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Berkeley ELENG 247A - Lecture Notes

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EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 1EE247Lecture 10Supplemental Material for S.C. FiltersEECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 2z-Domain Frequency Response•The jω axis maps onto the unit-circle•LHP singularities in s-plane map into inside of unit-circle in z-domain•RHP singularities in s-plane map into outside of unit-circle in z-domain•Particular values:– f = 0 Æz = 1– f = fs/2 Æz = -1f = 0f = fs /2LHP in s-domainimag. axis in s-domainz-planez = e s.TsEECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 3z-Domain Frequency Response• The frequency response is obtained by evaluating H(z) on the unit circle at:z = e jωT = cos(ωTs) + j sin(ωTs)• Once z=-1 (fs /2) is reached, the frequency response repeats, as expected• The angle to the pole is equal to 360° (or 2πradians) times the ratio of the pole frequency to the sampling frequency(cos(ωTs),sin(ωTs))2πffSz-planeEECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 4DDI IntegratorPole-Zero Map in z-Planez -1=0 Æz = 1on unit circlePole from f Æ0in s-plane mapped to z =+1As frequency increases zdomain point moves on unit circle (CCW)Once frequency gets to:z=-1 (f=fs/2)Æ frequency response repeatsz-planef = fs /2ff11(z-1)increasingEECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 5Example: 2ndOrder LDI Bandpass Filters-Plane versus z-Planeσjωs-planez-planeEECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 6EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 7Sixth-Order Elliptic LDI Bandpass FilterTransmissionZeroRef: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 8Use of T-Network High Q filter Æ large cap. ratio for Q & transmission zero implementationTo reduce large ratios required Æ T-networks utilizedRef: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).21 412134VC CVCCCC=− ×++2112VCVC=−EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 9Sixth Order Elliptic Bandpass FilterUtilizing T-Network•T-networks utilized for:• Q implemention• Transmission zero implementationQ implementationZero Ref: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 10EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 11More Realistic Switched-Capacitor Circuit Slew Scenario-+VinVoφ2CICsAt the instant Csconnects to input of opamp (t=0+)Æ Opamp not yet active at t=0+ due to finite opamp bandwidth Æ delayÆ Feedforward path from input to output generates a voltage spike at the output with polarity opposite to final Vo step- spike magnitude function of CI, CL , CsÆ Spike increases slewing periodÆ Eventually, opamp becomes active - starts slewing followed by subsequent settlingCLVoφ2CICsCLt=0+EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 12Switched-Capacitor Circuit Opamp not Active @ t=0+-+VinVoφ2CICsCLVoφ2CICsCLt=0+()()0000 000 0000 0ttILs Cs Cs s eq eqILtt tIsIout Cs CsIL seqILtt ttLs I eq L s Cs Cs s L Cs Cstt tsIout Cs CssLILCCCh arg e sharing : C V V C C where CCCCCCVV VCC CC CCAssumingCCCCCCVVCCVVCCVV VCC CCNote−+++ −−+ −++− −=+ =+Δ= = ×+++<< << → ≈ → ≈ + → ≈→Δ ≈ × ≈++00final t tssout Cs CsIICCthat V V VCC+−Δ≈− ≈−EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 13More Realistic Switched-Capacitor Circuit Slew Scenario-+VinVoφ2CICsCLVoφ2CICsCLt=0+()0000 0ttILs Cs Cs s eq eqILtt tssCs Cs CsILseqsILCCCh arg e sharing : C V V C C where CCCCCVV VCCCCCCC−++− −=+ =+==+++Notice that if CLis large Æ some of the charge stored on Cs is lost prior to opamp becoming effective Æ operation looses accuracyÆ Partly responsible for S.C. filters only good for low-frequency applicationsEECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 14More Realistic S.C. Slew ScenarioVo_realIncluding t=0+spikeSlewingLinearSettlingSlewing Spike generated at t=0+Vo_realVo_idealSlewingLinearSettlingSlewingLinearSettlingRef: R. Castello, “Low Voltage, Low Power Switched-Capacitor Signal Processing Techniques," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, Aug. ‘84 (ERL Memorandum No. UCB/ERL M84/67).EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 15EECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 16Extending the Maximum Achievable Critical Frequencyof Switched-Capacitor FiltersConsider a switched-capacitor resonator:Regular sampling:Each opamp is busy settling only during one of the two clock phasesÆ Idle during the other clock phaseφ1φ2φ2φ1Note: During φ1both opamps are idleEECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 17Switched-Capacitor Resonator Using Double-SamplingDouble-sampling:•2ndset of switches & sampling caps added to all integrators• While one set of switches/caps sampling the other set transfers charge into the intg. cap• Opamps busy during both clock phases• Effective sampling freq. twice the clock freq. while opamp bandwidth requirement remains the sameEECS 247 Lecture 10 Switched-Capacitor Filters © 2008 H. K. Page 18Double-Sampling Issuesfclockfs= 2fclockIssues to be aware of:- Jitter in the clock - Unequal clock phases -Mismatch in sampling caps.Æ parasitic passbands Ref: Tat C. Choi, "High-Frequency CMOS


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Berkeley ELENG 247A - Lecture Notes

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