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Unformatted text preview: EE321 Exam 1
Spring 2010 Notes: You must show work for credit.
The last page of exam has a piece of extra paper if needed. Good luck ! Consider a square loop of wire with 5 turns in a Cartesian coordinate system as shown
below. The x—, y, and zcomponents of the ﬂux density are denoted Bx, By, and 32 respectively and will be given in units of Tesla. la) y—axis Square loop
with 5 turns.
Corners marked
(x,y,z) wound
in direction
of arrows. xaxis zaxis is out of page 10 pts. Suppose the ﬂux density is constant in space and time with EX 2 0.5 T, By = —0.5 T, and 32 =1 T. What is the ﬂux, d) , through the coil ? What is the ﬂux linkage, A, of
the winding ? (P: f Bdls
a BI. : BZA 1b) 1c) 8 pts. Suppose the ﬂux density is constant in space but varies with time as Bx = sin(l 001‘)
T, By = sin(200t) T, and 32 = sin(300t) T. What is the voltage across at the terminals of the open—circuited coil at t = 0.23 . 6P = Fﬁz A « min (3001) (mm X 2 N cl) "v 20 9m{300(:) V: 2L7: : gt (2.0 Kt'nfaootl)
all: 6U: \) 6000 C98 (300 t) Vibe.“ :1 9000 09/1 (aooxoa—l 2 ~57IAV 7 pts. Suppose the Bﬁeld is constant in time but varies in space as function of the x
coordinate. In particular, Bx =O.25x2 T, By = 0.5x2 T, and 32 = 0.5x2 T. What is the ﬂux linkage associated with the coil (i.e. 2t ). a: n
9*
y?
A
Q
,__. u
L's.
)0
63
N
2>
X
i
“*5
>3
u
k
X
1
ii
I!
Po
2.
V Consider the UU core inductor below. This inductor has two coils. Each coil is wound
in a direction such that positive current will cause ﬂux to ﬂow in a clockwise direction. Each coil has N turns and has a packing factor of pf. The permeability of air is denoted ,uo , the permeability of the magnetic material is ,uHuO , where la]. is the relative
permeability of the material. The conductivity of the conductor in the coil is a. Side View Top View 2a) 15 pts. Using MEC techniques, derive an expression for the inductance of the winding
circuit consisting of the two coils connected in series in terms of N , yo , yr and the dimensions in the ﬁgure. Carr‘ent‘ through 601'! (#2
{Vch (K Cl>
NC NL‘ 1% 351551 % R k: 31” + 4 { 20B +du .1. W5+Wu § 0 L h WU LQ W
auro’gaf Mate Cay Wigwam/th
781006611766 05 “‘60”
W
Uwon nwwmw
L': :z Hm No cf. (turns “RN
‘ ‘5? = 3.19.4.
2 R
=> L: A N A c «z ((2d5+du+ mvout 4 WQ'MJU) Wu)
2.
ﬂo/M’Jr Wadi/g 9> L = 52mm Maw N2 (249 +61% +m\ om +Q/U51‘Wu) WU
2. 2b) 10 pts. Derive an expression for the resistance of the series connections of the two coils
in terms of pf , 0' , N and the dimensions in the figure. Rpst‘glzmca 075 9142 Leo/'1 K: j 2 V w aura” 0930" V: @wmdm I + a? Nada WU 4‘ [W61 dMDPF 3 WCaZd££ (QM/4+ 2wuf‘!‘ I‘M/(‘2) 0w = New—017m PF
N 737*“ wl‘g’mw Rt = W“ mez
We; def M2 f Rt. 2: 2 (2i +2ww x Whig W 14/1ng P; 0” 3.) 25 pts. Find an expression for force in terms of xand A, and 12 for the system with the following current equations. i1=511+11—0(11+12)3
+x i2=822+i°—(21+ﬂ2>3
1+x A! 3" 0“"??31 ,9t290 WA: fhld'h‘ + [fad/{2'0
a. 7) 3 f ‘3‘“ 7' j‘Seugasdﬁ,
O H
00
m
1.
5 I}
.D
go
.1.
'5 4.) 25 pts. The ﬂux linkage equations for a ceitain 2phase induction machine are shown
below. At a given instant of time 6 =7r/2 i =1, ibs =0, I'm, =2 , and i,” =—2. rm ’ as What is the electromagnetic torque at that instant of time. lax 4 0 3 COS 6m: 3 Sin 6/7" in:
x1,” _ O 4 —3 sin 9”" 3cos<9lm i,”
101' — 3 6H" _3 6/7” 5 0 illl‘
lb!“ 3 Sin 61711 3 COS 6m: 0 5 [hr
Farr Lima” A (Em 227,6
% W .T :5 .
c f .L L L L
2‘ i . inductance
Wivc
7‘92 9m, ; i (T a“: ‘v
(ang 2 39m
m , ft 0 2 ~23 o o "bﬂdnﬁrm 3C9§9rm l
2 0 ° »5 comm 34.5an 0
“95% 8th O o 2
Suawm ~3an 0 o L
: ,1. {mgaj ‘90 “30 l
2 0 O O "a O
'3 O o o 2
O "3 a O “'7‘
z i {to 2 “23 ~la
2 6
é ...
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 Spring '08
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