Chater 5: Root Locu
ey condton for Plottng Root Locu g n G Gven oen-loo tranfer functon G Charactertc equaton n g,,.., n Magntude Condton and Arguent Condton
5-3 Rule for Plottng Root Locu 5.3. Rule Rule : Startng and end ont For g =, we can get fro agntude equaton that g......,... n n>,... n end For g +, t ay reult n one of the followng fact tartng Rule #: The locu tart at a ole for g = and fnhe at a ero or nfnty when g =+. Ung Magntude Equaton 7//8 3
Pole and ero at nfnty G ha a ero at nfnty f G + G ha a ole at nfnty f G + + Exale G Th oen-loo tranfer functon ha three ole,,-,-. It ha no fnte ero. 3 For large, we can ee that. G So th oen-loo tranfer functon ha three n- ero at nfnty. 7//8 4
Rule : Nuber of egent Rule #: The nuber of egent equal to the nuber of ole of oen-loo tranfer functon. egent end at the ero, and n- egent goe to nfnty. Soete, ax{,n} Rule 3: Syetry rule Rule #3: The loc are yetrcal about the real ax nce colex root are alway n conugate ar. 7//8 5
Rule 4: Segent of the real ax Segent of the real ax to the left of an odd nuber of ole and ero are egent of the root locu, reeberng that colex ole or ero have no effect. S Ung Arguent Equaton 7//8 6
7//8 7 Exale Re I, I Re G n Arguent equaton 36 36 For colex ero and ole For real ero and ole on the rght Real-ax egent are to the left of an odd nuber of real-ax fnte ole/ero.
5. Aytote of locu a Aroache nfnty The aytote nterect the real ax at σ, where n n n n The ntercet σ can be obtaned by alyng the theory of equaton. The angle between aytote and otve real ax o 8 n,,, L To obey the yetry rule, the negatve real ax one aytote when n- odd. Ung Arguent Equaton 6 o 7//8 8
Exale G Th oen-loo tranfer functon ha three fnte ole and three ero at nfnty. n- egent go to ero at nfnty. G, g 3 o n 8,,, g 6, 6,8 L o o o Aue the root of cloed-loo yte at nfnty ha the ae angle to each fnte ero or ole. 7//8 9
Rule 6: Breaaway and Brea-n Pont on the Real Ax Breaaway ont Brea-n ont When the root locu ha egent on For two fnte ero or one fnte the real ax between two ole, ero and one at nfnty, the there ut be a ont at whch the egent are cong fro two egent brea away fro the colex regon and enter the real ax and enter the colex real ax. regon. Ung Magntude Equaton 7//8
Breaaway ont g tart wth ero at the ole. There a ont oewhere the g for the two egent ultaneouly reach a axu value. Brea-n ont The brea-n ont that the value of g a nu between two ero. How? Exre g a a functon of Dfferentatng the functon wth reect to equal to ero and olve for 7//8
Charactertc equaton Z g F G P P Aung there are r reeated root at the ont S, F can be rewrtten nto F P gz r nr dp dz df g d d d P P g Z Z Ue the followng neceary condton P Z P Z g P Z Wth the oluton of, we can get g. For otve g, the correondng ont ay be the breaaway or brea-n ont.
Exale G g 3 5 g Z P Z Z P P 6 6.45, 3.8 Alternatvely, we can olve for real. 7//8 3
Rule 7: The ont where the locu croe the agnary ax Rule #7: The ont ay be obtaned by ubttutng =ω nto the charactertc equaton and olvng for ω. Exale : G [ 6] 3.74 Charactertc equaton [ 6] Subttute = 5 3 4-3.74 = = 4 3.74, 6 =±3.74 7//8 4
Utle Routh Stablty Crteron 3 5 4 Charactertc equaton: Routh array 3 4 5 + 7 5 + = 5 7, = 6 3.74 7//8 5
8. The angle of eergence and entry The angle of eergence fro colex ole gven by 8 Angle of the vector fro all other oenloo ole to the ole n queton Angle of the vector fro the oen-loo ero to the colex ole n queton 7//8 6
The angle of entry nto a colex ero gven by 8 Angle of the vector fro all other oenloo ero to the ero n queton Angle of the vector fro the oen-loo ole to the colex ero n queton 7//8 7
Exale: Gven the oen-loo tranfer functon 3 G H 5[ 4] draw the angle of eergence fro colex ole. 33.5 o 63.5 o 35 o 9 o o o o o 63.5 35 9 33.5 8 o o 5 or 375 o 7//8 8
Rule 9: The gan at a elected ont t on the locu obtaned by alyng Magntude Equaton g n t t To locate a ont wth ecfed gan, ue tral and error. Movng t toward the ole reduce the gan. Movng t away fro the ole ncreae the gan. Rule : The u of real art of the cloed-loo ole contant, ndeendent of g, and equal to the u of the real art of the oen-loo ole. 7//8 9
Content Suary Rule Contnuty and Syetry Syetry Rule Startng and end ont Nuber of egent n egent tart fro n oen-loo ole, and end at oen-loo ero and n- ero at nfnty. 3 Segent on real ax On the left of an odd nuber of ole or ero 4 Aytote n- egent: 5 Aytote =, n n,,, n
6 Breaaway and brea-n ont d[ F] F P Z P d Z P Z n g 7 Angle of eergence and entry 8 Cro on the agnary ax Angle of eergence Angle of entry Subttute = to charactertc equaton and olve Routh forula n n
ero Brean Exale 5..: Gven the oen-loo tranfer functon, leae draw the root locu. =5.88-3 ole - ole - breaa way =.7 G H.Draw. 6.Breaaway and brea-n ont 3.Syetry 5.Aytote 4. Two Segent the egent oen-loo on real ole ax and ero n 3 7//8 8 8 3 a a a 3 n n a a a 3 a 3
Exale 5..:. 4. 6. 7.The 3. 5. No Breaaway Fnd Aytote Syetry ont ole where and and ero. 4 egent on real brea-n the locu ax ont 4 acro the agnary ax Breaaway - 3.6 = = -3.6 G H 3 8 d [ 4 8 4 45 或 8] 35 44 n d9 3 yeld 484-4 3 Soluton 4 3 n,3 4 9 4 4 4 4 9 7//8 3 3 8 4 8 9 4 4 3.6 484-4 = get =
Exale 5..3 G H. 3.. 4. 5. Aytote Segent Breaaway Pont Oen-loo acro on and ole real the brea-n ax agnary and ero ont ax - -.44 =6 =6 -.44 -.4 P Z 8 P Z yeld[3 6,8 3 6 ] 3 oluton 3.4.58 3 3 3 6 6 3 3 7//8 4 = =6
Exale 5..4 G *
Exale 5..5 R.5.5 C