ANALOG FILTERS. C. Sauriol. Algonquin College Ottawa, Ontario

Transcription

1 LOG ILT By. auril lgqui llege Ottawa, Otari ev. March 4, 003

2 TBL O OTT alg ilters TIO PI ILT. irst-rder lw-pass filter- -4. irst-rder high-pass filter ecd-rder lw-pass filter ecd-rder bad-pass filter ecd-rder lw-pass filter-l ecd-rder high-pass filter-l 9- TIO QUY PO O BI MPLII. Ope-lp respse f p amp -3. lsed-lp respse f -ivertig amplifier Op amp circuits with resistive feedback etwrks Trasiet respse f amplifiers t a square wave 7 TIO 3 TI ILT. irst-rder filter circuits 8-9. ecd-rder filter circuits Trasiet respse f first ad secd-rder filters th Order ilters 38-4 Desig example Lw-pass Butterwrth filter 4-45 Desig example High-pass Butterwrth filter Desig example 3 Lw-pass hebychev filter 5-54 Desig example 4 High-pass hebychev filter Desig example 5 Lw-pass Bessel filter Table f rmalized lw-pass filter ples 64 Ple cversi equatis 65 alculati f miimum rder f filters 66 ffect f cmpet variati frequecy respse Table f secd-rder trasfer fuctis auril alg ilters ev. 3/4/003 Page

3 . irst-rder lw-pass filter TIO PI ILT. alg ilters i() () Trasfer ucti ( ) ( ) i ( ) ( ) ( jω ) jω ( jω) i ( jω ) jω jω MG( um) ( jω) MG( de) jω / ( jω) / um / de 0 T requecy respse ( ω) ( ω) jω Imag (jω) θ eal Gai (/) Gai (db) Phase( ) xact xact Bde pprximati xact Bde pprximati ( ω ) 0 LOG ( jω ) L 0LOG( ) T ( ω) 45 LOG( 0. ) r. auril alg ilters ev. 3/4/003 Page r

4 alg ilters Gai espse 0 Gai (db) L asymptte db dw 3 db dw db dw c cutff frequecy H asymptte lpe -0 db/dec requecy (Hz) ctual Gai Bde apprximati Phase espse c c Bde apprximati -45 /decade Phase () dw up c requecy (Hz) ctual Phase Bde apprximati utff frequecy: defied as the frequecy where the atteuati is 3 db with respect t the pass-bad gai. Here the pass-bad is 0 t 000 Hz ad the pass-bad gai is 0 db, therefre whe the gai drps t 3 db we reach the cutff frequecy. T calculate the cutff frequecy fr a first-rder demiatr f the T, equate the real part t the imagiary part ad slve fr ω. ( jω) ( jω ) i ( jω ) jω ω ω ω c r c π π Hz ω first rder. auril alg ilters ev. 3/4/003 Page 3

5 Ples ad Zers alg ilters The rts f the umeratr f the T are called the zers f the T ad the rts f the demiatr are called the ples because they make the T equal t iifiity. The ples f the T determie the cutff frequecy f the filter. r the abve filter, the umeratr has rts, r zers ad the demiatr has e ple (rt), that is 0 P -/. The magitude f P is the cutff frequecy i r/s. Ple P ω P π w if we plt the gai ad the phase agaist frequecy fr a siewave iput, we btai the the frequecy respse f the circuit. eplacig with jω i the T we btai:. irst-rder high-pass filter Trasfer ucti i() () ( ) ( ) i ( ) ( ) ( ) ( jω ) jω ( jω) i ( jω ) jω MG( um) jω ( jω) MG( de) jω ( jω) ω / ( jω) / um / de 90 T ω ω ( ω) Imag jω um θ de eal utff frequecy: defied as the frequecy where the atteuati is 3 db with respect t the pass-bad gai. Here the pass-bad is 000 Hz t iifiity ad the pass-bad gai is 0 db, therefre whe the gai drps t 3 db we reach the cutff frequecy. T calculate the cutff frequecy fr a first-rder demiatr f the T, equate the real part t the imagiary part ad slve fr ω. ( jω) ( jω ) i ( jω ) jω jω ω ω ω ω c 000Hz π π first rder. auril alg ilters ev. 3/4/003 Page 4 r c

6 alg ilters Gai (/) Gai (db) Phase( ) (Hz) xact xact Bde pprximati xact Bde pprximati r ω 0 LOG ( jω ) 0LOG( ) H 90 T ( ω) r LOG 0. Gai espse 0 c cutff frequecy Gai (db) L asymptte lpe 0 db/dec db dw 3 db dw db dw H asymptte requecy (Hz) ctual Gai Bde apprximati. auril alg ilters ev. 3/4/003 Page 5

7 Phase espse alg ilters c Phase () dw - 45 c Bde apprximati -45 /decade 5.7 up c requecy (Hz) ctual Phase Bde apprximati Ples ad Zers ( ) ( ) i ( ) r the trasfer fucti f the high-pass filter, the umeratr has e rt, r e zer ad the demiatr has e ple, r e rt. Zers: umeratr, z 0 Ples: demiatr Ple P ω 0 P -/. The magitude f P is the cutff frequecy i r/s. P 3. ecd-rder lw-pass filter π i() Thevei X () TH i ( ) TH is the pe circuit vltage i ( ) X Z TH is derived by replacig the surce with a shrt circuit a ideal vltage surce has zer iteral resistace.. auril alg ilters ev. 3/4/003 Page 6

8 Z TH Z TH alg ilters TH () i () () i ( ) ( ) TH ( ) ( ) ( ) i ( ) ( ) ( ) i ( ) ( ( ) ( ) ( ) ( ( ( ) ) T fid the frequecy respse f the abve T, it is easier if e fids the rts f the demiatr (the ples f the T) ad expresses the T as fllws: ( ) i ( ) ( ) umerical example i() p p ( p ) ( p ) p 0 0 p () Let ad The T simplifies t the frm shw belw where the ples yield the tw break frequecies f each first-rder term f the demiatr. ( ) ( ) p p i ( ) auril alg ilters ev. 3/4/003 Page 7

9 ω p r / s ω c p c I Hz we have : 38 Hz ad 68. Hz where Hz ( ) ( ) i i ( jω) ( ) ( ) ( ) ( ) i ( 400) ( 6450) ( 400) ( 6450) ( jω) ( jω) jω jω p 6450 alg ilters 6450 r / s 400 p dB -0 db/dec 0dB -0 db/dec 38 Hz 68. Hz 0-45 /dec 380 Hz 0-45 /dec 68 Hz 38. Hz 6.8 Hz The idividual Bde plts fr each term is shw abve. The verall Bde plt apprximati is simply the additi f the idividual Bde plts. 0 LOG ( jω) 0 LOG jω jω LOG jω LOG jω 6450 / ( jω) G jω jω st term i db d term i db G G jω jω gle f st term agle f d term -T(ω/400) -T(ω/6450). auril alg ilters ev. 3/4/003 Page 8

10 structi f verall Bde plt gai respse alg ilters 0dB 38 Hz 68 Hz -0 db/dec T calculate the umber decades, use: dec LOG( )-LOG( ) dec LOG( / ) dec LOG(68 / 38) dec dB 0dB 6.7 db Overall gai respse Bde apprximati -0 db/dec dec -0 db/dec db slpe x dec db -0 db/dec x dec db db The L gai is bviusly the sum f the tw idividual L gais. T cstruct the verall Bde plt, add the slpes i each regi f the frequecy respse db -40 db/dec.58 dec 00 khz -80 db te: the actual gai is 3 db dw frm Bde plt at each break frequecy if they are at least e decade apart. If the chage i slpe is 40 db/dec ( frm 0 db/dec t 60 db/dec), the the actual gai is 6 db dw at the break frequecy. The actual gai is db dw frm Bde plt at 0.5 c ad c if c ad c are at least e decade apart. If the chage i slpe is 40 db/dec ( frm 0 db/dec t 60 db/dec), the the actual gai is db dw at 0.5 c ad c.. auril alg ilters ev. 3/4/003 Page 9

11 structi f verall Bde plt phase respse alg ilters Hz 6.8 Hz 380 Hz 68 Hz -45 / dec 0-45 / dec / dec 0 Overall phase respse Bde apprximati -45 / dec / dec / dec decade.64 decade decade -80 T cstruct the verall phase respse simply add the slpes i each regi f the frequecy respse d add the idividual phase agles t cmpute the L ad the H plateaus. The actual r exact phase respse always cuts iside the breakpits ad is 5.7 ff if the breakpits are at least e decade apart ad if the chage i slpe is 45 /dec. If the slpe chage is 90 /dec, say frm 45 /dec t 35 /dec, the the actual phase is.4 ff iside the breakpit.. auril alg ilters ev. 3/4/003 Page 0

12 ctual frequecy respse ad Bde apprximati alg ilters Gai espse 0 Gai (db) -0 ` requecy (Hz) Bde apprximati xact respse Phase espse 0 Phase () requecy (Hz) Bde apprximati xact respse s yu ca see, the phase respse has a Bde apprximati that is very clse t the actual respse except at the first ad last breakpits where it is ff by auril alg ilters ev. 3/4/003 Page

13 4. ecd-rder bad-pass filter alg ilters i() Thevei X () TH i ( ) i ( ) TH is the pe circuit vltage Z TH X Z TH is derived by replacig the surce with a shrt circuit a ideal vltage surce has zer iteral resistace. i TH () ( ) () ( ) ( ) TH i ( ) ( ) ( ) i ( ) ( ) ( ( ( ) ) ( ( ) ( ) ( ) i ( ) ( ) rm the abve T we ca derive the fllwig desig equatis: ce 0 max. auril alg ilters ev. 3/4/003 Page max ω ω ω max ω max ω ce max T d the Bde plt f the abve T, it is easier if e fids the rts f the demiatr ad expresses the T as shw beside. OT: the 3 db cutff frequecies are equal t the ple frequecies ly if they are at least decades apart. ( ) i ( ) ( ) ce ( p ) ( p ) ω max max gai at ce ω ce ω ω ω p ω p ω ω is the 3 db BW

14 Bad-pass filter qual s ad s alg ilters ( ) ( ) i ( ) 0 3 max ω ω ω ce p 0 LOG(0.38) p Z 0 Gai (db) p 0.38/(π) 3 db badwidth 3/(π) p 0.38/(π) 0 LOG(/3) at ce /(π) 0 db/dec -0 db/dec requecy (Hz) PH () phase shift at ce requecy (Hz) tice that the 3 db atteuati frequecies are t equal t the ple frequecies, this ccurs whe the ple frequecies are t far eugh frm each ther whe the ple frequecies are at least.5 t decades apart, fr uequal s ad s, the they crrespd t the 3 db atteuati frequecies which are rmally used t defie the badwidth. ls tice that the exact gai is /3 / ad the phaseshift is exactly 0 0 at the ceter frequecy which is give by ce /(π). T fid the 3 db atteuati frequecies e must slve the fllwig equatis: 3 ω ω ω ad ω ω ω ω ω ω ω 0 lve fr the rts f the quadratic equati ad btai ω ad ω use abslute value if rt is ve.. auril alg ilters ev. 3/4/003 Page 3

15 Bad-pass filter desig example alg ilters i() () Desig a secd-rder bad-pass filter with cutff frequecies f khz ad 9 khz ad a maximum gai f 0./. Oce desiged, fid the trasfer fucti ad the calculate the ples ad graph the actual gai respse ad the Bde apprximati. Use the fllwig desig equatis. max ω max ω ce max ce 0 max ω 9k k 8k ce ( ) / (9k*k) / 3 khz 3k 0. π µ k ( π 3000) 99.47µ 0. π 8000 std the std the std the std * 8.8µ/ 8.8µ/ ial selecti: k 6. ( ) max ω ω ω ce rm the T cefficiets, we have the fllwig results: ce π 988 Hz 00*3 *6k * Hz π max 0.0 / ctual cutff frequecies (at 3 db atteuati pits): slve fr rts f Hz ad Ples ad zers: z 0 p p Ple frequecies: Hz Hz Hz. auril alg ilters ev. 3/4/003 Page 4

16 We ca re-write the T i terms f the ples fud abve: alg ilters ( ) ( ) øöçèæ ( ) ( ) 6 ( ) 0dB Hz Hz -0 db/dec dB -0 db/dec -.7 db ( ) 6 0 db/dec -.7 db esultig Bde plt -0 db/dec G I (db) Hz -.7 db Hz db Hz 988 Hz Hz 0 db/dec -0 db/dec requecy (Hz) Bde apprx xact Gai. auril alg ilters ev. 3/4/003 Page 5

17 5. ecd-rder lw-pass filter L circuit alg ilters i() 4.47 L.5 mh 0. u () ( ) ω ( ) where ζ ad ω 684 r / s ad L gai ζω ω Q L ( ) ( ) i ( ) Trasfer ucti L L L L π L π 0 khz.5m 0.µ ζ ω L m 0. Q 5 ζ Ples ad zers The abve T has tw cmplex ples (secd-rder demiatr) which will result i frequecy respse quite differet frm the d rder T see befre with the duble etwrk where the ples were real. Usig the equati t fid the rts f the T demiatr, we have: 9 57 ± p ad p ± j / ± If yur calculatr des t hadle cmplex square rts, calculate ζ ad ω frm the T dematr cefficiets ad the use the fllwig frmulas: b ± b 4ac p, ± a If ζ <, the ples are cmplex ω /80 a cs( ) ζ 4, b ± b ac ± ζ a If ζ >, the ples are real ζω ω Defiiti f ζ ad ω p ω is the atural udamped frequecy f scillati dampig ccurs whe ζ 0 r 0 - that is the frequecy at which the circuit wuld aturally scillate if there were lss f eergy i the circuit. ζ is called the dampig factr ad is a idicati f the eergy lss i the circuit ad as we will see later, it determies hw fast scillatis are damped r dimiished.. auril alg ilters ev. 3/4/003 Page 6

18 alg ilters requecy respse ( ) ( jω) ω ζω ω ω ( ω ω ) ( ζω ω) ( jω) ω ( jω) ζω jω ω ( ω ω ) j( ζω ω) ζω ω / ( jω) T ω ω ω Usig the abve frmulas, we ca plt the gai ad phase respse f a d rder lw-pass filter. 0 Gai espse esat peaks Gai (db) db/dec asymptte fr ζ > ly 000 Hz H asymptte -40 db/dec requecy (Hz) ζ 0. ζ 0. ζ 0.5 ζ ζ ζ PH () Hz θ -90 Phase espse Bde asymptte except fr ζ >. Trasiti gets steeper as ζ gets smaller requecy (Hz) ζ 0. ζ 0. ζ 0.5 ζ ζ ζ. auril alg ilters ev. 3/4/003 Page 7

19 Bde Plt alg ilters Oly the gai respse has a Bde plt apprximati which has a break frequecy equal t, that is at the resat frequecy f the L circuit. fter, the Bde plt rlls dw directly at 40 db/dec withut gig thrugh a itermediary slpe f 0 db/dec whe ζ <. If ζ >, the the ples f the T are real ad determie the tw break frequecies f the Bde plt ad we have tw slpes, -0 db/dec ad 40 db/dec. The phase respse will have a Bde plt apprximati ly if ζ >, therwise the actual phase respse has t be cmputed ad pltted with a cmputer r a graphig calculatr. esat Peak The gai respse exhibits a peak ly whe ζ < T fid the peak f ( jω) ad differetiate it wrt ω ad equate the derivative t zer ad slve fr ω. 4 ω d d ( jω) 0 slve fr ω ω peak ω dω dω ( jω peak ) Peak( db) 0 LOG( ζ ζ ) ζ ζ, we ca use ( jω) ( ω ω ) ( ζω ω) The abve equati gives the magitude f the peak abve the L gai ( ). ζ If we measure the actual peak we ca the crrelate it t a ζ value by slvig the peak (db) equati fr ζ ad we btai: ζ meas 0 0. peak ( db) r small ζ values that is ζ < 0.3 the peak(db) equati ca be apprximated with: ( ζ ) 0 LOG( ζ ) 0 LOG( Q) Peak ( db) 0 LOG Measuremet f We ca measure by measurig the frequecy at which the phase shift betwee i ad is 90. We cat use the 3 db atteuati methd used fr st rder filters t measure the cutff frequecy because the actual gai at depeds the dampig factr ad is give by frequecy util the gai reaches /(ζ) that frequecy will be. ( jω peak ). If we vary the ζ ( jω) ( jω ) ( jω) ζω jω ω ( jω ) ω jζω ω ω j jζ ζ ( jω ) / 90 ζ 0 Q / 90 ω 0 ζω jω ω ω ω jζω ω ω. auril alg ilters ev. 3/4/003 Page 8

20 6. ecd-rder high-pass filter L circuit alg ilters i() u L.5 mh () ( ) ( ) where ζ ad ω 684 r / s ad H gai ζω ω Q L ( ) ( ) i ( ) Trasfer ucti L L L L π L π 0 khz.5m 0.µ ζ ω L m 0. Q 5 ζ Ples ad zers The abve T has tw zers ad tw ples, that is tw rts i the umeratr ad tw rts i the demiatr. The zers are: 0, z z 0 r Z Z 0 The tw cmplex ples will result i frequecy respse quite differet frm the d rder T see befre with the duble etwrk where the ples were real. Usig the equati t fid the rts f the T demiatr, we have: 9 57 ± p ad p ± j / ± If yur calculatr des t hadle cmplex square rts, calculate ζ ad ω frm the T demiatr cefficiets ad the use the fllwig frmulas: If ζ <, the ples are cmplex Ple-zer diagram b ± b 4ac p, ω /80 ± a cs( ζ ) a If ζ >, the ples are real p 4, b ± b ac ζω ± ω ζ a p Imag j657 graphical represetati f the ples ad zers i the cmplex plae is called a ple-zer diagram z z eal p -j657. auril alg ilters ev. 3/4/003 Page 9

21 requecy respse Usig the frmulas beside, we ca plt the gai ad phase respses ( ) ( jω) ζω ω ω ( ω ω ) ( ζω ω) ( jω) Gai espse ( jω) alg ilters ( jω) ζω jω ω ( ω ω ) j( ζω ω) / ( jω) 80 ω ζω ω T ω ω 0 H asymptte 40 db/dec esat peaks Gai (db) Hz 0 db/dec asymptte fr ζ > ly requecy (Hz) ζ 0. ζ 0. ζ 0.5 ζ ζ ζ PH () Hz θ 90 Phase espse Bde asymptte except fr ζ >. Trasiti gets steeper as ζ gets smaller requecy (Hz) ζ 0. ζ 0. ζ 0.5 ζ ζ ζ. auril alg ilters ev. 3/4/003 Page 0

22 Bde Plt alg ilters Oly the gai respse has a Bde plt apprximati which has a break frequecy equal t, that is at the resat frequecy f the L circuit. Befre, the Bde plt rlls dw directly at 40 db/dec withut gig thrugh a itermediary slpe f 0 db/dec whe ζ <. If ζ >, the the ples f the T are real ad determie the tw break frequecies f the Bde plt ad we have tw slpes, 0 db/dec ad 40 db/dec. The phase respse will have a Bde plt apprximati ly if ζ >, therwise the actual phase respse has t be cmputed ad pltted with a cmputer r a graphig calculatr. esat Peak The gai respse exhibits a peak ly whe ζ < T fid the peak f ( jω) ad differetiate it wrt ω ad equate the derivative t zer ad slve fr ω. d ( jω) dω d ω ( ω ω ) ( ζω ω) dω 4 0 slve fr ω ( ζ ) ( jω peak ) Peak( db) 0 LOG ζ ζ ζ The abve equati gives the magitude f the peak abve the H gai ( ). ω, we ca use ( jω) peak ω ζ If we measure the actual peak we ca the crrelate it t a ζ value by slvig the peak (db) equati fr ζ ad we btai: ζ meas 0 0. peak ( db) r small ζ values that is ζ < 0.3 the peak(db) equati ca be apprximated with: ( ζ ) 0 LOG( ζ ) 0 LOG( Q) Peak ( db) 0 LOG Measuremet f We ca measure by measurig the frequecy at which the phase shift betwee i ad is 90. We cat use the 3 db atteuati methd used fr st rder filters t measure the cutff frequecy because the actual gai at depeds the dampig factr ad is give by frequecy util the gai reaches /(ζ) that frequecy will be. ( jω peak ). If we vary the ζ ( jω) ( jω ) ( jω) ζω jω ω ( jω ) ω jζω ω ω j jζ ζ ( jω ) / 90 ζ 0 Q ω / 90 0 ζω jω ω ω ω jζω ω ω. auril alg ilters ev. 3/4/003 Page

23 alg ilters TIO QUY PO O BI MPLII. OP-LOOP PO O OP MP : : 3 db atteuati cutff frequecy lw frequecy r D differetial gai G I (db) 0 Lg( ) ctual gai 3 db ( ) ω 0 db c -0 db/dec GBW (rad/s) Mst p amps exhibit the abve frequecy respse because they are iterally stabilised r cmpesated t prevet self-scillatis whe egative feedback is used. This stability measure is geerally called frequecy cmpesati. tability f the egative feedback lp is achieved i mst p amp circuits by rllig dw the gai respse at -0 db/dec right thrugh the 0 db level. I sme circuits, this cmpesati methd is t sufficiet t stabilise agaist self-scillatis. Bde plt ( jω) jω ω ( j) j L asymptte: <<, 0 lg () H asymptte: >>, /j -0 db/dec The tw asympttes itersect at the frequecy where the gais give by the abve equatis are equal, that is: / OT: magitude f j is simply. Gai-badwidth prduct The H asymptte is / cs tat The abve result shws that the gai x frequecy prduct is a cstat whe a pit lies the -0 db/dec part f the respse. This result ly ccurs whe the slpe f the respse is exactly -0 db/dec, t 0 db/dec, t -40 db/dec, etc. The cstat gai-frequecy prduct is usually called the gai-badwidth prduct r GBW i shrt.. auril alg ilters ev. 3/4/003 Page

24 xample: 74 p amp alg ilters Op amp data: GBW MHz typical, L gai / typical. ) Draw the pe lp respse f the p amp. irst calculate all relevat pits f the respse. L gai: 0 lg (0 5 ) 00 db utff frequecy: x 0 5 GBW M, therefre M /0 5 0 Hz. 0 db crssig: x 0dB GBW M, therefre 0dB M/ MHz. 00 G I (db) db/dec K 0K 00K M (Hz) B) id the p amp gai at khz, 0 khz ad 35 khz usig the GBW prduct. Gai at khz : GBW/ M/k 000 r 60 db Gai at 0 khz : GBW/ M/0k 00 r 40 db Gai at 35 khz : GBW/ M/35k 8,57 r 9, db OT: r mst p amps the gai respse has the abve shape but the umbers will be differet. The decade: the decade is a lgarythmic uit f frequecy spa defied as fllws: #dec lg - lf lg ( / ) If 00 Hz ad 00 khz, #dec lg(00k/00) lg(000) 3 decades If 33 Hz ad 455 Hz, #dec lg(455/33) lg(3,79),4 decades xample: If we have tw pits lyig a slpe f -40 db/dec ad e f the pits lies at (0 Hz, 4 db), determie the gai f the secd pit lyig at 800 Hz. #dec lg(800/0),76 47,04 db 4 db -40 db/dec slpe db/ dec, therefre dbslpe x dec-40 db/dec x,76dec db-47,04 db at 800 Hz 4-47,04-3,04 db 0 Hz,76 decade 800 Hz -3,04 db. auril alg ilters ev. 3/4/003 Page 3

25 . LOD-LOOP PO O O-ITIG MPLII alg ilters i X - I 0 ideal β β I G I (db) 0 LOG( ideal) -0 db/dec p amp BW (Hz) s will be shw right after, the clsed-lp (with feedback) gai respse f the -ivertig amplifier is as shw abve where the badwidth f the amplifier is give by the fllwig expressi: BW β x GBW The abve frmula als applies t all p amp circuits with resistive feeback etwrks as will be shw later. Derivati f badwidth frmula I the "Itr t eedback" secti we derived the actual clsed-lp gai f the -iv amp t be: β β where is the pe-lp gai f the p amp. The asympttes f are determied as fllws: L asymptte: is csidered very large such as t have β >> /, therefre /β Ideal gai 0 lg ( β ) H asymptte: is csidered very small such as t have β << /, therefre Ope-lp gai -0 db/dec The itersecti pit f the tw asympttes will determie the BW f the circuit. t this pit we have /β ideal, ad sice this pit lies the -0 db/dec slpe f, we ca use the GBW f the p amp t fid the badwidth frequecy. BW GBW / GBW / (/β ) β GBW De! rm this basic result we ca w explai the rigi f the ame "gai-badwidth prduct". GBW BW / β BW x ideal BW x ( / ) rm the abve expressi we ca see that the prduct f the ideal gai times the badwidth f the circuit is a cstat duly amed "gai-badwidth prduct".. auril alg ilters ev. 3/4/003 Page 4

26 alg ilters umerical xample: -ivertig amplifier respse L 347 p amp is used i the circuit shw beside. Draw the typical pe-lp ad clsed lp gai respses fr feedback resistr values f 0K ad 00K. i L347 Op amp data: GBW 4 MHz typical L 0 db typical K ) Ope-lp respse db 0 ( ) 368 /, GBW/ 4M / 368,65 Hz 0 db crssig 0dB GBW/ ( Hz) GBW 4 MHz B) lsed-lp respse K β K 0K 6 β K K 00K 5 ideal 0K K 6 / r 5,56 db BW 4M 666,6 khz 6 00K ideal K 5 / r 34,5 db BW 4M 78,43 khz 5 G I (db) 0 db 00K 0K -0 db/dec p amp 34,5 db 5,56 db,6 78,4K 667K 4M (Hz) OT: The actual gai at the badwidth frequecy is 3 db dw frm the ideal gai. clusi: BW β x GBW GBW/ ideal This frmula ad the abve example shw that if we wat mre gai frm the amplifier this is de at the cst f a reduced badwidth. This result is geerally true fr all types f amplifiers.. auril alg ilters ev. 3/4/003 Page 5

27 3. OP MP IUIT WITH ITI DBK TWOK alg ilters It ca be shw that the actual gai f ay p amp circuit with a resistive feedback etwrk is give by the fllwig expressi whse Bde plt is shw as well. actual ( ideal β ) β The last term i the abve equati is the same as that f the -ivertig p amp ad it is multiplied by a cstat term ( ideal β ), therefre the badwidth f ay circuit with resistive feedback is give by the same expressi btaied with the -ivertig amplifier, that is: BW β x GBW where β must be calculated by determiig the rati / while replacig all surces by their iteral resistaces ad als by peig the feedback lp. OT: Ideal vltage surce it 0, ideal curret surce it. xample: ummig amplifier gai respse Determie the pe-lp ad the clsed-lp gai respses f the circuit shw beside assumig that the p amp has a GBW 5 MHz ad a D gai f 00 db. K 0K,8K 00K Ideal utput: r 34 db, -5 r 4 db alculati f β K 00K K 0K K 0K 00K 56 0, K - 0 X cut lp β /56 0,07857,8K Badwidth BW β x GBW (/56) 5M 89,3 khz. auril alg ilters ev. 3/4/003 Page 6

28 Gai respse alg ilters G I 00 db -0 db/dec p amp (db) 34 db 4 db / / 34,96 db 0 db 50 BW 89,4K 5M (Hz) tice that eve if ad are differet, the badwidths fr bth gais are the same which differs frm what we previusly btaied fr the -ivertig amplifier whse β chaged fr differet gais while it is the same here fr bth gais. ls tice that the clse-lp respses d t merge with the pe-lp respse f the p amp - there is a 0,96 db gap betwee ad. This ca be readily verified by calculatig f the p amp at 89,4 khz usig its GBW figure. 4. TIT PO O MPLII TO QU W Whe a square wave is applied t the iput, the p amp will utput a square wave with expetial risig ad fallig edges if the utput peak amplitude des t exceed /(4πBW). The 0%-90% rise ad fall times are the give by (t) 0 90% 0% P t - P t t 0,35/BW t r t f If P >> /(4πBW), the the edges f the utput square wave will be etirely liear ad the 0%-00% rise ad fall times will be give by t t PP / - if the egative ad psitive slew rates are differet the t t. (t) 0 P lpes equal t slew rate f p amp t - P t r t f OT: Whe psitive feedback is used r if the O/P is drive hard it saturati, the p amp will always switch at the slew rate ad will prduce liear edges at the utput. Whe feedback is preset, if the iput differetial vltage is large eugh (abut 00 m) the utput will als switch at the slew rate f the p amp.. auril alg ilters ev. 3/4/003 Page 7

29 TIO 3. irst-order ilter ircuits TI ILT alg ilters irst-rder -ivertig lw-pass filter irst-rder ivertig lw-pass filter i i x Trasfer fucti ( ) ( ) ilter parameters L ω Desig prcedure. elect stadard value. alculate ( ω ) ( ) 3. T btai desired gai calculate rati ad t miimize D O/P ffset ( ) vltage we must have. T meet bth f the abve cditis we have the fllwig: ( ) slve fr ( the.) Trasfer fucti ( ) ilter parameters L ω Desig prcedure. elect stadard value. alculate ( ω ) 3. alculate desired gai. ( ) t btai the 4. T miimize D O/P ffset vltage we must have. X elect clsest stadard value. Try several stadard values f arud (the) t achieve the desired rati ( ) as accurately as pssible with stadard values f ad. OT: If the p amp has T I/P's, it is t ecessary t balace the iputs fr miimum O/P ffset vltage, therefre step 3 takes it accut ly the rati ( ) OT: the ivertig filter allws fr gais less tha v/v which cat be achieve with the -ivertig cfigurati. OT: If the p amp has T I/P's, it is t ecessary t balace the iputs fr miimum O/P ffset vltage, therefre X is t ecessary.. auril alg ilters ev. 3/4/003 Page 8

30 alg ilters irst-rder -ivertig high-pass filter irst-rder ivertig high-pass filter i i B Trasfer fucti ( ) ( )( ) ( ) ilter parameters H ω Desig prcedure. elect stadard value. alculate ( ω ) ( ) 3. T btai desired gai calculate rati ad t miimize D O/P ffset ( ) vltage we must have. T meet bth f the abve cditis we have the fllwig: Trasfer fucti ( ) ( )( ) ( ) ilter parameters H ω Desig prcedure. elect stadard value ( ). alculate ( ω ) 3. alculate t btai the desired gai. 4. T miimize D O/P ffset vltage we must have. B ( ) slve fr ( the.) elect clsest stadard value. Try several stadard values f arud (the) t achieve the desired rati ( ) as accurately as pssible with stadard values f ad. OT: If the p amp has T I/P's, it is t ecessary t balace the iputs fr miimum O/P ffset vltage, therefre step 3 takes it accut ly the rati ( ) OT: the ivertig filter allws fr gais less tha v/v which cat be achieve with the -ivertig cfigurati. OT: If the p amp has T I/P's, it is t ecessary t balace the iputs fr miimum O/P ffset vltage, therefre B is t ecessary.. auril alg ilters ev. 3/4/003 Page 9

31 . OD-OD ILT IUIT alg ilters GL LOW-P LL-KY ILT i Trasfer fucti GG G G G ( ) GG 3 4 Parameter equatis ω 3 4 ζ 0.5 ab b a ( ) a where a adb 3 4 The abve equatis ca be used with ay f the fllwig specific lw-pass alle-key cfiguratis where cstats a, b ad differ frm e cfigurati t ather. T desesitize the filter t the p amp GBW, shuld, as a rule f thumb, make GBW 00 Q. The larger the GBW f the p amp is, the better, fr all types f filters - lw-pass, high-pass, etc. T this effect, maucfacturers f active filter I's fte specify a maximum Q X prduct that shuld t be exceeded i ay give applicati. B GL HIGH-P LL-KY ILT Trasfer fucti i G 3 ( ) G4 G3G 4 Parameter equatis ω ζ 0.5 ab ( ) b a 34. auril alg ilters ev. 3/4/003 Page 30 a where The abve equatis ca be used with ay f the fllwig specific high-pass alle-key cfiguratis where cstats a, b ad differ frm e cfigurati t ather. T desesitize the filter t the p amp GBW, shuld, as a rule f thumb, make GBW 00 Q. The larger the GBW f the p amp is, the better, fr all types f filters - lw-pass, high-pass, etc. T this effect, maucfacturers f active filter I's fte specify a maximum Q X prduct that shuld t be exceeded i ay give applicati. a ad b 4 3

32 LOW-P LL-KY WITH MTHD 's D UMTHD 's alg ilters Trasfer fucti i GG G ( ) G G G where 3 4 Parameter equatis ω ζ 0.5 a a 3 ( ) where a ad b 4 Desig prcedure Give, ω ad ζ,. elect stadard value fr. alculate frm ( ζ ζ ( ) ) 3. alculate frm ω ( ) 4. T btai desired gai calculate rati ( ) must have ( ω slve fr ) ad t miimize D O/P ffset vltage we. T meet bth f the abve cditis we have the fllwig: ( the.) ad calculate ( ) i rder t achieve the desired rati ( ) pssible. as accurately as OT: If the p amp has T I/P's, it is t ecessary t balace the iputs fr miimum O/P ffset vltage, therefre step 4 takes it accut ly the rati ( ). auril alg ilters ev. 3/4/003 Page 3

33 D HIGH-P LL-KY WITH MTHD 's D UMTHD 's alg ilters Trasfer fucti i G3( ) G where 4 G3G 4 Parameter equatis b ( ) ω ζ b 3 4 where a ad b 4 3 Desig prcedure Give, ω ad ζ,. elect stadard value fr. alculate 3 frm 3 ( ζ ζ ( ) ) 3. alculate frm 4 ω ( ) 4. T btai desired gai calculate rati ( ) must have ( ω slve fr ) 3 ad t miimize D O/P ffset vltage we. T meet bth f the abve cditis we have the fllwig: ( the.) alculate ( ) i rder t achieve the desired rati ( ) pssible. as accurately as OT: If the p amp has T I/P's, it is t ecessary t balace the iputs fr miimum O/P ffset vltage, therefre step 4 takes it accut ly the rati ( ). auril alg ilters ev. 3/4/003 Page 3

34 LL-KY WITH MTHD 's D 's alg ilters Lw-pass filter High-pass filter i Trasfer fucti GG G ( ) G G G i Trasfer fucti G3( ) G 4 G3G 4 Parameter equatis eferrig t the geeral equatis we have a ad b fr bth circuits, therefre the fllwig equatis apply t bth circuits. ω ζ ( 3 ) Give ω ad ζ, Desig prcedure. elect stadard value fr. alculate /( ω ) 3. T btai desired gai calculate rati ( ) ( ζ ) vltage we must have ( ) slve fr ( the.) ad t miimize D O/P ffset. T meet bth f the abve cditis we have the fllwig: ad calculate ( ) desired rati ( ) as accurately as pssible. i rder t achieve the OT: If the p amp has T I/P's, it is t ecessary t balace the iputs fr miimum O/P ffset. vltage, therefre step 3 takes it accut ly the rati ( ). auril alg ilters ev. 3/4/003 Page 33

35 UITY-GI LL-KY alg ilters Lw-pass filter High-pass filter i i Trasfer fucti Trasfer fucti i i Parameter equatis Parameter equatis ω ζ ω ζ Desig prcedure Give, ω ad ζ,. elect stadard values fr 3 ad 4 i rder t achieve accurate rati.. alculate /( ω 3 ) ζ 3. fr miimum O/P ffset vltage - select clsest stadard value. If p amp has T I/P's is t eeded. Desig prcedure Give, ω ad ζ,. elect stadard value fr.. alculate ( ζω ) 3. alculate ζ 4. 4 fr miimum O/P ffset vltage - select clsest stadard value. If p amp has T I/P's is t eeded. OT: The uity-gai alle-key circuit is the cfigurati with ζ beig the least sesitive t cmpet variatis f all pssible alle-key cfiguratis ad therefre shuld be used whe greater accuracy is eeded fr ζ - eve if verall filter gai is t 0 db, all that is eeded fr the extra gai is a plai -ivertig r ivertig amplifier.. auril alg ilters ev. 3/4/003 Page 34

36 G MULTIPL DBK ILT alg ilters Geeral trasfer fucti () s Y Y 3 4 Y 5 Y Y ( Y Y Y Y ) Where Y k are admittaces f the cmpets. Y G / fr a resistr Y fr a capacitr Type Of ilter Y Y Y 3 Y 4 Y 5 Lw-pass G G 3 G 4 5 High-pass G 3 4 G 5 Bad-pass G G G 5 ilter parameters Type Of ilter ω ζ Lw-pass High-pass Bad-pass G3G G 5 4 G G3 G4 ω G G5 ( 3 4 ) ( G G ) G5 3 ω G G G 3. auril alg ilters ev. 3/4/003 Page 35

37 alg ilters 3. TIT PO O IT D OD-OD ILT Iput utput st rder lw-pass filter If << squarewave gets thrugh uatteuated ad edges are expetial τ π t r t f 0.35 If >> squarewave is heavily atteuated ad distrted. ctually, the filter w itregrates the square wave ad it becmes triagular. The wavefrm shw beside is frm a PP I/P squarewave ad a filter gai f / O/P wavefrms fr tw differet frequecies st rder high-pass filter If >>, the square wave gets thrugh uatteuated but may have sme tilt at the peaks if is t high eugh. π ( e ) 00 % TILT O/P wavefrms fr tw differet frequecies Iput utput If <<, the edges f the I/P squarewave g thrugh but tp ad bttm f squarewave I/P are blcked by HP. xpetial spikes have fllwig parameters: τ π. auril alg ilters ev. 3/4/003 Page 36 t r t f 0.35

38 alg ilters O/P wavefrms fr tw differet dampig factrs d rder lw-pass filter If <<, squarewave gets thrugh uatteuated ad rigig ccurs if ζ<. mplitude f scillatis (rigig) decays expetially. e P settlig time t τ t t % 0.% τ τ L τ ( ζω ) ( 00) L( 000) If >>, I/P squarewave is heavily atteuated ad turs it a parablic wave that resembles a siewave. The higher is, the smaller the O/P amplitude O/P wavefrms fr tw differet frequecies d rder high-pass filter If >>, the square wave gets thrugh uatteuated but may have sme tilt at the peaks if is t high eugh O/P wavefrms fr tw differet frequecies If <<, edges f I/P squarewave get thrugh uatteuated but tp ad bttm f I/P squarewave are blcked by HP ad rigig ccurs if ζ <. mplitude f scillatis (rigig) decays expetially. e t τ P τ settlig time t τ L 00 % ( ζω ) ( ) O/P wavefrms fr tw differet dampig factrs. auril alg ilters ev. 3/4/003 Page 37

39 alg ilters 3. th Order ilters Oe eeds t use a high-rder filter i rder t btai mre atteuati past the cutff frequecy f the filter. Geerally, the rllff slpe fr a lw-pass filter f rder will be x 0 db/decade whe frequecy is at least half a decade away frm. lat respse Gai respse is the flattest f all filters. Trasiet respse exhibits sme rigig. Butterwrth Gai espse Butterwrth Uit-tep espse ipple i Pass-bad Gai respse exhibits ripple. ilter ca be desiged t specified ripple value. The mre ripple, the mre rigig i trasiet respse hebychev Gai espse hebychev Uit-tep espse Bessel has the best trasiet respse due t a flat delay respse- ee ext page Bessel Gai espse Bessel Uit-tep espse. auril alg ilters ev. 3/4/003 Page 38

40 alg ilters Bessel filters have by far the flattest delay respse which results i the least distrti f sigals see previus page fr trasiet respse t a step iput. Phase respse f 4 th rder filters Delay respse f 4 th rder filters Gai respse f 4 th rder filters Trasfer uctis Gai respse f each d rder stage f a 0th rder hebychev filter hebychev filters prvide mre atteuati tha Bessel r Butterwrth but als distrt the sigals the mst. We ca see that a 0 th rder hebychev filter made up f five d rder stages each havig their w resat peak will result i five peaks i the verall respse. th rder lw-pass filter ( s) (s) i (s) (s) ( s) (s) i L ( b b! b b b ) ( )( )(... )( ) p( ) th rder high-pass filter ( s) (s) K p( ) b b p i (s) p H ( b b! b b b ) Demiatr has plymial frm Demiatr has prduct frm where pk are the ples f the filter Demiatr has plymial frm (s) ( s) (s) i H ( )( )(... )( ) p( ) p( ) p p Demiatr has prduct frm where pk are the ples f the filter. auril alg ilters ev. 3/4/003 Page 39

41 alg ilters urth-rder lw-pass hebychev i irst d rder stage ecd d rder stage ( s) i ( s) ( s) ω ζ ω ω ω ζ ω ω L T I G I (db) TG- TG- OLL ζ ζ db ripple LTI QUY ( / c) hebychev filters are stagger-tued which meas that each stage is tued at differet value. By selectig apprpriate ζ values, e btais a equiripple gai respse which meas that the tw humps i the respse are equal ( db i the abve example). L T I G I (db) TG- TG- OLL db/dec -80 LTI QUY ( / c) -80 db/dec The H atteuati rate is - x 0 db/dec, but is higher ear the cutff frequecy. The mre pass-bad ripple, the mre the atteuati rate (r slpe) ear. Mre atteuati is btaied at the expese r mre distrti f the sigal.. auril alg ilters ev. 3/4/003 Page 40

42 alg ilters urth-rder Butterwrth lw-pass filter L T I G I (db) -6-8 ζ TG- TG- OLL ζ db LTI QUY ( / c) Butterwrth lw-pass ad high-pass filters are sychrus which meas that each stage is tued at the same frequecy. I Butterwrth filters the chice f ζ values prduces a very flat gai respse ad a 3 db atteuati at the cutff frequecy. urth-rder Butterwrth lw-pass filter 0 L T I G I (db) db/dec TG- -80 db/dec TG- OLL LTI QUY ( / c) The H atteuati rate is - x 0 db/dec, The atteuati rate ear is less tha x 0 db/dec ad is therefre t as great as fr a hebychev filter but is higher tha a Bessel filter. OT: requecy axis is t lgarythmic, this meas that the graphs d t shw slpes i db/dec.. auril alg ilters ev. 3/4/003 Page 4

43 alg ilters DIG XMPL LOW-P BUTTWOTH ILT. max Desig a lw-pass Butterwrth filter that meets the gai respse curve shw beside. G I (db) max mi 0 db mi atteuati. Miimum rder required (Hz) mi lg lg max s lg lg 500 c. ircuit diagram 3.38 Therefre we will use 4, c 500Hz ad LP 6 db. T miimize the umber f stages we must select the miimum reasable rder required, that is the ext higher iteger abve 3.38, which is 4. If it wuld have bee clser, say 3.5 t 3.99, it wuld have bee wiser t use 5 because f cmpet tlerace ad variatis with temperature. The alle-key circuit with umatched 's is chse because it allws fr a specific gai. i 3. ilter parameters ad trasfer fucti rm the rmalized lw-pass filter tabe we btai the fllwig filter parameters. OD( 4 ) LP /LP LP ω c LP ω LP ζ O /LP first stage ± /± secd stage ± /± ( ) ω ζ ω ω ω ζ ω ω auril alg ilters ev. 3/4/003 Page 4

44 alg ilters 4. alculati f filter cmpets irst stage 4.467, ζ , ω ) 6 stadard B) ω ( ζ ζ ( ) ) 39775Ω ) ( ω ) Ω D) fr desired gai ad Ω t miimize D O/P ffset vltage. T meet bth cditis we must have X Ω Ω elected cmpets: 6, Ω, 39775Ω, 64068Ω ad 4Ω ecd stage 4.467, ζ , ω ) 36 stadard B) ω ( ζ ζ ( ) ) 48Ω ) ( ) 30 ω D) fr desired gai ad Ω t miimize D O/P ffset vltage. T meet bth cdtis we must have X Ω Ω elected cmpets: 36, 30Ω, 48Ω, 35433Ω ad 847Ω. auril alg ilters ev. 3/4/003 Page 43

45 alg ilters IL IUIT i irst stage cmpet tlerace 0.35% ecd stage cmpet tlerace 0.88% fr ζ < 5% ad < % fr ζ < 5% ad < % TYPIL LOW QUY GI PO G I (db) K QUY (Hz). auril alg ilters ev. 3/4/003 Page 44

46 TYPIL WID G GI PO alg ilters G I (db) K 0K 00K M 0M QUY (Hz) TYPIL LOW QUY GOUP DLY PO.50m D L Y.0m 0.90m (sec) 0.60m 0.30m 0.00m K 5K QUY (Hz). auril alg ilters ev. 3/4/003 Page 45

47 G I (db) alg ilters DIG XMPL HIGH-P BUTTWOTH ILT. Desig a high-pass Butterwrth filter that meets the gai respse shw belw. 6 3 max mi TUL PO IDL PO 4 db mi atteuati BW BW (Hz) s we ca see frm the abve gai respse, a true high-pass filter is impssible t be implemeted with active filters because f the p amps' pe lp respses which cut it the ideal respse thus creatig high frequecy break pits i the actual respse. We ca determie thse break pits by assumig all capacitive reactaces f the circuit t be ull ad the aalizig fr β v (H) f each stage after which we ca calculate BW GBW* β v (H), BW GBW* β v (H), etc.. Miimum rder required mi lg lg max lg lg 400. ircuit diagram We will use 5, c 800 Hz ad LP 6 db. T miimize the umber f stages we must select the miimum reasable rder required, 4 wuld be t clse, let 5 which allws sme leeway fr cmpet tlerace ad variatis with temperature. The alle-key circuit with matched 's ad 's is chse because it prvides a easy desig prcedure with a lcked gai ad the first rder stage gai is a free parameter ad will thus prvide the extra gai required t achieve the verall gai.. auril alg ilters ev. 3/4/003 Page 46

48 3. ilter parameters ad trasfer fucti alg ilters rm the rmalized lw-pass filter table we first btai the rmalized lw-pass ples ad the cvert them t rmalized high-pass ples ad fially we de-rmalize high-pass ples. OD ( 5 ) first stage secd stage third stage LP ± ± / LP HP LP / /± /± HP ω c HP / /± /± ω HP ζ O/ HP / ( ) ( ) ζ ω ω ζ ω alculati f filter cmpets 3 ω ω ecd stage 3-ζ.380, ζ , ω ) 39 stadard B) ω ( ) 50 Ω ). 38 fr desired gai ad 0. K t miimize D O/P ffset vltage. T meet bth cditis we must have 7.58K.38 0.K Ω elected cmpets: 39, 50, 7580 ad 496 Third stage 3-ζ.380, ζ , ω ,B) ad will be the same fr all stages because ω is the same fr all three stages. ) fr desired gai ad 0. K t miimize D O/P ffset vltage. T meet bth cditis we must have K 36.9K Ω 0 elected cmpets: 39, 50, ad auril alg ilters ev. 3/4/003 Page 47

49 irst stage 6.05, ω alg ilters,b) ad will be the same fr all stages because ω is the same fr all three stages. ) The first stage gai makes up what is missig t achieve a verall gai f 0 db r 0 /. te that a first rder stage des t have ay dampig (ζ) specified which meas that its gai is a free parameter here determied by the desired verall gai. vtt ( )( ) v fr v v desired gai ad 50 t miimize D O/P ffset vltage. T meet bth cditis we must have 609Ω We w try stadard values f arud 609_ ad such that because the rati is mre imprtat (it affects the verall gai, but is t critical) tha miimizig D O/P ffset. stadard stadard elected cmpets: 39, 50, 4.7K ad 4K IL IUIT 4.7 K 4K i ll cmpets shuld have a tlerace f 0.5% r better.. auril alg ilters ev. 3/4/003 Page 48

50 TYPIL LOW QUY GI PO alg ilters TYPIL WID G GI PO. auril alg ilters ev. 3/4/003 Page 49

51 TYPIL LOW QUY GOUP DLY PO alg ilters. auril alg ilters ev. 3/4/003 Page 50

52 alg ilters DIG XMPL - 3 LOW-P HBYH ILT. Desig a lw-pass hebychev filter that meets the gai respse curve shw beside. G I 0 db (db) Hz 800 Hz. Miimum rder required 0 OH mi 0 0 max 0 OH s c OH OH Let us be cautius i chsig the rder ad use a furth rder filter agai t allw fr cmpet variatis.. ircuit diagram The alle-key circuit with matched 's ad uity gai is chse fr this applicati. OT: The uity-gai alle-key cfigurati is the mst stable with respect t ζ variatis ad shuld be chse whe better perfrmace is required i. auril alg ilters ev. 3/4/003 Page 5

53 3. ilter parameters ad trasfer fucti alg ilters rm the rmalized lw-pass filter table we btai the fllwig filter parameters. OD ( 4 ) LP /LP LP ω c LP ω LP ζ O /LP first stage ± /± secd stage ± /± ( ) ω ζ ω ω ω ζ ω ω alculati f filter cmpets irst stage, ζ0.0883, ω ) 3 stadard B) 4 ζ stadard ) ( ω 3 4 ) D) T miimize D O/P ffset vltage * K, 50K stadard. elected cmpets: 7538Ω, 300, 4. ad 50K ecd stage.0, ζ , ω ) 3 stadard B) 4 ζ stadard ) ( ω 3 4 ) D) T miimize D O/P ffset vltage * K, 00K stadard. elected cmpets: 50606Ω, 36, 48 ad 00K. auril alg ilters ev. 3/4/003 Page 5

54 alg ilters IL IUIT 50K 00K i ll cmpets shuld have a tlerace f % r better. TYPIL LOW QUY GI PO G I (db) QUY (Hz). auril alg ilters ev. 3/4/003 Page 53

55 TYPIL WID G GI PO alg ilters 0.00 G I (db) K 0K 00K M 0M QUY (Hz) TYPIL LOW QUY GOUP DLY PO 0.00m D L Y 8.00m 6.00m (sec) 4.00m.00m 0.00m QUY (Hz). auril alg ilters ev. 3/4/003 Page 54

56 alg ilters DIG XMPL - 4 HIGH-P HBYH ILT. Desig a high-pass Butterwrth filter that meets the gai respse shw belw. 0,5 db ripple TUL PO IDL PO G I (db) 0 0 max.5k 5K 0 db mi atteuati mi BW BW (Hz) s we ca see frm the abve gai respse, a true high-pass filter is impssible t be implemeted with active filters because f the p amps' pe lp respses which cut it the ideal respse thus creatig high frequecy break pits i the actual respse. We ca determie thse break pits by assumig all capacitive reactaces f the circuit t be ull ad the aalizig fr β v (H) f each stage after which we ca calculate BW GBW* β v (H), BW GBW* β v (H), etc.. Miimum rder required mi 0 OH max 0 OH 0 0 OH OH k.5k.959 We will use 3, 5 khz ad HP 0 db. T miimize the umber f stages we must select the miimum reasable rder required, is t clse, 3 allws sme leeway fr cmpet tlerace ad variatis.. ircuit diagram The Uity-gai alle-key circuit is chse fr the secd rder stage ad a simple -ivertig first rder stage whse gai is a free parameter that will be the verall gai f 0 db.. auril alg ilters ev. 3/4/003 Page 55

57 3. ilter parameters alg ilters rm the rmalized lw-pass filter table we first btai the rmalized lw-pass ples fr 0.5 db f ripple ad the cvert them t rmalized high-pass ples ad fially we de-rmalize high-pass ples. LP / LP first stage secd stage ± db High-Pass hebychev HP HP ω c HP ω HP ζ O / HP LP.5967 / / /±07.04 /± / ( ) 8 ζ ω ω ω alculati f filter cmpets ecd stage, ζ , ω ) 0 stadard B) 609Ω 4 ζω ζ ) 4 Ω D) T miimize D O/P ffset vltage, 4 K 3 elected cmpets: 0, 3 997, 4 609, ad K irst stage 0, ω ) 3.9 stadard B) ( ω ) ( ) 53 Ω ) The first stage gai makes up what is missig t achieve a verall gai f 0 db r 0 v/v. te that a first rder stage des t have ay dampig (ζ) specified which meas that its gai is a free parameter here determied by the desired verall gai. vtt 0 v 0 9 fr desired gai ad 53 v 5.68K 9 53 T meet bth cditis we must have We w try stadard values f arud 5.68K ad such that 9 because the rati is mre imprtat (it affects verall gai) tha miimizig D O/P ffset. stadard stadard elected cmpets: 3.9, 53, 6.K ad 56K t miimize D O/P ffset vltage.. auril alg ilters ev. 3/4/003 Page 56

58 alg ilters IL IUIT 6.K 56K K i ll cmpets shuld have a tlerace f % r better. TYPIL LOW QUY GI PO. auril alg ilters ev. 3/4/003 Page 57

59 TYPIL WID G GI PO alg ilters TYPIL LOW QUY GOUP DLY PO. auril alg ilters ev. 3/4/003 Page 58

60 DIG XMPL 5 LOW-P BL ILT. alg ilters Desig a lw-pass Bessel filter that meets the gai respse curve shw beside. G I (db) db mi. -0 max (Hz). Miimum rder required 0 O -0 M. -0 G I (db) OD d 3rd X 4th 5th 6th 7th 8th OMLIZD QUY ( /c ) ice we d't have a equati t calcultate miimum, we will prceed graphically by usig the rmalized curves shw beside. t /c600/4004, the atteuati must be greater tha 30 db which crrespds t a rmalized gai belw -30 db. It appears that 4 is sufficiet but t be safe let's use 5 t allw fr cmpet tlerace ad variatis with temperature.. auril alg ilters ev. 3/4/003 Page 59

61 . ircuit diagram alg ilters The alle-key circuit with matched 's ad 's is chse fr secd rder stages because f easy desig prcedure ad high ζ values which ca be achieved with reasable accuracy - if superir accuracy f ζ is desired, the the alle-key with 0 db is the best chice because it is the least sesitive. Here the gai (3-ζ) f the secd rder stages is determied by the ζ values but f the first rder stage is a free parameter which will be determied by the desired verall gai. r the first rder stage, a simple ivertig filter ca be used fr gais > 0 db ad a first rder ivertig filter ca be used istead if the gai < 0 db which cat be achieved with the -ivertig filter. i 3. ilter parameters ad trasfer fucti rm the rmalized lw-pass filter table we btai the fllwig filter parameters. OD ( 4 ) first stage secd stage third stage LP /LP LP ω c LP ω LP ζ O /LP 3 ζ /±80.0 v tt v v ± /± ± /±3.06 ( ) ( ) ω ζ ω ω ζ ω 7 ω ω 3ω ω auril alg ilters ev. 3/4/003 Page 60

62 4. alculati f filter cmpets alg ilters ecd stage.55, ζ0.8875, ω ) 6 stadard B) ω 5954Ω ) 0.55 fr desired gai ad 3908Ω t miimize D O/P ffset vltage. T meet bth cditis we must have Ω Ω elected cmpets: 6, 5954Ω, 73407Ω ad 3903Ω Third stage.9089, ζ , ω ) 6 stadard B) ω 445Ω ) fr desired gai ad 890Ω ffset vltage. T meet bth cditis we must have X X Ω Ω elected cmpets: 6, 445Ω, 5945Ω ad 5400Ω irst stage 4.747, ω ) 6 stadard B) ω 657Ω t miimize D O/P ) The first stage gai makes up what is missig t achieve a verall gai f 0 db r 0 /. te that a first rder stage des t have ay dampig (ζ) specified which meas that its gai is a free parameter here determied by the desired verall gai. v tt v v v fr desired gai. auril alg ilters ev. 3/4/003 Page 6