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# Communications Syst ECE 4600

WMU

GPA 3.61

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This 24 page Class Notes was uploaded by Lizeth Hegmann on Wednesday September 30, 2015. The Class Notes belongs to ECE 4600 at Western Michigan University taught by Bradley Bazuin in Fall. Since its upload, it has received 88 views. For similar materials see /class/216787/ece-4600-western-michigan-university in Engineering Electrical & Compu at Western Michigan University.

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Date Created: 09/30/15

Special Note Filter Design Methods Spectral Power Responses For Tjw Tjw expzTjw What is T jw Magnitude must be the same Phase 2 4 w M N 4Tjw i atanl R Z atanl Z atan 2 11 21 2 m4 Pm n1 1 w W 2 4quot w Q M N 4T jw Zatan WR Zatan WJ Zatan wquot2 11 21 2 m4 Pm n1 1 w W Therefore T jw Tjw and AT jw 4Tjw 0 Then what is TjwT jw Tjw eXpzTjwTjw eXp 4Tjw or TJIW T 1W ITOW12 This is a power term notice the square so we use 10log to create decibels It is the same result Note this works for ss too This is the generic form for de ning the magnitude response of a lter Why The poles and zeros are symmetric about the jw axis of the splane Therefore the LHP and RHP elements can be separated into Ts and Ts and guarantee marginal stability Page 1 The Butterworth Lowpass Filter ROW7km MN 7 1an2 1r Swn 171rnj Eunerwunh Huey Characteristic Eq Frequency normalized As AG 3 1 71 3quot 0 an Valkenburg Analog Filter Design Oxford Univ Press1982 Reference ME V ISBN 0195107349 Page 2 Solving for the Butterworth Filter poles Filter injw Tn jwTn jw 1 2 V 1 WO Laplace Tn sTn s 1 2n 1 2n 1 2n 1 woj 1 2quot 1 1quotV0 2V1 Characteristic Eq 1 01 j As A s 0 0 Normalize As A s1 1quotSZ 0 Forn odd AsA s1 s2 1s 1 s 0 Rootsat sz 1expj2m 9 Szexpm n Let As be the LHP poles and A s be the RHP poles For 11 even As A 3 1 32quot 1 jsquot 1 1393quot 0 Roots at 32quot 1 eXpj2m 7r 17239 9 s eXp mj n Let As be the LHP poles and A s be the RHP poles Page 3 Matlab Code BW Filter generation demonstration mo close all clear all Rinl Rloadl Rmatchl PBfregl PiWlog5paceloglOPBfreg72loglOPBfreg21024 I r I I muucr colorseg b ii0 PolesRange6ell for BWnPole5Range iimodii6l denProot571PBfreg 2 BWn zerosl2BWnel 1 Y sortrealdenP denPsortdenPD denpolydenPsortlBWn figurel plotrealdenPimagdenP5printf39cx39color5egii title39Power Magnitude Poles39 grid on hold on num PBfreg BWn zpiab5root5num ppiab5root5den BW5y5tfnumden PiMAG PiPHASEbodeBW5y5PiW figure2 5emilogxPiW dBV5gueezePiMAGcolor5egii title39Power vs Frequency39 xlabel39Freg rad5ec39 ylabel39Magnitude dB39 plotvaxi5 axi5plotvl plotv2 7120 10 figure3 5emilogxPiW 5gueezePiPHASEcolor5egii grid on hold on title39Pha5e v5 Freguency39 xlabel39Freg rad5ec39 ylabel39Pha5e39 axi5plotvl plotv2 emaxPole5Range90 15 pause end Page 4 Magmtude dB Phage Results PoweFMagmtude Po es Power vs Frequency Freq radsec Phase v5 Frequency Freq radsec Page 5 What if we want to change the frequency mom st 1 Hm WM Just change the natural frequency w0 27f0 the center frequency is simply scaled w TnsTVl S 2n 0 n in We 1 S Design approach 1 Determine the order of the lter you want What attenuation do you need at the 10 w0 point There are plenty of curves like those above if the value you need comes before t 10X the cutoff frequency 2 Generate the Butterworth Coefficients on the unit circle for wl 3 Scale the poles by the desired frequency remember that wl is in radianssec therefore multiply by w0 27f0 Page 6 Active Audio Frequency Filters An active lowpass lter implementation of a lSt order Butterworth lter CZ II II R2 WAVAVL R1 Vin W The transfer function for this circuit is Vouts R 2 Vins R1 1 sRZCZ MaxGaz39rzG R1 To tune the circuit Page 7 An alternate approach Vdc Vouts R0 Rb 1 Vins Rb 1 sRlCl R R MaxGainGa b w0 L Rb R101 To tune the circuit Page 8 A Second Order Butterworth Lowpass Filter Let s do the math for the second order system for n2 and w0 1 1 1 zltsgtzlt sgt n 1y 1 12j J39wo WO 1 1 T T M 2 s 1H4 1 1 T d T 2S 32 2s1i an 2 S isZ s1i For 713 T2S 4 1 32 2s1 322gs1 A second order underdamped system with 2 if xE or if XE 0707 i1 2 1 r 31 32 9 4 J5 J5 After frequency scaling 3132 gw0iw0 g2 1W ijw5 How to make a second order LPF for audio Page 9 SallenKey Circuit Lowpass Filter An active lowpass lter implementation of a unity gain Friend Circuit also referred to as a SallenKey circuit as described in Walter G Jung IC OPAmp Cookbook Howard W Sams Co Inc Indianapoli IN 1974 R1 V1 am The transfer function for this circuit is a generic second order f11ter equation is also shown Rum 1 Vout R3 C1C2R1R2 Kw2 Vls 1 1 R3 I 1 322 wsw2 2 S S C2R1 C2R2 C1R2 C1C2R1R2 VoutsR3R4 1 V1s R3 1sC1R2C1R1 R3C2R1s2 C1C2R1R2 Letting C1C2C and R1R2R and 6 MaxGaz39n G R3R4 1 R3 w R Page 10 Function Derivation The circuit derivation assumes a perfect opamp with in nite gain in nite input impedance and zero output impedance nonlimiting power supplies and voltage drops and no frequency response considerations The circuit derivation follows V2 iLsC2 EQV04CZ R1 R2 R1 R2 V2 VP LSC1 R2 R2 R3 Vo R3R4 Vn Letting VpVn VOLLHC1Q R3R4 R2 R2 R31sC1R2 V2 Vo R3R4 V2 iLsC2 ELVOSCZ R1 R2 R1 R21sC1R2 V2 iS39 C1SC2 EVosC2 R1 1sC1R2 R1 R31sC1R2 1 sCl V1 V0 SC2 VosC2 R3R4 R1 1sC1R2 R1 R3 1sC1R2sC1R1sC2R1szC1C2R1R2 V1 0 V0sC2 R3R4 R1 R1 R4 2 1sClR2sC1R1 sC2R1 s C1C2R1R2 R3 R3 V1 V0 R3R4 R1 R1 Page 1 1 V0R3R4 1 R3 Letting Resulting in Note that for a stable system Implying that V1 13C1R2sC1R1 sC2R1s2 C1C2R1R2 R3R4 1 R3 C1C2R1R2 1 L 1 Re3 L 1 C2R139C2R2 C1R2 39C1C2R1R2 amp V1 szs C1C2CandR1R2RandGM R3 1 E CRY V1 2 3 6 1 s 2 CR CR MaxGainszm 1 R3 w CR 3 G CI T 1 Glt3 0SR4lt2R3 Page 12 Multiple Feedback MFB Circuit Lowpass Filter An active lowpass lter implementation of a multiple feedback circuit MFB that is may also be referred to as a derivative of the SallenKey Filter R3 Vout Vdc V Figure 1 Sallen Key Lowpass Filter The transfer function for this circuit is E l yClC2R2R3 V1 R1 32 S391R11R21R31C2R2R3 E j 1 V1 R1 1sC2 RZ R1C2R3C2R2szC1C2R2R3 Resulting in R3 MaxGam G E W 1C2R2R3 Page 13 Function Derivation The circuit derivation assumes a perfect opamp with in nite gain in nite input impedance and zero output impedance nonlimiting power supplies and voltage drops and no frequency response considerations The circuit derivation follows V2 LLLSC1 EE R1 R2 R3 R1 R3 sC2VoQ0 R2 Combining Vo sC2R2iiisClZi2 R1 R2 R3 VOSCZR2R1R2R1R3R2R3sC1R1R2R3E V1 R1R2R3 R3E V0 R1sC2R1R2R1R3R2R3s2 C1C2R1R2R3 R1R3 R1 V0 R3 V1 R1sC2R1R2R1R3R2R3s2C1C2R1R2R3 E l yCLCZRZR3 V1 R1 325391R11R21R31C2R2R3 Resulting in R3 l MaxGam G R1 W C1C2 R2R3 And 4 W391R11R2139R3 Page 14 Higher Order Butterworth Lowpass Filters Take multiple stages and cascade them Remember 39 39 stage in As a ruleofthumb you should select the order for the stages ofyour lter Ifyou look at the output of each stage it will be the product ofthe transfer functions to that location So possible use those with damping factors closest to one before the smaller ones ColnpleeronjugmerPale Pairs Luwest Q gt Highesl 0 Optional Opricnal Figure 3 Building EvenOrder Filters by cascading SecondOrder Stages CumJleeroujugarerPole Pairs I R St 2 39 V39 A T C Real Pole Lowest n gt Highest u Optional Figure 4 Building OddOrder Filters by Cascading SecondOrder Stages and Audi 9 a Single Real Pole 11m Karla Texas Instruments Active LowrPass Filter Design Applicatl on Report SL0A049B September 2002 Note 1 Real elements may not exactly match the values you select 2 Components have a tolerance they are within some 3 rcr u r r What do RF designers do Why might it be different Page 15 Low Pass Filter 3ml Order The classic 3rd order LC Ladder Low Pass Filter L1 L3 Vin n m m n Vout CZ V Figure 2 LC Ladder 3rel Order Low Pass Filter The circuit derivation assumes a source and load resistance The source resistance is placed prior to the input voltage and the load is placed on the output For RSRLR and L1L2L M Vin L CR 2 CL 1s 1s s R 2 2 Page 16 Theoretical Derivation The circuit derivation assumes a source and load resistance The source resistance is placed prior to the input voltage and the load is placed on the output The circuit node equations follow VoutL 1 V2 1 RL sL3 sL3 V2 sC2 1 Vout 1 Vz39n RssLl sL3 sL3 RssLl Solving for V2 and substituting V2 Vow RL Vow M c V0m Wm 4 RL RssLl sL3 sL3 RssLl 1 2m L3 RssLl VWRLsL3RssL1sL3s2 C2L3RssL17RssL1RLVm RLsL3 RssL1sL3s2C2L3RssLl Vout Vout RL RssLlsL3 s sL3RL Vouti sL3RL Vin 7RLsL3RssL1sL3s2 C2L3RssL17RssL1RL Vouti sL3RL Vin RLsL3HZ CzL3RssL1sL3RssL1sL3s2 C2L3RssL1 Vouti RL Vin RLSC2RLRSS2 CzL1RLRssL1sL3s2 C2L3RssL1 Vouti RL Vin RLRssL1L3C2RLRssz C2L1RLC2L3Rss3 C2L3L1 Page 17 1 Vouti RL Vin RLRSJ L1L3 RLRs 2 C2L1RLC2L3Rs 3 C2L3L1 1s 7C2 3 RLRS RLRS RLRS RLRS ForRsRLR Vouti R Vin 2RsL1L3CzRZSZ C2L1L3Rs3 C2L3L1 0r Vouti 1 V 2s gczR s2CzR g s3C2R g R R R R R ForL1L2L Vouti RL Vin RLRss2LCRLRss2CLRLRss3CL2 RL Vout AL FRS 2 V 1s 2 L CRL39RS s2CLH3L RLRS RLRS RLRS ForRsRLRandL1L2L Vouti 2 7 7 2 Vm 1s C szCLsjCIL 1s 1sC Rsz 2 2R R 2 2 Page 18 Theoretical Derivation Pi Filter L2 Vin O m 0 Vout C1 CZ Figure 3 LC Ladder 3rel Order Low Pass Filter The circuit derivation assumes a source and load resistance The source resistance is placed prior to the input voltage and the load is placed on the output ForRsRLR and ClC3C Vour1 1 Vin 2 lsCRlsisz 2R 2 Page 19 Theoretical Derivation Pi Filter The circuit derivation assumes a source and load resistance The source resistance is placed prior to the input voltage and the load is placed on the output The circuit node equations follow Vout L 1 SC3 V2 1 RL 3 L2 sL2 V2 LSCl 1 Vout 1 Vz39n i Rs sL2 sL2 Rs Solving for V2 and substituting RLsL2szL2C3RLJ V2 Vout RL 2 Vow LHCH 1 V0m 1 V L RL Rs sL2 sL2 Rs RLsL2szL2C3RL RssL2szC1L2Rs l l V0ut 2 Vm R s s Vout RL 3 L2 Rs RLRssL2RsL2RLs2L2C3RLRsL2C1RsRLL22 33L22C3RLL22C1Rssquot L22C1C3RLRs RLRS Vm 1 Vout sLZRsRL Rs RsRLsC3RLRsC1RsRLL2 32L2C3RLL2C1Rss3L2C1C3RLRs Vm1 Vout Rs RL Rs Vout RL Vin RsRLsC3RLRsC1RsRLL2 s2 L2C3RLL2C1RSS3 L2C1C3RLRs Page 20 Vout RL Vin 1 RHRL 1sC3C1 RS39 RL L L2 LS2 L2C3RLL2C1RS RSRL 39 RsRL39 RSRL 33L2C1C3 RsRL For RSRLRI Vout 1 Vin 2 J1s RC3RC1E s 2R 1 2 L2C3C1s3 L2C1C35 For C1C3C Vout RL 1 Viquot RHRL 1s2C39RS39RLL2s2L2Cs3L2CC RS39RL RsRL RSRL ForRsRLR and C1C3C Voutl 1 Vm 21s CR Js2LCs3LC2 Vour1 1 Vm 2lsCR1sisz 2R 2 Page 21 15 MHz Low Pass Filter 73911 Order Cuilcra P7LP155 The 7m order elhpucal Lc Ladder Low Pass Frlter Figure 4 Cuilcran LC Ladder Luszss Film Manufacture frequmcy39 response Anenualiun ma Alummian Ida Amnuilian ma ma Frequency MHz Frequency MHz Cnilcmft Lc Ladder Luszss Finer Page 22 Test Analysis A test circuit was built and tested using the network analyzer R is assumed to be 50 ohms L1 and L3 were variable inductors in the range of0578 to 095 uH and C2 was 4 parallel 100pF 101K capacitors or 400 pF Network Analyzer Measurements 5mm 5mm LEYYER LEYYER spans spans am am spans spans msz msz um um pm pm smx DEV smx DEV mm mm mm mm 5mm 5mm LEYYER LEYYER spans spans am am spans spans msz msz um um pm pm smx DEV smx DEV mm mm mm mm Page 23 1 2 3 4 References Walter G Jung IC OPAmp Cookbook Howard W Sams Co Inc Indianapoli IN 1974 ME Van Valkenburg Analog Filter Design Oxford 1982 ISBN 019510734 9 httpwww 39 quot mm lterhtml httn39 fnm ti t n h nntentT phionr quot T 78ampnavSectionaDD notes TI Application Notes Slod006b Sloa093 TI Application Notes on Filtering Active Filter Design Techniques SLOA088 Analysis of the SallenKey Architecture Rev B SLOA024 FilterPro MFB and SallenKey LowPass Filter Design Program SBFAOOlA Active LowPass Filter Design Rev A SLOA049 Using the Texas Instruments Filter Design Database SLOA062 Filter Design in Thirty Seconds SLOA093 Filter Design on a Budget SLOA065 More Filter Design on a Budget SLOA096 Page 24

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