EE40 Lec 13Filters and ResonanceFilters and ResonanceProf. Nathan Cheung10/13/2009Reading: Hambley Chapter 6.6-6.8Slide 1EE40 Fall 2009 Prof. CheungChapter 14.10,14.5Common Filter Transfer Function vs. Freq()Hf()HfHigh PassLow PassFrequencyFrequency()Hf()Hf()HfBdPBdRjt()HfFrequencyBand PassFrequencyBand RejectSlide 2EE40 Fall 2009 Prof. CheungFrequencyFilters• Passive: resonant circuits that contain only passive elements, namely resistors (R), capacitors (C), and inductors (L)• Active: resonant circuits that contain op amps transistors and/or other activeamps, transistors, and/or other active devices, in addition to the passive elementselementsSlide 3EE40 Fall 2009 Prof. CheungFilter Order• The order of a filter is equal to the absolute value of the highest power ofωinabsolute value of the highest power of ωin its transfer functionSlide 4EE40 Fall 2009 Prof. CheungFirst-Order Filter CircuitsLow PassHigh Pass+RLow Pass+RHigh PassL+–VSCRHigh Pass+–VSRLow PassHR= R / (R + jωL)HR= R / (R + 1/jωC)HL= jωL / (R + jωL)HC= (1/jωC) / (R + 1/jωC)Slide 5EE40 Fall 2009 Prof. CheungFirst Order Low Pass FilterSlide 6EE40 Fall 2009 Prof. CheungSecond-Order RLC Filter CircuitsBand PassZ = R + 1/jωC + jωLC+VSRLow PassBandHBP= R / ZHLP= (1/jωC) / Z–VSLHigh PassBand RejectLP(j)HHP= jωL / ZPassHBR= HLP+ HHPSlide 7EE40 Fall 2009 Prof. CheungSecond Order BandPass Filter• Write the expression for VR:• Now find HBP(ω):()RCjLCRCjωωωω+==21)(RBPVVHSlide 8EE40 Fall 2009 Prof. Cheung()RCjLCωω+−1SVPassive Filter: BandpassLinear-Linear plots()RCjωω==)(RVHSlide 9EE40 Fall 2009 Prof. Cheung()RCjLCωωω+−==21)(SBPVHResonance Frequency Defined as frequency when the total impedance is liti(i ii t)purely resistive ( i.e. zero imaginary component)jCjLjR)(Zω−ω+=ωCωThereforeSlide 10EE40 Fall 2009 Prof. CheungThe Quality Factor Q• The Quality Factor (Q) characterizes the degree of selectivity of the circuitdBof the circuit.• It is determined by:•For a bandpass filter:For a bandpass filter:•Notice that Q•Notice that Q depends on the resonant frequencySlide 11EE40 Fall 2009 Prof. CheungLinear ω scaleResonanceBandwidth / fo=1/√2At resonance with large Q,VLand VC>>VRSlide 12EE40 Fall 2009 Prof. CheungLCR* This is a Linear-Linear plotAnother Way to write H(f) (See Hambley text )1()11cRHfL==1111cLRjL jjCjRC Rωωωω++ + +001()1( )csHfjQωωωω=+−RLRC1QooSω=ω=2201()1( )cHfffQ=+LC12o=ω01( )sQff+−LCSlide 13EE40 Fall 2009 Prof. CheungTo Generate the Bode Plots2020011: dominates ( ) ; ( )ccssfLow f H f H ffffjQQff==−210log ( ) 20log : 20 /() 90sccffyHfKfSlope dB decHf=≈+⇒∠=°2200() 901: dominates ( ) , ( )cccssfffHigh f H f H fffQfjQf≈0210log() 20log:20 /() 90scjQfyHfKfSlope dB decHf=≈−⇒−∠=− °() 90cHf∠20:()1,()1 0ccatf f Hf Hf y dB== =⇒=Slide 14EE40 Fall 2009 Prof. Cheung() 0cHf=°Bode Plots+20dB/dec-20dB/dec+20dB/decSlide 15EE40 Fall 2009 Prof. CheungSecond Order LowPass Filter()C1V)(H==ω()()[](){}2/1222LP2SLP1)(MRCjLC1V)(H=ωω+ω−==ω()[](){}2/12020Q//1ωω+ωω−12L1ωSlide 16EE40 Fall 2009 Prof. CheungLC12o=ωRLRC1Qooω=ω=Another Way to write H(f) (See Hambley)1111()111cjC jRCHfLRjL jjC jRC Rωωωω==++ + +20000011 1,;sjC jRC RLLet Hence L define QLC C R R Cωωωωωωω== ==00000 0011,ssLjLj jjQ QRR jRCjRCjQωωωωωωωωωωω ωω== = =−000()1( )scsjQHfjQωωωωω−=+−02022()scfQfHf=Slide 17EE40 Fall 2009 Prof. Cheung2001( )csffQff+−Second Order HighPass Filter()221)(ωωωωRCjLCLC+−−==SLHPVVH()()()[](){}2/12022020//1/)(ωωωωωωωQMjHP+−=SSlide 18EE40 Fall 2009 Prof. Cheung()[](){}00QSecond Order Bandreject Filter)(1)(ωωBPCLBRHVVH−=+=)(1)(ωωBPSBRHVHSlide 19EE40 Fall 2009 Prof. CheungParallel LC Circuit (See Hambley 6.7)IRLC111pZjCRjLωω=++IinRLC0011()111pRjLIZHfRIRjRCjCRjLRjLωωωωω== =++++000000,1ppjLRjLjjRR jQjRCjRC QjL Lωωωωωωωωωωω ω ω ω== = =001()1( )1cpHfffjQff=+−RCLRQpω=ω=22001()1( )cpHfffQff=+−Slide 20EE40 Fall 2009 Prof. CheungActive Filters• Contain less components (no inductors)• Transfer function that is insensitive to component tolerance or load variationscomponent tolerance or load variationsEil djtd•Easily adjusted• Allow a wide range of useful transfer functionsSlide 21EE40 Fall 2009 Prof. CheungSingle Pole Lowpass Filter−=−==fffRCjRωω11||)(outLPVH+ffsCRjRRω1)(ssLPV=Gω1)(HfRG1Slide 22EE40 Fall 2009 Prof. Cheung+=LPLPjGωωω/1)(LPHffLPsfLPCRRG;=−=ωSingle Pole Highpass Filterω−=−==ωffoutPHC/jRRZZVV)(Hω−ssssC/jRZVωω=ωHP/jG)(HfRG1Slide 23EE40 Fall 2009 Prof. Cheungωω+=ωHPHPPH/j1G)(HssHPsfHPCRRG;=−=ωCascaded Active FiltersSlide 24EE40 Fall 2009 Prof. CheungCascaded Active FiltersSlide 25EE40 Fall 2009 Prof. CheungCascaded Active FiltersSlide 26EE40 Fall 2009 Prof. CheungCascaded Active FiltersSlide 27EE40 Fall 2009 Prof. CheungCascaded Active FiltersSlide 28EE40 Fall 2009 Prof. CheungCascaded Active FiltersSlide 29EE40 Fall 2009 Prof. CheungAppendix (For reference only)• See Hambley 14.5The open loop gain of an Op Amp decreases with frequencyThe open loop gain of an Op Amp decreases with frequencyThe unity-gain bandwidthft~ several MHz for typical op ampsSlide 30EE40 Fall 2009 Prof. CheungAppendix (For reference only)Circuit supposed to have a closed lop gain of 10(20db)GfGiven: ft= 4MHz, circuit can operate with f < 400kHzwith f < 400kHzSlide 31EE40 Fall 2009 Prof.
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