 C ),/,, 3,/,/, Bridged-T Feedback,/ /3,/, /6 Two-Integrator Loop,/,/, -KRC less than or equal to / less than or equal to /
Are these active sensitivities acceptable? Passive RLC Active Sensitivity Comparisons Δω ω Δ +KRC Equal R, Equal C (K=3-/) - 3- ω - 3- ω Equal R, K= (C =4 C ) Bridged-T Feedback Two-Integrator Loop -KRC - ω ω 3 - ω ω - ω 4ω 5 ω 5 3 ω
Are these sensitivities acceptable? Passive Sensitivities: Δω ω S ω x Δx x In integrated circuits, R/R and C/C due to process variations can be K 3% or larger due to process variations Many applications require Δω /ω <. or smaller and similar requirements on Δ/ Even if sensitivity is around ½ or, variability is often orders of magnitude too large Active Sensitivities: All are proportional to τω Some architectures much more sensitive than others Can reduce τω by making GB large but this is at the expense of increased power and even if power is not of concern, process presents fundamental limits on how large GB can be made
What can be done to address these problems?. Predistortion Design circuit so that after component shift, correct pole locations are obtained Predistortion is generally used in integrated circuits to remove the bias associated with inadequate amplifier bandwidth Predistortion does not help with process variations of passive components Tedious process after fabrication since depends on individual components Temperature dependence may not track Difficult to maintain over time and temperature Over-ordering will adversely affect performance Seldom will predistortion alone be adequate to obtain acceptable performance Bell Labs did to this in high-volume production (STAR Biquad)
What can be done to address these problems?. Predistortion Design circuit so that after component shift, correct pole locations are obtained Desired Pole Actual Pole Location due to parameter variations Im Actual Response Desired Response Re Pole shift due to parametric variations (e.g. inadequate GB)
What can be done to address these problems?. Predistortion Design circuit so that after component shift, correct pole locations are obtained Pre-distored Pole Location Actual Pole Location due to parameter variations Desired Pole Im Desired Response Actual Response Predistorted Response Re Pre-distortion concept
What can be done to address these problems?. Predistortion Design circuit so that after component shift, correct pole locations are obtained Im Over-order pole Desired Response Actual Response Predistorted Response Re Over-ordering Limitations with Pre-distortion Parasitic Pole Affects Response Predistortion almost always done even if benefits only modest Not effective is significant deviations exist before predistortion
What can be done to address these problems?. Trimming a) Functional Trimming trim parameters of actual filter based upon measurements difficult to implement in many structures manageable for cascaded biquads b) Deterministic Trimming (much preferred) Trim component values to their ideal value Continuous-trims of resistors possible in some special processes Continuous-trim of capacitors is more challenging Link trimming of Rs or Cs is possible with either metal or switches If all components are ideal, the filter should also be ideal R-trimming algorithms easy to implement Limited to unidirectional trim Trim generally done at wafer level for laser trimming, package for link trims Filter shifts occur due to stress in packaging and heat cycling c) Master-slave reference control (depends upon matching in a process) Can be implemented in discrete or integrated structures Master typically frequency or period referenced Most effective in integrated form since good matching possible Widely used in integrated form
End of Lecture