6.685 Electric Machines, Quiz 1 Solutions

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science G.GS5 Elee'tric l\lachiner:-.-' Quiz 1 One Crib Sheet Allowed Novel her 6, 2013 There is space for you to write your answers on this quiz. There are four problems on this quiz. They have equal weight Problem 1: Induction Motors The single phase equivalent circuit of a three-pha.::;e, four pole induction motor is shown in Figlll'e 1. For th purpo::;(;'s of this problem we will assume armat.llI'e resistance is negligible. Th machine is connected to a till ~-plli e voltage source with line-Ilcutral voltage of 200 volts, Rl\lS (346 volts, line-line) and a fn'qucncy of 400 radians/..econd. The motor reaches peak torquf' at a speed of 160 Radians/second (about 1528 RPl\I). c~- 2t>O X ~ I'Z.~ = IbD o=J j 2.5Q 2~- + loI/2'>~ '1..'~?- 200: oJ? 2.L\D j2 Q _ ~ ~2 J (0 ~ ~~9 n" Figure 1: Induction l\lotor Equivalent Circuit 1. What. is the \'alue of that peak torquE' (in N-m)? f'e~..fc.. -ft:>tl-r..e :::;: > IhCJ T-- 3.:b. >< ~()O -t? =3-.>< W ~ I-fL..f vl-;< Jey; - ~ X 4 :: Lx ~'1. 4()O (~ () x4 ~ 6)< 16 x16() IOt:J X "3 ~ 32 JX J.,6l(/~O= '3)( 160:: YlJi /6 ,yl-1' d~ 2. \\"hat is the vallIe /6 OJ( f R:!.? (in n) 1 II) !~-F-N-~-' l Probl m 2 DC' l\Iachines A perrnanent magnet DC motor is connect,cd to a 250 Volt (DC) source. Running :light" (no mechanical load). it draws negligible CUlT nt and turns at a speed of 200 Radians/second. (about. 1910 RPM). The armature circuit of the macbine betS a resistance of one fl. Now tbe machine is loadeJ so that it is driving a load torque of 100 N-m. still connecteu to the 250 VDC source. 1. Ho\\' much cmrent is it drn\Ving? I.. 2> W~ k I -: : d"-/X ICD =}loA7 :2. How fast is it turni ll"? Kn. :; ZS--O \1:= RI = l5'D- ~O 6>. J? X 1"10 :2 J ~O V [n c IeJh"C] = s 1'1" .3 ­ I 36.() Problem 3 Time constants J.1-ro N turns h I ~ w Figure 2: Pot Core \\'ith Coil and Shell Inside Shown in Figure 2 is a pot core wit.h t.wo elements inside: a coil with N t.urns is located at the out.er radius of the cavity \\'ithin the core. The other thing is a conductive shell of radius R and height h, \\'hidl jllst fits inside the axial dimension of the ntvit.\, The' shell is thin ami Ims surface condllCl,ivity (J.~. For (·he I.Jllrpose of this probleul.. WE' are ill a universe iu which . l = ~OO, 000. The coil has 1.000 tlll'lli', the cylinder radius is R r• = lOelll and the coil mclius jf; R = 20 'Ill. Hc'ight of both elements and of the inside of the pot core is h = 10('[11, I~ 1. Assuming the coil is radially 't.hin', and ignoring the conducti\'e cylinder, what is the coil ind uctance? L~ 1.. P Q 4 ~ :: ", ~ X I~ 8~rc!J'" 7T X 't -:r /~ '11~ ') =- rr ;: :' ~ I. >11 If ~O\\': as:mrninf!; the conductive cylinder is 'perfcctlv dial1lnguetic', rncanillg zero flux can penetrak it.. as vou \\'Ould haH~ with infinite cylinder couducti\'ity, \\'hat is the coil ind ud allce? p~~ 1U~.1 L';~f'[E] :3 :3. Assuming that the c~'linder has surface conductivity of 80.000 S (reciprocal ohms). and the coil is driven by a step of 10 A, what is the magnetic field inside of the cylinder. as a function of time? IOOD )< (0 j;q ()D () J'" .. , 3' lJ.; to ()D X ~00 OOD X ~ , 2.. -.:: • 12 S- T Problem 4 Synchronous 1\Iac:hine A three-phase, two pole synchronous machine hFts the following characteristics: Pu Rated Power Rated Volt,age Synchronous Inductance Field to Phase 1\Iutual Inductance Synchronolls Frequency 3 1 :2.5 25 'ol00 VB Ln - Lab ?II UJo I . ". 1\IVA kV (Peak) mH (3 mH Radians/Second Assunw that armature resistance is negligible. 1. On no-load. open-circuit. te::;t. at. rated speed. what field current. is required to prod lice rated voltage? I OD 0 ,. .::: /fOD t< .. J];~ (DD ~ 6)1. to 7 j/ :2. On short.-circuit test, what field current is required to produce rated current? 2. := CJ ()O ~ u,.; t.- 'r.. ~ ~r ..r.=F ':' ~- ~ ) -:z-£;-D-A-/.. . L 2£),00)< :z !? 3. Running at rated speed. wit.h field current. If = :300A and with rated armature terminal volt.age (1.000 V, Peak, what is tlw peak torque t.he machine can produce? 1:f == "7 ~ ..L. V&e.-f X X -::.L '''00 ,>< :? DO 0 X - - - - , i.. ): 40v I = < f )( !!?- f ;:: Se>D = 1. X / 0 9 qat) 2 LfDL).k '" 0 0 2. >" EQ.f- ~ ~ ~:: ... I -:: ()Q D ~ II"J 2 $'0 A.J-v'-t 4. RUlllling at rated speed with field ClllTent. If = 300A and with the armature driven by a bFtlanced three-pha.s· current of 10 = ::\. OOOA.. Peal.:. what is the maximulll peak t.orque that t.he 111 ac:hill(' can prodlice? =- .1 X 2 ­ ;::; 3~ 3d~O'x 4 2 j;h- X L.j )1 5 ., 1.:r ~ :r ~ ± 4 :?x .. D2. ~ X 5 .1. ~ 3 (Joo >s 3 t:> 0 30 '27 If) 'I:: -c:r -x .. J? 0 - X 10 < f :::: ~ 5 . 3 ~5- X to :5 5/ 1~iJ 5.!3r./7 '5 )( ID '1 /l/-~