EECS 247 Lecture 3: Filters © 2009 H.K. Page 1EE247Administrative• 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 HallEECS 247 Lecture 3: Filters © 2009 H.K. Page 2EE247Course 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 listedEECS 247 Lecture 3: Filters © 2009 H.K. Page 3EE247 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 © 2009 H.K. Page 4Higher-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 ΝCascade 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 2ndorderFilterEECS 247 Lecture 3: Filters © 2009 H.K. Page 5Integrator 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 © 2009 H.K. Page 6What 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 7What 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 State-space description SFG ZZVoinIinIVoIZVin o×=EECS 247 Lecture 3: Filters © 2009 H.K. Page 8Signal Flowgraph (SFG)Examples1SLCircuit State-space description SFG RLRoVC1SCVinVooIVoinIinIinIinIVoVinoIIRVin o1VIin oSL1IVin oSC×=×=×=EECS 247 Lecture 3: Filters © 2009 H.K. Page 9Useful Signal Flowgraph (SFG) Rules1Va2Vba+b1V2Va.b1V2V3Vab1V2Va.V1+b.V1=V2(a+b).V1=V2a.V1=V3 (1)b.V3=V2(2)Substituting for V3 from (1) in (2)(a.b).V1=V2• Two parallel branches can be replaced by a single branch with overall BMF equal to sum of two BMFs• 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 BMFsEECS 247 Lecture 3: Filters © 2009 H.K. Page 10Useful Signal Flowgraph (SFG) Rules• An intermediate node can be multiplied by a factor (k). BMFs for incomingbranches have to be multiplied by k and outgoing branches divided by k3Vab1V2Vk.ab/k1V2V3.Vka.V1=V3(1)b.V3=V2(2)Multiply both sides of (1) by k(a.k) . V1= k.V3(1)Divide & multiply left side of (2) by k(b/k) . k.V3 = V2(2)EECS 247 Lecture 3: Filters © 2009 H.K. Page 11Useful Signal Flowgraph (SFG) RuleshiV2VaoVbg-b3VhiV2Va/(1+b)oVbg3V3VciV2VaoVbd-bciV2VaoV-1-b1d3V• Simplifications can often be achieved by shifting or eliminating nodes• Example: eliminating node V4• A self-loop branch with BMF y can be eliminated by multiplying the BMFof incoming branches by 1/(1-y)V4EECS 247 Lecture 3: Filters © 2009 H.K. Page 12Integrator Based Filters1st Order LPF• Conversion of simple lowpass RC filter to integrator-based type by using signal flowgraph techniquesinV1osCV1R=+oVRsCinVEECS 247 Lecture 3: Filters © 2009 H.K. Page 13What is an Integrator?Example: Single-Ended Opamp-RC Integrator∫inVoVCinV-+R-Note: Practical integrator in CMOS technology has input & output both in the form of voltage and not current Consideration for SFG derivationoVRCτ=a ≈ ∞insC ,ooin o inV11VVV ,V VdtR sRC RC= −= =−× =−∫EECS 247 Lecture 3: Filters © 2009 H.K. Page 14Integrator Based Filters1st Order LPF1. Start from circuit prototype-Name voltages & currents for allcomponents2. Use KCL & KVL to derive state space description in such a way to have BMFs in the integratorform: Capacitor voltage expressed as function of its current VCap.=f(ICap.) Inductor current as a function of its voltage
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