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" 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" ;' 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:) {V-"-",'thtD J- S,j"" fj'1 CUf\Jt'VA,f~/""- 01 1{t{~ {'lotj "L/l~cil) J t""'-)j~j ~J,,L- (c4('iyt'j 1kt,t,s v"t, ~;t 1J1~(j"", ~) D'\{; D'1( t, /J/yOi'\)c,j- zlj X: /Vl~ /)0 J/j ~:)lcj/}", ~ /j-j-j ~ J-uc ~)r,~ L),)-V-"l ()c + tx ~ ) ~ /1,/(,,' ~ =- ~!:::L- ~t>,}6/t ~/),- -+ ~cx A -::: N (~c/-},yo/t A):: ~c_~~~ L-~ )Jol,-J- ~,,--x (Blank page follows this page) 2 -,~'o, --"
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,~ - Problem 2 (25 pts) (AjJr~ "» A single-phase rotating machine has one coil on the stator with current is and one coil on the rotor with current if, The inductances for this machine are (assume linear magnetic core): Lss = Ls Lrs = Mcos(e) Lsr = Mcos(e) Lrr= Lr The machine is being operated such that the currents is and ir can be assumed to be constants at s and r respectively while the shaft is rotated from e equals zero to e equals 1[/2 For this change from "point a" to point b", find: a) The energy transferred from the mechanical system into the coupling field as the system moved from point a to point b with constant currents b) The energy transferred from the electrical system into the coupling field as the system moved from point a to point b with constant current /f \ ~ - L t'( - N\{"s O " {\ -1 \ ( "L- A > --(" v r \,1 '" ~ t '5 -'--/11\( 0 S l! (t-- /' (' m ~ 1- S T Jr; r -r""i: Lr, bj ~ r -: 1""("5 8cc: J- L, (' r J e " TJ(-z T :: _ft'\sj\fj-c,str '";fr L f r; ;: -(-V' SJ~ fj) ~1r j c9- ;: -f\') (';5 ~ r ~ ) ~ T T :5~r -"'P-;- r () 0 Ls ; l r, ~ -~ f, Ft -:- it; j ~ + r J ~r :: -~ 5Tr _Y\ 1J;; :: -Z M ~1', l)!j f My- " J + L, "tr D'( - -1 7- "7 -/ -j (J:~, f la",h~ W~b -zls">f-o,+1lrt'" WM~~~""A -7 S s 7",~rr"'Tlr~ \A-~, D -~ hl( :: --i-'\ 1: S r f- -:J -j:: r t;'~ ~ + ' r f " Nt = -~,J:-sJ r,' V '" ' /A -r " -fa r " L C ;<:J (Blank page follows this page) c f \',! ' J /, " - \", ~~--"~ 4
~- Problem 2 (25 pts) (;)fj~-;1 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) i3 A2 = (b/x) it + d i2 + (e/x) i3 A3 = (c/x) it + (e/x) i2 + f i3 The machine is being operated such that the currents il, i2, and i3 can be assumed to be constants while the movable mechanical member position is changed from x equals 010 to 02 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 (L /"' 0 \ "l ) ;r')-z At, + -:;- 4-- -2 d'~2 f- L (, (~ e l2(? J f', --f -+- 2 0 7"-- ]<, f e b (', ( 2 C l ' co " :::- --i? ~ ~l (~ L ~ )(2,,"2- "L,"""l f(n),: -)-~,C~l~il-f((~(3+-t'(Z~)Jx-=: -d[b(;(2+-((;(j-:l~"l(jj f / ' 3 ' ~~~!:~~=~~~;JJ b) WN:' Lv,! :: i ~c'12+-)bl~(2j1j: f J{t;)'( s-e(l i; -4+(j) 1-,L Z (3 Jj{; L t,,/v :: """,,[ = ~ ~(~2+(O '(,(2 -:-{Ji2l+{Dt~';rl{; L',,,, '- ' ) '2 ] Wr""l,1 -C-"-""" -= -S-[ b (: i j--((~( J - ~ lz rj) :: 1 f E + S-[ bc: (2 +-Cf; (3 +- ~() ~ (Blank page follows this page) -( D [b ; ~ +- { ~ ~ + ~;2 (j J () ~ 111 S () lls L' J ";\ f 5ec '\ljt't S"tl1-"
, ':} ~3 (/~ 1),""1-,2- f f f::- v-h ) J,,~ J -' f}- j l2 J -1 L,- (J J -\ f ~/, ~2 ':)3,, :-, h l-~ rl(j) 1" (,'2 (b (~ f ~ lj) + ~ (Ct( + t'c2 ~(S- --/6) L L-~L-JL-- tt--j L 1 f Fe = -{ D ( b "f ""2- -} L t ~ i 3 + ~ J t c1"» V t~cl s 5
,, Problem 3 (25 pts) An electromechanical system is described by the following flux-linkage vs current characteristic: A= -~ i x- 01 t is operated on the closed cycle a -b -c -d -e as indicated below, with x constant during a -b and also c -d The current is constant during b -c and also d -e a b c d e i (Amps) 0 ib ib 0 0 (Wb turns) 0 8!c 0 0 x (meters) 03 03 02 02 03 Find the following things: a) ib and!c b) The energy stored in the coupling field at points band c c) The force of electric origin at points band c d) Sketch this cycle in the vs i plane (label points a, b, c, d, and e) e) Sketch this cycle in the force vs x plane (label points a, b, c, d, and e) f) s this is a motor or a generator? if ) DC{', -L {, () f,, V\ t: ~ ~ "b -~'J '>rc -:: --tb -= / b ~ iiv',()l- ) L,~1-~/' b f D 2t CJ1 X / ~ - [, T W'VtL lnt'f' ~ /'r) :: --W~b :: -b 0 t -X-, vi ' ()L,(,f -:: --= 32 :r t, e,()'z}(/& () f t, OLiL J(- "!!-~- -<'DO'" ~ =- = -[DOt 3 71,- -,Ol~,p~--O ( ~"()f < S' e)! \,,0 1,,0 L 0 3 (X-,()(\"2 01- J) ">r /b c- :>l-;~;o? f- ~ \ f C -~ Dr) -:;( d e bur) - L ~ "z yo v (Blank page follows this page) \ _11CO (" }) ffe:: + '0 6 (~~(t 10)CJ p"tofd/' J!
«, " "'" \\;,,,-"~~ Problem 4 (25 pts) A system can be described by the following equations ~ ~ X' -t~ Xl -X2 Y' 2 ';:: ~ 2 X2 =XX2 +16-x d-ta) Write the equivalent second order differential equation that matches the given state space equations b) Find all equilibrium points of this system c) f Xl(t = 0) = 05,X2(t = 0) = 10, find the values of X and X2 at t=0001 and 0002 s L r:? \ ~ -"j ~~, ( 'L- ) d 1- -A 011 ;- f b -X f b) X'l q : 0 0 ':: / b -X ~'2 X, ~-ill ~ e- e ':)(' = -LJ X r ;:- J- V ~ e ':i 2- =-- 0 't -::- 0 c) ', ( DOl) ;;: 0 S- + ( 0 ~, 0 () =:O ~ S- :)(l(,oo):::' /VJ- (0)~(Q-l-'-;r1)j(}O/ -::::!Of 0'20 g- )(,(001)::: O,Sf 4- (10/0 LoK)~' ou( ~ 0, S-Z 11 (,OOL) -::: D,o[ - {j/xo,ol-j /(,_',)"(1))(0001-= 10/ VL-r 8