EECS 247 Lecture 4: Active Filters © 2010 H.K. Page 1EE247 Lecture 4• Active ladder type filters –For simplicity, will start with all pole ladder type filters• Convert to integrator based form- example shown–Then will attend to high order ladder type filters incorporating zeros• Implement the same 7th order elliptic filter in the form of ladder RLC with zeros– Find level of sensitivity to component mismatch – Compare with cascade of biquads• Convert to integrator based form utilizing SFG techniques–Effect of integrator non-Idealities on filter frequency characteristicsEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 2Summary Lecture 3• Active Filters– Active biquads• 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 4: Active Filters © 2010 H.K. Page 3RLC Ladder FiltersExample: 5thOrder Lowpass Filter• Made of resistors, inductors, and capacitors• Doubly terminated or singly terminated (with or w/o RL)RsC1C3L2C5L4inVRLoVDoubly terminated LC ladder filters Lowest sensitivity to component mismatchEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 4LC Ladder Filters• First step in the design process is to find values for Ls and Cs based on specifications:– Filter graphs & tables found in:• A. Zverev, Handbook of filter synthesis, Wiley, 1967.• A. B. Williams and F. J. Taylor, Electronic filter design, 3rdedition, McGraw-Hill, 1995.– CAD tools• Matlab, • Agilent ADS (includes Filter package does the job of the tables)• SpiceRsC1C3L2C5L4inVRLoVEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 5LC Ladder Filter Design ExampleDesign a LPF with maximally flat passband:f-3dB = 10MHz, fstop = 20MHzRs >27dB @ fstop• Maximally flat passband ButterworthFrom: Williams and Taylor, p. 2-37Stopband AttenuationNormalized w• Find minimum filter order :• Here standard graphs from filter books are usedfstop / f-3dB = 2Rs >27dBMinimum Filter Orderc5th order Butterworth1-3dB2-30dBPassband Attenuation EECS 247 Lecture 4: Active Filters © 2010 H.K. Page 6LC Ladder Filter Design ExampleFrom: Williams and Taylor, p. 11.3Find values for L & C from Table:Note L &C values normalized tow-3dB=1Denormalization:Multiply all LNorm, CNormby:Lr= R/w-3dBCr= 1/(RXw-3dB)R is the value of the source and termination resistor (choose both 1W for now)Then: L= LrxLNormC= CrxCNormEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 7LC Ladder Filter Design ExampleFrom: Williams and Taylor, p. 11.3Find values for L & C from Table:Normalized values:C1Norm=C5Norm=0.618C3Norm= 2.0L2Norm= L4Norm=1.618Denormalization:Since w-3dB =2px10MHzLr= R/w-3dB= 15.9 nHCr= 1/(RXw-3dB)= 15.9 nFR =1cC1=C5=9.836nF, C3=31.83nFcL2=L4=25.75nHEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 8Last Lecture:Example: 5thOrder Butterworth FilterRs=1WC19.836nFC331.83nFL2=25.75nHC59.836nFL4=25.75nHinVRL=1WoVSpecifications:f-3dB = 10MHz, fstop = 20MHzRs >27dBUsed filter tables to obtain Ls & CsFrequency [MHz]Magnitude (dB)01020 30-50-40-30-20-10-50-6 dB passband attenuationdue to double termination30dBSPICE simulation ResultsEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 9Low-Pass RLC Ladder FilterConversion to Integrator Based Active Filter1I2VRsC1C3L2C5L4inVRL4V6V3I5I2I4I6I7I• To convert RLC ladder prototype to integrator based filer:Use Signal Flowgraph techniqueName currents and voltages for all componentsUse KCL & KVL to derive equationsMake sure reactive elements expressed as 1/s term V(C ) =f(I) & I(L)=f(V)Use state-space description to derive the SFGModify & simply the SFG for implementation with integrators e.g. convert all current nodes to voltage1V 3V 5V 1sCoVEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 10Low-Pass RLC Ladder FilterConversion to Integrator Based Active Filter1I2VRsC1C3L2C5L4inVRL4V6V3I5I2I4I6I• Use KCL & KVL to derive equations:1V 3V 5V 1sCoV1 in 21313256574I2V V V , V , V V V2 3 2 4sC1II46V , V V V , V V V4 5 4 6 6 o 6sC sC35VVI , I I I , I2 1 3RssLVVI I I , I , I I I , I4 3 5 6 5 7sL RL 7IEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 11Low-Pass RLC Ladder FilterSignal Flowgraph1sCSFG1Rs11sC2I1I2VinV1111VoV11131sC51sC21sL41sL1RL11111113V4V5V6V3I5I4I6I7I1 in 21313256574I2V V V , V , V V V2 3 2 4sC1II46V , V V V , V V V4 5 4 6 6 o 6sC sC35VVI , I I I , I2 1 3RssLVVI I I , I , I I I , I4 3 5 6 5 7sL RL EECS 247 Lecture 4: Active Filters © 2010 H.K. Page 12Low-Pass RLC Ladder FilterNormalize1sC11*RRs*11sCR'1V2VinV111VoV11*2RsL11111113V4V5V6V'3V'2V'4V'5V'6V'7V*31sCR*4RsL*51sCR*RRL1Rs11sC2I1I2VinV1111VoV11131sC51sC21sL41sL1RL11111113V4V5V6V3I5I4I6I7IEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 13Low-Pass RLC Ladder FilterSynthesize1sC111*RRs*11sCR'1V2VinV111VoV11*2RsL11111113V4V5V6V'3V'2V'4V'5V'6V'7V*31sCR*4RsL*51sCR*RRLinV1+ -- +- ++ -+ -*RRs*RRL21st31st41st51st11stoV2V4V6V'3V'5VEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 14Low-Pass RLC Ladder FilterIntegrator Based Implementation* * * * *2**L L4C C C C.R , .R , .R , .R , C .R11 2 2 3 3 4 4 5 5RRt t t t t Main building block: IntegratorLet us start to build the filter with RC& Opamp type integratorinV1+ -- +- ++ -+ -*RRs*RRL21st31st41st51st11stoV2V4V6V'3V'5VEECS 247 Lecture 4: Active Filters © 2010 H.K. Page 15Opamp-RC Integrator 12o o in1 in1in2 in21V V V VsR CI1VVsR CI oVCIin1V-+R1Note: Implementation with single-ended integrator requires extra circuitry for sign inversion whereas in differential case both signal polarities are availableDifferential topologies additional advantage of immunity to parasitic signal injection & superior
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