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Spring2004_Final

# Spring2004_Final - 50[Duffy/i Name(Print Name...

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Unformatted text preview: 50 [Duffy/i Name (Print Name) Section: (circle one) 10 MWF 2 MWF (Sauer) (Kimball) ECE330 Final Exam, Spring 2004 Tuesday, May 11, 2004, 1:30 — 4:30 PM One sheet (2-sided) provided Problem 1 Problem 2 gamble.“ 3 i Receiving or giving aid in a I final examination is a cause for Problem 4 ._ dismissal from the University Problem 5 Problem 6 TOTAL: Problem 1 (25 pts.) You have three wires of equal length sufﬁcient to supply a load consisting of 3 resistors (12 Ohms each). The impedance of each wire is 0.1 +jO.l Ohms. You can provide service to these resistors using three possible techniques: a) Use two of the wires connected to a 120Volt, 60Hz, single—phase, 2—wire source and then connecting the 3 resistors in parallel at the load end b) Use all three wires connected to a 120/208 Volt, 60Hz, three—phase, three—wire, wye— connected source and then connecting the 3 resistors in a wye conﬁguration at the load end. 0) Use all three wires connected to a 69/120 Volt, 60Hz, three—phase, three—wire, wye— connected source and then connecting the 3 resistors in a delta configuration at the load end. For each case, compute the magnitude of the voltage across each resistive load, the magnitude of the source line current, and the total real power lost in the lines. I o! J‘s f . f“ I 0 - IZDL‘) i L/n' L/rZ-LJIL v: HL/V. ﬁes; : “2'; W '~: 12.0}: 1:: ill-hid v L [oldst W Problem 2 (25 pts.) In the ﬁgure below, N1 = 100, N2 = 25. The reluctances of the left and right branches of the core are each 1x105 H". The gap, g, in the center post is 1 cm, and the cross—sectional area of the center post is 4 cm2. Neglect fringing and the effect of the steel in the center post. The resistance of the left coil (primary) is 1 ohm and the resistance of the right coil (secondary) is 0.05 ohm. a) Label voltages v1 and v; and currents i1 and i; so that all self and mutual inductance terms are positive. Assign dot polarities. Assign v1 and i1 to the primary and v; and i2 to the secondary. b) Find L1 (self inductance of the primary), L; (self inductance of the secondary), and M (mutual inductance between coils). c) Treating this structure as a non-ideal transformer, draw the equivalent circuit with all impedances referred to the primary (assume a 60Hz supply and use the frequency domain). I“! " qrryrﬂyllx (0’7 ’r : Zorn/05H 20, mm mm», =- m a 5 .t * azooxmsﬂ iUlwo ¢Z - 2572 » {005,-} (0545, + 20 array/é'éz) : (2 Ilfv‘z ~f [0:461 +Zoowof/lfzrdﬂ90 ,I Z‘f‘t'suoifl‘z + [7‘75 4), H (I); 3 Blank page for work space 70/yr054'l411t0f¢,= loci/+21% WK; 2)”ng T [005} {AZ/41,3952, J, : ,0005‘4} +,006/2 7!; at: F;% I ,Ofé; 7‘w0/24/[Z \$2 7 JZWX/E [l 4 ,7?f(.oaof[, + ,000/27‘23 > INN/€757} +. , Obof’lfté ’ ~ .. : 0025 Lmi: (952%: 2:3?» : ‘Otr47g Ll]- L) La, o0 l (00 ~ 1 ‘ ,003(2,f[Z/5,)r 05" 50 L12: ,QODZ‘IH Problem 3 (25 pts.) Consider the machine below. The two gray boxes represent permanent magnets, sized so that the air gap ﬂux density is 1.0 T (constant DC ﬁeld) when the rotor current is zero. The air gap is 1 mm long, 3 cm wide (arc length), and extends 3 cm into the page. The rotor self-inductance is measured as having a maximum value of 0.3 H and a minimum value of 0.1 H. The rotor coil has 100 turns. -Lf kc 100x ix ‘Iuo :m”! a) Using the general form of the ﬂux linkage given below, ﬁnd the constants L0, 11 and K . This flux linkage 9» is defined with respect to the voltage labeled as positive where the current enters the page (on the right in the ﬁgure) and v = dk/dt. x1=(Lo+Llcos(26))i+Kcosl9 b) Find the co-energy, Wm’, as a function of current i and angle 6. c) Find the torque of electric origin, Te. d) Suppose the machine goes through the following four transitions along the given paths: -_ “mu m n“ Sketch the cycle in the 7t—i plane and determine EFEcycle and EFMcycle. Path Constant Current Constant Position “ Constant Current Constant Position ’. ' A : (,1 +,( £05269); 4*, 04 {0563 Blank page for work space ’2“, + lo? (055 L‘ ’9' 'r {M- ~Z—’~/,Z+.//u;29)£ I .2 \$1343 a Problem 4 (25 pts.) A dynamic system is modeled as: J'cl = —3xl +2x2 3'62 =le--2x2 +2 a) Find all equilibrium points. b) Linearize the system at each equilibrium point. 0) Determine the eigenvalues at each equilibrium point. Determine which points are stable and which are unstable. d) Starting near the stable equilibrium point at Ax] = 0.01, sz = —0.01, and using At = 0.001, find the values of AX] and Ax; at the ﬁrst three time steps after zero. Use the linearized state— space form and Euler’s method. ' 69% e, 60 Q1 Misﬁt? "2&5: ———s> 9‘2 e24“; ‘ f ’, X?a_ ZK: +Z )[fz'gxfgvzﬂa X] z [,2 x“ 3 3 L'Z) ,5 '2. D ,3 a I"; Z a M; ’2 {'4 9+? to . ='-W,’ Lts‘e ' Shﬂé " i “Z 3M2 ‘ r f ‘ e M; ,2 ,' at“? ,2 :¢ :3 s 37/ '53? Mnéﬁél )4 w v 01[-;o3'.ozl= ,0 0474' X (100/) f ’0‘ +’0 Z) ‘ (mt-hall: 'vaoq‘“ ' I. ISA} ZAXZ 6l\ bx! {’90I\ : n’0(%,00{ ’- Ah a 20.x, r2 Ari Blank page for work space Problem 5 (25 pts.) A three—phase, 60 Hz, 2 pole, wye connected synchronous generator has a rating of 6OOMVA and 23KV (line-line). When it delivers 400MW at unity power factor, the torque angle is 45 degrees. (neglect armature resistance) a) What is the synchronous reactance of this machine? b) What is the maximum reactive power this machine can deliver when it is delivering SOOMW of real power at rated voltage? 0) What is the minimum reactive power this machine can deliver when it is delivering SOOMW p of real power at rated voltage? d) Repeat b) and 0) when it is delivering 400MW of real power at rated voltage. ‘5? ﬂ) Momma 1: gmir’f” : int, 20; 1‘s k 2 {33 at? D“ ‘5 alwtﬁ'o éir'ém 43ma‘3/352mmx, or ﬁ'!’ V 5 Wélbllr‘ﬁ? IIW’ g fag 5M“ @2524: W7! m mazhmm. ,3 ..-.. £7; 00 m axe/e? Problem 6 (25 pts.) A 3-phase, 4—pole, 60 Hz, 440 V (line—line), wye~connected squirrel cage induction motor is running at 1620 RPM and draws 15 Amps at 0.85 power factor lagging. Neglect all equivalent circuit parameters except the magnetizing reactance, rotor resistance, and slip. Find the following things: a) The frequency of the rotor currents b) The efﬁciency of the machine c) The magnetizing reactance and the rotor resistance as seen by the stator d) The torque to the shaft e) The load on the machine in HP - 157004530: 0,, 4X 79: M5 ...
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