Question

$T(s) =k \frac{s^{2}-8s+32}{(s^{2})}$

--- k is in the set of non-negative Real Numbers.

PLEASE SHOW ALL WORK.

A.) Provide an ACCURATE HAND SKETCHED ROOT LOCUS

B.) Asymptote Intersection:

C.) Asymptote angles :

D.) Breakaway point(s):

E.) Departure/Arrival angles:

F.) Imaginary axis Intercept(s):

Answer 1

The detailed solution of the given question is shown in the image below:

(0) Root Locus sketch 3k OK=oo K=0 K=0 T() = K. 5²85 +32 SP CLOZ = 9285 +32=0 → 5= 8164-4(32) – 4+ C.L.P two at in number at origin P-Z B Asymptote intersection: 6A = {real parts of Poles - Real part of zeros P-Z = (0+0) – (4 +1) = undefined 2-2 (O Asymptote angle, A = 2K+1°180° wher, K=0,1,2,..., CP-7-1) = undefined (as P-z=o) (D) Breakaway fornits / Saddle points: For areakaway point we need to find characteristic equation, gove 1+T() = +K (SX_85 +32). Numerator is the characteristics can , 9(6) = CORIRN) 59+k (s®-85+32) = 0 Now, Saddle point is obtained by equating dk o the value of 's thus obtain are saddle point

88+32 S +32 m ut K = - dk --TG2-88492) 26 - 5" (25-3)] = 0 as (52.88 +32) 2 => 98-85 +32 = 5(+4) = 5X-88+32 = 82-45 = - 48 =-32 → S = 8 The breakaway point is at s=8 as it doesn't lie on roof locus we will not show in sketch (E) Departwee and arrival angle: The vangle of departure of the root locus branch from Multiple pole is given by, Pm= (2x+1) 180° ; K=0,1,2,...(4-1) where, er is No. of multiple pole here, K=0,1 for K=0, Pm = 180 = 90° for K=1, 0m = 270° Angle of varrival of the root locus Branch from complex conjugate pair of pole is genin by. Par = (2K+1) 180° - OJ' (wchove kus any mtegoy ) Dar = + [180°-07 where 0 is Net angle conforbution at the complex zero due to all other poles and wios. = Lzvros - Lpoles = L 90°- 490° = 0° Dar (180° ] - So, Porr = +180°, -180° O * 67 90°

(F) Imaginary caxis intercept The intercept is obtained from Routh- Hurwitz vriterion, We know, ou Ws) =s&+K(82-85+32) = 0 8PC1+K) -8 KS +32K = 0 g2 ltko 32k s -8K o sº 32K As frist coloumn nove mode should by of same sign ander e the first coloumn is For No value of K of same sign. So, the root do not intersect ju aris