EECS 247 Lecture 10: SC Filters © 2006 H. K. Page 1EE247Lecture 10• Switched-capacitor filters– Switched-capacitor network electronic noise– Switched-capacitor integrators• DDI integrators• LDI integrators• Effect of parasitic capacitance• Bottom-plate integrator topology– Resonators– Bandpass filters– Lowpass filters• Termination implementation• Transmission zero implementation– Switched-capacitor filter design considerations– Switched-capacitor filters utilizing double sampling technique – Effect of non-idealitiesEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 2Summary of last lecture• Continuous-time filters continued– Various Gm-C filter implementations– Comparison of continuous-time filter topologies• Switched-capacitor filters– Emulating resistor via switched-capacitor network– 1st order switched-capacitor filter– Switch-capacitor filter considerations:• Issue of aliasing and how to avoid it– Sample at high enough frequency so that the entire range of signals including the parasitics are at freqs < fs /2– Use of anti-aliasing prefilters• Tradeoffs in choosing sampling rate• Effect of sample and holdEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 3Switched-Capacitor Network Noise•During φ1high: Resistance of switch S1 (RonS1) produces a noise voltage on C with variance kT/C (lecture 1- first order filter noise)• The corresponding noise charge is:Q2 = C2V2 = C2. kT/C = kTC•φ1low: S1 opens ÆThis charge is sampledvINvOUTCS1 S2φ1φ2RonS1CEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 4Switched-Capacitor NoisevINvOUTCS1 S2φ1φ2RonS2C•During φ2high: Resistance of switch S2 contributes to an uncorrelated noise charge on C at the end of φ2: with variance kT/C• Mean-squared noise charge transferred from vINto vOUT per sample period is:Q2=2kTCEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 5• The mean-squared noise current due to S1 and S2’s kT/C noise is :• This noise is approximately white and distributed between 0 and fs /2 (noise spectra Æ single sided by convention) The spectral density of the noise is:ÆS.C. resistor noise = a physical resistor noise with same value!Switched-Capacitor Noise()2Q22Since i then i Qf 2k TCfsBst=→==222k TCfiBs4k TCf Bsffs224k T1iBSince R then: EQfC f RsEQ==Δ==ΔEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 6Periodic Noise AnalysisSpectreRFPSS pss period=100n maxacfreq=1.5G errpreset=conservativePNOISE ( Vrc_hold 0 ) pnoise start=0 stop=20M lin=500 maxsideband=10SpectreRF PNOISE: checknoisetype=timedomainnoisetimepoints=[…]as alternative to ZOH.noiseskipcount=largemight speed up things in this case.ZOH1T = 100nsZOH1T = 100nsS1R100kOhmR100kOhmC1pFC1pFPNOISE Analysissweep from 0 to 20.01M (1037 steps)PNOISE1Netlistahdl_include "zoh.def"ahdl_include "zoh.def"Vclk100nsVrc Vrc_holdSampling Noise from SC S/HC11pFC11pFC11pFC11pFR1100kOhmR1100kOhmR1100kOhmR1100kOhmVoltage NOISEVNOISE1NetlistsimOptions options reltol=10u vabstol=1n iabstol=1psimOptions options reltol=10u vabstol=1n iabstol=1psimOptions options reltol=10u vabstol=1n iabstol=1psimOptions options reltol=10u vabstol=1n iabstol=1pEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 7Sampled Noise SpectrumDensity of sampled noise including sinc distortionSampled noise normalized density corrected for sinc distortionEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 8Total NoiseSampled noise in 0 … fs/2: 62.2μV rms(expect 64μV for 1pF)EECS 247 Lecture 10: SC Filters © 2006 H. K. Page 9Switched-Capacitor Integrator-+VinVoφ1φ2CICsssignal samplings0Is0sIfor f ffCVVdtinCCfCω×=<<→=×∫-+∫φ1φ2T=1/fsMain advantage: No tuning needed Æ critical frequency function of ratio of capacitors & clock freq.EECS 247 Lecture 10: SC Filters © 2006 H. K. Page 10Switched-Capacitor Integrator-+VinVoφ1φ2CICs-+VinVoφ1CICs-+VinVoφ2CICsφ1φ2T=1/fsφ1 High ÆCsCharged to Vinφ2 HighÆCharge transferred fromCsto CIEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 11Continuous-Time versus Discrete Time Analysis ApproachContinuous-Time• Write differential equation• Laplace transform (F(s))• Let s=jω Æ F(jω)•Plot |F(jω)|, phase(F(jω) Discrete-Time • Write difference equation Æ relates output sequence to input sequence• Use delay operator z -1to transform the recursive realization to algebraic equation in Z domain• Set z= ejωT• Plot mag./phase versus frequency[]() ()os is1oiV (nT ) V ..........(n 1)TV z V .......ZZ−=−−=EECS 247 Lecture 10: SC Filters © 2006 H. K. Page 12Discrete Time Design Flow• Transforming the recursive realization to algebraic equation in z domain:– Use Delay operator z :s1s1/2s1s1/2snT ..................... 1............. z(n 1)T.......... z(n 1/2)T............. z(n 1)T.......... z(n 1/2)T⎡⎤⎣⎦⎡⎤⎣⎦⎡⎤⎣⎦⎡⎤⎣⎦−−++→→−→−→+→+* Note: z = e jωTs = cos(ωTs )+ j sin(ωTs)EECS 247 Lecture 10: SC Filters © 2006 H. K. Page 13Switched-Capacitor IntegratorOuput Sampled on φ1φ1φ2φ1φ2φ1VinVoVsClockVo1-+VinVo1φ1φ2CICsφ1VoEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 14Switched-Capacitor Integratorφ1φ2φ1φ2φ1VinVoVsClockVo1Φ1 Æ Qs [(n-1)Ts]= CsVi [(n-1)Ts] , QI [(n-1)Ts] = QI[(n-2)Ts] Φ2Æ Qs [(n-1/2) Ts] = 0 , QI[(n-1/2) Ts] = QI[(n-2) Ts] + Qs[(n-1) Ts] Φ1 _Æ Qs[nTs] = CsVi [nTs] , QI[nTs] = QI[(n-1) Ts] + Qs[(n-1) Ts]Since Vo1= - QI/CI& Vi= Qs/ CsÆCIVo1(nTs) = CIVo1[(n-1) Ts] -CsVi[(n-1) Ts] (n-1)TsnTs(n-1/2)Ts(n+1)Ts(n-3/2)Ts(n+1/2)TsEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 15Switched-Capacitor IntegratorOutput Sampled on φ1sIsI1s1Ios o sssIIinCosossinCC11ooinCCC1inCV(nT) CV CV(n 1)T (n 1)TV(nT)V V(n 1)T (n 1)TV (Z) Z V (Z) Z V (Z)VoZ(Z)ZV⎡⎤⎡⎤⎣⎦⎣⎦⎡⎤ ⎡⎤⎣⎦ ⎣⎦−−−−−=−−−=−−−=−=− ×DDI (Direct-Transform Discrete Integrator)-+VinVo1φ1φ2CICsφ1VoEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 16z-Domain Frequency Response•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•The jω axis maps onto the unit-circle•Particular values:– f = 0 Æz = 1– f = fs/2 Æz = -1f = 0f = fs /2LHP in s-domainimag. axis in s-domainz-planeEECS 247 Lecture 10: SC Filters © 2006 H. K. Page 17z-Domain Frequency Response• The frequency
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