EECS 247 Lecture 3: Filters © 2006 H.K. Page 1EE247 Lecture 3• Last lecture’s summary• Active Filters– Active biquads• Sallen- Key & Tow-Thomas• Integrator based filters– Signal flowgraph concept– First order integrator based filter– Second order integrator based filter & biquads– High order & high Q filters• Cascaded biquads– Cascaded biquad sensitivity to component variations• Ladder type filtersEECS 247 Lecture 3: Filters © 2006 H.K. Page 2Summary of Last Lecture– Nomenclature– Filter specifications • Quality factor• Frequency characteristics• Group delay– Filter types• Butterworth• Chebyshev I• Chebyshev II• Elliptic• Bessel– Group delay comparison example– RLC filtersEECS 247 Lecture 3: Filters © 2006 H.K. Page 3Integrated Filters• Implementation of RLC filters in CMOS technologies requires on-chip inductors– Integrated L<10nH with Q<10 – Combined with max. cap. 10pFÆLC filters in the monolithic form feasible: freq>500MHz • Analog/Digital interface circuitry require fully integrated filters with critical frequencies << 500MHz• Hence:F Need to build active filters built without inductorsEECS 247 Lecture 3: Filters © 2006 H.K. Page 4Filters2ndOrder Transfer Functions (Biquads)• Biquadratic (2ndorder) transfer function:PQjHjHjHP====∞→=ωωωωωωω)(0)(1)(022PPPP2PP1P21H(s)ss1QBiquad poles @: s 1 1 4Q2Q for Q poles are real, complex otherwiseωωω=++⎛⎞=− ± −⎜⎟⎝⎠≤EECS 247 Lecture 3: Filters © 2006 H.K. Page 5Biquad Complex PolesDistance from origin in s-plane:()22221412PPPPQQdωω=−+⎟⎟⎠⎞⎜⎜⎝⎛=122Complex conjugate poles:s1412PPPPQjQQω>→⎛⎞=− ± −⎜⎟⎝⎠polesωjσdS-planeEECS 247 Lecture 3: Filters © 2006 H.K. Page 6s-PlanepolesωjσPω radius =P2Q- part realPω=PQ21arccosEECS 247 Lecture 3: Filters © 2006 H.K. Page 7Implementation of Biquads• Passive RC: only real poles - can’t implement complex conjugate poles• Terminated LC– Low power, since it is passive– Only fundamental noise sources Æ load and source resistance– As previously analyzed, not feasible in the monolithic form for f <500MHz• Active Biquads– Many topologies can be found in filter textbooks! – Widely used topologies:• Single-opamp biquad: Sallen-Key• Multi-opamp biquad: Tow-Thomas• Integrator based biquadsEECS 247 Lecture 3: Filters © 2006 H.K. Page 8Active Biquad Sallen-Key Low-Pass Filter• Single gain element• Can be implemented both in discrete & monolithic form• “Parasitic sensitive”• Versions for LPF, HPF, BP, …Æ Advantage: Only one opamp used Æ Disadvantage: Sensitive to parasitic – all pole no zerosoutVC1inVR2R1C2G22112 211 21 2 2()11111PPPPPPGHsssQRCRCQGRCRCRCωωωω=++==−++EECS 247 Lecture 3: Filters © 2006 H.K. Page 9Addition of Imaginary Axis Zeros• Sharpen transition band• Can “notch out” interference• High-pass filter (HPF)• Band-reject filterNote:Always represent transfer functions as a product of a gain term,poles, and zeros (pairs if complex). Then all coefficients have a physical meaning, and readily identifiable units.2Z2PP P2PZs1H(s) Kss1QH( j ) Kωωωωωωω→∞⎛⎞+⎜⎟⎝⎠=⎛⎞++⎜⎟⎝⎠⎛⎞=⎜⎟⎝⎠EECS 247 Lecture 3: Filters © 2006 H.K. Page 10Imaginary Zeros• Zeros substantially sharpen transition band• At the expense of reduced stop-band attenuation at high frequencyPZPPffQkHzf32100===104105106107-50-40-30-20-10010Frequency [Hz]Magnitude [dB]With zerosNo zerosReal AxisImag Axis-2 -1.5 -1 -0.5 0 0.5 1 1.5 2x 106-2-1.5-1-0.500.511.52x 106Pole-Zero MapEECS 247 Lecture 3: Filters © 2006 H.K. Page 11Moving the ZerosPZPPffQkHzf===2100104105106107-50-40-30-20-1001020Frequency [Hz]Magnitude [dB]Pole-Zero MapReal AxisImag Axis-6 -4 -2 0 2 4 6-6-4-20246x105x105EECS 247 Lecture 3: Filters © 2006 H.K. Page 12Tow-Thomas Active BiquadRef: P. E. Fleischer and J. Tow, “Design Formulas for biquad active filters using three operational amplifiers,” Proc. IEEE, vol. 61, pp. 662-3, May 1973.• Parasitic insensitive• Multiple outputsEECS 247 Lecture 3: Filters © 2006 H.K. Page 13Frequency Response()( )()( )012100102001301201222012002112211asasbabasabbakVVasasbsbsbVVasasbabsbabkVVinoinoino++−+−−=++++=++−+−−=• Vo2implements a general biquad section with arbitrary poles and zeros•Vo1and Vo3realize the same poles but are limited to at most one finite zeroEECS 247 Lecture 3: Filters © 2006 H.K. Page 14Component Values87217328211112173280682748168111217538011RRkCRRCRRkCRaCCRRRRaRRbRRRRRRCRbCCRRRRb=====⎟⎟⎠⎞⎜⎜⎝⎛−==82728620015112124102132012111111111RkRbRRCbakRCbbakRCakkRCakRCaR===−====821 and , ,,, given RCCkbaiii thatfollowsit 11217328CRQCCRRRRPPPωω==EECS 247 Lecture 3: Filters © 2006 H.K. Page 15Higher-Order Filters• Higher-order filters (N>2) can be built with cascade of 2ndorder biquads, e.g. Sallen-Key,or Tow-Thomas2ndorderFilter 2ndorderFilter 2ndorderFilter ………Nx 2ndorder sections Æ Filter with 2N order1 2 ΝAs will be shown later:High-Q high-order filters built with cascade of 2ndorder sections Æ Highly sensitive to component variationsÆ Good alternative: Integrator-based ladder type filtersEECS 247 Lecture 3: Filters © 2006 H.K. Page 16Integrator Based Filters• Main building block for this category of filters Æintegrator• By using signal flowgraph techniques Æconventional filter topologies can be converted to integrator based type filters• Next few pages:– Signal flowgraph techniques– 1st order integrator based filter– 2nd order integrator based filter– High order and high Q filtersEECS 247 Lecture 3: Filters © 2006 H.K. Page 17What is a Signal Flowgraph (SFG)?•SFG Æ Topological network representation consisting of nodes & branches- used to convert one form of network to a more suitable form (e.g. passive RLC filters to integrator based filters)• Any network described by a set of linear differential equations can be expressed in SFG form• For a given network, many different SFGs exists • Choice of a particular SFG is based on practical considerations such as type of available components*Ref: W.Heinlein & W. Holmes, “Active Filters for Integrated Circuits”, Prentice Hall, Chap. 8, 1974. EECS 247 Lecture 3: Filters © 2006 H.K. Page 18What is a Signal Flowgraph (SFG)?• Signal flowgraph
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