EECS 247 Lecture 5: Filters © 2006 H.K. Page 1EE247 Lecture 5• Summary last lecture• Today– Effect of integrator non-idealities on continuous-time filter behavior• Integrator frequency characteristics influence on filter response• Effect of component non-linearities on filter behavior– Various integrator topologies utilized in monolithic filters• Resistor + C based filters• Transconductance (gm) + C based filters• Switched-capacitor filters– Continuous-time filters• Facts about monolithic Rs & Cs and its effect on integrated filter characteristics• Opamp MOSFET-C filters• Opamp MOSFET-RC filters• Gm-C filtersEECS 247 Lecture 5: Filters © 2006 H.K. Page 2Summary of Lecture 4• Ladder Type Filters – All pole ladder type filters• Convert RLC ladder filters to integrator based form•Example: 5thorder Butterworth filter– High order ladder type filters incorporating zeros• 7th order elliptic filter in the form of ladder RLC with zeros– Sensitivity to component mismatch – Compare with cascade of biquadsÆDoubly terminated LC ladder filters Lowest sensitivity to component variations• Convert to integrator based form utilizing SFG techniques• Example: Single-ended & differential high order filter implementationEECS 247 Lecture 5: Filters © 2006 H.K. Page 3Effect of Integrator Non-Idealities on Filter Frequency Characteristics• In the section on passive filter design (RLC filters)– Reactive element (L & C) non-idealities Æ expressed in the form of Quality Factor (Q)– Filter impairments due to component non-idealities explained in terms of component Q• In the context of active filter design (integrator-based filters)– Integrator non-idealities Æ Translated to the form of Quality Factor (Q)– Filter impairments due to integrator non-idealities explained in terms of component QEECS 247 Lecture 5: Filters © 2006 H.K. Page 4Effect of Integrator Non-Idealities on Filter Performance• Ideal integrator characteristics• Real integrator characteristics:– Effect of finite DC gain– Effect of integrator non-dominant polesEECS 247 Lecture 5: Filters © 2006 H.K. Page 5Effect of Integrator Non-Idealities on Filter PerformanceIdeal IntegratorIdeal Intg.oVCinV-+RIdeal Intg.0ωψ-90o()logHsψopamp DC gainSingle pole @ DC no non-dominant polesoH(s)s1/ RCoωω=→−==∞0dBEECS 247 Lecture 5: Filters © 2006 H.K. Page 6Ideal Integrator Quality Factor()() ()()()1HjRjXXQRωωωωω=+=Since Q is defined as:Then:intg.Qideal=∞1ooH(s)sjjoωωωωω−−== =−Ideal intg. transfer function:EECS 247 Lecture 5: Filters © 2006 H.K. Page 7Real Integrator Non-IdealitiesIdeal Intg.Real Intg.()()()oossap2 p3aH(s) H(s)111ss...ωω−−=≈+++0ωa-90oψP2P30P1aω=-90o()logHsψ0ωψ-90o()logHsψEECS 247 Lecture 5: Filters © 2006 H.K. Page 8Effect of Integrator Finite DC Gain on Q -90-89.5ωoωoP1P1o(in radian)Arctan2oPhase lead@ωπωω−+→∠Example: P1/ ω0 = 1/100Æ phase error ≅ +0.5degree0P1aω=0ωa-90o()logHsψEECS 247 Lecture 5: Filters © 2006 H.K. Page 9Effect of Integrator Finite DC Gain on Q• Phase lead @ ω0ÆDroop in the passbandNormalized FrequencyMagnitude (dB)1Droop in the passbandIdeal intgIntg with finite DC gainEECS 247 Lecture 5: Filters © 2006 H.K. Page 10Effect of Integrator Non-Dominant Poles -90-90.5ωoωoioipi2opi2(in radian)Arctan2Phase lag @ωωπω∞=∞=−−→∑∑∠Example: ω0 /P2 =1/100 Æ phase error ≅ −0.5degree0ω-90o()logHsψP2P3EECS 247 Lecture 5: Filters © 2006 H.K. Page 11Effect of Integrator Non-Dominant PolesNormalized FrequencyMagnitude (dB)1•Phase lag @ ω0ÆPeaking in the passbandIn extreme cases could result in oscillation!Peaking in the passbandIdeal intgOpamp with finite bandwidthEECS 247 Lecture 5: Filters © 2006 H.K. Page 12Effect of Integrator Non-Dominant Poles & Finite DC Gain on Q -90ωoωP1Arctan2ooArctanpii2πωω∠− +∞−∑=0ωa-90o()logHsψP2P30P1aω=-90o()logHsψNote that the two terms have different signs Æ Can cancel each other’s effect!EECS 247 Lecture 5: Filters © 2006 H.K. Page 13Integrator Quality Factor()()()ossap2 p3aH(s)111s...ω−≈+++Real intg. transfer function:o1&a1p2,3,.....intg.1Qreal11oapii2ωω<< >>≈∞−∑=Based on the definition of Q and assuming that:It can be shown that in the vicinity of unity-gain-frequency:Phase lead@ω0Phase lag@ω0EECS 247 Lecture 5: Filters © 2006 H.K. Page 14Example:Effect of Integrator Finite Q on Bandpass Filter BehaviorIntegrator DC gain=100Integrator P2 @ 100.ωoIdealIdeal0.5οφlead@ ωointg0.5οφexcess@ ωointgEECS 247 Lecture 5: Filters © 2006 H.K. Page 15Example:Effect of Integrator Q on Filter BehaviorIntegrator DC gain=100 & P2 @ 100. ωοIdeal( 0.5οφlead−0.5οφexcess ) @ ωointgÆ φerror@ ωointg ~ 0EECS 247 Lecture 5: Filters © 2006 H.K. Page 16SummaryEffect of Integrator Non-Idealities on Q• Amplifier DC gain reduces the overall Q in the same manner as series/parallel resistance associated with passive elements• Amplifier poles located above integrator unity-gain frequency enhance the Q! – If non-dominant poles close to unity-gain freq. Æ Oscillation• Depending on the location of unity-gain-frequency, the two terms can cancel each other out!i11opi2intg.idealintg.1realQQaω∞==≈−∞∑EECS 247 Lecture 5: Filters © 2006 H.K. Page 17Effect of Integrator Non-Linearities on Overall Integrator-Based Filter Performance• Maximum signal handling capability of a filter is determined by the non-linearities associated with its building blocks• Filter specifications wrt linearity are given in terms of :– Maximum allowable harmonic distortion– Maximum tolerable intermodulation distortionEECS 247 Lecture 5: Filters © 2006 H.K. Page 18Effect of Component Non-Linearities on Overall Filter PerformanceIdeal Components() ()Ideal DC transfer characteristics:sin sinVout VinIfVin A t Vout A tαωαω==→=VinVoutf1ff1fEECS 247 Lecture 5: Filters © 2006 H.K. Page 19Effect of Component Non-Linearities on Overall Filter LinearityReal Components including Non-LinearitiesVinVoutf1ff1f2f13f1()() ()()()()()() ( )()33331333222222211Real DC transfer characteristics:........sinsin......sin..........sin1cossin3si ...nsin3422Vout VinIf Vin A tVout A tVinAtVinAtAor Vout A ttAttαωαωααωαωαωααωαωω=+ +=→= +++−+=+−++EECS 247 Lecture 5: Filters © 2006 H.K. Page 20Effect of Component Non-Linearities on
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