EE247 Administrative Due to office hour conflict with EE142 class New office hours Tues 4 to 5pm same as before Wed 10 30 to 11 30am new Thurs no office hours Office hours held 567 Cory Hall EECS 247 Lecture 3 Filters 2009 H K Page 1 EE247 Course Reading Material Note that the class website includes a section named Reading Material including List of books on reserve in the library List of articles as pdf files Articles for Filters Articles for Nyquist Rate Data Converters Articles for Oversampled Data Converters You will be asked to read some of the articles and answer questions If you plan to embark on a career in Mixed Signal Circuit Design consider reading all the publications listed EECS 247 Lecture 3 Filters 2009 H K Page 2 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 2009 H K Page 3 Higher Order Filters in the Integrated Form One way of building higher order filters n 2 is via cascade of 2nd 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 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 2009 H K Page 4 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 2009 H K Page 5 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 2009 H K Page 6 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 space description Simple example Circuit I in State space description Vo Z EECS 247 Iin Z Vo SFG I in Vo Z Lecture 3 Filters 2009 H K Page 7 Signal Flowgraph SFG Examples Circuit I in Vin I in EECS 247 State space description Vo R Io L Vo C Iin R Vo Vin Iin 1 SL 1 SC Io Vo Lecture 3 Filters SFG I in Vo R Vin 1 Io SL Vo I in 1 SC 2009 H K Page 8 Useful Signal Flowgraph SFG Rules Two parallel branches can be replaced by a single branch with overall BMF equal to sum of two BMFs b V1 a a b V1 V2 a V1 b V1 V2 V2 a b V1 V2 A node with only one incoming branch one outgoing branch can be eliminated replaced by a single branch with BMF equal to the product of the two BMFs a V1 V3 b V2 V1 a V1 V3 1 b V3 V2 2 Substituting for V3 from 1 in 2 EECS 247 a b V2 a b V1 V2 Lecture 3 Filters 2009 H K Page 9 Useful Signal Flowgraph SFG Rules An intermediate node can be multiplied by a factor k BMFs for incoming branches have to be multiplied by k and outgoing branches divided by k V1 a V1 V3 b V3 V2 a b V2 V1 V3 k a b k V2 k V3 1 2 Multiply both sides of 1 by k a k V1 k V3 1 Divide multiply left side of 2 by k 2 b k k V3 V2 EECS 247 Lecture 3 Filters 2009 H K Page 10 Useful Signal Flowgraph SFG Rules Simplifications can often be achieved by shifting or eliminating nodes Example eliminating node V4 V4 1 Vi c a V2 b b 1 d Vo V3 Vi b c V2 Vo d a V3 A self loop branch with BMF y can be eliminated by multiplying the BMF of incoming branches by 1 1 y b Vi b b h V2 g a Vo V3 EECS 247 Vi g Vo h V2 a 1 b Lecture 3 Filters V3 2009 H K Page 11 Integrator Based Filters 1st Order LPF Conversion of simple lowpass RC filter to integratorbased type by using signal flowgraph techniques Vo Rs Vin EECS 247 Vo 1 s Vin 1 R C C Lecture 3 Filters 2009 H K Page 12 What is an Integrator Example Single Ended Opamp RC Integrator C Vin a R Vo Vin Vo RC Vin Vo s C R Vo Vin 1 sRC Vo 1 RC V in dt Note Practical integrator in CMOS technology has input output both in the form of voltage and not current Consideration for SFG derivation EECS 247 Lecture 3 Filters 2009 H K Page 13 Integrator Based Filters 1st Order LPF 1 Start from circuit prototypeName voltages currents for all components V1 Rs I 1 Vin C I2 Vo VC 2 Use KCL KVL to derive state space description in such a way to have BMFs in the integrator form Capacitor voltage expressed as function of its current VCap f ICap Inductor current as a function of its voltage IInd f VInd 3 Use state space description to draw signal flowgraph SFG see next page EECS 247 Lecture 3 Filters 2009 H K Page 14 Integrator Based Filters First Order LPF V1 Vin VC 1 VC I2 sC Vo VC 1 I1 V1 Rs I2 I1 Inte V1 gra tor form Vo Rs I 1 C I2 Vin VC SFG Vin All voltages currents 1 1 Rs nodes of SFG Voltage nodes on top corresponding current nodes below each voltage node EECS 247 VC 1 Vo 1 1 V1 I1 sC 1 Lecture 3 Filters I2 2009 H K Page 15 Normalize Since integrators are the main building blocks in the form of voltage not current require in out signals Convert all currents to voltages by multiplying current nodes by a scaling resistance R Corresponding BMFs should then be scaled accordingly V1 Vi n Vo V I1 1 Rs Vo I2 sC I2 I1 EECS 247 …
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