Question

All of these problems should be done by hand. You may use a computer or calculator to find roots of polynomials with constant coefficients. All hand calculations can of course be verified with a computer. Show calculations for asymptote angles, center of mass, angles of departure, etc. for full credit.

Answer 1

. hien (s) R(S) + k (+60) s(s+20) (8+20) [ Negative feedback Closed loop Transfer function (S+60) al Serum ] s(s+20) RCS2 (S+60) lt -] [size] s(s+20) k(5+60) 4(5) SCS720) RUS) s(s+20) (S420) + k (S460) s(s+2) (+20) It's US PLAN } k (S+60) (S+20) s(s+20)² h (3+60) RCS2 a) Denominator Equation of the Transfer function is s(s+20)2+k(S+60) = 0 s[$*+405+400]+ k [s+ Go) = 0 3+405²+ you stk st bokeo 0 534 4052+5(k+ 400] + Guk = 0 - Applying Routh - Hurwitz untuia to Equation ;

Routh- Hursity Method: g k+400 - 54 40 Gok 40(k+400)-60k s' о so Gok for the system to be stable, All the elements along the first column must be higher than 'Zeus! чо (k+ Чор) - Gok & Gol> o 40 (k+400) Jook Thoo >0 D 40 4 . mara ya 22 it ☺ de kt-3465 hok+ 16000) Gok 16000) 20k anda l6ooo ka 20 Tk < 8000 " From Equations ② & 0; The Range & stability for system to be stable is ocka <80ol The limiting value of gain 'k for a stable ik = 0 system is --

20 b) for marginal stability, the first element under 's low should be zero. holk +400) - Gok 40 40k+ 16000 - Gokzo 201 - 16000 k=800 Substituting 't value in 405² Gok=0 40 ljur + box sw=0 406-609 = - Gox fro mou 16?=, GO x PUO . g2 low ho wa sizeo ... W = + 34.64 | rad free are the imaginary routs + s = + j(34.64) & S = _ j (34.64) are the which touch jo' anis. . Magnitude & Imaginary lasts: ||Wl= 34.64 lova c from Equation 0; closed loop equation is 33+ 405²+ s (k+ho) + Gok=0 hiven, k= 400 $24.4052 s ( 400+400]* Go x 400 - 0 8²7 40524 8008+ vrouo=0

SE-36.21 S = -1.89+ j(25.67) si-1.89 - j (2567) النم ها 34.84 25.67 -36.21 -1189 -26.12. ,-34.64 As 여 36.21 > 5x1.89. The pole -36.21 can be ignded by applying dominant pole approximation. . At the value of k= 400, the System qualifies for 'Dominent pole approximation: Finding Equation for Conjugate pair of closed loop poles. {s-[-1.89 + j(251»}} {S- [-1-54-j(266333=0 5? [-1-89+j (25-67) – 1. 49- j (26.67)] [-1.8i+j (26-677) (-189-1 (21-621) - 0 8² +3.787 662.52=0 Compluing it with perun'st oneo (986 8บ & wwn - 3.78 Wn = 25.74 rad/rec 18 = 0.07 j + ㅕ

Y() RCSI 2 k (+60) (s+2017 s(strol? h(+60) LE no k (60] (20 n. c gain = SO نے 0+ k (6) 7 (D.c Cain 20 Apprecinaate second order ga Transfer henchon; UCS k Mer (36.41 [5273.785 +662:52] + Damping Ratio, f= 0.07 Natural frequency, Wn= 25.74 lad/sec and the Required second order Transfer function is approximated