EECS 247 Lecture 8: Filters© 2006 H.K. Page 1EE247 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 filtersEECS 247 Lecture 8: Filters© 2006 H.K. Page 2Summary 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 filtersEECS 247 Lecture 8: Filters© 2006 H.K. Page 3()Hjω()HjωLowpassHighpassω()HjωωωQ<5Q>5• Bandpass Filters:– Low Q (Q < 5) Æ Combination of lowpass & highpass– High Q or narrow-band (Q > 5)Æ Direct implementationω()Hjω+Bandpass FiltersBandpassBandpassEECS 247 Lecture 8: Filters© 2006 H.K. Page 4Narrow-Band Bandpass FiltersDirect Implementation• Narrow-band BP filters Æ Design based on lowpass prototype• Same tables used for LPFs are used for BPFsLowpass Freq. Mask Bandpass Freq. Maskccss2 s1cB2B1ssQsωωΩΩ−ΩΩΩ−Ω⎡⎤×+⎢⎥⎣⎦⇒⇒EECS 247 Lecture 8: Filters© 2006 H.K. Page 5Lowpass to Bandpass TransformationLowpass pole/zero (s-plane) Bandpass pole/zero (s-plane)From: Zverev, Handbook of filter synthesis, Wiley, 1967- p.156.PoleZeroEECS 247 Lecture 8: Filters© 2006 H.K. Page 6Lowpass to Bandpass Transformation TableFrom: Zverev, Handbook of filter synthesis, Wiley, 1967- p.157.''''1111rrrrrrrrCQCRRLQCRLQLCRQCωωωω=×=×=×=×CLC’LP BP BP ValuesLCL’Lowpass filter structures & tables used to derive bandpass filters''C &L are normilzed LP valuesfilterQQ=EECS 247 Lecture 8: Filters© 2006 H.K. Page 7Lowpass to Bandpass TransformationExample: 3rdOrder LPF Æ 6thOrder BPF• Each capacitor replaced by parallel L& C• Each inductor replaced by series L&CoVL2C2RsC1C3inVRLL1L3RsC1’C3’L2’inVRLoVLowpass BandpassEECS 247 Lecture 8: Filters© 2006 H.K. Page 8Lowpass to Bandpass TransformationExample: 3rdOrder LPF Æ 6thOrder BPF'1101'012'02'220'3303'03111111CQCRRLQCCRQLRLQLCQCRRLQCωωωωωω=×=×=×=×=×=×oVL2C2RsC1C3inVRLL1L3Where:C1’, L2’, C3’Æ Normalized lowpass valuesQ Æ Bandpass filter quality factor ω0Æ Filter center frequencyEECS 247 Lecture 8: Filters© 2006 H.K. Page 9Lowpass to Bandpass TransformationSignal FlowgraphoVL2C2RsC1C3inVRLL1L31- Voltages & currents named for all components2- 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 SFG5- Convert all current nodes to voltage EECS 247 Lecture 8: Filters© 2006 H.K. Page 10Signal Flowgraph6thOrder Bandpass Filter1*RRs−*11sCR1−*1RsL−1−1*RRL−*31sCR*3RsL−*21sCR−*2RsL1−Note: each C & L in the original lowpass prototype Æ replaced by a resonatorSubstituting the bandpass L1, C1,….. by their normalized lowpass equivalent from page 8The resulting SFG is:1V1’V2V3’V1V2’VoutVinV3EECS 247 Lecture 8: Filters© 2006 H.K. Page 11Signal Flowgraph6thOrder Bandpass Filter1*RRs−01'QCsω1−'10QCsω−1−1*RRL−'30QCsω'30QCsω−20'QLsω−02'QLsω1−• Note the integrators Æ different time constants• Ratio of time constants for two integrator in each resonator ~ Q2Æ Typically, requires high component ratiosÆ Poor matching• Desirable to convert SFG so that all integrators have equal time constants for optimum matching.• To obtain equal integrator time constant Æ scale nodes1V1’V2V3’V1V2’VoutVinV3EECS 247 Lecture 8: Filters© 2006 H.K. Page 12Signal Flowgraph6thOrder Bandpass Filter'11QC−'21QL*'1R1RsQC−×0sω1−0sω−'21QL−'31QC*3R1RLQC−×0sω0sω−0sω−0sω• All integrator time-constants Æ equal• To simplify implementation Æ chooseRL=Rs=R*1V1’/(QC1’)V2 /(QL2’)V3’/(QC3’)V1V3V2’VinVoutEECS 247 Lecture 8: Filters© 2006 H.K. Page 13Signal Flowgraph6thOrder Bandpass Filter'21QL'11QC−0sω1−0sω−'21QL−'31QC'31QC−0sω0sω−0sω−0sω'11QC−Let us try to build this bandpass filter using the simple Gm-C structure 1VinVoutEECS 247 Lecture 8: Filters© 2006 H.K. Page 14Second Order Gm-C FilterUsing Simple Source-Couple Pair Gm-Cell• Center frequency:• Q function of: Use this structure for the 1stand the 3rdresonatorUse similar structure w/o M3, M4 for the 2ndresonatorHow to couple the resonators?M1,2mointgM1,2mM3,4mg2CgQgω=×=EECS 247 Lecture 8: Filters© 2006 H.K. Page 15Coupling of the Resonators1- Additional Set of Input DevicesCoupling of resonators:Use additional input source coupled pairs for the highlighted integrators For example, the middle integrator requires 3 sets of inputs'21QL'11QC−0sω1−0sω−'21QL−'31QC'31QC−0sω0sω−0sω−0sω'11QC−1VinVoutEECS 247 Lecture 8: Filters© 2006 H.K. Page 16Example: Coupling of the Resonators1- Additional Set of Input Devicesint gCAdd one source couple pair for each additional input Coupling level Æ ratio of device widthsDisadvantage Æ extra power dissipationoVmaininV+-+-M1 M2M3M4-+couplinginV+--++-MainInputCouplingInputEECS 247 Lecture 8: Filters© 2006 H.K. Page 17Coupling of the Resonators2- Modify SFG Æ Bidirectional Coupling Paths''121QCL'11QC−0sωinV1−0sω−''321QCL−'1''32CQC L31QC'−0sω0sω−0sω−0sω1''QCL12−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=10outV1EECS 247 Lecture 8: Filters© 2006 H.K. Page 18Sixth Order Bandpass Filter Signal Flowgraphγ1Q−0sωinV1−0sω−1Q−0sω0sω−0sω−0sωoutV1γ−γγ−1Q2114γγ=≈• Where for a Butterworth shape• Since Q=10 then:EECS 247 Lecture 8: Filters© 2006 H.K. Page 19Sixth Order Bandpass Filter Signal FlowgraphSFG Modification1Q−0sωinV1−0sω−1Q−0sω0sω−0sω−0sωoutV1γ−20sωγ⎛⎞⎜⎟⎝⎠×γ−20sωγ⎛⎞⎜⎟⎝⎠×EECS 247
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