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1 EE47 Administrtive Finl exm group hs een chnged to group 3 Finl exm new dte/time Dec. 3 th, 5pm to 8pm Homework # hs een posted on course wesite nd is due Sept th (next Thurs.) Sumissions cn e on pper or vi emil Pper sumission in clss or during sme dy office hours Plese show your derivtions nd explin your work EES 47 Lecture 3: Filters 8 H.K. Pge EE47 Lecture 3 Active Filters Active iquds Sllen Key & TowThoms Integrtorsed filters Signl flowgrph concept First order integrtorsed filter Second order integrtorsed filter & iquds High order & high Q filters scded iquds & first order filters scded iqud sensitivity to component mismtch Ldder type filters EES 47 Lecture 3: Filters 8 H.K. Pge
2 Filters nd Order Trnsfer Functions (Biquds) Biqudrtic ( nd order) trnsfer function: H(s) s s ω Q ω H( j ω ) P P P ω ω ω P ωpqp H ( jω) ω H ( jω) H ( jω) ω ω ωp Q P ωp Biqud s 4Q ± P QP Note: for Q P poles re rel, complex otherwise EES 47 Lecture 3: Filters 8 H.K. Pge 3 Q P > omplex conjugte poles: splne jω ω s P j 4QP ± QP rccos Q P poles σ rdius ω P rel prt ω P Q P EES 47 Lecture 3: Filters 8 H.K. Pge 4
3 Implementtion of Biquds Pssive : only rel poles cn t implement complex conjugte poles Terminted L Low power, since it is pssive Only fundmentl noise sources lod nd source resistnce As previously nlyzed, not fesile in the monolithic form for f < few s of MHz Active Biquds Mny topologies cn e found in filter textooks! Widely used topologies: Singleopmp iqud: SllenKey Multiopmp iqud: TowThoms Integrtor sed iquds EES 47 Lecture 3: Filters 8 H.K. Pge 5 Active Biqud SllenKey LowPss Filter G H() s s s ω PQP ωp G ω V V P in out ω Q P P G Single gin element n e implemented oth in discrete & monolithic form Prsitic sensitive Versions for LPF, HPF, BP, Advntge: Only one opmp used to otin poles Disdvntge: Sensitive to prsitic ll pole no zeros EES 47 Lecture 3: Filters 8 H.K. Pge 6
4 Addition of Imginry Axis Zeros Shrpen trnsition nd n notch out interference Highpss filter (HPF) Bndreject filter s ω H(s) K Z s s ωpq P ω P H( j ω ) K ω ωp H( j ω ) K ω ω Z Note: Alwys represent trnsfer functions s product of gin term, poles, nd zeros (pirs if complex). Then ll coefficients hve physicl mening, nd redily identifile units. EES 47 Lecture 3: Filters 8 H.K. Pge 7 Mgnitude [db] Imginry Zeros Zeros sustntilly shrpen trnsition nd At the expense of reduced stopnd ttenution t high frequency 3 4 With zeros No zeros Frequency [Hz] Img Axis x f Q f P P Z khz 3 f el Axis x 6 P PoleZero Mp EES 47 Lecture 3: Filters 8 H.K. Pge 8
5 Moving the Zeros fp khz QP fz fp Bnd reject filter x 5 6 PoleZero Mp Mgnitude [db] Frequency [Hz] Img Axis el Axis x 5 EES 47 Lecture 3: Filters 8 H.K. Pge 9 TowThoms Active Biqud Prsitic insensitive Multiple outputs ef: P. E. Fleischer nd J. Tow, Design Formuls for iqud ctive filters using three opertionl mplifiers, Proc. IEEE, vol. 6, pp. 663, My 973. EES 47 Lecture 3: Filters 8 H.K. Pge
6 EES 47 Lecture 3: Filters 8 H.K. Pge Frequency esponse ( ) ( ) ( ) ( ) 3 s s s k V V s s s s V V s s s k V V in o in o in o implements generl iqud section with ritrry poles nd zeros nd 3 relize the sme poles ut re limited to t most one finite zero EES 47 Lecture 3: Filters 8 H.K. Pge omponent Vlues k k k k k k k k it follows tht Q P P P ω ω zeros poles desired from the exmple for k i i i / nd,,,, given 8
7 HigherOrder Filters in the Integrted Form One wy of uilding higherorder filters (n>) is vi cscde of nd order iquds, e.g. SllenKey,or TowThoms nd order Filter nd order Filter Nx nd order sections Filter order: nn nd order Filter Ν scde of nd order iquds: Esy to implement Highly sensitive to component mismtch good for low Q filters only Good lterntive: Integrtorsed ldder type filters EES 47 Lecture 3: Filters 8 H.K. Pge 3 Integrtor Bsed Filters Min uilding lock for this ctegory of filters Integrtor By using signl flowgrph techniques onventionl L filter topologies cn e converted to integrtor sed type filters Next few pges: Introduction to signl flowgrph techniques st order integrtor sed filter nd order integrtor sed filter High order nd high Q filters EES 47 Lecture 3: Filters 8 H.K. Pge 4
8 Wht is Signl Flowgrph (SFG)? SFG Topologicl network representtion consisting of nodes & rnches used to convert one form of network to more suitle form (e.g. pssive L filters to integrtor sed filters) Any network descried y set of liner differentil equtions cn e expressed in SFG form For given network, mny different SFGs exists hoice of prticulr SFG is sed on prcticl considertions such s type of ville components ef: W.Heinlein & W. Holmes, Active Filters for Integrted ircuits, Prentice Hll, hp. 8, 974. EES 47 Lecture 3: Filters 8 H.K. Pge 5 Wht is Signl Flowgrph (SFG)? Signl flowgrph technique consist of nodes & rnches: Nodes represent vriles (V & I in our cse) Brnches represent trnsfer functions (we will cll the trnsfer function rnch multipliction fctor or BMF) To convert network to its SFG form, KL & KVL is used to derive stte spce description Simple exmple: ircuit Sttespce description SFG I in I Z V Z in o I in Z EES 47 Lecture 3: Filters 8 H.K. Pge 6
9 Signl Flowgrph (SFG) Exmples ircuit Sttespce description SFG Iin Iin Vo I in I o L Io SL SL I o I in Iin Vo S I in S EES 47 Lecture 3: Filters 8 H.K. Pge 7 V Useful Signl Flowgrph (SFG) ules Two prllel rnches cn e replced y single rnch with overll BMF equl to sum of two BMFs V V V 3 V V V.V.V V ().V V A node with only one incoming rnch & one outgoing rnch cn e eliminted & replced y single rnch with BMF equl to the product of the two BMFs.V V3 ().V 3 V () Sustituting for V3 from () in () (.).V V. V V EES 47 Lecture 3: Filters 8 H.K. Pge 8
10 Useful Signl Flowgrph (SFG) ules An intermedite node cn e multiplied y fctor (k). BMFs for incoming rnches hve to e multiplied y k nd outgoing rnches divided y k V V V k. /k V V 3 k.v 3.V V 3 ().V 3 V () Multiply oth sides of () y k (.k). V k.v 3 () Divide & multiply left side of () y k (/k). k.v 3 V () EES 47 Lecture 3: Filters 8 H.K. Pge 9 Useful Signl Flowgrph (SFG) ules Simplifictions cn often e chieved y shifting or eliminting nodes Exmple: eliminting node V 4 V i c V V 4 V 3 d V i c V V 3 d A selfloop rnch with BMF y cn e eliminted y multiplying the BMF of incoming rnches y /(y) V i h V V 3 g V i h V /() V 3 g EES 47 Lecture 3: Filters 8 H.K. Pge
11 Integrtor Bsed Filters st Order LPF onversion of simple lowpss filter to integrtorsed type y using signl flowgrph techniques s Vo s EES 47 Lecture 3: Filters 8 H.K. Pge Wht is n Integrtor? Exmple: SingleEnded Opmp Integrtor τ Vo, Vo dt s Note: Prcticl integrtor in MOS technology hs input & output oth in the form of voltge nd not current onsidertion for SFG derivtion EES 47 Lecture 3: Filters 8 H.K. Pge
12 Integrtor Bsed Filters st Order LPF. Strt from circuit prototype Nme voltges & currents for ll components V s I I V. Use KL & KVL to derive stte spce description in such wy to hve BMFs in the integrtor form: pcitor voltge expressed s function of its current V p. f(i p. ) Inductor current s function of its voltge I Ind. f(v Ind. ) 3. Use stte spce description to drw signl flowgrph (SFG) (see next pge) EES 47 Lecture 3: Filters 8 H.K. Pge 3 V V V I s Vo V I V s I I Integrtor Bsed Filters First Order LPF Integrtor form All voltges & currents nodes of SFG Voltge nodes on top, corresponding current nodes elow ech voltge node V I s s I I SFG V V V s I EES 47 Lecture 3: Filters 8 H.K. Pge 4
13 Normlize Since integrtors re the min uilding locks require in & out signls in the form of voltge (not current) onvert ll currents to voltges y multiplying current nodes y scling resistnce orresponding BMFs should then e scled ccordingly V Vo V I s I Vo s I I V Vo I V s I Vo s I I I x Vx V Vo V V s V Vo s V V EES 47 Lecture 3: Filters 8 H.K. Pge 5 st Order Lowpss Filter SGF Normlize V s s V s s V s s I I I I V V EES 47 Lecture 3: Filters 8 H.K. Pge 6
14 st Order Lowpss Filter SGF Synthesis V s V V s hoosing s V V τ s V τ s, onsolidte two rnches V τ s V EES 47 Lecture 3: Filters 8 H.K. Pge 7 First Order Integrtor Bsed Filter V τ s V H ( s) τ s EES 47 Lecture 3: Filters 8 H.K. Pge 8
15 st Order Filter Built with Opmp Integrtor Singleended Opmp integrtor hs sign inversion from input to output onvert SFG ccordingly y modifying BMF EES 47 Lecture 3: Filters 8 H.K. Pge 9 st Order Filter Built with Opmp Integrtor To void requiring n dditionl opmp to perform summtion t the input node: EES 47 Lecture 3: Filters 8 H.K. Pge 3
16 st Order Filter Built with Opmp Integrtor (continued) V V s o in EES 47 Lecture 3: Filters 8 H.K. Pge 3 k vo m Opmp st Order Filter Noise Identify noise sources (here it is resistors & opmp) Find trnsfer function from ech noise source to the output (opmp noise next pge) H m( f ) S m( f ) df S i( f ) Noise spectrl density of m noise source H(f) H(f) ( π f) v v 4KTΔf n n th v n v n kt vo α α Typiclly, α increses s filter order increses EES 47 Lecture 3: Filters 8 H.K. Pge 3
17 Opmp Filter Noise Opmp ontriution So fr only the fundmentl noise sources re considered In relity, noise ssocited with the opmp increses the overll noise For welldesigned filter opmp is designed such tht noise contriution of opmp is negligile compred to other noise sources v n v n vopmp Vo The ndwidth of the opmp ffects the opmp noise contriution to the totl noise EES 47 Lecture 3: Filters 8 H.K. Pge 33 Stte spce description: V VL V Vo I V s V Integrtor form I VL IL sl I Iin I IL Integrtor Bsed Filter nd Order L Filter I in V I V L L V I L SFG V V s I V L sl Drw signl flowgrph (SFG) I I in I I L EES 47 Lecture 3: Filters 8 H.K. Pge 34
18 nd Order L Filter SGF Normlize onvert currents to voltges y multiplying ll current nodes y the scling resistnce V V s V L sl I x Vx V V s sl I I in I I L V V V 3 EES 47 Lecture 3: Filters 8 H.K. Pge 35 nd Order L Filter SGF Synthesis V V V s V 3 sl τ sτ sτ τ L EES 47 Lecture 3: Filters 8 H.K. Pge 36
19 Second Order Integrtor Bsed Filter Filter Mgnitude esponse V BP Mgnitude (db) 5 5 sτ sτ V HP VLP. Normlized Frequency [Hz] EES 47 Lecture 3: Filters 8 H.K. Pge 37 Second Order Integrtor Bsed Filter τ s βτ VBP ττ s s VLP ττ s βτs VHP ττ s ττ s βτs τ L β τ ω τ τ Q β τ τ L From mtching pointof viewdesirle: τ Q τ V BP sτ sτ V HP VLP EES 47 Lecture 3: Filters 8 H.K. Pge 38
20 Second Order Bndpss Filter Noise k vo m H m( f ) S m( f )df Find trnsfer function of ech noise source to the output Integrte contriution of ll noise sources Here it is ssumed tht opmps re noise free (not usully the cse!) vn vn 4KTdf V BP v n sτ sτ v n vo kt Q α Typiclly, α increses s filter order increses Note the noise power is directly proportion to Q EES 47 Lecture 3: Filters 8 H.K. Pge 39 Second Order Integrtor Bsed Filter Biqud By comining outputs cn generte generl iqud function: V ττ s τs 3 ττ s βτs 3 V BP jω splne sτ sτ σ V HP V LP EES 47 Lecture 3: Filters 8 H.K. Pge 4
21 Summry Integrtor Bsed Monolithic Filters Signl flowgrph techniques utilized to convert L networks to integrtor sed ctive filters Ech rective element (L& ) replced y n integrtor Fundmentl noise limittion determined y integrting cpcitor vlue: For lowpss filter: Bndpss filter: vo vo kt α kt α Q where α is function of filter order nd topology EES 47 Lecture 3: Filters 8 H.K. Pge 4 Higher Order Filters How do we uild higher order filters? scde of iquds nd st order sections Ech complex conjugte pole uilt with iqud nd rel pole with st order section Esy to implement In the cse of high order high Q filters highly sensitive to component mismtch Direct conversion of high order ldder type L filters SFG techniques used to perform exct conversion of ldder type filters to integrtor sed filters More complicted conversion process Much less sensitive to component mismtch compred to cscde of iquds EES 47 Lecture 3: Filters 8 H.K. Pge 4
22 Higher Order Filters scde of Biquds Exmple: LPF filter for DMA cell phone send receiver LPF with fpss 65 khz pss. db fstop 75 khz stop 45 db Assumption: n compenste for phse distortion in the digitl domin Mtl used to find minimum order required 7th order Elliptic Filter Implementtion with cscded Biquds Gol: Mximize dynmic rnge Pir poles nd zeros In the cscde chin plce lowest Q poles first nd progress to higher Q poles moving towrds the output node EES 47 Lecture 3: Filters 8 H.K. Pge 43 Overll Filter Frequency esponse Bode Digrm Phse (deg) Mgnitude (db) kHz MHz Frequency [Hz] 3MHz Mg. (db). EES 47 Lecture 3: Filters 8 H.K. Pge 44
23 PoleZero Mp (pzmp in Mtl) Img Axis X splne PoleZero Mp.5.5 el Axis x 7 Q pole f pole [khz] f zero [khz] EES 47 Lecture 3: Filters 8 H.K. Pge 45 DMA Filter Built with scde of st nd nd Order Sections st order Filter Biqud Biqud3 Biqud4 st order filter implements the single rel pole Ech iqud implements pir of complex conjugte poles nd pir of imginry xis zeros EES 47 Lecture 3: Filters 8 H.K. Pge 46
24 Biqud esponse.5 LPF Biqud Biqud Biqud EES 47 Lecture 3: Filters 8 H.K. Pge 47 Individul Biqud Mgnitude esponse Mgnitude (db) 3 LPF Biqud 4 Biqud 3 Biqud Frequency [Hz] EES 47 Lecture 3: Filters 8 H.K. Pge 48
25 Mgnitude (db) Mgnitude (db) khz Intermedite Outputs LPF Mgnitude (db) LPF Biqud LPF Biquds,3 LPF Biquds,3,4 Biquds,, 3, & Mgnitude (db) khz MHz 6 MHz khz khz MHz MHz Frequency [Hz] Frequency [Hz] EES 47 Lecture 3: Filters 8 H.K. Pge 49 Sensitivity to eltive omponent Mismtch omponent vrition in Biqud 4 reltive to the rest (highest Q poles): Increse ω p4 y %.db Decrese ω z4 y % Mgnitude (db) 3 3dB 4 5 khz 6kHz Frequency [Hz] MHz High Q poles High sensitivity in Biqud reliztions EES 47 Lecture 3: Filters 8 H.K. Pge 5
26 High Q & High Order Filters scde of iquds Highly sensitive to component mismtch not suitle for implementtion of high Q & high order filters scde of iquds only used in cses where required Q for ll iquds <4 (e.g. filters for disk drives) Ldder type filters more pproprite for high Q & high order filters (next topic) Will show lter Less sensitive to component mismtch EES 47 Lecture 3: Filters 8 H.K. Pge 5 Ldder Type Filters For simplicity, will strt with ll pole (no finite zero) ldder type filters Strt with L ldder type nd find vlues for Ls & s onvert to integrtor sed form exmple shown Next will ttend to high order ldder type filters incorporting zeros Implement the sme 7th order elliptic filter in the form of ldder L with zeros Find level of sensitivity to component mismtch ompre with cscde of iquds onvert to integrtor sed form utilizing SFG techniques Effect of integrtor nonidelities on filter frequency chrcteristics EES 47 Lecture 3: Filters 8 H.K. Pge 5
27 L Ldder Filters s L 3 L4 5 L Mde of resistors, inductors, nd cpcitors Douly terminted or singly terminted (with or w/o L ) Douly terminted L ldder filters Lowest sensitivity to component mismtch EES 47 Lecture 3: Filters 8 H.K. Pge 53 L Ldder Filters s L 3 L4 5 L First step in the design process find vlues for Ls & s Filter tles A. Zverev, Hndook of filter synthesis, Wiley, 967. A. B. Willims nd F. J. Tylor, Electronic filter design, 3 rd edition, McGrw Hill, 995. AD tools Mtl Spice EES 47 Lecture 3: Filters 8 H.K. Pge 54
28 L Ldder Filter Design Exmple Design LPF with mximlly flt pssnd: f3db MHz, fstop MHz s fstop Mximlly flt pssnd Butterworth Find minimum filter order : Here stndrd grphs from filter ooks re used fstop / f3db s >7dB Minimum Filter Order 5th order Butterworth Pssnd Attenution 3dB Νοrmlized ω From: Willims nd Tylor, p. 37 Stopnd Attenution 3dB EES 47 Lecture 3: Filters 8 H.K. Pge 55 L Ldder Filter Design Exmple Find vlues for L & from Tle: Note L & vlues normlized to ω 3dB Denormliztion: Multiply ll L Norm, Norm y: L r /ω 3dB r /(Xω 3dB ) is the vlue of the source nd termintion resistor (choose oth Ω for now) Then: L L r xl Norm r x Norm From: Willims nd Tylor, p..3 EES 47 Lecture 3: Filters 8 H.K. Pge 56
29 L Ldder Filter Design Exmple Find vlues for L & from Tle: Normlized vlues: Norm 5 Norm.68 3 Norm. L Norm L4 Norm.68 Denormliztion: Since ω 3dB πxmhz L r /ω 3dB 5.9 nh r /(Xω 3dB ) 5.9 nf nF, 33.83nF LL45.75nH From: Willims nd Tylor, p..3 EES 47 Lecture 3: Filters 8 H.K. Pge 57 Mgnitude esponse Simultion sohm L5.75nH 9.836nF L45.75nH nF nF LOhm 5 SPIE simultion esults 6 db pssnd ttenution due to doule termintion Mgnitude (db) 3 4 3dB 5 3 Frequency [MHz] EES 47 Lecture 3: Filters 8 H.K. Pge 58
EE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form- example shown
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