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66. ELECTROMECHANICALENEI\GYCONVEI\SION
. FIGURE2.1:1
DvnamicSwem
Giv.enin .
:.xcmple2.7
Theinductanceis
= N~ '" IOOO~ = 5.655/ 10-. H
L 'di. 1.768X In"\" x
At x ,=.I mm= 0.001m.F'I ='-~32.74/~.Thl:rct'orc.:H2.74/~
0.595A.
. . EXAMPLE2.7 .
100'or/.=
An electromechanicalsyst.einis shownin Fig.2.13.Developa set01'differential
..,equationsgoverningitsbehav~oraftertheswitchis c1o\ed.
Solurion
After theswitchis closed.lheequ.ati<)n()fmotioncanbewdttenas
du
m dt =Fd - VI/ - kU - xl
whcrcm.u, v,and.\"arethemass,velocity.coefficient01'friclion.anâdistance
or thcmovablepartfromtheelectromainet.respectively.Thespringc()n~tantisdenotedbyk. However.
d.\"
dt
.wherethenegativesignaccountsfor~hedecreasein.\'.
1/ =.
.\'-/
__ F...I--l- I.' 1
I I F.
I '1-
I 1---
1 I
\Iovable part
.,
I
RELUCTANCEMOTOR
Thecquàtionsatisfyingtheclectriccircuilis
I
f
.
whcre
IJ =dÀ .,.Ri
dI
.\=Li. .
;-Jeglc:clingfringingandleakage.toe'inductanceo( themagncticcircuitcant
cxpressedas " . : .
L = . 1M'.v~A'.
J.l.f)r\:illn,- ~x
where9lmis thereluctanceof themagncticmaterial.Theforc.edevelopedis
F =_ c3W!(À.xf = _.~ !!..:(~.). =_ ~. c3x'. 2 óx L J.l.o.AN~ .
. ,':".'":', :..-,::. ,..-,) " . ,.' " ", . . .'
Finally.thesetof (iitfé'r~ntlalequatióíi:sihatgc)~ernthebehavior'ofthe.gi~e
.systemO!!:!1besumm~fjudas'foHQv.:s:. . .
du
dt =
. .
I " v k k
---;:;À.- - /I+-.\" - - I .
mf.LoAN- m m m
d.\"
dt =
dÀ- :.
dI
-/I
Thesedin~rei1IÜ\1equationsarenonlinearin termsaf theindependentvariable
.t-andÀündcanbesolvcdnurnerically..
2.G RELUCTANCEMOTOR
.ToilIustrawth::applicationofelectromechanicalenergy-conversionprincipies11
rotatingmachines.cúnsideran clementary.sínglyexcited.!-polereluctanc.
motorasshownmFig:~.14.II con~ists01'twoparts.a freei;rOlalingmembe
(rolor)revolvingunderthe.poles01'a slationarymagncticcircuit(SlalprJ.whicl
carriestheexcitaliQnwindingasdepicted. .
If theangulardisplacemenl11betwcenthtrotorillídlhestatormagncticaxe
is zero(whichisreferredtoasthedirp.ctord.axisposition).theeffectiveairga
andtherebytheteluctance01'themagneticsystemwillbeminimuin.risiseviden
fromFig. ;.l~. To simplifythe.theoreticaldeve1'opment.,assumethattheslatO
.. '.