EE247 Lecture 8 Summary of last lecture Continuous time filters Bandpass filters Lowpass to bandpass transformation Example Gm C BP filter using simple diff pair Linearity noise issues Various Gm C Filter implementations Comparison of continuous time filter topologies Switched capacitor filters EECS 247 Lecture 8 Filters 2006 H K Page 1 Summary Last Lecture Automatic on chip filter tuning continued from last lecture Continuous tuning Reference integrator locked to a reference frequency Error due to integrator DC offset and cancellation method DC tuning of resistive timing element Periodic digitally assisted tuning Systems where filter is followed by ADC DSP existing hardware can be used to periodically update filter freq response Continuous time filters High pass filters EECS 247 Lecture 8 Filters 2006 H K Page 2 Bandpass Filters Bandpass Filters Low Q Q 5 Combination of lowpass highpass Bandpass Highpass Lowpass H j H j H j Q 5 H j Bandpass Q 5 High Q or narrow band Q 5 Direct implementation 2006 H K Page 3 EECS 247 Lecture 8 Filters Narrow Band Bandpass Filters Direct Implementation Narrow band BP filters Design based on lowpass prototype Same tables used for LPFs are used for BPFs Lowpass Freq Mask s s Q c s c EECS 247 Lecture 8 Filters Bandpass Freq Mask s c s2 s1 B2 B1 2006 H K Page 4 Lowpass to Bandpass Transformation Lowpass pole zero s plane Bandpass pole zero s plane Pole Zero From Zverev Handbook of filter synthesis Wiley 1967 p 156 2006 H K Page 5 EECS 247 Lecture 8 Filters Lowpass to Bandpass Transformation Table Lowpass filter structures tables used to derive bandpass filters LP BP BP Values 1 Rr r C QC C C L L 1 QC Rr r Q Q filter L L From Zverev Handbook of filter synthesis Wiley 1967 p 157 EECS 247 Lecture 8 Filters C L QL C 1 QC Rr r 1 Rr r C L are normilzed LP values 2006 H K Page 6 Lowpass to Bandpass Transformation Example 3rd Order LPF 6th Order BPF Lowpass Bandpass L2 Vo L2 Rs Vin C1 C2 Vo Rs C3 RL Vin C1 C3 L1 RL L3 Each capacitor replaced by parallel L C Each inductor replaced by series L C 2006 H K Page 7 EECS 247 Lecture 8 Filters Lowpass to Bandpass Transformation Example 3rd Order LPF 6th Order BPF C1 QC1 L1 1 R 0 1 R QC1 0 1 1 C2 QL2 R 0 L2 QL 2 C3 QC3 L3 L2 C2 Rs Vin C1 L1 Vo C3 L3 RL R 0 1 R 0 1 R QC3 0 EECS 247 Lecture 8 Filters Where C1 L2 C3 Normalized lowpass values Q Bandpass filter quality factor 0 Filter center frequency 2006 H K Page 8 Lowpass to Bandpass Transformation Signal Flowgraph C2 L2 Vo Rs Vin C1 C3 L1 RL L3 1 Voltages currents named for all components 2 Use KCL KVL to derive state space description 3 To have BMFs in the integrator form Cap voltage expressed as function of its current VC f IC Ind current as a function of its voltage IL f VL 4 Use state space description to draw SFG 5 Convert all current nodes to voltage 2006 H K Page 9 EECS 247 Lecture 8 Filters Signal Flowgraph 6th Order Bandpass Filter Vin R Rs 1 V1 R s L1 V1 1 R 1 sC1R 1 1 sC2 R s L2 V2 V3 1 V2 1 R s L3 1 1 sC3R Vout R RL V3 Note each C L in the original lowpass prototype replaced by a resonator Substituting the bandpass L1 C1 by their normalized lowpass equivalent from page 8 The resulting SFG is EECS 247 Lecture 8 Filters 2006 H K Page 10 6th Vin R Rs Signal Flowgraph Order Bandpass Filter 1 V1 0 Q C1 0 0 s Q L 2 s Q C1 s 1 V1 V3 1 V2 1 0 s Q C3 Q C3 0 Q L 2 0 s Vout R RL s 1 V2 1 V3 Note the integrators different time constants Ratio of time constants for two integrator in each resonator Q2 Typically requires high component ratios Poor matching Desirable to convert SFG so that all integrators have equal time constants for optimum matching To obtain equal integrator time constant scale nodes 2006 H K Page 11 EECS 247 Lecture 8 Filters Signal Flowgraph 6th Order Bandpass Filter 1 Vin R 1 Rs QC1 1 V1 0 s QL 2 V2 QL2 0 s 0 s V1 QC1 1 Q C1 V2 1 0 s Q L 2 0 s 1 QC 3 V3 1 Vout 0 s R 1 R L QC3 V3 QC3 All integrator time constants equal To simplify implementation choose RL Rs R EECS 247 Lecture 8 Filters 2006 H K Page 12 Signal Flowgraph 6th Order Bandpass Filter 1 Vin 1 Q C1 1 0 s 1 QL 2 0 0 s 1 s Q C1 Q L 2 0 s 0 s 1 1 0 Vout s 1 QC3 QC 3 Let us try to build this bandpass filter using the simple Gm C structure EECS 247 Lecture 8 Filters 2006 H K Page 13 Second Order Gm C Filter Using Simple Source Couple Pair Gm Cell Center frequency o gmM 1 2 2 Cint g Q function of Q gmM 1 2 gmM 3 4 Use this structure for the 1st and the 3rd resonator Use similar structure w o M3 M4 for the 2nd resonator How to couple the resonators EECS 247 Lecture 8 Filters 2006 H K Page 14 Coupling of the Resonators 1 Additional Set of Input Devices 1 Vin 1 Q C1 1 0 s 1 QL 2 0 0 s s 1 Q C1 Q L 2 0 s 0 s 1 Vout 1 0 s 1 QC3 QC 3 Coupling of resonators Use additional input source coupled pairs for the highlighted integrators For example the middle integrator requires 3 sets of inputs 2006 H K Page 15 EECS 247 Lecture 8 Filters Example Coupling of the Resonators 1 Additional Set of Input Devices Add one source couple pair for each additional input Cint g Coupling level ratio of device widths Disadvantage extra power dissipation Coupling Main Input Input EECS 247 Lecture 8 Filters coupling Vin M3 M1 Vinmain Vo M4 M2 2006 H K Page 16 Coupling of the Resonators 2 Modify SFG Bidirectional Coupling Paths 1 Vin 1 1 0 s Q C1 1 Q C1 L 2 0 0 s 1 s Q C 3 L 2 0 s 1 Vout 0 0 s s 1 Q C 3 C1 Q C1 L 2 Q C 3 L 2 Modified signal flowgraph to have equal coupling between resonators In most filter cases C1 C3 Example For a butterworth lowpass filter C1 C3 1 L2 2 Assume desired overall bandpass filter Q 10 2006 H K Page 17 EECS 247 Lecture 8 Filters Sixth Order Bandpass Filter Signal Flowgraph Vin 1 Q 1 0 s 0 0 s s Where for a Butterworth shape Since Q 10 …
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