Summary of Last Lecture
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1 EE47 Lecture 3 Lst lecture s summry Active Filters Active iquds Sllen Key & TowThoms Integrtor sed filters Signl flowgrph concept First order integrtor sed filter Second order integrtor sed filter & iquds High order & high Q filters scded iquds scded iqud sensitivity to component vritions Ldder type filters EES 47 Lecture 3: Filters 6 H.K. Pge Summry of Lst Lecture Nomenclture Filter specifictions Qulity fctor Frequency chrcteristics Group dely Filter types Butterworth heyshev I heyshev II Elliptic Bessel Group dely comprison exmple L filters EES 47 Lecture 3: Filters 6 H.K. Pge
2 Integrted Filters Implementtion of L filters in MOS technologies requires onchip inductors Integrted L<nH with Q< omined with mx. cp. pf L filters in the monolithic form fesile: freq>5mhz Anlog/Digitl interfce circuitry require fully integrted filters with criticl frequencies << 5MHz Hence: Need to uild ctive filters uilt without inductors EES 47 Lecture 3: Filters 6 H.K. Pge 3 Filters nd Order Trnsfer Functions (Biquds) Biqudrtic ( nd order) trnsfer function: H(s) s s ωpqp ωp H ( jω) ω H ( jω) H ( jω) ω ω ωp Q P ωp Biqud s 4Q ± P QP for Q P poles re rel, complex otherwise EES 47 Lecture 3: Filters 6 H.K. Pge 4
3 Biqud omplex Poles Q P > omplex conjugte poles: s P j 4QP ± QP Distnce from origin in splne: d ω ω P Q P ω P ( 4Q ) P d Splne jω poles σ EES 47 Lecture 3: Filters 6 H.K. Pge 5 splne jω rdius ω P rccos Q P poles σ ωp rel prt Q P EES 47 Lecture 3: Filters 6 H.K. Pge 6
4 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 <5MHz 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 6 H.K. Pge 7 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 Disdvntge: Sensitive to prsitic ll pole no zeros EES 47 Lecture 3: Filters 6 H.K. Pge 8
5 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 ωpqp ωp ωp H( j ω ) K ω ωz 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 6 H.K. Pge 9 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 6 H.K. Pge
6 Moving the Zeros f Q P P khz f Z f P x 5 6 PoleZero Mp Mgnitude [db] Frequency [Hz] Img Axis el Axis x 5 EES 47 Lecture 3: Filters 6 H.K. Pge 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 6 H.K. Pge
7 EES 47 Lecture 3: Filters 6 H.K. Pge 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 EES 47 Lecture 3: Filters 6 H.K. Pge 4 omponent Vlues k k k k k k k k 8 nd,,,, given k i i i it follows tht Q P P P ω ω
8 HigherOrder Filters Higherorder filters (N>) cn e uilt with cscde of nd order iquds, e.g. SllenKey,or TowThoms nd order Filter nd order Filter Nx nd order sections Filter with N order nd order Filter Ν As will e shown lter: HighQ highorder filters uilt with cscde of nd order sections Highly sensitive to component vritions Good lterntive: Integrtorsed ldder type filters EES 47 Lecture 3: Filters 6 H.K. Pge 5 Integrtor Bsed Filters Min uilding lock for this ctegory of filters integrtor By using signl flowgrph techniques conventionl filter topologies cn e converted to integrtor sed type filters Next few pges: Signl flowgrph techniques st order integrtor sed filter nd order integrtor sed filter High order nd high Q filters EES 47 Lecture 3: Filters 6 H.K. Pge 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 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 6 H.K. Pge 7 Wht is Signl Flowgrph (SFG)? Signl flowgrph 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 Z Iin Z Vo I in Z EES 47 Lecture 3: Filters 6 H.K. Pge 8
10 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 6 H.K. Pge 9 Signl Flowgrph (SFG) ules Two prllel rnches cn e replced y single rnch with overll BMF equl to sum of two BMFs V V A node with only one incoming rnch & one outgoing rnch cn e replced y single rnch with BMF equl to the product of the two BMFs V V V. V V 3 An intermedite node cn e multiplied y fctor (x). BMFs for incoming rnches hve to e multiplied y x nd outgoing rnches divided y x V V V x. /x V V 3 x.v 3 V V EES 47 Lecture 3: Filters 6 H.K. Pge
11 Signl Flowgrph (SFG) ules Simplifictions cn often e chieved y shifting or eliminting nodes V i V 4 / V V 3 V i V / / V 3 A selfloop rnch with BMF y cn e eliminted y multiplying the BMF of incoming rnches y /(y) / / V i V / V 3 V i V /() V 3 EES 47 Lecture 3: Filters 6 H.K. Pge Integrtor Bsed Filters st Order LPF onversion of simple lowpss filter to integrtorsed type y using signl flowgrph technique s Vo s EES 47 Lecture 3: Filters 6 H.K. Pge
12 Wht is n Integrtor? Exmple: SingleEnded Opmp Integrtor Vo, Vo dt s τ 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 6 H.K. Pge 3 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 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 6 H.K. Pge 4
13 V V Integrtor form V I s Vo V I V s I I Integrtor Bsed Filters First Order LPF All voltges & currents nodes of SGF 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 6 H.K. Pge 5 Normlize Since integrtors the min uilding locks require in & out signls in the voltge form (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 6 H.K. Pge 6
14 Normlize V I s s I V I s s I V V V s s EES 47 Lecture 3: Filters 6 H.K. Pge 7 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 6 H.K. Pge 8
15 First Order Integrtor Bsed Filter V τ s V H ( s) τ s EES 47 Lecture 3: Filters 6 H.K. Pge 9 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 6 H.K. Pge 3
16 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 6 H.K. Pge 3 st Order Filter Built with Opmp Integrtor (continued) V V s o in EES 47 Lecture 3: Filters 6 H.K. Pge 3
17 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 ) Noisespectrl densityof m noisesource H(f) H(f) ( π f) v v 4KTΔf n n th v n v n vo α kt α Typiclly, α increses s filter order increses EES 47 Lecture 3: Filters 6 H.K. Pge 33 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 << contriution of 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 6 H.K. Pge 34
18 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 6 H.K. Pge 35 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 6 H.K. Pge 36
19 Synthesis V V V s V 3 sl τ sτ sτ τ L EES 47 Lecture 3: Filters 6 H.K. Pge 37 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 6 H.K. Pge 38
20 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 6 H.K. Pge 39 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 6 H.K. Pge 4
21 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 6 H.K. Pge 4 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 6 H.K. Pge 4
22 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 vritions 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 vritions compred to cscde of iquds EES 47 Lecture 3: Filters 6 H.K. Pge 43 Higher Order Filters scde of Biquds Exmple: LPF filter for DMA 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 7th order Elliptic Filter Implementtion with cscded Biquds Gol: Mximize dynmic rnge Pir poles nd zeros Highest Q poles with closest zeros is good strting point, ut not necessrily optimum Ordering: Lowest Q poles first is good strt EES 47 Lecture 3: Filters 6 H.K. Pge 44
23 Filter Overll Frequency esponse Bode Digrm Phse (deg) Mgnitude (db) kHz MHz Frequency [Hz] 3MHz Mg. (db). EES 47 Lecture 3: Filters 6 H.K. Pge 45 PoleZero Mp 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 6 H.K. Pge 46
24 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 pole nd pir of imginry zero EES 47 Lecture 3: Filters 6 H.K. Pge 47 Biqud esponse.5 LPF Biqud Biqud Biqud EES 47 Lecture 3: Filters 6 H.K. Pge 48
25 Biqud esponse Bode Mgnitude Digrm Mgnitude (db) 3 LPF Biqud 4 Biqud 3 Biqud Frequency [Hz] EES 47 Lecture 3: Filters 6 H.K. Pge 49 Mgnitude (db) Mgnitude (db) khz Intermedite Outputs LPF Mgnitude (db) LPF Biqud 4 6 LPF Biquds,3 LPF Biquds,3,4 Biquds,, 3, & Mgnitude (db) 8 khz MHz 6 MHz khz khz MHz MHz Frequency [Hz] Frequency [Hz] 8 EES 47 Lecture 3: Filters 6 H.K. Pge 5
26 Sensitivity omponent vrition in Biqud 4 (highest Q pole): Increse ω p4 y % Decrese ω z4 y %.db Mgnitude (db) 3 3dB 4 5 khz 6kHz Frequency [Hz] MHz High Q poles High sensitivity in Biqud reliztions EES 47 Lecture 3: Filters 6 H.K. Pge 5 High Q & High Order Filters scde of iquds Highly sensitive to component vritions 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) L ldder filters more pproprite for high Q & high order filters (next topic) Less sensitive to component vritions EES 47 Lecture 3: Filters 6 H.K. Pge 5
Correction & Clarification From Last Lecture (2)
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