EE247 Administrative Homework 1 will be posted on EE247 site and is due Sept 9th Office hours held 201 Cory Hall Tues and Thurs 4 to 5pm EECS 247 Lecture 3 Filters 2010 H K Page 1 EE247 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 filters EECS 247 Lecture 3 Filters 2010 H K Page 2 Filters 2nd Order Transfer Functions Biquads Biquadratic 2nd order transfer function 1 H s 1 s s2 PQP P2 H j 0 1 1 H j H j 0 2 2 2 1 2 PQP P Biquad pole s s Note for QP 1 H j QP P P 2 1 1 4QP 2QP pole s are re al c om ple x othe rwise 2 EECS 247 Lecture 3 Filters 2010 Page 3 Biquad Complex Poles QP 12 s Complex conjugate poles P 2 1 j 4QP 1 2QP j d S plane poles Distance from origin in s plane 2 d 2 P 1 4QP2 1 2QP P2 EECS 247 Lecture 3 Filters 2010 Page 4 s Plane j radius P arccos poles 1 2QP P real part 2Q P EECS 247 s P 2 1 j 4QP 1 2QP Lecture 3 Filters 2nd 2010 Page 5 Example Order Butterworth j arccos 1 2QP Since 2nd order Butterworth 45 deg ree Cos 4 EECS 247 2 2 Qp 2 2 Lecture 3 Filters 2010 Page 6 Implementation 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 biquads EECS 247 Lecture 3 Filters 2010 Page 7 Active Biquad Sallen Key Low Pass Filter H s C1 R1 R2 G Vin C2 Vout P QP G s s2 1 PQP P2 1 R1C1R2C2 P 1 1 1 G R C R C R 1 1 2 1 2C2 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 zeros EECS 247 Lecture 3 Filters 2010 Page 8 Addition of Imaginary Axis Zeros Sharpen transition band Can notch out interference H s K Band reject filter s 1 Z 1 High pass filter HPF 2 s PQP P s H j K P Z 2 2 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 EECS 247 Lecture 3 Filters 2010 Page 9 Imaginary Zeros f P 100kHz Zeros substantially sharpen transition band At the expense of reduced stop band attenuation at high frequencies QP 2 fZ 3 fP 6 x 10 2 Pole Zero Map 1 5 With zeros No zeros 1 0 Imag Axis Magnitude dB 10 0 5 10 20 0 0 5 30 1 40 50 4 10 1 5 5 10 6 10 7 10 Frequency Hz EECS 247 2 2 1 5 1 0 5 0 0 5 Real Axis Lecture 3 Filters 1 1 5 2 6 x 10 2010 Page 10 Moving the Zeros f P 100kHz QP 2 5 x10 fZ fP Pole Zero Map 6 4 Imag Axis Magnitude dB 20 10 0 10 2 0 2 20 4 30 6 40 6 50 4 10 105 106 107 Frequency Hz EECS 247 Lecture 3 Filters 4 2 0 2 4 Real Axis 6 5 x10 2010 Page 11 Tow Thomas Active Biquad Parasitic insensitive Multiple outputs Ref 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 EECS 247 Lecture 3 Filters 2010 Page 12 Frequency Response b a b s b2a0 b0 Vo1 k2 2 1 2 1 Vin s a1s a0 Vo 2 b2 s 2 b1s b0 2 Vin s a1s a0 Vo 3 1 b0 b2a0 s a1b0 a0b1 Vin s 2 a1s a0 k1 a0 Vo2 implements a general biquad section with arbitrary poles and zeros Vo1 and Vo3 realize 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 13 Component Values b0 R8 R3 R5 R7C1C2 b1 1 R8 R1R8 R1C1 R6 R4 R7 R b2 8 R6 a0 R8 R2 R3 R7C1C2 a1 1 R1C1 k1 R2 R8C2 R3 R7C1 R k2 7 R8 EECS 247 given ai bi ki C1 C2 and R8 R1 R2 1 a1C1 k1 a0 C 2 R3 1 k1k2 1 a0 C1 R4 1 1 1 k2 a1b2 b1 C1 R5 k1 a0 b0C2 R6 R8 b2 it follows that P R8 R2 R3 R7C1C2 QP P R1C1 R7 k2 R8 Lecture 3 Filters 2010 Page 14 Higher Order Filters in the Integrated Form One way of building higher order filters n 2 is via cascade of 2 nd order biquads 1st order e g Sallen Key or Tow Thomas RC 1st or 2nd order Filter 1 2nd order Filter 2nd order Filter 2 N Nx 2nd order sections Filter order n 2N Cascade of 1st and 2nd order filters Easy to implement Highly sensitive to component mismatch good for low Q filters only For high Q applications good alternative Integrator based ladder filters EECS 247 Lecture 3 Filters 2010 H K Page 15 Integrator 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 filters EECS 247 Lecture 3 Filters 2010 H K Page 16 What 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 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 2010 H K Page 17 What is a Signal Flowgraph SFG Signal flowgraph technique consist of nodes branches Nodes represent variables V I in our case Branches represent transfer functions we will call the transfer function branch multiplication factor or BMF To convert a network to its SFG form KCL KVL is used to derive state …
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