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8
M.Barański,J.Kołowrotkiewicz
Inthispaperthefield-circuitequationshavebeenformulatedandanalgorithm
hasbeenelaboratedforsolvingthem.Thisalgorithmtakesthenonlinearityofthe
magneticmaterialandskewedslotsintherotorintoaccount.Thedevelopedcom-
putersoftwarehasbeenusedforanalysisofstaticanddynamicoperationofa
squirrelcageinductionmotor.
2.FIELD-CIRCUITMATHEMATICALMODEL
Incaseof2-Danalysis,theequationgoverningthemagneticfieldcanbewrit-
teninformof
1
l
[
{
[
∂
∂
x
(
|
k
ν
∂
∂
φ
x
N
|
)
+
∂
∂
y
(
|
|
k
ν
∂
∂
φ
y
N
|
|
)
]
}
J
=
−
J
(1)
J
=
γ
∂
∂
V
z
e
−
γ
l
d
d
φ
t
(2)
Intheseexpressions:
φ
(
t,
x,
y
)
=
lA
z
(
t,
x,
y
)
isaquantityrepresentingelement-
edgevaluesofthemagneticvectorpotentialand
A
z
(
t,
x,
y
)
isanaxial
()
z
compo-
nentofmagneticvectorpotential
A
,
l
istherotorlength,
J
isanaxial
()
z
componentofconductioncurrentdensityvector
J
,
ν
isamagneticreluctivity,
γ
isanelectricconductivity,and
V
e
isanelectricscalarpotential.
Equations(1)and(2)describethemagneticfielddistributionintheairgapof
aninductionmachineandtakethenonlinearityofthemagneticcircuitintoac-
count.Generally,whendealingwithvoltage-excitedfieldsinaninductionmachine
containingnonlinearelementsthecurrentsinwindingarenotknowninadvance.
Hence,thecurrentdensity
J
isalsoanuknown.Therefore,oneneedstotakethe
electriccircuitequationsofthesedevicesintoconsideration.Asetofindependent
loopcircuitequationscanbeexpressedas
u
=
R
i
+
p
Ψ
(3)
where
isavectorofloopcurrents,
u
isavectorofloopvoltages,
p
isadifferentialoperator
R
isadiagonalmatrixofloopresistances,
(
p
=
d
/
d
t
)
,and
Ψ
isa
i
vectoroflinkagefluxesofthewinding.
Intheconsideredmathematicalmodel,theequations(1),(2)and(3)aresolved
simultaneouslywithequationofmotion[3,5,6and8]thatisapproximatednumeri-
callybyanexplicitfinite-differenceformula
