Chapter Frequency Response Response Techniques

Transcription

1 Chater 0 Frequecy Ree Techique

2 Itrducti Rereetati f cmlex umber A B M φ where M A B & φ ta B / A -lae Im Me φ where φ e φ c i φ B M φ A Ay trafer fucti Re A it the uit circle Magitude: c x i x Phae: iφ φ ta cφ Nte. Euler frmula: x x e e c x x x e e i x N L σ N σ σ σ σ L σ A B cmlex! M φ σ σ Ay trafer fucti ca be rereeted a magitude ad hae agle

3 Frequecy ree: teady-tate ree f a liear ytem t a iuidal iut A. The Ccet f Frequecy Ree Frequecy ree: teady tate ree f a liear ytem t a iuidal iut σ 0 Fr frequecy ree Liear ytem C B R t B t r i N C Aumig : table liear ytem havig -ditict le. i.e. { } 0 Re > i t B t r i N R C L N B B C L L A A A K K C term traiet term tate teady t t t t t Takig ivere Lalace trafrm give t t t t t A e A e A e e K K e t c L { } 0 Re > i Q t t e K K e c t c

4 B B B Uig artial fracti exai methd, B B B e B B B K φ * K e B B B K φ Cmlex cugate f K φ where Re Im φ Re i t t t t t B e e B e K K e t c φ φ φ B it d fi t i id i i t Y t B φ φ Frequecy ree f a ytem B: magitude f iut iuid : magitude f trafer fucti Y: magitude f utut B Frequecy ree f a ytem

5 Frequecy ree f a liear ytem Bit Liear ytem B i t φ Cmmet - Outut ha ame frequecy with iut frequecy! φ - Outut deed the magitude & hae f the ytem C - Amlitude rati: Outut iuid magitude R Iut iuid magitude - Phae hift C φ C R R Iut과 utut의 magitude와 hae를알면 ytem 즉 를재구성가능 φ : hae hift φ > 0 < 0 φ : hae lead : hae lag B B Outut t iut iuid C t B i t φ T π Iut iuid rt Bi t

6 B. Plttig Frequecy Ree - Frequecy ree f a ytem - Frequecy ree characteritic 주파수응답특성 : Siuidal iut의 frequecy 의값에따라ytem utut의 magitude 와 hae agle 이변화하는 ytem 의성질 - rahical rereetati f frequecy ree characteritic Bde lt lgarithmic lt: - lttig magitude db ad hae f a ytem v. frequecy lg earately φ Nte. db decibel 0 lg Plar lt Nyquit lt: - lttig magitude ad hae i cmlex lae

7 Ex0. Fid aalytical exrei fr the frequecy ree fr a ytem /. Ad lt them i Bde ad lar lt Frequecy ree fr a ytem Real art imagiary art M φ magitude 4 4 M Methd I Aalytical exrei fr the frequecy ree f a ytem ta 4 ta φ g hae 4 Methd II Magitude: 분모 / 분자의크기를따로계산후나누기 4 M k k M M Phae: 분자항들의 hae 의합 분모항들의 hae 의합 ta ta 0 ta φ k k z Phae: 분자항들의 hae 의합 분모항들의 hae 의합

8 Bde lt 0lg dB 0lg M 0lg 4 lg φ ta -45 rad/ lar lt M φ ta 4-45

9 Iterretati f the Bde lt Filter - Lw a filter - High a filter - Bad a filter M-C-K ytem

10 Aymttic Arximati: Bde Plt Simlifyig the ketchig Bde lt by traight-lie arximati k z z z K L m L Magitude frequecy ree m k z z z K L L Multilicati/divii cvertig it db by takig the lgarithm z z K L 0lg 0lg 0lg lg 0 Al b i / b i m L 0lg lg 0 Algebraic um/ubtracti Phae ree Much eaier fr grahic! m z z K L L Advatage f the Bde lt lgarithmic lt

11 Baic factr f Bde lt. ai K m T m. ai K. Itegral ad derivative factr 3. Firt-rder factr [ ] m / / ζ 4. Quadratic factr * Oce lgarithmic lt f thee baic factr are kw, Bde lt f cmlicated fucti ca be Oce lgarithmic lt f thee baic factr are kw, Bde lt f cmlicated fucti ca be ctructed eaily by addig idividual curve f each factr k z z z K L [ ] m ζ L * Nte. Alway make trafer fucti i rmalized frm whe t draw Bde lt k z z z K L L a a a m k z z z K ζ L L

12 a. Bde lt fr ai K K db 0 lg 0lg φ K 0 K ctat N agle ctributi db 0lg K φ lg Nte. If K >, db>0 K<, db<0 Effect f varyig gai K: raie r lwer ther lg-magitude curve f the trafer fucti by 0lg K, but effect the hae curve!

13 b. Bde Plt fr Itegral ad Derivative factr ± ± db 0lg ± 0lg φ Bde lt uig traight lie aymtte db ± ± L le ± ± i ± time [ L ] ta db/decade db/decade 90 φ lg - -0 db/decade db/decade lg * Decade: 0 time the iitial frequecy

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15 Frm ctat a ± C. Bde Plt fr firt-rder factr a let db 0lg ± 0 lg a lg ± ± a a a ± a 0 φ a ta ± ta ± a i a i a i Nrmalize it firt! Straight lie aymtte If << i. e. << a db ± 0lga ta φ 0 a a If r a db ± 0 lg a lg φ ta ± 45 a a If >> r >> a db 0lg a ± 0lg φ ta ± a 90 a a Frm ctat a db 0lga 40 Break crer frequecy 40 db/decade 0 db/decade φ 80 Rugh etimate ta 0 lga lga 0 0.0a 0.a a 0a 00a lg 0.0a 0.a a 0a 00a lg 0lga l lga db/decade Decade: db/decade 0 time the iitial frequecy Actual v. aymtte? a

16 a ± Cmari betwee ad a a ± ± a ± a a ± ± Magitude hift by 0lga Bde lt f lwer magitude lt f f b by 0 lga Same hae lt Fr 0 0lga 0lga 0lga 0 lga 0lga db Fr Break crer frequecy 0.0a 0.a a 0a 00a lg 40 db/decade 0 db/decade - -0 db/decade db/decade φ Rugh etimate 0.0a 0.a a 0a 00a lg

17 Bde lt f T T * t rder ytem 0dB/decade le

18 Ex draw Bde lt fr the fllwig ytem where 3 K Bde lt uig e-l TF [ ] g um f each firt-rder term 3 3 K N li f it l f i i 0dB 3 Nrmalize fr uity lw-frequecy gai, i.e. 0 db

19 D. Bde Plt fr d -erder factr ± ± ζ ζ ± ζ Nrmalized frm ± lg lg 0 0lg db ζ [ ] ± ± ± ± ta ζ ζ ζ φ Straight lie aymtte If Straight lie aymtte e i << << ta φ db lg 40 lg lg 0 ± ± If 80 0 t ± ± φ 90 0 ta ± ± φ If e i >> >> 40l 0l db ± ± e i.. ζ lg lg 0 ± db 80 0 ta ± ± φ e i >> >>.. lg 40 lg 0 db ± ± Fucti f damig rati

20 ± 40lg ± 40lg db Break frequecy 80 db/decade 40 db/decade ± 40lg lg ± 40lg -40 ± 40lg db/decade db/decade φ 360 Rugh etimate lg * d rder ytem 40dB/decade le 0 fr tadard frm ζ ζ db Fucti f damig φ *magitude hift by ± 40lg lg db/decade lg

21 Actual ree v. aymttic arximati f ζ - reater diarity deedig ζ - Crrecti factr fr actual hae 0lg ζ ζ g R f e i > ζ There reat eak fr Reat frequecy e i.. T be exact ζ ζ Shuld be reat frequecy r 0 r d d r ζ reat frequecy

22 t rder v. d rder ytem db lg lg -40 차 ytem -90 차 ytem 차 ytem 차 ytem φ

23 Ex draw Bde lt fr the fllwig ytem Ste Firt rmalize each cmet Ste draw Bde magitude lt fr each rmalized cmet dB dB dB Ste 3 Add each lt

24 Ste 4 draw each Bde hae lt fr each rmalized cmet Ste 5 Add each lt

25 Nyquit Stability Criteri determie the tability f a cled-l ytem frm it e-l frequecy ad e-l le N N Baic defiiti D N D N H H H let N N H H O l TF D D H H Oe-l T.F. H H H N N D D N N H Characteritic equati Cled-l T.F. H H D D D D H Characteritic equati H N N D D D N H H T R C H H N N D D H R Ple f H Ple f e-l ytem H H y zer f H Ple f cled-l ytem T

26 Maig A. Maig f a cmlex umber F: ay cmlex fucti Ay cmlex umber σ -lae Ather cmlex umber F-lae Plug i F S-lae σ 4 3 F Maig F-lae Im F 6 30 Re F F i -lae are maed it i F-lae thrugh the fucti F F B. Maig f a ctur Ctur cled-curve: cllecti f it Maig ctur A t ctur B thrugh fucti F

27 a e b Ctur maig? F d 90 Ple F zer F 크기 : 반비례 크기 : 비례 Phae: 반대 /- Phae: 비례 /- 시계 반시계 시계 시계

28 Prertie f ctur maig Aumig clckwie directi f ctur A. If the le r zer f F i ecled by ctur A, a cuterclckwie directi if the le f F i ecled by ctur A Ctur B ma i a clckwie directi if the zer f F i ecled by ctur A. If the le r zer f F i ecled by ctur A, Ctur B ecircle the rigi f the F-lae 3 If ct r A ecle bth the er ad the le the le ad er rtati cacel 3. If ctur A ecle bth the zer ad the le, the le ad zer rtati cacel maig de t ecma the rigi But till mve i cuterclckwie directi due t the le

29 Fr examle F H z z Vectr rereetati f maig A we mve arud ctur A i a clckwie directi, each vectr iide ctur A Uderge a cmlete rtati r 360 agle chage utide ctur A cillate ad retur t it iti r a et 0 agular chage VV R F V V V Clckwie rtati F V V V3 V4 V Cuter-clckwie rtati 0 Oe clckwie rtati 360 Z P 360 Z-P Z : Number f zer f H iide ctur A P : Number f le f H iide ctur A N : Number f clckwie rtati f ctur B abut the rigi Fr thi cae Z, P N If N i related t cuterclckwie rtati, the bviuly N P - Z N Z P

30 Z : Number f zer f H iide ctur A #fple f cled-l l ytem T Nt kw P : # f le f H iide ctur A # f le f e-l ytem H Z N P # f cled-l le iide the ctur # f clckwie rtati f maig abut the rigi # f e-l le f H iide the ctur kw A ytem tability i determied by cled-l le i RHP, let cider the etire RHP a the ctur i.e. Z Z > 0 0 at leat e cled-l le exit i RHP utable N cled-ll le exit i RHP table Nyquit Stability Criteri P Eaily fud by examiati f e-l TF H A lg a N i fud, the ytem tability ca be determied Z N P # f cled-l le i RHP Z N P # f clckwie rtati f maig abut the rigi # f e-l le f H i RHP

31 Maig thrugh H itead f H Tralate the ctur B made by H e uit t the left cider the rtati abut -0 itead f abut the rigi σ - σ Ctur B made by H Ctur B made by H Nyquit Stability Criteri # f cled-l l le i RHP Z N P # f clckwie rtati f maig abut -0 # f e-l le f H i RHP

32 Ex N 0 P 0 ZNP 0 Stable! N P0 ZNP Utable!

33 Sketchig the Nyquit Diagram Nyquit diagram btaied by ubtitutig it alg the ctur i.e. etire RHP it the fucti H Ex 500 H 3 0 ctur i.e. etire RHP Im H -lae Maig Thrugh H H-lae Nyquit Diagram? Re H Nyquit diagram??sb Subtitute every it the ctur it i H Let : itive imagiary axi : ifiite emicircle : egative imagiary axi Fr cveiece, let draw Nyquit diagram fr earately ad cect them later

34 A. Nyquit diagram fr art itive imagiary axi Pit itive axi where it betwee A t C 500 H 3 0 H 0 ~ Methd I Vectr arithmetic V,V,V 3 : Vectr f H draw t ay it the ctur Pit B i thi cae 500 Reultat Vectr H V V V V3 zer legth le legth Magitude f reultat Vectr: 500 H V V V3 Phae f reultat t Vectr 0 H ta ta V ta V ta V3 500 z k A we mve i a clckwie directi i i.e. a it B mve frm A t C k agle f each vectr ge frm 0 t 90 reultat agle ge frm 0 t -3x90-70 Pit A Pit C magitude f each vectr ge frm fiite value t ifiite value Reultat magitude ge frm a fiite value t zer

35 Methd II Aalytical methd ta ta At it A, lw frequecy, i.e. 0 0 it A 30 At middle rage frequecy arud it B Cre the egative Im-axi 80 Cre the egative Re-axi At it high frequecy, i.e it C

36 B. Nyquit diagram fr art it betwee C t D, i.e. arud the ifiite emicircle Methd I Vectr arithmetic Vectr V,V,V 3 rtate clckwie 80 each reultat vectr uderge a cuterclckwie rtati f 3x 80.5 tur cuterclckwie Vectr frm Ple frm C t D Methd II Aalytical methd ifiite emicircle At it C, At it D, θ 90 θ 90 θ e θ θ θ e 0 C D e it C it D Nte θ e cθ iθ c θ i θ ta iθ θ cθ C. Nyquit diagram fr art it egative imagiary axi Maig f the egative imagiary axi mirrr image f the maig f the itive imagiary axi -Fiihed-

37 Secial cae: Oe-l le exit the axi Detur arud t the left r right the le the axi Detur mut be ifiiteimally mall t t iclude me cled-l le Detur ath ifiiteimal emicircle θ e Ex Sketch the Nyquit diagram f the fllwig uity-feedback ytem where V V Byaed ctur Nyquit diagram

38 I Aalytical methd At lw frequecie 0 80 it A At high frequecie 0 90 it B At middle frequecie rage Bth real ad imagiary art f are egative e huld be i 3 rd quadrat ifiite emicircle At it C, θ e θ 0 θ θ e e 0 θ θ θ θ e e e e it C Maig it betwee D ad E mirrr image f the maig f A t B D t E ifiiteimal emicircle At it F, At it A, ε e θ θ 0 θ 90 θ εe θ θ θ ε e εe εe θ 0 0 it F εe εe it A At it E, θ it E 90 εe

39 Methd I Vectr arithmetic At it A V Vectr V ctribute -90 x -80, Vectr V ctribute 0 reultat vectr ctribute Magitude f V ~ 0 ttal magitude ifiite V it A ma it it A At it B Vectr V ctribute 90, Vectr V ctribute 90 reultat vectr ctribute it B ma it it B Magitude f V & V ttal magitude 0 0 Pit betwee B ad D 0 ttal magitude 0 Vectr V & V ctribute -80 clckwie each reultat vectr uderge a 80 cuterclckwie BD ma it it B D Maig it betwee D ad E mirrr image f the maig f A t B D t E Ctur EFA Vectr V ctribute 0 becaue f ifiiteimal emicircle, but vectr V ctribute 80 each reultat vectr ctribute ttal magitude 0 Ctur EFA ma it E F A V V V ε

40 Stability via the Nyquit Diagram Fr cled-ll ytem that ha a variable gai i the l, fr what rage f gai i the ytem table?. Rt lcu. Ruth-Hurwitz 3. Nyquit criteri Z N P Fr variu value f gai K, hw t draw the Nyquit diagram? Set the l gai K Draw the Nyquit diagram Multily the gai aywhere alg the Nyquit diagram ice gai i imly a multilyig factr - A K icreae, Nyquit diagram exad - A K decreae, Nyquit diagram hrik Or Nyquit diagram remai tatiary ad cider the critical it /K, mre -0 mve alg the real axi

41 Exaded Nyquit diagram with large gai K N -, P, Z0 Stable! Shriked Nyquit diagram with mall gai K N 0, P, Z- Utable! - ctur Nyquit diagram ai K huld be large t make N- ice P Stable! If K i mall, the critical i it /K wuld be lcated utide the Nyquit diagram N 0 Utable! K > /.33

42 Ex fid the rage f gai K t make the uity feedback ytem table where Set the l gai K ad Draw the Nyquit diagram Multily the gai aywhere alg the Nyquit diagram H 3 K Pit A where the Nyquit diagram iterect the egative real axi Set the imagiary art t be zer 5 Value f real art? K 3 5 Pit A where the Nyquit diagram iterect the egative real axi Sice P0, N huld be zer t be table Pit A huld tay left t the -0 Fr tability K<0.5 Nyquit diagram ca be exaded befre it A meet -0 gai K ca multilied r icreaed / time befre the ytem becme utable At K0.5, it A meet critical it -0, ytem becme margially table ad ytem cillate at the frequecy f 5 rad/ec

43 Stability via Maig Oly the Pitive axi H 80 e-l magitude at the frequecy where the hae agle i 80 r -80 H 35 e-l magitude at the frequecy where the hae agle -35 Ecirclemet f critical it Ca be determied frm the e-l magitude at the frequecy where the hae agle i 80 r -80 Ca be determied frm the maig f the itive -axi ale, i.e. A B Pitive -axi AB A B Draw the Nyquit diagram crredig t the -axi rti f the ctur

44 Ex 0-7 determie the tability rage f K fr the uity feedback ytem Prti f ctur axi ly Firt, draw the rti f the ctur ly alg the itive imagiary axi H K Iterecti with the egative real axi imagiary art 0 6 Put it back t equati real art magitude f e-l K Nyquit diagram draw frm maig f -axi ly with the aumti K Nyquit diagram ca be exaded 0 time befre it becme utable /0 K ca be icreaed 0 time mre tha the curret value K befre it becme utable K<0 : table K0 : margially table cillati frequecy 6 K>0 : utable

45 ai Margi ad Phae Margi via the Nyquit Diagram ai margi M r K g - Amut f e-l gai variati allwed at 80 hae agle befre the cled-l ytem becme utable - recircal f the real-axi crig it e-l magitude exreed i decibel db Phae margi Φ M r γ - Amut f additial e-l hae hift lag allwed at uity gai befre the cled-l ytem becme utable Aume N huld be 0 fr the tability Nyquit diagram ca be exaded a time befre it becme utable /a K ca be icreaed a time mre tha the curret value K befre it becme utable ai margi a r 0lg a Rereet ytem rximity t itability At uity gai, the ytem becme utable if a hae hift f α degree ccur hae margi α

46 Pitive gai margi H H Negative gai margi Pitive hae margi - H Stable Utable Ex 0-8 Fid the gai ad hae margi fr the ytem f Examle 0-7ifK6 K Simly multily 6 t all the reult f Ex 0-7 i which K H ai margi real axi crig Phae margi 6 - [ 6 ] [ 6 ] 6 4 H 6 6 Negative hae margi ai margi 0 lg / dB Slve fr fr which the magitude.53 rad /ec [ 6 ] H.3 Phae margi

47 ai Margi ad Phae Margi via the Bde Plt Bde lt - Eaily draw withut lg calculati required fr the Nyquit diagram - but give the ame ifrmati tability, gai ad hae margi, rage f gai fr tability btaied by Nyquit diagram - Viable alterative t Nyquit lt M ad PM via Bde lt

48 Stability via Bde lt Ex 0-9 Fid the rage f gai K fr the tability f the uity feedback ytem where K K 4 5 Requiremet f tability frm Nyquit tability ccet - N0 becaue P0 - magitude f e-l < at 80 i.e. 0 lg < 0dB 0 lg 40K 0 lg 40K-0 0 lg 40K-40 0 lg 40K-60 0 lg 40K-80 0 lg 40K-00 0 lg 40K 0 M 0 lg at 80, 7rad / ec 0 lg 40K-0 0 lg 0 lg 40K -0 <0 Fr tability: 0<K<400 PM Nte. Same reult ca be btaied by Nyquit diagram which require lg cmutati

49 Ex 0-9 Fid the M ad PM whe K00 fr the ame ytem i Ex 0-8 M 6.0dB ai margi: gai required t raie the magitude curve t 0 db PM 5 Remark M ad PM - Fr table ytem, M db >0 & PM >0 - fr utable ytem, M db <0 & PM <0 - Otimal M> 6dB - Otimal PM : 30 ~60

50 Relati betwee cled-l traiet ad cled-l frequecy ree Secd rder cled-l l ytem C T R ζ Frequecy ree f the cled-l ytem Subtitutig it the T Magitude f the cledl Frequecy ree T 4ζ Peak value f the cled-l magitude ree d T Slve 0 fr d ζ Peak frequecy M ζ ζ Peak value Remark - M i a fucti f ζ - - N eak exit if ζ > 0.707

51 Badwidth f the cled-l frequecy ree BW -Frequecy at which the magitude ree curve i 3dB dw frm it value at zer frequecy - idicate hw well the ytem will track a iut iuid - idicatr f ytem eed - Calculati f BW 0lg T 0lg 4ζ 3 BW 4 ζ 4ζ 4ζ - By ubtitutig 4/ Tζ, i related t ettlig time T BW 4 4 BW ζ 4ζ 4ζ T ζ - By ubtitutig π / ζ, i related t ettlig time T T BW BW π 4 ζ 4ζ 4ζ T ζ T r - By uig Fig.4.6NISE, BW i related t ettlig time T r

52 Relati betwee cled-l traiet ad Oe-l frequecy ree Relatihi betwee the hae margi ad the damig rati - eable u t evaluate the %OS frm the hae margi btaied frm the e-l frequecy ree A uity feedback ytem with a e-l TF C.L.T.F. C T R ζ ζ Firt fid the frequecy fr which t evaluate the PM ζ ζ ζ 4 4 Phae agle f at thi frequecy Phae margi Φ M 90 ta ζ 90 ta 80 ζ ζ 4ζ 4 90 ta ta ζ ζ ζ ζ 4ζ 4ζ 4 4 PM i a fucti f damig rati!

53 Φ M ta ζ ζ 4ζ 4 We kw that there i eak fr ζ > frm ζ Imlie that PM larger tha 65 5 i required t eure there i eak i Imlie that PM larger tha 65.5 i required t eure there i eak i the cled-l frequecy ree

54 Steady-State Errr Characteritic frm Frequecy Ree Static errr cefficiet frm Bde magitude lt cider a e-l TF K N zi K zi - At lw-frequecy regi lim 0 N i zi Tye 0, N 0 0 lg 0lg K 0lg K i Tye, N Tye, N i le 0 db / dec K z i K 0lg 0lg 0lg 0lg K 0lg le 0dB / dec i t K z z i iterect the frequecy axi at i K Q K v i K z i K 0lg 0lg 0lg 0lg 40lg K i zi iterect the frequecy axi at K K a i i Q le K z i Sytem tye dt determied dfrm the le at the lw frequecy regi Tye Sle K 저주파영역에서의 magitude 0lg K K v a 0dB/dec-le curve 가 frequecy axi 와교차하는주파수 -40dB/dec-le curve 가 frequecy axi 와교차하는주파수 i 40dB / dec K -40

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56 Ex Fid the ytem tye ad the arriate tatic errr ctat fr the Bde magitude lt Iitial le: 0 db/dec tye 0 0lg K 5 K 7.78 Iitial le: -0 db/dec tye I 0.55 Iitial -0dB/dec curve cre 0 db at 0.55 rad/ec K v 0.55 Iitial le: -40 db/dec tye II Iitial le curve cre 0 db at 3 rad/ec K a 3 9 3

57 Obtaiig Trafer Fucti Exerimetally Ctructi f the trafer fucti - uig te ree data - uig iuidal frequecy ree data hw? - frcig the ytem with iuidal iut - meaure the utut teady-tate iuid amlitude ad hae agle -reeatig thee rce at a umber f frequecie yield data fr a frequecy ree lt - btai the frequecy ree lt Bde magitude ad hae lt uig the fllwig relati C magitude: Outut iuid magitude R Iut iuid magitude Phae hift C φ C R R Outut iuid hae - Iut iuid hae -Oce bde trafer fucti f the ytem ca be etimated frm the break frequecie ad le

58 uidelie:. Determie the ytem tye by examiig the iitial le.. Differece betwee # f le ad # f zer determied by examiig the hae excuri 3. See if rti f the Bde lt rereet bviu firt- r ecd-rder rder le r zer frequecy ree lt 4. See if there i ay telltale eakig r derei i the magitude ree lt that idicate a uderdamed ecd-rder le r zer, reectively 5. If ay le r zer ree ca be idetified, verlay arriate ±0 r ±40 db/decade lie the magitude curve r ±45/decade lie the hae curve ad etimate the break frequecie. Fr ecd rder le r zer, etimate the damig rati ad atural lfrequecy frm the tadard dbd Bde lt lt 6. Frm a TF f uity gai uig the le ad zer fud. Obtai the frequecy ree f thi TF ad ubtract thi ree frm the reviu frequecy ree. Nw yu have a frequecy ree f reduced cmlexity

59 Ex Fid the trafer fucti f the ubytem whe Bde lt are hw belw verlay -0 db/decade lie the magitude lt ad a -45/decade lie the hae lt Break frequecy at 90 rad/ eak i the magitude curve extract uderdamed le Arximate 5rad / eak 6.5dB ζ 0.4 frm tadard d rder Bde lt Thu, uity gai d rder ytem TF 5 ζ.4 5 ubtract the Bde lt f frm the rigial Bde lt Subtract the Bde lt f

60 imilar t that f lag cmeatr Draw -0dB/decade d lie cectig tw 0dB/dec lie at lw/high frequecy regi break frequecie at 9 ad 30 rad/ ubtract agai Yield DC gai f -9dB 0. almt flat magitude ad flat hae: ±db ad -3 ±5 fluctuati ti Fially, ur etimate f the TF c.f. actual TF f the rigial i Bde lt