EE247 Lecture 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 idealities EECS 247 Lecture 10 SC Filters 2006 H K Page 1 Summary 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 hold EECS 247 Lecture 10 SC Filters 2006 H K Page 2 Switched Capacitor Network Noise During 1 high Resistance of switch S1 RonS1 produces a noise voltage on C with variance kT C lecture 1 first order filter noise vIN 1 2 S1 S2 vOUT C The corresponding noise charge is RonS1 Q2 C2V2 C2 kT C kTC C 1 low S1 opens This charge is sampled EECS 247 Lecture 10 SC Filters 2006 H K Page 3 Switched Capacitor Noise During 2 high Resistance of switch S2 contributes to an uncorrelated noise charge on C at the end of 2 with variance kT C Mean squared noise charge transferred from vIN to vOUT per sample period is vIN 1 2 S1 S2 vOUT C RonS2 C Q2 2kTC EECS 247 Lecture 10 SC Filters 2006 H K Page 4 Switched Capacitor Noise The mean squared noise current due to S1 and S2 s kT C noise is S i n ce i Q t 2 t hen i2 Qfs 2kBT C fs2 This noise is approximately white and distributed between 0 and fs 2 noise spectra single sided by convention The spectral density of the noise is 2 i2 2kBT C fs 4k T C f B s fs f 2 i2 4kBT f REQ Si nce REQ 1 t hen fsC S C resistor noise a physical resistor noise with same value EECS 247 Lecture 10 SC Filters 2006 H K Page 5 Periodic Noise Analysis SpectreRF Sampling Noise from SC S H Netlist ahdl include zoh def Netlist simOptions options reltol 10u vabstol 1n iabstol 1p Vclk 100ns 100kOhm R Vrc Vrc hold ZOH1 S1 PNOISE Analysis PNOISE1 sweep from 0 to 20 01M 1037 steps C 1pF ZOH1 T 100ns SpectreRF PNOISE check noisetype timedomain noisetimepoints C1 as alternative to ZOH 1pF noiseskipcount large might speed up things in this case PSS pss period 100n maxacfreq 1 5G errpreset conservative PNOISE Vrc hold 0 pnoise start 0 stop 20M lin 500 maxsideband 10 100kOhm R1 Voltage NOISE VNOISE1 EECS 247 Lecture 10 SC Filters 2006 H K Page 6 Sampled Noise Spectrum Density of sampled noise including sinc distortion Sampled noise normalized density corrected for sinc distortion EECS 247 Lecture 10 SC Filters 2006 H K Page 7 Total Noise Sampled noise in 0 fs 2 62 2 V rms expect 64 V for 1pF EECS 247 Lecture 10 SC Filters 2006 H K Page 8 Switched Capacitor Integrator 1 2 Vin 1 CI 2 T 1 fs Cs f signa l f sampling for Vo V0 f s Cs CI Vin dt Cs 0 fs C I Main advantage No tuning needed critical frequency function of ratio of capacitors clock freq EECS 247 Lecture 10 SC Filters 2006 H K Page 9 Switched Capacitor Integrator Vin 1 CI 2 Cs Vin 2 Vo T 1 fs CI 1 Cs High Cs Charged to Vin EECS 247 Lecture 10 SC Filters Vo CI 2 Vin Cs 1 1 2 Vo High Charge transferred from Cs to CI 2006 H K Page 10 Continuous Time versus Discrete Time Analysis Approach Continuous Time Discrete Time Write differential equation Laplace transform F s Let s j F j Write difference equation relates output sequence to input sequence Plot F j phase F j Use delay operator z 1 to transform the recursive realization to algebraic equation in Z domain Vo nTs Vi n 1 Ts Vo Z z 1Vi Z Set z e j T Plot mag phase versus frequency EECS 247 Lecture 10 SC Filters 2006 H K Page 11 Discrete Time Design Flow Transforming the recursive realization to algebraic equation in z domain Use Delay operator z nTs 1 n 1 Ts z 1 n 1 2 Ts z 1 2 n 1 Ts z 1 n 1 2 Ts z 1 2 Note z e j Ts cos Ts j sin Ts EECS 247 Lecture 10 SC Filters 2006 H K Page 12 Switched Capacitor Integrator Ouput Sampled on 1 Vin 1 2 CI Cs 1 Vo 2 1 1 Vo1 1 2 Clock Vin Vs Vo Vo1 EECS 247 Lecture 10 SC Filters 2006 H K Page 13 Switched Capacitor Integrator n 3 2 Ts n 1 Ts 1 n 1 2 Ts 2 1 nTs 2 n 1 2 Ts n 1 Ts Clock 1 Vin Vs Vo Vo1 1 Qs n 1 Ts Cs Vi n 1 Ts QI n 1 Ts QI n 2 Ts 2 Qs n 1 2 Ts 0 QI n 1 2 Ts QI n 2 Ts Qs n 1 Ts 1 Qs nTs Cs Vi nTs QI nTs QI n 1 Ts Qs n 1 Ts Since Vo1 QI CI Vi Qs Cs CI Vo1 nTs CI Vo1 n 1 Ts Cs Vi n 1 Ts EECS 247 Lecture 10 SC Filters 2006 H K Page 14 Switched Capacitor Integrator Output Sampled on 1 Vin 1 2 CI Cs 1 Vo Vo1 CI Vo nTs CI Vo n 1 Ts Cs Vin n 1 Ts Cs V n 1 T Vo nTs Vo n 1 Ts C s in I Cs V Z Vo Z Z 1Vo Z Z 1 C in I Vo Cs Z 1 Z C 1 Z 1 I Vin DDI Direct Transform Discrete Integrator EECS 247 Lecture 10 SC Filters 2006 H K Page 15 z Domain Frequency Response LHP singularities in s plane imag axis in map into inside of unit circle s domain in z domain RHP singularities in s plane map into outside of unitcircle in z domain The j axis maps onto the f fs 2 unit circle Particular values z plane f 0 f 0 z 1 f fs 2 z 1 LHP in s domain EECS 247 Lecture 10 SC Filters 2006 H K Page 16 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 z plane EECS 247 Lecture 10 SC …
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