ME 343 Cl Sy cu 8 Ocb 8, 009 ME 343 Cl Sy Fall 009 343 R cu Cll Pla R E U Y C - H C D S Y C C R C H Wg h l ga a w a d acg h cld-l l a ga va ME 343 Cl Sy Fall 009 344
Chaacc Equa: R cu 0 Th f h chaacc qua a h cld-l l f h fdbac y!!! Th cld-l l a a fuc f h ga Wg h l ga a b a b a b a b a Th cld l l a gv dcly by h lu f: b 0, 0, a b 0, a ME 343 Cl Sy Fall 009 345 R cu { } { d u } R wh va f 0 v R cu f 0 - gav R cu > 0 : 80 Magud cd Pha cd < 0 : 0 Magud cd Pha cd ME 343 Cl Sy Fall 009 346
ME 343 Cl Sy Fall 009 Pha ad Magud f a Taf Fuc 0 0 a a a b b b b Th fac, - ad - a clx ub:,, 347 ME 343 Cl Sy Fall 009 Pha ad Magud f a Taf Fuc Nw ay gv h ha ad agud f h af fuc: [ ], 348
R cu by Pha Cd Exal: R E U Y 5 4 8-5 4 8 5 3 blg h lcu? ME 343 Cl Sy Fall 009 349 R cu by Pha Cd 36.87 45 90 08.43 78.70 [ 08.43 36.87 45 78.70 ] 80 90 3 blg h lcu! N: Chc cd lcu_hacd. ME 343 Cl Sy Fall 009 350
R cu by Pha Cd 3 W d a yac aach l h cld-l l a fuc f h ga ROOT OCUS ME 343 Cl Sy Fall 009 35 Bac P: R cu { } { d u } R wh va f 0 v R cu f 0 - gav R cu 0 a b 0 Nub f bach ub f -l l R bg a -l l 0 a 0 R d a -l ay b 0 0 > 0 R ycal abu R-ax ME 343 Cl Sy Fall 009 35
R cu Rul : Th bach f h lcu a a h l f ad f h bach d h f. : d f h da f : d f h ua f Rul : Th lcu h al ax h lf f ad dd ub f l ad. I h wd, a val h al ax blg h lcu f h al ub f l ad h gh dd. Th ul c f h ha cd!!! ME 343 Cl Sy Fall 009 353 R cu Rul 3: A, f h cld-l l aach h -l, ad - f h aach - ay wh agl ad cd a π l l, l 0,,, b a l α, l 0,,, ME 343 Cl Sy Fall 009 354
R cu Rul 4: Th lcu c h ax l ably wh h Ruh c hw a a f h lf half-la h gh-half la. Exal: 5 4 5 0, ± 5 ME 343 Cl Sy Fall 009 355 R cu Dg dag vald by h R cu: Hgh lav dg: F - 3 w hav cld l ably du ay. 4 3 3 7 6 4 Nu ha : Thy aac cld l l h RHP N: Chc cd lcu. ME 343 Cl Sy Fall 009 356
ME 343 Cl Sy Fall 009 R cu V fula: Wh h lav dg -, h u f h cld l l ca l cld l a a a a b b b a b 357 ME 343 Cl Sy Fall 009 R cu- Magud ad Pha Cd { } { } u d R wh va f 0 v R cu f 0 - gav R cu > 0 : < 0 : [ ] 80 0 358
R cu Slcg f dd cld l l R cu: If blg h lcu, u af h chaacc qua f valu f Th w ca ba a ME 343 Cl Sy Fall 009 359 R cu Pha lad ca Pu dvav cl ally accal bcau f h alfca f h du h dffa ad u b axad: D, > D, > 5 Pha lad COMPENSATOR Wh w udy fqucy w wll udad why w call Pha ad h ca. Hw d w ch ad lac h cld l l a -75? ME 343 Cl Sy Fall 009
Exal: R cu Pha lad ca D, > 5 Pha lad COMPENSATOR.04 u ch 0.80 40.9? 80 80 40.9.80.04 453.03 93.03 6. 735 ME 343 Cl Sy Fall 009 36 Exal: R cu Pha lad ca 6.735 D 0 5 Pha lad COMPENSATOR 7 5 7 ME 343 Cl Sy Fall 009 36
R cu Pha lad ca Slcg ad a al a cdu. I gal: Th lacd h ghbhd f h cldl aual fqucy, a dd by - lg qu. Th l lacd a a dac 5 0 h valu f h lca. Th l fa ugh avd dfyg h da l bhav. Th xac f h l a c bw: N u w wa a all valu f Ca ffcv w wa lag valu f ME 343 Cl Sy Fall 009 363 Exal: R cu Pha lag ca 6.735 D 0 6.735 l l D l 0 0 0 0 5 5 6.735 0 Wha ca w d ca? Su w wa 0. 6.735 D, < 0 5 W ch: 0 3 48. 48 6.735 Pha lag COMPENSATOR ME 343 Cl Sy Fall 009 364
Exal: R cu Pha lag ca 0.4848 6.735 D 0.00 0 5 6.94 5.03 8.3 ME 343 Cl Sy Fall 009 365 R cu Pha lag ca Slcg ad a al a cdu. I gal: Th a /l ch bad h ca cfca. W c ad all avd affcg h da dyac f h y avd dfyg h a f h lcu g h da dyac Slw a du h all al caclld by a all. Th a /l ca b vy bg. Th xac f ad a c bw: Sady a w wa a lag valu f / Th a w wa h l lacd fa f h g ME 343 Cl Sy Fall 009 366
R cu - Ca Pha lad ca: Pha lag ca: D, < D, > W wll why w call ha lad ad ha lag h ca wh w udy fqucy ME 343 Cl Sy Fall 009 367 Fqucy R W w w hw aaly ad dg y va -da hd whch yld dyacal fa Th a dcbd by h xal d Th d a dd by h l f h alac Taf W x wll l a dcbg y fac va fqucy hd Th gud u cfyg h y l ad ME 343 Cl Sy Fall 009 368
ME 343 Cl Sy Fall 009 Sudal Sady-Sa R c A U A u Cd a abl af fuc wh a udal u: Wh h aual y d l Th a h lf half la R<0 A h u d ad - Oly h du h l h agay ax a af a uffcly lg Th h udal ady-a Th alac Taf f h ha l A U Y 369 ME 343 Cl Sy Fall 009 Sudal Sady-Sa R Iu Taf R Taf R Sgal Sudal Sady Sa R c c c A A A u c A A U N N U Y * 4444 4 3 4444 4 4 4 3 4 4 aual fcd * N N y SS y * fcd aual 0 370
ME 343 Cl Sy Fall 009 Sudal Sady-Sa R Calculag h SSS Rdu calcula Sgal calcula [ ] [ ] c c l l l A A A A U Y c A u y c * c A y 37 ME 343 Cl Sy Fall 009 Sudal Sady-Sa R R Ouu fqucy u fqucy Ouu alud u alud Ouu ha u ha Th Fqucy R f h af fuc gv by valua a a fuc f a clx vaabl a W a f h alud ad f h ha Thy ca ddly b vad» Bd la f aalyc fuc hy c c A y A u 37
Fqucy R u Ac Ac y Sabl Taf Fuc Af a a, h uu l a ud wh a alud agfd by ad ha hfd by. Sc all gal ca b d by ud Fu ad af, h qua ad a xly a. Bd dvld hd f qucly fdg ad f a gv ad f ug h cl dg. ME 343 Cl Sy Fall 009 373 Bd Daga [ ] Th agud ad ha f wh gv by:, Nla h agud a h ha ME 343 Cl Sy Fall 009 374
Why d w x ME 343 Cl Sy Fall 009 Bd Daga dcbl? 0lg By f h lgah w ca w: 0lg 0lg 0lg 0lg 0lg 0lg 0lg 0lg Th agud ad ha f wh gv by: a h agud? a h ha 375 ME 343 Cl Sy Fall 009 Bd Daga Dcad: Ay fqucy ag wh d hav a 0: a A cuff fqucy ccu wh h ga ducd f axu abad valu by a fac / : 0lg T 0lgT 0lg 0lgT 3 MAX MAX MAX Badwh: fqucy ag ad by h ga abad hav a l a u xal: T R 0 R / T T / Th a lw-a fl!!! Pabad ga, Cuff fqucyr/ Th Badwh R/! 376
al Taf Fuc Bd Daga ζ τ Th agud ha h u f h agud ha f ach f h. W la hw l ach, w la hw l h whl agud ad ha Bd Pl. q Cla f : - - 3-4- ME 343 Cl Sy Fall 009 τ ζ q 377 al Taf Fuc: DC ga Magud Magud ad Pha: 0 40 35 30 5 0 5 0 5 0lg 0 ± π Pha dg 00 80 60 40 0 00 80 60 40 0 f f > 0 < 0 0 0-0 0 0 0 Fqucy ad/c ME 343 Cl Sy Fall 009 0 0-0 0 0 0 Fqucy ad/c 378
al Taf Fuc: Pl/ a g Magud ad Pha: 0, 0lg π 0 0 0-0 Magud 0-0 -0-30 0 dc Pha dg -40-60 -80-00 -40 0-0 0 0 0 Fqucy ad/c ME 343 Cl Sy Fall 009-0 0-0 0 0 0 Fqucy ad/c 379 al Taf Fuc: Ral l/ Magud ad Pha: τ 0lg τ a τ Ayc bhav: 0 << / τ τ >> / τ 0lg 0 << / τ 90 >> / τ ME 343 Cl Sy Fall 009 380
al Taf Fuc: Ral l/ Magud 0 5 0-5 -0-5 -0-5 -30-35 3 0 dc -40 0-0 0 0 0 0 3 Fqucy ad/c, τ /0 0 / τ 0 0 3 g ME 343 Cl Sy Fall 009 38 al Taf Fuc: Ral l/ Pha dg 0 0-0 -0-30 -40-50 -60-70 -80-90 45-00 0-0 0 0 0 0 3 Fqucy ad/c, τ /0 0 0 0 / τ 45 90 ME 343 Cl Sy Fall 009 38
al Taf Fuc: Clx l/ Magud ad Pha: Ayc bhav: ζ q 0lg ζ / / q a q ζ 0 << q >> q 40lg 0 << q 80 >> ME 343 Cl Sy Fall 009 383 Magud -0-40 -60-80 -00 al Taf Fuc: Clx l/ 0 0-0 0-0 0 0 0 0 3 Fqucy ad/c MAX ME 343 Cl Sy Fall 009 q ζ q 40 dc q,, ζ 0.05 0 0 q ζ ζ q ζ 0. 3 ζ g q 384
Pha dg -0-40 -60-80 -00-0 -40-60 -80 al Taf Fuc: Clx l/ 0 0 q 90-00 0-0 - 0 0 0 0 Fqucy ad/c q,, ζ 0.05 0 0 0. / τ q 90 q 80 ME 343 Cl Sy Fall 009 385 Fqucy R: Pl/Z h RHP Sa. Th ffc ha h abl ca. A uabl l bhav l a abl A uabl bhav l a abl l Exal: Th fqucy ca b fud xally bu ca b cud ad ud f cl dg. ME 343 Cl Sy Fall 009 386
Nual Sably U Y - R lcu cd:, 80 A f ual ably R cd hld f, 80 ME 343 Cl Sy Fall 009 Sably: A < > 80 If lad ably If lad ably 387 Sably Mag Th AIN MARIN M h fac by whch h ga ca b ad bf ably ul. M < M < 0 M qual / wh 80. UNSTABE SYSTEM a h fqucy Th PHASE MARIN PM h valu by whch h ha ca b ad bf ably ul. PM < 0 UNSTABE SYSTEM PM h au by whch h ha f -80 wh 0 xcd ME 343 Cl Sy Fall 009 388
Sably Mag M PM ME 343 Cl Sy Fall 009 389 Fqucy R u Ac Ac y Sabl Taf Fuc BODE l { } I{ } R NYQUIST l ME 343 Cl Sy Fall 009 390
R Nyqu Daga { } I{ } Hw a h Bd ad Nyqu l lad? Thy a w way h a fa I { } R { } ME 343 Cl Sy Fall 009 39 Nyqu Daga al cdu f chg Nyqu Daga: Fd 0 Fd Fd * uch ha R{*}0; I{*} h c wh h agay ax. Fd * uch ha I{*}0; R{*} h c wh h al ax. Cc h ME 343 Cl Sy Fall 009 39
Nual Sably U Y - R lcu cd:, 80 A f ual ably R cd hld f, 80 ME 343 Cl Sy Fall 009 Sably: A < > 80 If lad ably If lad ably 393 Sably Mag Th AIN MARIN M h fac by whch h ga ca b ad bf ably ul. M < M < 0 M qual / wh 80. UNSTABE SYSTEM a h fqucy Th PHASE MARIN PM h valu by whch h ha ca b ad bf ably ul. PM < 0 UNSTABE SYSTEM PM h au by whch h ha f -80 wh 0 xcd ME 343 Cl Sy Fall 009 394
Sably Mag M PM ME 343 Cl Sy Fall 009 395 Sably Mag /M PM ME 343 Cl Sy Fall 009 396
Nyqu Sably C Ca : N l/ wh cu Ca : Pl/ wh cu Agu Pcl: A cu a f a clx fuc wll ccl h g Z-P, wh Z h ub f ad P h ub f l f h fuc d h cu. ME 343 Cl Sy Fall 009 397 Nyqu Sably C u cd h cu ad cld-l y U Y - Th cld-l l a h lu f: Th valua f H wll ccl h g ly f H ha a RHP l ME 343 Cl Sy Fall 009 0 398
Nyqu Sably C u aly h agu cl h fuc H. If h l f ccl h g, h l f ccl - h al ax. ME 343 Cl Sy Fall 009 399 Nyqu Sably C By wg b a b a a w ca cclud ha h l f a al h l f. Aug l f h RHP, a ccl f h - by dca a f h RHP, ad hu a uabl l f h cld-l y. A clcw cu f C clg a f wll ul cclg h - h clcw dc. A clcw cu f C clg a l f wll ul cclg h - h cuclcw dc. Th ub f clcw ccl f h -, N, qual h ub f cld-l l h RHP, Z, u h ub f l -l l h RHP, P: ME 343 Cl Sy Fall 009 N Z P 400
Nyqu Sably C U Y - Wh h af fuc Sabl? NYQUIST: Th cld l aycally abl f h ub f cuclcw ccl N gav f h -0 by h Nyqu cuv f qual h ub f l f wh v al a uabl l P. Cllay: If h -l y abl P0, h h cld-l y al abl vdd a ccl f h -0 N0. ME 343 Cl Sy Fall 009 40 Scfca h Fqucy Da. Th cv fqucy c, whch d badwh BW, ad lg.. Th ha ag PM, whch d h dag cffc ζ ad h vh M. 3. Th lw-fqucy ga, whch d h ady-a chaacc. ME 343 Cl Sy Fall 009 40
Scfca h Fqucy Da Th cv fqucy: c BW c ME 343 Cl Sy Fall 009 403 Fqucy R Pha ad Ca T D, α < αt α lg MAX MAX MAX lg T lg αt I a hgh-a fl ad axa PD cl. I ud whv ubaal v dag dd. I d ca h d f f a y f a fxd lwfqucy ga. ME 343 Cl Sy Fall 009 404
Fqucy R Pha ad Ca. D h -l ga afy badwdh qu: - T qu, c afy ca, v, a ha cfca. - T badwdh qu, c ha h -l cv fqucy a fac f w blw h dd cld-l badwdh.. D h dd ha lad αbad h PM cfca. MAX α MAX 3. Pc MAX b a h cv fqucy. 4. D h ad l f h ca. /T MAX α / / α T MAX α -/ 5. Daw h cad fqucy ad chc PM. 6. Ia h dg. Add addal ca f dd. ME 343 Cl Sy Fall 009 405 Fqucy R Pha ag Ca T D α, α > αt I a lw-a fl ad axa PI cl. I ud ca h lw fqucy ga f h y ad v ady a f fxd badwdh. F a fxd lw-fqucy ga, wll dca h d f f h y. ME 343 Cl Sy Fall 009 406
Fqucy R Pha ag Ca. D h -l ga ha wll h PM qu whu ca.. Daw h Bd l f h ucad y wh cv fqucy f ad valua h lwfqucy ga. 3. D α h lw fqucy ga qu. 4. Ch h c fqucy /T h f h ca b dcad blw h w cv fqucy c. 5. Th h c fqucy h l f h ca h / α T. 6. Ia h dg ME 343 Cl Sy Fall 009 407