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
~
~r
'r..
~
u,.; t.-
..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
f
LfDL).k '" 0 0 2. >"
;::
Se>D
,>< :? DO 0
X - - - - , i.. ): 40v
I
=
= 1. X
<f )( !!?-
9
qat)
2
/ 0
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~
:?x
3d~O'x
4
2
j;h- X
L.j
)1 5
., 1.:r
~
:r ~ ±
4
.. 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/-~
MIT OpenCourseWare
http://ocw.mit.edu
6.685 Electric Machines
Fall 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.