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Unformatted text preview: EE321 Exam 3
Spring 2010 Notes: You must show work for credit.
Trigonometric identities are towards the back. The last page of exam is blank for extra paper if needed. 1) 18 pts. Consider the aphase of a distributed winding machine. The number of
conductors in each slot is given by Nas=[0440—4—40440—4—4] The total ﬂux through each of the teeth is given by
(D:[—1—2—1121—1—2—1121] where the ﬁrst element is the ﬂux through the ﬁrst tooth and so forth. What is the winding function (12 pts)? What is the ﬂux linking the a—phase winding (i.e. what
is A“) (6 pts)? In Nets (04L, 04px. pailZWi mews mice, :5. £29. 2; pelt Maw—1J2 (ownﬂea Was,» 2 New ~ Mag: 1 1+ so on. => WMa Elmowvao wondro} w.
.4 ”a” +0~L\HQ"+O ~L\—<I; +0 ~Lt~2+~c> 7V1; '3 "Hg V9 2) 18 pts. The a— and bphase winding functions of a twophase machine may be expressed
was 2 SOsin (4%")
W = 50 cos(4¢m)
where €155», is measured in the counterclockwise direction (as is our custom). The a and b—
phase currents may be expressed
ins :IOOsin (SOt)
ibs = —lOOcos(50t)
The machine has a uniform airgap of 7r mm, a stator radius of 2 cm, and a length of
50/ 7: cm, and there are no circuits on the rotor, PM materials, or other complications.
Part A. Find an expression for the radial component of the B —ﬁeld in terms of
¢m and t. (9pts)
Part B. What is the speed of the B ﬁeld traveling wave in the counterclockwise
direction? (3 pts)
Part C. Neglecting the resistance of the aphase winding, what is the voltage across
the aphase in terms of time. (6 pts)
Part A L“: Wag/qr, + ”4,5155
3 5000 (55’? 4112M cm sot —~ 69!; Mun use Got)
Wing Wig ID 09$ (A 1‘ a) > MA 695 e 2‘ mm 5m 12
F a e 5000 6% (AQSSM Mot) 6“ 5000 09% (4955M Soﬁa)
A
PM?" I}
2 ~7
g MO H 5‘ Mag/3 : Ami/o xgggo 69$ (Aug/twp hex)
{mod
”‘e 52 09% (ark/M Set“ + 7r) T
“(if)” Hm Awgrm— germ 2K wN 15W (extra paper for problem 2) Pearl: C 2/7
M: 21 f m, dtﬁgm
O 2W : Axw’z'x 150 MOQ‘XQX 00 ‘
76m ‘50 69% ( 579F+4¢m1~W) 9m 4’7““ 493m
0 Uga’ng 93mg 49 Sa'nA ass/2.: = )1 (gave/4mm) + 5mm. 7323.)) 1‘." 2r
w?” M .i .
1r 2; Mn (WM +4¢M+20 + xii/’7 (”éorw'f) mm 2.7?“
:o—1 "
0.2}! ¢naot+m mam: ”W3 45m] 27‘“ 0 2A.“ ~.~. ~ X60 (goHVﬂ V, Va; 2 i244 :; 0% "’50 Caz)» (got—HT) \/ 3.) 18 pts. A two—phase permanent magnet synchronous machine has the following flux
linkage equation. Take it to be correct even though it doesn’t make complete sense. A“ 5 ~2 ins 5
= +
21,3 —2 5 i,” 3
Now consider the following two—phase qd transformation to the ‘weird’ reference frame
[f5] 2 K: [f]
ds fl): 52 NF”, ’“2
K::[21l ﬁg“; I” 7 ["2 5 Express the machine flux linkage equations in the weird reference frame. 4.) 18 pts. A three phase brushless DC machine has the following parameters: rs =39 , LSS = 10 mH, xi," = 0.17 Vs, P = 4 . At is desired to operate at obtain a torque of 2 Nm at a speed of 4000 RPM. What is the highest machine efﬁciency that can be obtained to achieve the desired
torque at the desired speed. 492 0,0 h‘wwm (9523 1‘65me Ira/t" {a}, .n. p norm: 1mm; x35 {3997f mail/i
be 3 i) ll
)0
X
J.“ G 0
>7
‘1) $9
Q 0 >; i ) 90 o: :4 \I
0‘ n 2 row a. as 93776 : Mm 025“ W“
Firm 6106ch E) 7... 7, h‘lﬂqa “n T173 5.) 28 pts. Answer true or false. Grading on each is as follows: Correct(2pts), Incorrect(
lpt), No response (0 pts).
Amway? c 7: Fave, A ~ pause) a.)
b.) c.) d.) f.) g.) h.) i.) l)
k.) 1.) Distributed windings are used to create a smoothly rotating MMF waves. Continuous and discrete winding descriptions are both used to represent distributed
windings. In a distributed winding machine, the conductor density function serves a similar role
to turns in our lumped winding analysis. In many ac machines, the rotor moves at the same speed as the MMF wave. The speed of the stator MMF wave is controlled by the frequency of the currents and
the number of poles. In a distributed winding machines with a uniform airgap and a round iron rotor (as
we discussed in class), the radial ﬂux density is a sinusoidal function in space (i.e. ¢sm ) In a distributed winding machine with a uniform airgap and a round iron rotor (as we
discussed in class), the radial ﬂux density is a sinusoidal function of time. In practice, “singlephase” machines are actually twophase machines. All other factors being as equal as possible, a threephase machine fed from a three
phase bus will be more efﬁcient than a singlephase machine. Flux is your friend. Obtaining a sinusoidal back emf in a brushless dc machine requires a speciﬁc magnet
shape. Setting 473v = atan(a),_Lm / ’1) yields maximum efﬁciency. m.) Voltage controlled brushless dc motor drives are, in general, more efﬁcient than n.) current—sourced drives. Permanent magnets contribute to coenergy. ...
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 Spring '08
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