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.r " #### ECE430 Name 5 () ( '-'1-+/~Or"- f w.s #### Exam #2 (print Name) #### Spring 2005 #### Section: (Circle One) 10MWF 12:30TuTh #### (Sauer) (Liu) #### Problem1 Problem2 Problem3 Problem #### TOTAL: #### USEFULINFORMATION ~ f};;JR, -), :; ("4;";D J;jC.. (^) ~6.---"""" JI ;:: -J (^1) 8.'" J~ ~ 13 .A\ .J~"'- == 0 ~".. .-cLj.-~ "", J ..,,~ M" r:=^ I!^ IVb" -= I,f'It::AI n^ R'- -^ ."u A---~^ A1<r ::!)^0 ,4^ !;) ../-'t?J, -1"'; I"t "/I;:^ N^ ."A. J ~ } f f .i I WM ;:; L. J.). w,.,::: fA }J; /;.AI""f- w..,:: ;::}L ' )C= /ti"J1 ~ " (11". #### ft~ -~ .f c.: ~ for rt1~+;p", X -.". ($ #### J.,t. )X f ~ T ",),&, )('/: [FE ::' (^) r t~ / JI\ ffyil\::::- r-f~ (^) IJx ~ -" t"- -& ""'" r4;t'J.. ...A. I..:t~ ..JVVh'~ (^) -vv""~.., (^) -C-rp£ r Err?'! :X/ff()t)~ Xli'.J,.~~ d d... (^1) Dt p.-b J--{, -t-a I , ";. ' .. r. " #### Problem 1 (25 pts.) A coupled electro-mechanical system is diagrammed below. The current through the coil is designated as i. The cross-section of the magnetic core is A. The total length of the dotted line is lc. The number of turns in the coil is N. The permeability of the magnetic core is Pc. f--X ~ r tJ;r" ;'I ~ N a) Find the equation that relates the current and the flux linkage (you define the current direction and flux linkage polarity). b) Find the analytical expression of the force of electric origin. c) Write the second-order dynamic equation that governs the displacement of the movable piece. Suppose that the spring force constant is K (with a zero-force distance of C) and the damping factor is B. The mass of the movable piece is M. d) Write the state space equations that correspond to the mechanical dynamic equation found in c) above. #### p.) D{.f,~-( -{-liP O'/--('°'-( ;\-.J + f4 J A:) {VI-"-" ,'thtD .J- S,j"". fJ'1 CUf\JtI'VA,.fI~/""- 01 1{t.{~ {'lotj "L./l~cil) J It""'-)J~J ~J,...,L-. (c.4('iyt'J 1kt.,.t. ,s v"t, ~;...t 1.J1~(j"",~...) 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N (~c.../-},Yo/tA)::. ~c_~~~ L-~ #### .)Jo..l,--J-~,,--.x (Blank page follows this page) #### I -.,~'o , --".. . .. - #### Problem2 (25 pts) (AjJr~ I "» A single-phaserotating machinehasone coil on the statorwith current is and one coil on the rotor with current if, The inductancesfor this machineare (assumelinear magneticcore): Lss= Ls Lsr = Mcos(e) Lrs = Mcos(e) Lrr= Lr The machineis being operatedsuchthat the currentsis and ir canbe assumedto be constantsat Is and Ir respectivelywhile the shaft is rotated from e equalszero to e equals1[/2. .For this changefrom "point a" to point b", find: a) The energytransferredfrom the mechanicalsysteminto the coupling field as the systemmoved from point a to point b with constantcurrents. b) The energytransferred from the electrical systeminto the coupling field as the systemmoved from point a to point b with constantcurrent. 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Problem 2 (25 pts) (;)fJ~-;1 II 8 J An electromechanical device his the following flux-linkage vs current relationship with constants a through f and translational displacement x: Al = a it + (b/x) i2 + (c/x) i A2 = (b/x) it + d i2 + (e/x) i A3 = (c/x) it + (e/x) i2 + f i The machine is being operated such that the currents il , i2 , and i3 can be assumedto be constants while the movable mechanical member position is changed from x equals 0.10 to 0.2. For this change in operating conditions, find: a) The energy transferred from the mechanical system into the coupling field. b) The energy transferred from the electrical system into the coupling field. ~ \ \A/ ( -J .'"2 h ,'of lZ I (L /"' 0 \ "l.. ### ) ;.r') -Z At, + -:;- 4---2 d.'~2 .f- L (, (~ e l2(? J f' ,... --f -+- 2. ...,' 0 7"-- ]<. 3 f e (^) .:::- --i? b (', ( 2 C l^ '^ co^ ~ ~l" (~ ~ L .)(.2 ,,"2- "L ,"""l.. I' #### f(N),: -)-~,C~l~il-f((~(3+-t'(Z~)Jx-=: -d[b(;(2+-((;(j-:l~"l(JJI .f ./ ~~~!:~~=~~~;JJ b) WNI:' Lv,! .::. i ~c'12+-)bl~(2.J..1JI: f J{t;I)'( s-e(l i; -I4+(j) .1- ,L t,,/VI ::.. """,,[ = ~ ~(~2+(O '(,(2.. -:I- {Ji2.l+{Dt~II';rl{; L',. Z^ (3^ .J.j...{;^ L.. , , ., '- 'I ) '2 ] ### Wr""l,1.-C-"-""" -= -S-[ b (: Ii .j--((~( J -I ~ lZ rj) .:: 1 f E + S-[ bc: (2 +-Cf;(3 +-~() ~ (Blank page follows this page) -( D [b I; ~ +-{ I~~ + ~;2 (jJ () ~ 111 S() lIl.s.. fL' J ";\ ### 5ec. I'\lJt't S"tl1-" ,.. Problem 3 (25 pts.) An electromechanicalsystemis describedby the following flux-linkage vs currentcharacteristic: A= -~ i x-. It is operatedon the closedcycle a -b -c -d -e as indicated below, with x constantduring a -b and also c -d. The current is constantduring b -c and also d -e. a b c d e i (Amps) 0 ib ib 0 0 I..(Wb turns) 0 8 !..c 0 0 x (meters) .03 .03 .02 .02. Find the following things: a) ib and !..c. b) The energystored in the coupling field at points band c. c) The force of electric origin at points band c. d) Sketchthis cycle in the I..vs i plane (label points a, b, c, d, and e) e) Sketchthis cycle in the force vs x plane (label points a, b, c, d, and e) f) Is this is a motor or a generator? if ) DC{' , -L { , ()If, , #### V\ t.: ~£..~ "b -~I'.J '>rc -:: --tb -= / b ~ iiV' ,()L- ,(,If ) .L ,~1-~/' b f. D 2t I CJ1X / ~ - (^) [ , T W'IVtL-::--.= 32 .:r lNt'f'I ~ I.../I'r) :: --W~.b .::. -b .0 t -X-, vi ' ()L t I I, e ,()'Z}(/& () f t , OLiL J(- ".!!-~- -<'DO'" ~ =- I .= -[DOt 3 71,.- -,Ol~,p~--O (.. .~".()f (X-,()(\"2... 01- ### J) ">r /b c.- :>l-;.~;IO? .f- <£S' e) (^) .! \ ,,0 1 ,,0 L .0 3 J!. #### \ If C -~ Dr) -:;( ### e Ibur) ..- ### L ~ "zyo^ v ## d (Blank pagefollows this page) #### \ _11CO (" }) ffe:: + '0 6 (~~(t. ...10)CJ p"tofd/' "'" \\;, ,.,-"~~ « ,^ ... .." .. #### Problem4 (25 pts.) A systemcanbe describedby the following equations ~ ~ X' Xl -X2^ .-t~ Y'2. ';::. ~ .. X2=X.X2 +16-x. d-t- a) Write the equivalentsecondorder differential equationthat matchesthe given statespace equations. b) Find all equilibrium points of this system. c) If Xl(t = 0) = 0.5,X2(t = 0) = 10, find the values of XI and X2at t=0.001 and 0.002 s. ### L r:?.\ ~ -"j ~~ , ( 'L- ) d 1- -A 011 ;- f b -X #### f b) X'lq....: 0 ### 0 ':: / b -X I ~'2...^ X, ~-..ill~ I e.- e ':)(' I =. -LJ X r ;:- .J- V ~ e ### ':i 2- =-- 0 'It -::- 0 c) 'I, (. DOl);;:. 0 I S-+ ( I 0~ , 0 ()I=:O ~S-I :)(l(,OOI):::' /V.J- (0)~(Q-l-I'-;r1)j..(}O/ -.:::: !Of 0'20 g- ### )(,(.001):::.. O,Sf 4- (10/0 LoK)~' ou( ~ 0, S-Z ### 11. (,OOL) -:::I D,o[ -I {.j/xIO,OL-J /(,_',.)"(1))(0001-= 10/ VL-r
