Question

Using the Nyquist criterion finds the range of K for stability for each of the systems in Fig. 2. RS CIS (s + 2) (s + 4)(8 + 6) System 1 RES K(52-45 +13) (s + 2)(8 + 4) System 2 Kis-1) (s+1) System 3

Answer 1

from System W; let u assume Gló)= K and H(s) = 1 (5+4) (5+6) (5+2) Therefore tore, Openloob system function, K (2) G(s)H(s) = (5+2) (5+4) (5+6). Put s=jw from (2); Gj. HW) K (a+jw)(4+1w) (6+jW) Rationalizing the equn ei); G(16) Hw) = K (2-10) (4-6) (6-6) 4+62) (16+62)(36+62) Gliw-Hliw) = K (2-jw) 24-10w-621 (4+03 (16+w)(36+64 Giliw Hljw) = K ( 48-2016-2W2-24jw* 10w2+3637 (+62) (16+62)(36+62) Gliwo) Hljw) - k[ 48-2W2-1062] + jk[W3_-441W). 144+62)(16762)(36+62) Seperate real and imaginary parts. - lii) Gliw.hilic) = K[ 48 - 12w1] + j K[W2.446] Scanned with 14762)/16+62)(36+w) (4+6) (16+61) (36+62) CamScanner

2 Equale the real bait to Zen, we get from equn (2) kr 48-1202] (4+62) (16+62)(36+64) 48-12.02-0 01, or w: 48 OY, W= Yod/sec. toom 0); Vy palie).uties we2 * shoreJacow. wsz ar laivutumwa na ho faria o, laluwstelholms - The X DOCHO I Gliwiwws2 or, lactiu) cia) w 22 for stable system, On, K< 80 cs Scanned with Suun the gain Ku Lus than 80, the point (-1+0) not encircled by GW()) plot and the system is stable.

1) b). from the System (2); let us assume. Gls)= k154-45413) 75+2) (5+4) H13) = { 5o. Obenloob system funchon, G(s)H(s) = k152-45+13) 5(5+2)(5+4) - u Put s= jw. G(jw). Hiw) le (V) k l jw) - 4jW+13) jw (jw+2) (jw+4). Rationalizing the egen (21); G(jw) H1)W): kl 13-62- 4jw) (-iw)(a-iw)(4-jw) w.(4+63) (62+16) Glju"Hliw) - K.(13-62- 4jw) kiw) (8-60w-62) W.(4+62)(W2+16) Gliw} Hliw) - K(13-62-416) (-81W-662+W3) W(4+62) (16+63) Gliw.hliw) - K(-104jW- 78W2+ 13003-8jwB+604-jw? 3262243w1404) W14762)(167692) Gliw.HlW) - - Kl 10w4-11062) + j k ( 28w3_1040) W (4+2) (16+62) (*) (V) c Equalan the health best to CamScanner Zero, are get

trom (V); K. (J0w4-110W) = 0 01, (10W2-110)W?=0 01, 100-110 01, W23.11 Or, wenu = 3.31 Yad/sec. trom (V); . Gliw.Hl/wW-2.31 = K. (13-w2)+(46)2 augusti . Set wit. / 1601 /w= 8:04 OT, W IE 01, I Gliw.bliwwsvio - K. 113-(8.31)2) = (4 *3-31) 2 - 26741 - K/13 - 11)++ 476 Or, | Gliw.klw w- Vio 3.31. 15x27 or 1619. Hliww.id K./180 3.31405 or I Gliw). Hll)wali Khuss or Gliw-Hiw) cs Scanned with amScanner nos i altro su to the

for stable system, or [k< 5] Ans When the gain k tk less than 5, the point (-1,+jo) is encircled by Gliw plot. and the system is stable... (16) b) c). from System (3); let us assume that Gls) = x(5-1) 5+1) Á H15)= (3-2) 5+2). Oben loob System Tunchon, G(3) H(s) K(S-1)(S-2) (5+1)(5+2) Put s=iw from; G(W.H(160) = K (jw-) ( jw-2) (jw+) (+2) - Rationalizing the equn (21); GC1w).Alim) Kliw-1) (jw-2) (jw-) (1-2) (2+1)(63+4) 01, Glj) Hlim) - K.((w-u)?. (19-232) []+62)(4+62) op, Gliw). Hliw) - K161-62- ajw)(4-624W)] (1+w2)(4+62) 01 Giw) nije svith = K[ 4-62-416-46276444363-8jw+2jW2-8627 CamScanner (1+w?) (4+W?)

or, Glimillim) = K[4-130*7604] + jk [ 663_120] (1+2) (4+02) Se perale reale. Imaginary bart in the above. equin: Or, Gliw.hija)= K[4-13w2764] (1+w") (4+62) j k l 66.2.120) (1+0?) (4+ (2) Equate the real part to kem, we get, k [4-13w1764= 0 Os, w9-13w++4 0. let =x; . x?_132+4 =0 X = 12-68, 0.3L WPEX w?: 12.68 Wa 0.31 W=112.68 Á W10.31 w - 3.56 A W : 0.55 rad/sec for (1) | Gliw) klju) mes . 16116) Hliw)w=0.55 K. J1w2+1)/6+4).K 16241644) k./16241)81844, k. Vw2)(W444) Scanned with CamScanner

For stable System, Ikal When the gain k is lus then t, the point (-1+lo) is not encircled by Gi(jw). Hl/W) plot and the systems is stable. Scanned with CamScanner