EE247 Lecture 10 Supplemental Material for S C Filters EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 1 z Domain Frequency Response The j axis maps onto the imag axis in unit circle s domain LHP singularities in s plane map into inside of unit circle in z domain RHP singularities in s plane map into outside of unitf fs 2 circle in z domain Particular values z e s Ts z plane f 0 f 0 z 1 f fs 2 z 1 LHP in s domain EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 2 z Domain Frequency Response cos Ts sin Ts The frequency response is obtained by evaluating H z on the unit circle at z e j T cos Ts j sin Ts 2 f fS Once z 1 fs 2 is reached the frequency response repeats as expected The angle to the pole is equal to 360 or 2 radians times the ratio of the pole frequency to the sampling frequency EECS 247 Lecture 10 z plane Switched Capacitor Filters 2008 H K Page 3 DDI Integrator Pole Zero Map in z Plane f z 1 0 z 1 on unit circle f1 Pole from f 0 in s plane mapped to z 1 increasing z 1 As frequency increases z domain point moves on unit circle CCW f fs 2 Once frequency gets to z 1 f fs 2 frequency response repeats EECS 247 Lecture 10 Switched Capacitor Filters 1 z plane 2008 H K Page 4 Example 2nd Order LDI Bandpass Filter s Plane versus z Plane s plane z plane j EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 5 EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 6 Sixth Order Elliptic LDI Bandpass Filter Transmission Zero 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 7 Use of T Network V2 C C4 1 V1 C2 C1 C3 C4 V2 C 1 V1 C2 High Q filter large cap ratio for Q transmission zero implementation To reduce large ratios required T networks utilized 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 8 Sixth Order Elliptic Bandpass Filter Utilizing T Network Q implementation Zero T networks utilized for Q implemention Transmission zero implementation 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 9 EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 10 More Realistic Switched Capacitor Circuit Slew Scenario CI 2 Vin Cs CI 2 CL CL Cs Vo Vo t 0 At the instant Cs connects to input of opamp t 0 Opamp not yet active at t 0 due to finite opamp bandwidth delay 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 Cs Spike increases slewing period Eventually opamp becomes active starts slewing followed by subsequent settling EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 11 Switched Capacitor Circuit Opamp not Active t 0 CI 2 Vin Cs CL CL Cs Vo Vo t 0 t 0 t 0 Ch arg e sharing CsVCs VCs Cs Ceq where Ceq t 0 t 0 Vout VCs CI 2 CI CL CI C L CI Cs CI t 0 VCs CI C L Cs Ceq CI CL t 0 t 0 VCs Assu min g CL Cs CI Ceq CL CsVCs Cs CL VCst 0 VCst 0 t 0 t 0 Vout VCs Cs CI t 0 VCs Cs C L C I C L final Note that Vout EECS 247 Lecture 10 Cs t 0 C t 0 VCs s VCs CI CI Switched Capacitor Filters 2008 H K Page 12 More Realistic Switched Capacitor Circuit Slew Scenario CI 2 Vin Cs CI 2 CL CL Cs Vo Vo t 0 Notice that if CL is large some of the charge stored on Cs is lost prior to opamp becoming effective operation looses accuracy CC t 0 t 0 Ch arg e sharing CsVCs VCs Cs Ceq where Ceq C I CL I L Cs Cs t 0 t 0 t 0 VCs VCs VCs CC Cs Ceq Cs I L CI CL Partly responsible for S C filters only good for low frequency applications EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 13 More Realistic S C Slew Scenario Vo ideal Vo real Slewing Linear Settling Slewing Linear Settling Spike generated at t 0 Vo real Including t 0 spike Slewing Linear Settling Slewing Ref 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 14 EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 15 Extending the Maximum Achievable Critical Frequency of Switched Capacitor Filters Consider a switchedcapacitor resonator Regular sampling Each opamp is busy settling only during one of the two clock phases 1 2 2 1 Idle during the other clock phase Note During 1 both opamps are idle EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 16 Switched Capacitor Resonator Using Double Sampling Double sampling 2nd set 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 same EECS 247 Lecture 10 Switched Capacitor Filters 2008 H K Page 17 Double Sampling Issues fclock fs 2fclock Issues to be aware of Jitter in the clock Unequal clock phases Mismatch in sampling caps parasitic passbands 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 18 Double Sampled Fully Differential S C 6th Order All Pole Bandpass Filter 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 19 Sixth Order Bandpass Filter Signal Flowgraph Vin 1 Q 1 0 s 0 0 s s 0 s EECS 247 Lecture 10 1 0 s 0 s Vout 1 Q Switched Capacitor Filters 2008 H K Page 20 Double Sampled Fully Differential 6th Order S …
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