EECS 247 Lecture 3: Filters © 2010 H.K. Page 1EE247Administrative– Homework #1 will be posted on EE247 site and is due Sept. 9th– Office hours held @ 201 Cory Hall:• Tues. and Thurs.: 4 to 5pmEECS 247 Lecture 3: Filters © 2010 H.K. Page 2EE247 Lecture 3• Active Filters– Active biquads-–How to build higher order filters?• Integrator-based filters– Signal flowgraph concept– First order integrator-based filter– Second order integrator-based filter & biquads– High order & high Q filters• Cascaded biquads & first order filters– Cascaded biquad sensitivity to component mismatch• Ladder type filtersEECS 247 Lecture 3: Filters © 2010 Page 3Filters2ndOrder Transfer Functions (Biquads)• Biquadratic (2ndorder) transfer function:PQjHjHjHP)(0)(1)(022PPP2222PPPP2PP1P21H(s )ss1Q1H( j )1QBiquad poles @: s 1 1 4Q2Q Note: for Q poles are real, complex otherwise EECS 247 Lecture 3: Filters © 2010 Page 4Biquad Complex PolesDistance from origin in s-plane: 22221412PPPPQQd122Complex conjugate poles:s 1 4 12PPPPQjQQ polesjdS-planeEECS 247 Lecture 3: Filters © 2010 Page 5s-PlanepolesjP radius P2Q- part realPPQ21arccos2s 1 4 12PPPjQQ EECS 247 Lecture 3: Filters © 2010 Page 6Example2ndOrder Butterworthj 45 deg224 2 2Since 2nd order Butterworth:reeCospQPQ21arccosEECS 247 Lecture 3: Filters © 2010 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 load and source resistance– As previously analyzed, not feasible in the monolithic form for f <350MHz• 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 © 2010 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, … Advantage: Only one opamp used Disadvantage: Sensitive to parasitic – all pole no finite zerosoutVC1inVR2R1C2G221 1 2 21 1 2 1 2 2()111 1 1PPPPPPGHsssQR C R CQGR C R C R CEECS 247 Lecture 3: Filters © 2010 Page 9Addition of Imaginary Axis Zeros• Sharpen transition band• Can “notch out” interference– Band-reject filter• High-pass filter (HPF)Note: 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.2Z2P P P2PZs1H(s) Kss1QH( j ) KEECS 247 Lecture 3: Filters © 2010 Page 10Imaginary Zeros• Zeros substantially sharpen transition band• At the expense of reduced stop-band attenuation at high frequenciesPZPPffQkHzf32100104105106107-50-40-30-20-10010Frequency [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 © 2010 Page 11Moving the ZerosPZPPffQkHzf2100104105106107-50-40-30-20-1001020Frequency [Hz]Magnitude [dB]Pole-Zero MapReal AxisImag Axis-6 -4 -2 0 2 4 6-6-4-20246x105x105EECS 247 Lecture 3: Filters © 2010 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 © 2010 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 zero• Possible to use combination of 3 outputs EECS 247 Lecture 3: Filters © 2010 Page 14Component Values87217328211112173280682748168111217538011RRkCRRCRRkCRaCCRRRRaRRbRRRRRRCRbCCRRRRb82728620015112124102132012111111111RkRbRRCbakRCbbakRCakkRCakRCaR821 and , ,,, given RCCkbaiii thatfollowsit 11217328CRQCCRRRRPPPEECS 247 Lecture 3: Filters © 2010 H.K. Page 15Higher-Order Filters in the Integrated Form• One way of building higher-order filters (n>2) is via cascade of 2ndorder biquads & 1storder , e.g. Sallen-Key,or Tow-Thomas, & RC2ndorderFilter ……Nx 2ndorder sections Filter order: n=2N 1 2 NCascade of 1stand 2ndorder filters: Easy to implement Highly sensitive to component mismatch -good for low Q filters only For high Q applications good alternative: Integrator-based ladder filters2ndorderFilter 1stor 2ndorderFilter EECS 247 Lecture 3: Filters © 2010 H.K. Page 16Integrator Based Filters• Main building block for this category of filters Integrator• By using signal flowgraph techniques Conventional RLC filter topologies can be converted to integrator based type filters• How to design integrator based filters?– Introduction to signal flowgraph techniques– 1st order integrator based filter– 2nd order integrator based filter– High order and high Q filtersEECS 247 Lecture 3: Filters © 2010 H.K. Page 17What is a Signal Flowgraph (SFG)?• SFG 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
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