EECS 247 Lecture 11: SC Filters/ DACs © 2005 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: SC Filters/ DACs © 2005 H. K. Page 2Summary Last Lecture• Switched-capacitor filter design considerations– DDI & LDI Integrator characteristics– Bottom-plate LDI integrator à 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 (this lecture)– Finite slew rate of the opamp (this lecture)EECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 3Effect 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 à does not slew (will be revisited)Opamp has only one pole à exponential settlingEECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 4Effect of Opamp Non-IdealitiesFinite Opamp BandwidthactualidealIkk1IsItIsstsC1eeZH(Z)H(Z)CCCfwherekCCffOpampunitygainfrequency,fClockfrequencyπ−−−−+×≈+=××+→−−→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: SC Filters/ DACs © 2005 H. K. Page 5Effect of Opamp Finite Bandwidth on Filter Magnitude ResponseMagnitude deviation due to finite opamp unity-gain-frequencyExample: 2ndorder bandpass with Q=25fc/ft|Τ|non-ideal/|Τ|ideal(dB)fc/fs=1/32fc/fs=1/12Active RCRef: 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.EECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 6Effect of Opamp Finite Bandwidth on Filter Magnitude ResponseExample:For 1dB magnitude response deviation:1- fc/fs=1/12fc/ft~0.04à ft>25fc2- fc/fs=1/32fc/ft~0.022à ft>45fc3- Cont.-Timefc/ft~1/700à ft >700fcfc/ft|Τ|non-ideal/|Τ|ideal(dB)fc/fs=1/32fc/fs=1/12Active RCEECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 7Effect of Opamp Finite Bandwidth Maximum Achievable Qfc/ftMax. allowable biquad Q for peak gain change <10%C.T. filtersRef: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of SwitchedCapacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.Oversampling RatioEECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 8Effect of Opamp Finite Bandwidth Maximum Achievable Qfc/ftC.T. filtersExample:For Q of 40 required Max. allowable biquad Q for peak gain change <10%1- fc/fs=1/32fc/ft~0.02à ft>50fc2- fc/fs=1/12fc/ft~0.035à ft>28fc3- fc/fs=1/6fc/ft~0.05à ft >20fcRef: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of SwitchedCapacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.EECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 9Effect of Opamp Finite Bandwidth on Filter Critical FrequencyCritical frequency deviation due to finite opamp unity-gain-frequencyExample: 2ndorder filterfc/ft∆ωc/ωcfc/fs=1/32fc/fs=1/12Active RCRef: 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.EECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 10Effect of Opamp Finite Bandwidth on Filter Critical Frequencyfc/ft∆ωc/ωcfc/fs=1/32fc/fs=1/12Active RCExample:For maximum critical frequency shift of <1%1- fc/fs=1/32fc/ft~0.028à ft>36fc2- fc/fs=1/12fc/ft~0.046à ft>22fc3- Active RCfc/ft~0.008à ft >125fcC.T. filtersRef: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of SwitchedCapacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.EECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 11Double-Sampled Fully Differential 6thOrder S.C. All-Pole Bandpass FilterRef: B.S. Song, P.R. Gray "Switched-Capacitor High-Q Bandpass Filters for IF Applications," IEEE Journal of Solid State Circuits, Vol. 21, No. 6, pp. 924-933, Dec. 1986.-Cont. time termination (Q) implementation-Folded-Cascode opamp with fu = 100MHz used-Center freq. 3.1MHz, filter Q=55-Clock freq. 12.83MHz à effective oversampling ratio 8.27-Measured dynamic range 46dB (IM3=1%)EECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 12Sources of Distortion in Switched-Capacitor Filters• Distortion induced by finite slew rate of the opamp• Opamp output/input transfer function non-linearity• Distortion incurred by finite setting time of the opamp• Capacitor non-linearity• Distortion due to switch clock feed-through and charge injectionEECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 13What is Slewing?oVVi+Vi-φ2Cs-+VinVoφ2CICsCICLAssume opamp is of a simple differential-pair class A transconductance typeIssEECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 14What is Slewing?IoVinVmax-Iss/2Slope ~ gmVin<+-Vmaxà Io=gmVin Vin>+-Vmax à Io=Imax Class A amplifiers where the maximum output current is limited to Iss: oVVi-IssVi+Class A amplifier input stageIss/2oIEECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 15What is Slewing?oVVi+Vi-φ2CsCICLIf at the rising edge of φ2: Vcs > Vmaxà Output current constant Io=Iss/2à SlewingAfter Vcs is discharged enough to have Vcs< Vmaxà Linear settlingIssIoIoVinVmax-Iss/2Slope ~ gmIss/2EECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 16Distortion Induced by Opamp Finite Slew RateEECS 247 Lecture 11: SC Filters/ DACs © 2005 H. K. Page 17Ideal Switched-Capacitor Integrator Output Waveformφ1φ2VinVoVcsClock-+VinVoφ1CICs-+VinVoφ2CICsφ2 High àCharge
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