Direct Torque Control with Space Vector Modulation (DTC-SVM) of ...

Modeling and control modes of PM synchronous motor drives
Ld + Lq Lq −Ld j2( θr θK) j( θr θK)
Ψ
sK
= ( ) IsK − ( ) I ∗ sK
e − +Ψ
PM
e
− (2.20)
2 2
Stator fixed system ( α,
β )
Taking the angular speed of the reference frame to be Ω
K
= 0 and θ
K
= 0 , the set of
synchronous machine vector equations (2.19) and (2.20) my be written as:
U
d Ψ
sαβ
sαβ
= Rs
Isαβ
+ (2.21)
dt
Ld + Lq Lq −Ld ∗ j2θr
jθ
r
Ψ
sαβ = ( ) I
sαβ − ( ) Isαβ
e +Ψ
PM
e
(2.22)
2 2
Substituting to above equations the following expressions for complex vectors
U U jU
= s sα
+ , αβ sβ
s sα
sβ
I αβ = I + jI , Ψ αβ =Ψ
sα
+ jΨ sβ
and splitting into real and
s
imaginary parts one can obtain the scalar form of the machine equations in stationary
α,
β reference frame:
U
U
dΨ
dt
sα
sα
= RsIsα
+ (2.23a)
dΨ
sβ
sβ
= RsIsβ
+ (2.23b)
dt
Ld + Lq Lq −Ld Lq −Ld
( cos2 θ ) Is
( )sin2θ Is
cosθ
2 2 2
Ψ = − − +Ψ (2.24a)
sα r α r β PM r
Lq − Ld Ld + Lq Lq −Ld
Ψ
sβ =− ( )sin 2 θrIsα + [( ) + ( )cos2 θr] Isβ
+Ψ
PM
sinθ
(2.24b)
r
2 2 2
Note, that in the flux-current equations (2.24a and b) still we can observe that value of
inductances depends on rotor position θ
r
.
Stator flux fixed system ( x,
y )
In order to take advantage of the set of equations (2.19) and (2.20) in rotating coordinate
system, one assumes that the coordinate system rotates with the stator flux linkage
angular speed Ω
K
=Ω
Ψ
and θK
= θ Ψ
. As a
sx s
Ψ =Ψ , δ = −( θ − θ )
Ψ
r
Ψ
d Ψ
U R I j
dt
sxy
sxy
=
s sxy
+ + ΩΨsΨ (2.25)
sxy
Ld + Lq Lq −Ld ∗ − j2δΨ
− jδΨ
Ψ
sxy
= ( ) Isxy − ( ) Isxy
e +Ψ
PM
e
(2.26)
2 2
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