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1 EE47 Administrtive Homework # will e posted on EE47 site nd is due Sept. 9 th Office hours Cory Hll: Tues. nd Thurs.: 4 to 5pm EECS 47 Lecture 3: Filters H.K. ge Active Filters EE47 Lecture 3 Active iquds How to uild higher order filters? Integrtorsed filters Signl flowgrph concept First order integrtorsed filter Second order integrtorsed filter & iquds High order & high Q filters Cscded iquds & first order filters Cscded iqud sensitivity to component mismtch Ldder type filters EECS 47 Lecture 3: Filters H.K. ge
2 EECS 47 Lecture 3: Filters ge 3 Filters nd Order Trnsfer Functions (Biquds) Biqudrtic ( nd order) trnsfer function: Q j H j H j H ) ( ) ( ) ( H( s ) s s Q H( j ) Q Biqud s 4Q Q Note: for Q poles re rel, comple x otherwise EECS 47 Lecture 3: Filters ge 4 Biqud Complex oles Distnce from origin in splne: 4 Q Q d Complex conjugte poles: s 4 Q j Q Q poles j d Splne
3 slne j rdius rccos Q poles rel prt Q s j 4Q Q EECS 47 Lecture 3: Filters ge 5 Exmple nd Order Butterworth j rccos Q Since nd order Butterworth: 45 deg ree Cos Q p 4 EECS 47 Lecture 3: Filters ge 6
4 Implementtion of Biquds ssive C: only rel poles cn t implement complex conjugte poles Terminted LC 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 <35MHz Active Biquds Mny topologies cn e found in filter textooks! Widely used topologies: Singleopmp iqud: SllenKey Multiopmp iqud: TowThoms Integrtor sed iquds EECS 47 Lecture 3: Filters ge 7 Active Biqud SllenKey Lowss Filter G H() s C s s Q G V V in C out C C Q G Single gin element C C C Cn e implemented oth in discrete & monolithic form rsitic sensitive Versions for LF, HF, B, Advntge: Only one opmp used Disdvntge: Sensitive to prsitic ll pole no finite zeros EECS 47 Lecture 3: Filters ge 8
5 Mgnitude [db] Img Axis Addition of Imginry Axis Zeros Shrpen trnsition nd Cn notch out interference Bndreject filter Highpss filter (HF) s H( s ) K Z s s Q H( j ) K 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. EECS 47 Lecture 3: Filters ge 9 Imginry Zeros Zeros sustntilly shrpen trnsition nd At the expense of reduced stopnd ttenution t high frequencies 3 4 With zeros No zeros Frequency [Hz] x f Q f Z khz 3 f el Axis x 6 olezero Mp EECS 47 Lecture 3: Filters ge
6 Img Axis Moving the Zeros f Q khz f Z f x 5 6 olezero Mp Mgnitude [db] Frequency [Hz] el Axis x 5 EECS 47 Lecture 3: Filters ge TowThoms Active Biqud rsitic insensitive Multiple outputs ef:. E. Fleischer nd J. Tow, Design Formuls for iqud ctive filters using three opertionl mplifiers, roc. IEEE, vol. 6, pp. 663, My 973. EECS 47 Lecture 3: Filters ge
7 EECS 47 Lecture 3: Filters ge 3 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 ossile to use comintion of 3 outputs EECS 47 Lecture 3: Filters ge 4 Component Vlues k C C k C C C C C C k C k C k C k k C k C 8 nd,,,, given C C k i i i it follows tht C Q C C
8 HigherOrder Filters in the Integrted Form One wy of uilding higherorder filters (n>) is vi cscde of nd order iquds & st order, e.g. SllenKey,or TowThoms, & C st or nd order Filter nd order Filter Nx nd order sections Filter order: n=n nd order Filter N Cscde of st nd nd order filters: Esy to implement Highly sensitive to component mismtch good for low Q filters only For high Q pplictions good lterntive: Integrtorsed ldder filters EECS 47 Lecture 3: Filters H.K. ge 5 Integrtor Bsed Filters Min uilding lock for this ctegory of filters Integrtor By using signl flowgrph techniques Conventionl LC filter topologies cn e converted to integrtor sed type filters How to design integrtor sed filters? Introduction to signl flowgrph techniques st order integrtor sed filter nd order integrtor sed filter High order nd high Q filters EECS 47 Lecture 3: Filters H.K. ge 6
9 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 LC 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 Choice of prticulr SFG is sed on prcticl considertions such s type of ville components ef: W.Heinlein & W. Holmes, Active Filters for Integrted Circuits, rentice Hll, Chp. 8, 974. EECS 47 Lecture 3: Filters H.K. ge 7 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, KCL & KVL is used to derive stte spce description Simple exmple: Circuit Sttespce description SFG I in I Z V Z in o I in Z EECS 47 Lecture 3: Filters H.K. ge 8
10 Signl Flowgrph (SFG) Exmples Circuit Sttespce description SFG Iin Iin Vo I in I o L Vin SL Io SL I o I in C Iin SC Vo I in SC EECS 47 Lecture 3: Filters H.K. ge 9 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 EECS 47 Lecture 3: Filters H.K. ge
11 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 () yk (.k). V = k.v 3 () Divide & multiply left side of () y k (/k). k.v 3 = V () EECS 47 Lecture 3: Filters H.K. ge 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 EECS 47 Lecture 3: Filters H.K. ge
12 Integrtor Bsed Filters st Order LF Conversion of simple lowpss C filter to integrtorsed type y using signl flowgrph techniques s C Vo Vin sc EECS 47 Lecture 3: Filters H.K. ge 3 Wht is n Integrtor? Exmple: SingleEnded OpmpC Integrtor I C I V x C Node x: since opmp hs high gin V x = / Node x is t virtul ground No voltge swing t V x comined with high opmp input impednce No input opmp current Vo sc, Vo, Vo Vin dt sc C EECS 47 Lecture 3: Filters H.K. ge 4
13 Wht is n Integrtor? Exmple: SingleEnded OpmpC Integrtor Vin C Vo V in sc Idel Mgnitude & phse response log H db hse 9 o C Note: rcticl integrtor in CMOS technology hs input & output oth in the form of voltge nd not current Considertion for SFG derivtion EECS 47 Lecture 3: Filters H.K. ge 5 st Order LF Convert C rototype to Integrtor Bsed Version. Strt from circuit prototype Nme voltges & currents for ll components V s I C I V C. Use KCL & KVL to derive stte spce description in such wy to hve BMFs in the integrtor form: Cpcitor voltge expressed s function of its current V Cp. =f(i Cp. ) 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) EECS 47 Lecture 3: Filters H.K. ge 6
14 V Vin VC VC I sc Vo VC IV s I I Integrtor Bsed Filters First Order LF All voltges & currents nodes of SFG Voltge nodes on top, corresponding current nodes elow ech voltge node V I s s I C I SFG Vin V V C V C sc I EECS 47 Lecture 3: Filters H.K. ge 7 Normlize Since integrtors re the min uilding locks require in & out signls in the form of voltge (not current) Convert ll currents to voltges y multiplying current nodes y scling resistnce Corresponding BMFs should then e scled ccordingly V Vin Vo V I s I Vo sc I I V Vin Vo I V s I Vo sc I I I x Vx V Vin Vo V V s V Vo sc V V EECS 47 Lecture 3: Filters H.K. ge 8
15 st Order Lowpss Filter SGF Normlize Vin V s sc Vin V s sc Vin V s sc I I I I V V EECS 47 Lecture 3: Filters H.K. ge 9 st Order Lowpss Filter SGF Synthesis Vin V s V V sc Choosing s Vin V V s V s, C Consolidte two rnches Vin V s V EECS 47 Lecture 3: Filters H.K. ge 3
16 First Order Integrtor Bsed Filter Vin V s V H s s EECS 47 Lecture 3: Filters H.K. ge 3 st Order Filter Built with OpmpC Integrtor Singleended OpmpC integrtor hs sign inversion from input to output Convert SFG ccordingly y modifying BMF Vin Vin EECS 47 Lecture 3: Filters H.K. ge 3
17 st Order Filter Built with OpmpC Integrtor To void requiring n dditionl opmp to perform summtion t the input node: Vin Vin EECS 47 Lecture 3: Filters H.K. ge 33 st Order Filter Built with OpmpC Integrtor (continued) C V V sc o in EECS 47 Lecture 3: Filters H.K. ge 34
18 k vo m OpmpC 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 i noise sou rce H ( f ) H ( f ) fc vn vn 4KTf th v n v n C kt vo C Typiclly, increses s filter order increses EECS 47 Lecture 3: Filters H.K. ge 35 OpmpC Filter Noise Opmp Contriution 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 C Vo The ndwidth of the opmp ffects the opmp noise contriution to the totl noise EECS 47 Lecture 3: Filters H.K. ge 36
19 Stte spce description: V VL VC Vo IC VC sc V I IL VL sl IC Iin I IL Integrtor Bsed Filter nd Order LC Filter Integrtor form I in V V C I C I C SFG V V C sc V L L I L V L sl Drw signl flowgrph (SFG) I I in I C I L EECS 47 Lecture 3: Filters H.K. ge 37 nd Order LC Filter SGF Normlize Convert currents to voltges y multiplying ll current nodes y the scling resistnce V V C sc V L sl I x Vx V V sc sl I I in I C I L V V V 3 EECS 47 Lecture 3: Filters H.K. ge 38
20 Mgnitude (db) nd Order LC Filter SGF Synthesis V V V sc V 3 sl s s C L EECS 47 Lecture 3: Filters H.K. ge 39 Second Order Integrtor Bsed Filter Filter Mgnitude esponse V B 5 s s V H VL 5. Normlized Frequency [Hz] EECS 47 Lecture 3: Filters H.K. ge 4
21 Second Order Integrtor Bsed Filter s V V B in s s VL V in s s VH s V in s s C L Q LC From mtching point of view desirle : Q V B s s V H VL EECS 47 Lecture 3: Filters H.K. ge 4 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 B v n s s v n vo kt Q C Typiclly, increses s filter order increses Note the noise power is directly proportion to Q EECS 47 Lecture 3: Filters H.K. ge 4
22 Second Order Integrtor Bsed Filter Biqud By comining outputs cn generte generl iqud function: V s s 3 Vin s s 3 V B j splne s s V H V L EECS 47 Lecture 3: Filters H.K. ge 43 Summry Integrtor Bsed Monolithic Filters Signl flowgrph techniques utilized to convert LC networks to integrtor sed ctive filters Ech rective element (L& C) replced y n integrtor Fundmentl noise limittion determined y integrting cpcitor vlue: For lowpss filter: Bndpss filter: vo vo kt C kt Q C where is function of filter order nd topology EECS 47 Lecture 3: Filters H.K. ge 44
23 Higher Order Filters How do we uild higher order filters? Cscde 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 LC 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 EECS 47 Lecture 3: Filters H.K. ge 45 Higher Order Filters Cscde of Biquds Exmple: LF filter for CDMA cell phone send receiver LF with fpss = 65 khz pss =. db fstop = 75 khz stop = 45 db Assumption: Cn 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 ir poles nd zeros In the cscde chin plce lowest Q poles first nd progress to higher Q poles moving towrds the output node EECS 47 Lecture 3: Filters H.K. ge 46
24 Img Axis X 7 Mg. (db) Overll Filter Frequency esponse Bode Digrm hse (deg) Mgnitude (db) kHz MHz Frequency [Hz] 3MHz. EECS 47 Lecture 3: Filters H.K. ge 47 olezero Mp (pzmp in Mtl).5.5 slne olezero Mp.5.5 el Axis x 7 Q pole f pole [khz] f zero [khz] EECS 47 Lecture 3: Filters H.K. ge 48
25 CDMA Filter Built with Cscde 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 EECS 47 Lecture 3: Filters H.K. ge 49 Biqud esponse.5 LF Biqud Biqud Biqud EECS 47 Lecture 3: Filters H.K. ge 5
26 Mgnitude (db) Mgnitude (db) Mgnitude (db) Mgnitude (db) Mgnitude (db) Individul Stge Mgnitude esponse LF Biqud 4 Biqud 3 Biqud Frequency [Hz] EECS 47 Lecture 3: Filters H.K. ge Intermedite Outputs LF 4 6 LF Biqud LF Biquds,3 LF Biquds,3,4 Biquds,, 3, & khz khz MHz 6 MHz Frequency [Hz] 8 khz khz MHz MHz Frequency [Hz] EECS 47 Lecture 3: Filters H.K. ge 5
27 Mgnitude (db) Sensitivity to eltive Component Mismtch Component vrition in Biqud 4 reltive to the rest (highest Q poles): Increse p4 y %.db Decrese z4 y % 3dB khz 6kHz Frequency [Hz] MHz High Q poles High sensitivity in Biqud reliztions EECS 47 Lecture 3: Filters H.K. ge 53 High Q & High Order Filters Cscde of iquds Highly sensitive to component mismtch not suitle for implementtion of high Q & high order filters Cscde 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 EECS 47 Lecture 3: Filters H.K. ge 54
28 Ldder Type Filters Active ldder type filters For simplicity, will strt with ll pole ldder type filters Convert to integrtor sed form exmple shown Then will ttend to high order ldder type filters incorporting zeros Implement the sme 7th order elliptic filter in the form of ldder LC with zeros Find level of sensitivity to component mismtch Compre with cscde of iquds Convert to integrtor sed form utilizing SFG techniques Effect of integrtor nonidelities on filter frequency chrcteristics EECS 47 Lecture 3: Filters H.K. ge 55 LC Ldder Filters Exmple: 5 th Order Lowpss Filter s L L4 C C3 C5 L Mde of resistors, inductors, nd cpcitors Douly terminted or singly terminted (with or w/o L ) Douly terminted LC ldder filters Lowest sensitivity to component mismtch EECS 47 Lecture 3: Filters H.K. ge 56
29 ssnd Attenution LC Ldder Filters s C L C3 L4 C5 L First step in the design process is to find vlues for Ls nd Cs sed on specifictions: Filter grphs & tles found in: A. Zverev, Hndook of filter synthesis, Wiley, 967. A. B. Willims nd F. J. Tylor, Electronic filter design, 3 rd edition, McGrw Hill, 995. CAD tools Mtl Spice EECS 47 Lecture 3: Filters H.K. ge 57 LC Ldder Filter Design Exmple Design LF 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 c5th order Butterworth 3dB Normlized From: Willims nd Tylor, p. 37 Stopnd Attenution 3dB EECS 47 Lecture 3: Filters H.K. ge 58
30 LC Ldder Filter Design Exmple Find vlues for L & C from Tle: Note L &C vlues normlized to 3dB = Denormliztion: Multiply ll L Norm, C Norm y: L r = / 3dB C r = /(X 3dB ) is the vlue of the source nd termintion resistor (choose oth W for now) Then: L= L r xl Norm C= C r xc Norm From: Willims nd Tylor, p..3 EECS 47 Lecture 3: Filters H.K. ge 59 LC Ldder Filter Design Exmple Find vlues for L & C from Tle: Normlized vlues: C Norm =C5 Norm =.68 C3 Norm =. L Norm = L4 Norm =.68 Denormliztion: Since 3dB =xmhz L r = / 3dB = 5.9 nh C r = /(X 3dB )= 5.9 nf = cc=c5=9.836nf, C3=3.83nF cl=l4=5.75nh From: Willims nd Tylor, p..3 EECS 47 Lecture 3: Filters H.K. ge 6
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