EE47 Lecture 3 Lst week s summry Active Filters Active biquds Sllen Key & TowThoms Integrtor bsed filters Signl flowgrph concept First order integrtor bsed filter Second order integrtor bsed filter & biquds High order & high Q filters Cscded biquds Cscded biqud sensitivity Ldder type filters EECS 47 Lecture 3: Filters 5 H.K. ge Summry Lst Week Mjor success in CMOS technology scling: Inexpensive DSs technology esulted in the need for high performnce Anlog/Digitl interfce circuitry Min Anlog/Digitl interfce building blocks includes Anlog filters D/A converters A/D converters EECS 47 Lecture 3: Filters 5 H.K. ge
Monolithic Filters Monolithic inductor in CMOS tech. Integrted L<nH with Q< combined with mx. cp. pf LC filters in the monolithic form fesible: freq>5mhz Anlog/Digitl interfce circuitry require fully integrted filters with criticl frequencies << 5MHz Good lterntive: Active filters built without the need for inductors EECS 47 Lecture 3: Filters 5 H.K. ge 3 nd Order Trnsfer Functions (Biquds) Biqudrtic ( nd order) trnsfer function: H(s) s s ωq ω H( jω) ω H ( jω) H ( jω) ω ω ω Q ω Biqud poles @: s 4Q ± Q for Q poles re rel, complex otherwise EECS 47 Lecture 3: Filters 5 H.K. ge 4
Biqud Complex oles Q > ω s Q ( ± j 4Q ) Distnce from origin in splne: d ω Q ω ( 4Q ) EECS 47 Lecture 3: Filters 5 H.K. ge 5 slne jω rdius ω rccos Q poles σ ω rel prt Q EECS 47 Lecture 3: Filters 5 H.K. ge 6
Implementtion of Biquds ssive C: only rel poles cn t implement complex conjugte poles Terminted LC Low power, since it is pssive Only noise source lod nd source resistnce As previously nlyzed, not fesible in the monolithic form for f <5MHz Active Biquds Mny topologies cn be found in filter textbooks! Widely used topologies: Singleopmp biqud: SllenKey Multiopmp biqud: TowThoms Integrtor bsed biquds EECS 47 Lecture 3: Filters 5 H.K. ge 7 Active Biqud SllenKey Lowss Filter in C C G ut G H( s) s s ω Q ω ω C C Single gin element Cn be implemented both in discrete & monolithic form rsitic sensitive ersions for LF, HF, B, Advntge: Only one opmp used Disdvntge: Sensitive to prsitics ω Q G C C C EECS 47 Lecture 3: Filters 5 H.K. ge 8
Imginry Axis Zeros Shrpen trnsition bnd notch out interference Highpss filter (HF) Bndreject filter s ω H(s) K Z s s ωq ω ω H( j ω ) ω K ωz Note: Alwys represent trnsfer functions s product of gin term, poles, nd zeros (pirs if complex). Then ll coefficients hve physicl mening, resonble mgnitude, nd esily checkble unit. EECS 47 Lecture 3: Filters 5 H.K. ge 9 Imginry Zeros Mgnitude [db] 3 4 Zeros substntilly shrpen trnsition bnd At the expense of reduced stopbnd ttenution t high frequency With zeros No zeros 5 4 5 6 7 Frequency [Hz] Img Axis x 6.5.5.5.5 f Q f Z olezero Mp khz 3 f.5.5.5.5 el Axis x 6 EECS 47 Lecture 3: Filters 5 H.K. ge
Moving the Zeros f Q khz f Z f Mgnitude [db] 3 4 5 4 5 6 7 Frequency [Hz] Img Axis x 5 olezero Mp 6 4 4 6 6 4 4 6 5 x el Axis EECS 47 Lecture 3: Filters 5 H.K. ge TowThoms Active Biqud rsitic insensitive Multiple outputs ef:. E. Fleischer nd J. Tow, Design Formuls for biqud ctive filters using three opertionl mplifiers, roc. IEEE, vol. 6, pp. 663, My 973. EECS 47 Lecture 3: Filters 5 H.K. ge
EECS 47 Lecture 3: Filters 5 H.K. ge 3 Frequency esponse ( ) ( ) ( ) ( ) 3 s s b b s b b k s s b b s s b s s b b s b b k in o in o in o implements generl biqud section with rbitrry poles nd zeros nd 3 relize the sme poles but re limited to t most one finite zero EECS 47 Lecture 3: Filters 5 H.K. ge 4 Component lues 8 7 7 3 8 7 3 8 6 8 7 4 8 6 8 7 5 3 8 k C C k C C C b C b C C b 8 7 8 6 5 4 3 k b b C k C b b k C k k C k C 8 nd,,,, given C C k b i i i tht it follows 7 3 8 C Q C C ω ω
Integrtor Bsed Filters Min building block for this ctegory of filters integrtor By using signl flowgrph techniques conventionl filter topologies cn be converted to integrtor bsed type filters Next few pges: Signl flowgrph techniques st order integrtor bsed filter nd order integrtor bsed filter High order nd high Q filters EECS 47 Lecture 3: Filters 5 H.K. ge 5 Wht is Signl Flowgrph (SFG)? SFG Topologicl network representtion consisting of nodes & brnches used to convert one form of network to more suitble form (e.g. pssive LC filters to integrtor bsed filters) Any network described by set of liner differentil equtions cn be expressed in SFG form. For given network, mny different SFGs exists. Choice of prticulr SFG is bsed on prcticl considertions such s type of vilble components. ef: W.Heinlein & W. Holmes, Active Filters for Integrted Circuits, rentice Hll, Chp. 8, 974. EECS 47 Lecture 3: Filters 5 H.K. ge 6
Wht is Signl Flowgrph (SFG)? SFG nodes represent vribles ( & I in our cse), brnches represent trnsfer functions (we will cll these trnsfer functions brnch multipliction fctor BMF) To convert network to its SFG form, KCL & KL is used to derive stte spce description: Exmple: Circuit Sttespce SFG description I I in in o o Iin o in I o L in Io SL in SL I o I in C Iin o SC I in SC EECS 47 Lecture 3: Filters 5 H.K. ge 7 Signl Flowgrph (SFG) ules Two prllel brnches cn be replced by single brnch with overll BMF equl to sum of two BMFs b b A node with only one incoming brnch & one outgoing brnch cn be replced by single brnch with BMF equl to the product of the two BMFs b.b 3 An intermedite node cn be multiplied by fctor (x). BMFs for incoming brnches hve to be multiplied by x nd outgoing brnches divided by x b x. b/x 3 x. 3 EECS 47 Lecture 3: Filters 5 H.K. ge 8
Signl Flowgrph (SFG) ules Simplifictions cn often be chieved by shifting or eliminting nodes i 4 /b 3 i /b /b 3 A selfloop brnch with BMF y cn be eliminted by multiplying the BMF of incoming brnches by /(y) /b /b i /b 3 i /(/b) 3 EECS 47 Lecture 3: Filters 5 H.K. ge 9 Integrtor Bsed Filters st Order LF Strt from C prototype Use KCL & KL to derive stte spce description: in s I C I o in sc C Use stte spce description to drw signl flowgrph (SFG) EECS 47 Lecture 3: Filters 5 H.K. ge
Integrtor Bsed Filters First Order LF KCL & KL to derive stte spce description: I C sc in C I s I I o C Use stte spce description to drw signl flowgrph (SFG) in I s s I C I SFG in C C sc I EECS 47 Lecture 3: Filters 5 H.K. ge Normlize Since integrtors the min building blocks require in & out signls in the voltge form (not current) Convert ll currents to voltges by multiplying current nodes by scling resistnce Corresponding BMFs should then be scled ccordingly in o I s I o sc I I Ix x in o I s I o sc I I in o s o sc EECS 47 Lecture 3: Filters 5 H.K. ge
Normlize in s sc in s sc in s sc I I I I EECS 47 Lecture 3: Filters 5 H.K. ge 3 Synthesis in s sc in τ s Consolidte two brnches τ s, C in τ s EECS 47 Lecture 3: Filters 5 H.K. ge 4
First Order Integrtor Bsed Filter in τ s H ( s) τ s in EECS 47 Lecture 3: Filters 5 H.K. ge 5 OpmpC SingleEnded Integrtor C in o o in dt, C in sc τ C EECS 47 Lecture 3: Filters 5 H.K. ge 6
st Order Filter Built with OpmpC Integrtor Singleended OpmpC integrtor hs sign inversion from input to output Convert SFG ccordingly by modifying BMF in in in in EECS 47 Lecture 3: Filters 5 H.K. ge 7 st Order Filter Built with OpmpC Integrtor (continued) C in in o in sc EECS 47 Lecture 3: Filters 5 H.K. ge 8
n 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(f)df i S(f) i Inputreferrednoisespectrldensity H(f) H(f) ( π fc) vn vn 4KT f v n v n C vo α kt C α Typiclly, α increses s filter order increses EECS 47 Lecture 3: Filters 5 H.K. ge 9 OpmpC Filter Noise Opmp Contribution So fr only the fundmentl noise sources re considered. In relity, noise ssocited with the opmp increses the overll noise. The bndwidth of the opmp ffects the opmp noise contribution to the totl noise v n C v n vopmp o EECS 47 Lecture 3: Filters 5 H.K. ge 3
Stte spce description: L C o IC C sc I IL L sl IC Iin I IL Integrtor Bsed Filter nd Order LC Filter Integrtor form I in I SFG C L L C I L sc C I C L sl Drw signl flowgrph (SFG) I I in I C I L EECS 47 Lecture 3: Filters 5 H.K. ge 3 Normlize Convert currents to voltges by multiplying ll current nodes by the scling resistnce C sc L sl I x x sc sl I I in I C I L in 3 EECS 47 Lecture 3: Filters 5 H.K. ge 3
Synthesis sc in 3 sl τ sτ sτ τ in C L EECS 47 Lecture 3: Filters 5 H.K. ge 33 Second Order Integrtor Bsed Filter Filter Mgnitude esponse B Mgnitude (db) 5 5 sτ sτ H in L. Normlized Frequency [Hz] EECS 47 Lecture 3: Filters 5 H.K. ge 34
B τs in ττ βτ s L in ττ βτ s H ττ s in ττ βτ s τ C τ L β ω ττ LC Q β Second Order Integrtor Bsed Filter B sτ sτ H L τ τ in Frommtchingpointofviewdesirble: τ τ Q EECS 47 Lecture 3: Filters 5 H.K. ge 35 Second Order Bndpss Filter Noise n vo m Find trnsfer function of ech noise source to the output Integrte contribution of ll noise sources Here it is ssumed tht opmps re noise free (not usully the cse!) vn vn 4KTdf H m(f) S(f)df i B v n sτ sτ v n in vo kt Q C α Typiclly, α increses s filter order increses Note the noise power is directly proportion to Q EECS 47 Lecture 3: Filters 5 H.K. ge 36
Second Order Integrtor Bsed Filter Biqud By combining outputs cn generte generl biqud function: ττ s τs 3 in ττ s βτs 3 B jω splne sτ sτ σ H in L EECS 47 Lecture 3: Filters 5 H.K. ge 37 Summry Integrtor Bsed Monolithic Filters Signl flowgrph techniques utilized to convert LC networks to integrtor bsed ctive filters Ech rective element (L& C) replced by n integrtor Fundmentl noise limittion determined by integrting cpcitor: For lowpss filter: Bndpss filter: vo vo kt α C kt α Q C where α is function of filter order nd topology EECS 47 Lecture 3: Filters 5 H.K. ge 38
Higher Order Filters How do we build higher order filters? Cscde of biquds nd st order sections Ech complex conjugte pole built with biqud 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 LC filters SFG techniques used to perform exct conversion of ldder type filters to integrtor bsed filters More complicted conversion process Much less sensitive to component vritions compred to cscde of biquds EECS 47 Lecture 3: Filters 5 H.K. ge 39 Higher Order Filters Cscde of Biquds Exmple: LF filter for CDMA bsebnd receiver LF with fpss 65 khz pss. db fstop 75 khz stop 45 db Assumption: Cn compenste for phse distortion in the digitl domin 7th order Elliptic Filter Implementtion with Biquds Gol: Mximize dynmic rnge ir poles nd zeros highest Q poles with closest zeros is good strting point, but not necessrily optimum Ordering: Lowest Q poles first is good strt EECS 47 Lecture 3: Filters 5 H.K. ge 4
Filter Frequency esponse Bode Digrm hse (deg) Mgnitude (db) 4 6 8 8 36 54 3kHz MHz Frequency [Hz] 3MHz Mg. (db). EECS 47 Lecture 3: Filters 5 H.K. ge 4 olezero Mp Img Axis X 7.5.5 slne olezero Mp.5.5 el Axis x 7 Q pole f pole [khz] 6.79 659.496 3.659 6.744.6 473.643 39.568 f zero [khz] 97.5 836.6 744. EECS 47 Lecture 3: Filters 5 H.K. ge 4
Biqud esponse.5 LF 4 4 5 6 7 Biqud 4 4 5 6 7.5.5 4 Biqud 3 4 5 6 7 4 Biqud 4 4 5 6 7 EECS 47 Lecture 3: Filters 5 H.K. ge 43 Biqud esponse Bode Mgnitude Digrm.5.5.5 Mgnitude (db) 3 4 LF Biqud Biqud 3 Biqud 4 5 4 5 6 7 Frequency [Hz] EECS 47 Lecture 3: Filters 5 H.K. ge 44
5 6 7 4 5 6 Mgnitude (db) Mgnitude (db) 4 6 8 4 6 8 khz Intermedite Outputs LF Mgnitude (db) Mgnitude (db) 4 6 8 LF Biqud LF Biquds,3 LF Biquds,3,4 Biquds,, 3, & 4 4 6 8 khz MHz 6 MHz khz khz MHz MHz Frequency [Hz] Frequency [Hz] EECS 47 Lecture 3: Filters 5 H.K. ge 45 Sensitivity Component vrition in Biqud 4 (highest Q pole): Increse w p4 by % Decrese w z4 by %.db Mgnitude (db) 3 3dB 4 5 khz 6kHz Frequency [Hz] MHz High Q poles High sensitivity in Biqud reliztions EECS 47 Lecture 3: Filters 5 H.K. ge 46
High Q & High Order Filters Cscde of biquds Highly sensitive to component vritions not suitble for implementtion of high Q & high order filters Cscde of biquds only used in cses where required Q for ll biquds <4 (e.g. filters for disk drives) LC ldder filters more pproprite for high Q & high order filters (next topic) Less sensitive to component vritions EECS 47 Lecture 3: Filters 5 H.K. ge 47 Ldder Type Filters For simplicity, will strt with ll pole ldder type filters Convert to integrtor bsed form Exmple shown Then will ttend to high order ldder type filters incorporting zeros Implement the sme 7 th order elliptic filter in the form of ldder type Find level of sensitivity to component vritions Compre with cscde of biquds Convert to integrtor bsed form utilizing SFG techniques Exmple shown EECS 47 Lecture 3: Filters 5 H.K. ge 48
LC Ldder Filters in s C L C3 L4 C5 L Mde of resistors, inductors, nd cpcitors Doubly terminted or singly terminted (with or w/o L ) Doubly terminted LC ldder filters Lowest sensitivity to component vritions EECS 47 Lecture 3: Filters 5 H.K. ge 49 LC Ldder Filters in s C L C3 L4 C5 L Design: CAD tools Mtlb Spice Filter tbles A. Zverev, Hndbook of filter synthesis, Wiley, 967. A. B. Willims nd F. J. Tylor, Electronic filter design, 3 rd edition, McGrwHill, 995. EECS 47 Lecture 3: Filters 5 H.K. ge 5
LC Ldder Filter Design Exmple Design LF with mximlly flt pssbnd: f3db MHz, fstop MHz s >7dB Mximlly flt pssbnd Butterworth Determine minimum filter order : Use of Mtlb or Tbles Here tbles used fstop / f3db s >7dB 3dB Stopbnd Attenution db Minimum Filter Order 5th order Butterworth Νοrmlized w From: Willims nd Tylor, p. 37 EECS 47 Lecture 3: Filters 5 H.K. ge 5 LC Ldder Filter Design Exmple Find vlues for L & C from Tble: Note L &C vlues normlized to w 3dB Denormliztion: Multiply ll L Norm, C Norm by: L r /w 3dB C r /(Xw 3dB ) is the vlue of the source nd termintion resistor (choose both Ω for now) Then: L L r xl Norm C C r xc Norm From: Willims nd Tylor, p..3 EECS 47 Lecture 3: Filters 5 H.K. ge 5
LC Ldder Filter Design Exmple Find vlues for L & C from Tble: Normlized vlues: C Norm C5 Norm.68 C3 Norm. L Norm L4 Norm.68 Denormliztion: Since w 3dB πxmhz L r /w 3dB 5.9 nh C r /(Xw 3dB ) 5.9 nf CC59.836nF, C33.83nF LL45.75nH From: Willims nd Tylor, p..3 EECS 47 Lecture 3: Filters 5 H.K. ge 53 Mgnitude esponse Simultion in sohm L5.75nH C 9.836nF L45.75nH C3 3.83nF C5 9.836nF LOhm 5 SICE simultion esults 6 db pssbnd ttenution due to double termintion Mgnitude (db) 3 4 3dB 5 3 Frequency [MHz] EECS 47 Lecture 3: Filters 5 H.K. ge 54
LC Ldder Filter Conversion to Integrtor Bsed Active Filter in 3 4 5 6 s I L I3 L4 I 5 C C3 C5 I I4 I 6 I 7 L Use KCL & KL to derive equtions: I in,, 3 sc 4 I I 4 6 4, 5 4 6, 6 o 6 sc 3 sc 5 3 I, I I I 3, I s 3 sl 5 6 I 4 I 3 I 5, I 5, I 6 I 5 I 7, I7 sl4 L EECS 47 Lecture 3: Filters 5 H.K. ge 55 in I s sc I LC Ldder Filter Signl Flowgrph I in,, 3 sc 4 I 4 I,, 6 4 5 4 6 6 o 6 sc 3 sc 5 3 I, I I I 3, I3 s sl 5 6 I 4 I 3 I 5, I 5, I 6 I 5 I 7, I7 sl4 L 3 4 5 6 sl sc3 sl4 sc5 I3 I4 I 5 I 6 I 7 SFG EECS 47 Lecture 3: Filters 5 H.K. ge 56 o L
LC Ldder Filter Signl Flowgrph in 3 4 5 6 s I L I3 L4 I 5 C C3 C5 I I4 I 6 I 7 L in I s sc I 3 4 5 6 sl sc3 sl4 sc5 o I3 I4 I 5 I 6 I 7 SFG L EECS 47 Lecture 3: Filters 5 H.K. ge 57 in I s sc I LC Ldder Filter Normlize 3 4 5 6 sl sc3 sl4 sc5 o I 3 I4 I 5 I 6 I 7 L in s 3 4 5 6 sc sc sl 3 sl sc 4 5 3 4 5 6 o 7 L EECS 47 Lecture 3: Filters 5 H.K. ge 58
in s LC Ldder Filter Synthesize 3 4 5 6 sc sc sl 3 sl sc 4 5 3 4 5 6 o 7 L in s sτ sτ sτ 3 sτ 4 sτ 5 L EECS 47 Lecture 3: Filters 5 H.K. ge 59 in s LC Ldder Filter Integrtor Bsed Implementtion sτ sτ sτ 3 sτ 4 sτ 5 L L L4 C., C., C., C., C. 3 3 4 4 5 5 τ τ τ τ τ Building Block: C Integrtor sc EECS 47 Lecture 3: Filters 5 H.K. ge 6
Negtive esistors o o o EECS 47 Lecture 3: Filters 5 H.K. ge 6 Synthesize EECS 47 Lecture 3: Filters 5 H.K. ge 6
Frequency esponse EECS 47 Lecture 3: Filters 5 H.K. ge 63 Scle Node oltges Scle by fctor s EECS 47 Lecture 3: Filters 5 H.K. ge 64
Noise Totl noise:.4 µ rms (noiseless opmps) Tht s excellent, but the cpcitors re very lrge (nd the resistors smll). Not possible to integrte. Suppose our ppliction llows higher noise in the order of 4 µ rms EECS 47 Lecture 3: Filters 5 H.K. ge 65 Scle to Meet Noise Trget Scle cpcitors nd resistors to meet noise objective s 4 Noise: 4 µ rms (noiseless opmps) EECS 47 Lecture 3: Filters 5 H.K. ge 66
Completed Design EECS 47 Lecture 3: Filters 5 H.K. ge 67 Sensitivity C mde (rbitrrily) 5% (!) lrger thn its nominl vlue.5 db error t bnd edge 3.5 db error in stopbnd Looks like very low sensitivity EECS 47 Lecture 3: Filters 5 H.K. ge 68