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Elektriske likninger og momentbalanse i
den transformerte modell……….
■ Transformerte modell på komponent form:
k
k
k dΨsα
k
U sα
= R s ⋅ Isα
+ − ωk
⋅ Ψsβ
dt
k
dΨ
k
k sβ
k
U sβ
= R s ⋅ Isβ
+ + ωk
⋅ Ψsα
dt
k
dΨ
k
k sγ
U sγ
= R s ⋅ Isγ
+
dt
k
k
k dΨrα
k
U rα
= R r ⋅ Irα
+ − ωr
⋅ Ψrβ
dt
k
dΨ
k
k rβ
k
U rβ
= R r ⋅ I rβ
+ + ωr
⋅ Ψrα
dt
k
dΨ
k
k rγ
U rγ
= R r ⋅ Irγ
+
dt
k
k
k
Ψsα
= ( 3 / 2 ⋅ Lsh
+ L sσ
) ⋅ Isα
+ 3/
2 ⋅ L h ⋅ I rα
k
k
k
Ψrα
= ( 3 / 2 ⋅ L rh + L rσ
) ⋅ I rα
+ 3 / 2 ⋅ L h ⋅ I sα
k
k
k
Ψsβ
= ( 3 / 2 ⋅ Lsh
+ L sσ
) ⋅ Isβ
+ 3 / 2 ⋅ L h ⋅ I rβ
k
k
k
Ψrβ
= ( 3/
2 ⋅ L rh + L rσ
) ⋅ I rβ
+ 3 / 2 ⋅ L h ⋅ Isβ
k
k
Ψsγ
= Lsσ
⋅ I sγ
k
k
Ψrγ
= L rσ
⋅ I rγ
dθk
= ω
dt
k
dθr
= ωr
= ωk
− p ⋅ Ω
dt
Elektriske likninger og momentbalanse i
den transformerte modell……….
■ Moment uttrykket:
SR
SR T ∂�
SR
( I ) ⋅ I
p
M e = ⋅
2 ∂θ
⋅
p
M e =
2
∂θ
2
∂θ
�
mek
SR
SR
−k
k T ∂�
−k
k p k T
−k
T ∂�
−k
k
⋅ ( � I ) ⋅ ⋅ � I = ⋅ ( I ) ( � ) ⋅ ⋅ I
k k k k 3 k T k 3
Ψs
( Ψ ⋅ I − Ψ ⋅ I ) = ⋅ p ⋅ ( Is
) Ψ s = ⋅ p ⋅ Ψ ⋅ I ⋅ sin ε
3
M e = ⋅ p ⋅ sα
sβ
sβ
sα
�
2
2
k
k
⎡ ⎤ ⎡ ⎤
k Isα
k Ψsα
Is = ⎢ ⎥ Ψ =
k
s ⎢ k ⎥
⎢⎣
Isβ
⎥⎦
⎢⎣
Ψsβ
⎥⎦
⎡0 −1⎤
= ⎢ ⎥
⎣1
0 ⎦
�
Skalert modell - pu-modell
■ Ønsket form på induktansmatrisen:
⎡x
h + x sσ
0 0 x h 0 0 ⎤
⎢
⎥
⎢
0 x h + x sσ
0 0 x h 0
⎥
⎢
⎥
k 0 0 x sσ
0 0 0
� = ⎢
⎥
⎢ x h 0 0 x h + x rσ
0 0 ⎥
⎢ 0 x
+ ⎥
h 0 0 x h x rσ
0
⎢
⎥
⎢⎣
0 0 0 0 0 x rσ
⎥⎦
x s = x h + x sσ
x r = x h + x rσ
2
s
s
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Ser bort i fra γ-systemet ……….
■ Kan da benytte to-dimensjonale romvektorer:
k
r
dθk
= ω
dt
k
rh
k
s
k
k dΨ
U s = R s ⋅ Is
+ + �⋅
ωk
⋅ Ψ
dt
k
r
rσ
r
k
s
k
k dΨ
k
U r = R r ⋅ I r + + �⋅
( ωk
− p ⋅ Ω mek ) ⋅ Ψ r
dt
k
k
k
Ψ s = ( 3/
2 ⋅ Lsh
+ Lsσ
) ⋅ Is
+ 3/
2 ⋅ L h ⋅ I r
Ψ = ( 3/
2 ⋅ L + L
k ) ⋅ I + 3/
2 ⋅ L
k
⋅ I
dθr
= ωr
= ωk
− p ⋅ Ω
dt
h
s
mek
Skalert modell - pu-modell
■ Grunner for å innføre pu-modell:
➨ Det er lettere å se om motoren er overbelastet
➨ Man kan lettere trekke erfaringer fra andre motorytelser.
Parametrene i pu endrer seg ikke så mye.
➨ Når man skal implementere regulatorer må man allikevel
skalere de variable.
■ Tilleggskrav for pu-modell for asynkronmaskinen:
➨ Velge basiser slik at man får en enkel modell
➨ Alle ikke-diagonale ledd skal ha verdien xh ➨ Alle pu-egeninduktanser skal kunne skrives som xh pluss
en lekkinduktans
Skalert modell - pu-modell
■ Valg av basis-verdier for stator viklinger:
I s , basis = Î n U s,
basis = Û n
Û n Û n
Ψs,
basis = =
ω 2π
⋅ f
n
n
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■ Valg av basis-verdier for rotor viklingene er gitt av
forholdet mellom antall turn i stator og rotorvikling
■ Skaleringsmatrisene:
−1
Sψ
= diag[
Ψs,
basis Ψs,
basis Ψs,
basis
−1
Su
= diag[
U s,
basis U s,
basis U s,
basis
−1
S = diag[
I I I I
Ψr,
basis
U r,
basis
I
Ψr,
basis
U r,
basis
I
Ψr,
basis ]
U r,
basis ]
]
i
s,
basis
s,
basis
s,
basis
r,
basis
r,
basis
r,
basis
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