A#19_Solutions - Assignment#19 W gfl‘ikfiw’b‘...

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Unformatted text preview: Assignment #19 W gfl‘ikfiw’b‘ “’“‘ f 'l ECSE-2410 Signals & Systems — Fall 2006 ‘ Fri 11/17/06 1(2 5). The root locus below is for a system with the closed-E001) transfer function 1&2” 3 WW. X(s) I + K G6?) The breakaway point is ab 2 -4.44. Note: All the poles of 6(5) have real and imaginary parts that are integers (no complicateé fractions or decimals!) Root Locus imaginary Axis —5 «4.5 44ft 73.5 ~23 -2.5 —2 4 (3.5 Real Axis a.) (8) Find the gain K so that one of the closed-loop poles is at S, x — 2 . ,_ flit. __,, fl st$+leS~rSl a. Wagzz l’” as!“ “GEE—“l "“ m WW1 b)(2) Find the otheffvgez- 1 Scan?” ' gg'i' €32 gZngé “aw . 1 p0 es. {again} Wufifids “i” S wax-K HE: 34?» lS3+Qsl+3as+36 Titan $2475~H5 :0 1;. (gal 2+2; §g+Z$k a ‘i‘ l7§1+3a$¢3é EL ¢snmgfigfimfil eefiega £95 +3.3 Assignment #19 — Solutions — p.2 ECSE-24IO Signals & Systems - Fall 2006 Fri 11/17/06 K 2(10). The root locus below is for a system with the closed-100p transfer function E = fl. X(s) 1 + K 6(5) Find K as the root locus crosses the ja) axis. Note: All the poles of G(S) have real and imaginary parts that are integers (no complicated fractions or decimals!) Root Locus 3’ s 1 g 0 {i Reaixxis 2 1 cflaamm‘s‘ém w .3. :1 KW ‘1 3., 1+sz 0 -t- (figjfiwzfij '. sg+$sa+ais+(a§-tlc3;a EM: a 3 a H sfissw =77 3 “film $ $33 -W1 $2, 5 {m w MP *-\szfii-y S} % §§;1$"14— G :7 Pgng rag-93mg law-+9 E i 50 g Kit-{cm Assignment #19 —- Solutions — p.3 ECSE—2410 Signals & Systems - Fa112006 Fri 11/ 17/06 3(10). Shown below is a DC motor connected directly to a rotating load which has an inertia of 0.1 kg — m2 . From the motor data sheet, the resistance of the armature is R m 5 ohms, the moment of inertia of the armature and the armature inductance are negtigible, the motor constant is Kg 2 0.05 Newton— meter/amp, and the motor back-emf constant is Kb 3 1.0 volt—sec/rad. Assume frictional losses are zero. The angular speed at the motor shaft is a) (I) and the inertia is J . This is a compatible set of units. t 60(1‘) R J m (3%» — O . CU __ Kt ‘ The transfer functlon mm w W where a) (s) 18 the angular speed of the motor and load. V65.) RJS + Kr Kb Find the dc voltage V so that in steadywstate the toad will rotate at a constant speed of 2 radians/see. The final answer should be expressed as a number in volts. is: as... batsman» ‘9" m} §+§tfl ewe-l :23" Assignment #19 m Solutions —— p.43 ECSE~2410 Signals & Systems - Fall 2006 Fri 11/17f06 4(30). Use the Routh array to determine the number of roots of the following characteristic polynomials, if any, that lie in the right half of the s—piane. Use the roots command in Matlab to check your answers. (a) s4+s3+232+53+8 5% E '2— 8 ‘3? g 5 g...“ M? s3 M”; 8 M % «tr—8 2} w S “33"”; «'37 5° 8 (b) 35+s4+233+s+5 rmfi Q, gigs ms: :15} Assignment #19 -- Solutions — 13.5 ECSE—2410 Signals & Systems - Fall 2006 Fri 11/17/06 5(20). For the following characteristic equations, use the Routh array to determine the range of K > O (and the corresponding values of smjm) that result in a stable system. Then use MATLAB to plot the root Eoeus and verify your previous values of K andjco—axis crossings. (a) sé+s3+3sz+23+K a 3 K Z. '= t { ga+2210 2:7 6:3”? 3W1 *‘ «W t 25k“ sf? 1%“ ZHK>Oqfifi fies—‘2. l W . . . . K MATLAB check of Root Locus. Charactemstlc equation is I + W = 0 s + s + 35‘ + 25 Root Locus 4 51’ System: eye ‘3 Gain:2.02 g g Po§e:0.{)03‘19+‘§.41i Z a x s s s a a a ‘ a s s a a x a .n it; 2 Overshootwq): 101 Frequency (radisec): 3.41 num=[i]; den={1 13 2 0]; rlocus(nu1n,den) Real Axis Assignment #19 — Solutions — p.6 ECSE—2410 Signals & Systems — Fall 2006 Fri 11/17/06 5(20). For the following characteristic equations, use the Routh array to determine the range of K > 0 (anti the corresponding values of srjw) that result in a stable system. Then use MATLAB to plot the root locus and verify your previous values ofK andjayaxis crossings. (b) 36 + 635 + 1334 + 3053 + 4452 + 243 + K. 5% :6 es I dg‘lgw Wig-“Law g Me: glam. *léw mi:- jaw We) em? fifffifle‘ djgaacfiw $10M W? gems, in Assignment #19 — Solutions —— p.37 ECSE~2410 Signals & Systems ~ Fall 2006 Fri 31/ 17/06 5(20). Continued. 36 + 655 + ms“ + 3053 + 4452 + 243 + K Continued. Let’s check the root locus. Char. eqn. is 1+—g-——5——m;1—~M§MWW = 0 3 +63 +135 +305 +445 +245 Root Locus g E E _-E ' g E “MM-em ” g g nummfi]; é den:{16133044240]; fl ‘ g f _ riocusmumfien) V . § .2. .. . . .4 as g .1 0 “a z 3 4 5 Real Axis Roct Locus Systemsys Gain: 35.2 : Pore: 43.0206 + $99; .2 g Damping: 0.0103 E 2 Overshoot(%): 96.8 E Frsguenpyead/sac}:_1.99 ................................................... .. g System: sys . : Gain: 35.2 "2 _ m Pore: 0.03376 - 1.84i if ; Damping: —{).036 f Overshoot {%): 112 . _ 3 Frequency (radlsec): 1.04 u3 E WES 2 5i: 3 U3 05 T !\J n 4:? Real Axis Assignment #19 — Solutions — p.8 ECSE~2410 Signals & Systems - Fall 2006 Fri 11/17/06 6(5). The following method allows for rapid deceleration of an arma‘wre controlled d’c motor with censtant field current. Suppose the switch has been in position “A” long enough for the motor to reach a constant speed. Then at fit} the switch is moved from position “A” to position “B”, short circuiting the armature. Find the equation for the motor shaft speed at), for z 2 0. H wag Assume the shaft has inertia J, but no vjsgousfi‘flitiom Also, assume the armature inductance is negligible. Express your answ pi—rr’te/rnis of the motor constants, KI, Kb, the electrical circuit parameters, V, fiWnical shaft parameter, J. / xaé:E(—3; 01w Haws}: 332+ %mwmw= gm) w Ff \ State ief‘d“ V [r a mutual} 4%; firm 03.9. images a {sweat sieazfiwlaf) mg, fl.“ "F our flwmt'fég‘vfi? gnu-7H ...
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This note was uploaded on 04/10/2008 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.

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A#19_Solutions - Assignment#19 W gfl‘ikfiw’b‘...

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