EE247 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 filters EECS 247 Lecture 3 Filters 2006 H K Page 1 Summary 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 filters EECS 247 Lecture 3 Filters 2006 H K Page 2 Integrated Filters Implementation of RLC filters in CMOS technologies requires on chip inductors Integrated L 10nH with Q 10 Combined with max cap 10pF LC 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 inductors EECS 247 Lecture 3 Filters 2006 H K Page 3 Filters 2nd Order Transfer Functions Biquads Biquadratic 2nd order transfer function H j 0 1 1 H s 1 s PQP H j 0 s2 P2 H j QP P Biq ua d p ole s for QP 1 2 EECS 247 Lecture 3 Filters s P 2 1 1 4QP 2QP po les are rea l co m p le x o the rwi se 2006 H K Page 4 Biquad Complex Poles QP 1 2 s Complex conjugate poles P 2 1 j 4QP 1 2QP j S plane poles d Distance from origin in s plane 2 d 2 P 1 4QP2 1 2QP P2 EECS 247 Lecture 3 Filters 2006 H K Page 5 s Plane j radius P arccos poles 1 2QP real part P 2Q P EECS 247 Lecture 3 Filters s P 2 1 j 4QP 1 2QP 2006 H K 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 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 biquads EECS 247 Lecture 3 Filters 2006 H K Page 7 Active Biquad Sallen Key Low Pass Filter R1 1 R2 G Vin C2 Vout P QP G H s C1 s s2 PQP P2 1 R1C1R2C2 P 1 1 1 G R1C1 R2C1 R2C2 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 zeros EECS 247 Lecture 3 Filters 2006 H K Page 8 Addition of Imaginary Axis Zeros Sharpen transition band Can notch out interference High pass filter HPF Band reject filter s 1 Z H s K 1 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 2006 H K Page 9 Imaginary Zeros f P 100kHz Zeros substantially sharpen transition band At the expense of reduced stop band attenuation at high frequency 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 Frequency Hz EECS 247 Lecture 3 Filters 7 10 2 2 1 5 1 0 5 0 0 5 Real Axis 1 1 5 2 6 x 10 2006 H K 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 50 4 10 6 105 106 Frequency Hz EECS 247 Lecture 3 Filters 107 4 2 0 2 4 Real Axis 6 5 x10 2006 H K 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 2006 H K 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 EECS 247 Lecture 3 Filters 2006 H K Page 13 Component Values given ai bi ki C1 C2 and R8 b0 R8 R3 R5 R7C1C2 b1 1 R8 R1R8 R1C1 R6 R4 R7 R2 b2 R8 R6 R3 a0 R8 R2 R3 R7C1C2 1 k1k2 R4 a1 1 R1C1 1 1 1 k2 a1b2 b1 C1 R5 k1 a0 b0C2 R6 R8 b2 k1 k2 R2 R8C2 R3 R7C1 R7 R8 R1 1 a1C1 k1 a0 C2 1 a0 C1 it follows that P R8 R2 R3 R7C1C2 QP P R1C1 R7 k2 R8 EECS 247 Lecture 3 Filters 2006 H K Page 14 Higher Order Filters Higher order filters N 2 can be built with cascade of 2nd order biquads e g Sallen Key or Tow Thomas 2nd order Filter 2nd order Filter 1 2 2nd order Filter Nx 2nd order sections Filter with 2N order As will be shown later High Q high order filters built with cascade of 2nd order sections Highly sensitive to component variations Good alternative Integrator based ladder type filters EECS 247 Lecture 3 Filters 2006 H K Page 15 Integrator Based Filters Main building block for this category of filters integrator By using signal flowgraph techniques conventional 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 filters EECS 247 Lecture 3 Filters 2006 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 2006 H K Page 17 What is a Signal Flowgraph SFG Signal flowgraph 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 …
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