Question

16.1 For the following systems, sketch the Bode diagram, and from the straight-line approximations to the gain and phase plots, estimate the maximum value of K for which the system is stable: a. GH(s) = s(s + 1) (s + 4) b. GH(s) = = s(1 + s) KS c. GH() = 6 *21 к d. GH(s) = s(s? + 2s + 16) 5K(1 + s) e. GH(S) = f'( + s/352

Answer 1

I sali Gus= k S(5+1) (stu) and share The magnitude one given by 16(sw) la 2o los,o [wrutti sultat ] dis {alow) = -98-ban wacan we des The Cornen frequencies one wala. The slope for a pole is -20 dol decode and to zero it is +20 dig/decod. The Bode plot i r -20 dilder. 1G15w) 1-40 dolder. > -60 dBldec. - 9 1a5w) t 480 -270+ Oll LT I sw 9 10 The gown magin is defined at phase Goss over frequeny ... 14 (5w) = -1800 C -18° = -900-Gonow-ban w S = w= 2 ead/sec The gain at Waz is 1 Q (sw) = 2010s. S a la -26 dB ( 252²1 J 2²+ 2J

& gain magin a 2018. The Ronge of 'k' to stability is - 20/20 S 2o los k= 26 dB =) k= 10 = 19.9526 I The system will be stable to [k < 19.9526 b) G His) = K 3 (5+1) | 9H (sw)1= 20.109. [ w owon] 1 GH (sw) = The Corner -180- Can'w pequeny is wal -40 dBldec. I la nowy -60 doldec. 186 | GH(5w). 2) s -110 + 0.11 I sw 10 tn 9(JW) = -180° Wro gain at wzo i r ind ie Gaun magin us a injing . so The system is unstable troll values of 'K' fram 'ö to 'a' Å His) = ks (5+2)4 c)

| au cas ) = 20 les. I can saya ] La H(SW) a The Colna Tano wa uxsañl w fuquing is w=2 I IGH (ws) 20 doldec . -60 doldec. i + 20 dolded LCHEUS -2700 2 aw 10 The phase Goss over pequency i 1G15w) = -18.01. Dant w-abon = -1800 » w= 9.66 eaalsec. - The gan at wa 9.66 is - la t(sw) l =-619 8 Goun malgin is 61.90 - ew=a.66 The Ronge g 'ks to stabilit is. as 61910 2olos k=61.9 2) K210 = 12465 The System is stable for kalaukis I a Gissa k s 132+25+16) I The second order system is s²+25+ 16 in the from

8² + 25onst want a wnau and {op 0.25 wp= wn Sraga = 3.7407 and the peak. magnitude is. R 2010 (22 Jaga Ja 6 dB. The Corner tuquery is w=u., The Bode plot is. -20 at der I laiswa -60 dBldec. -90° I GL5w = 20109.. [w Joshuis being [9 (5w)= -98 Gantzw 16(JW) 16-w2. -270 t 3.74 wo The phase Goss over ~180=-9- fuqueng san zw is 2 wa 4 ladlec 16 - W2 Gain of wou eaalsec - -30.1 dB 08 Grown margin is 30.1 dis and (30.1/20) ? $1.989 k= 10