EECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 1EE247Lecture 11• Switched-Capacitor Filters (continued)–Effect of non-idealities–Bilinear switched-capacitor filters–Filter design summary • Comparison of various filter topologies• Data ConvertersEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 2Summary Last Lecture• Switched-capacitor filter design considerations– DDI & LDI Integrator characteristics– Bottom-plate LDI integrator Æ overcomes parasitic sensitivity issues– Continuous-time and complex conjugate terminations– Use of T-networks to implement high capacitor ratios• Switched-capacitor filters utilizing double sampling technique • Effect of non-idealities– Opamp finite gain– Opamp finite bandwidth– Finite slew rate of the opamp (this lecture)EECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 3Switched-Capacitor Direct-Transform Discrete (DDI) Integrator1s1Is1ICC1inCC1Voz(z)zVz−−−=− ×=− ×−-+VinVoφ1φ2CICsφ1φ1φ2T=1/fsEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 4DDI Switched-Capacitor IntegratorCIIdeal IntegratorMagnitude ErrorPhase Error()()1s1IjT/2ssjT jT/2 jT/2IIsIsICjTC1injjCCCC1CjT/2CCjT/2CjTVoz(z) , z ezVeee1since: sin2jeee1je2sinT/2T/21esinT/2ωωωωωααωωαωωωω−−−−−−−−−−=− × =−=× = × ==− × ×=− × ×-+VinVoφ1φ2CICsφ1EECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 5DDI Switched-Capacitor IntegratorExample: Mag. & phase error for:1- f maxsig ./ fs = 1/12 ÆMag. error = 1% or 0.1dBPhase error=-ωT/2= -πf / fs= -π/12 [radian] Æ15 [degree]QintgÆ1/( phase error @ωο in radian ) (Lecture 5 page 1)Æ Qintg= -12/π = -3.82- f / fs=1/32 ÆMag. error=0.16% or 0.014dBPhase error = ω T/2= -πf / fs= -π/32 [radian] Æ 5.6 [degree]Qintg= -32/π = -10.2CI-+VinVoφ1φ2CICsφ1DDI Integrator:Æ Magnitude error no problemPhase error major problem EECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 6LDI Switched-Capacitor Integrator()()1/2s1IjT/2se s1jT j T/2 j T/2IIsIsICjTC1zinCCCC1CCCCjTVo2z(z) , z eVeee1j2sinT/2T/21sinT/2ωωωωωωωωω−−−−−+−−−=− × ==− × = ×=− ×=− ×CIIdeal IntegratorMagnitude ErrorNo Phase Error! For signals at frequencies << sampling freq. Æ Magnitude error negligible-+VinVo2φ1φ2CICsφ2LDILDI (Lossless Discrete Integrator) Æsame as DDI but output is sampled ½clock cycle earlierEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 7Switched-Capacitor IntegratorParasitic SensitivityEffect of parasitic capacitors:1- Cp1 - driven by opamp o.k.2- Cp2 - at opamp virtual gnd o.k.3- Cp3– Charges to Vin & discharges into CIÆ Problem parasitic sensitivity-+VinVoφ1φ2CICsCp3Cp2Cp1EECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 8Parasitic InsensitiveBottom-Plate Switched-Capacitor IntegratorSensitive parasitic cap. Æ Cp1Æ rearrange circuit so that Cp1does not charge/dischargeφ1=1 Æ Cp1groundedφ2=1 Æ Cp1at virtual groundCp2 Æ driven by a low impedance sourceSolution: Bottom plate capacitor integratorVi+Cs-+VoCICp1Cp2φ1φ2Vi-EECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 9Bottom Plate Switched-Capacitor Integrator12121ss11IIss11IICCzzCC1z 1zCCz1CC1z 1z−−−−−−−−−−−−−Note: Different delay from Vi+ &Vi- to either outputÆ Special attention needed for input/output connectionsVi+Cs-+VoCIφ1φ2Vi-Vi+on φ1Vi-on φ2Vo1on φ1Vo2on φ2φ1φ2Vo2Vo1Output/Inputz-TransformEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 10Bottom Plate Switched-Capacitor Integratorz-Transform Model121211111zz1z 1zz11z 1z−−−−−−−−−−−−121z1z−−−12z−Input/Output z-transformVi+Cs-+VoCIφ1φ2Vi-φ1φ2Vo2Vo1Vi+Vi-12z+Vo2Vo1LDIsICCsICC−EECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 11LDI Switched-Capacitor Ladder FilterCsCI121z1z−−−- ++ -+ -31sτ41sτ51sτ12z−12z+12z+12z+12z−CsCI−CsCI−CsCI−CsCICsCIDelay around integrator loop is (z-1/2. z+1/2=1) Î LDI function121z1z−−−121z1z−−−12z−EECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 12Effect of Opamp Nonidealities on Switched Capacitor Filter Behaviour• Opamp finite gain• Opamp finite bandwidth• Finite slew rate of the opamp• Non-linearities associated with opamp output/input characteristicsEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 13Effect of Opamp Non-IdealitiesFinite DC GainInput/Output z-transformVi+Cs-+VoCIφ1φ2Vi-DC Gain = asIs1I1CsCCsCaooa1H(s) fsfH(s)sQaωω≈−+×−≈+×⇒ ≈ÆFinite DC gain same effect in S.C. filters as for C.T. filtersÆIf DC gain not high enough Æ causes lowing of overall Q & droop in passbandEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 14Effect of Opamp Non-IdealitiesFinite Opamp BandwidthInput/Output z-transformVi+Cs-+VoCIφ1φ2Vi-Unity-gain-freq.= ftVoφ2T=1/fssettlingerrortimeRef: K.Martin, A. Sedra, “Effect of the OPamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.Assumption-Opamp Æ does not slew (will be revisited)Opamp has only one pole Æ exponential settlingEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 15Effect of Opamp Non-IdealitiesFinite Opamp Bandwidthactual idealIkk 1IsItIsstsC1e e ZH(Z)H(Z)CCCfwhere kCC ffOpamp unity gain frequency , f Clock frequencyπ−− −⎡⎤−+×≈⎢⎥+⎣⎦=× ×+→−− →Input/Output z-transformVi+Cs-+VoCIφ1φ2Vi-Unity-gain-freq.= ftRef: K.Martin, A. Sedra, “Effect of the OPamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.Voφ2T=1/fssettlingerrortimeEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 16Effect of Opamp Finite Bandwidth on Filter Magnitude ResponseExample:For 1dB magnitude response deviation:1- fc/fs=1/12fc/ft~0.04Æ ft>25fc2- fc/fs=1/32fc/ft~0.022Æ ft>45fc3- Cont.-Timefc/ft~1/700Æ ft >700fcfc/ft|Τ|non-ideal/|Τ|ideal(dB)fc/fs=1/32fc/fs=1/12Active RCEECS 247 Lecture 11: S.C. Filters/ Data Converters © 2006 H. K. Page 17Effect of Opamp Finite Bandwidth on Filter Critical
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