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ELECTRIC DRIVES
SPACE VECTORS
Dr. Nik Rumzi Nik Idris
Dept. of Energy Conversion, UTM
2013
Space Vector
WHY space vectors?
Representation of 3-phase equations (for 3-phase AC motor) is more
compact: only one equations is needed
Space vectors can also be represented in using d and q axes. If
windings are transform into d-q phases, magnetic coupling between
them is avoided (since they are quadrature)
Transformation between frames is conveniently performed using space
vectors equations.
Space Vector
Definition:
Space vector representation of a three-phase quantities xa(t), xb(t) and
xc(t) with space distribution of 120o apart is given by:

2
x  x a ( t )  ax b ( t )  a 2 x c ( t )
3

a = ej2/3 = cos(2/3) + jsin(2/3)
a2 = ej4/3 = cos(4/3) + jsin(4/3)
x – can be a voltage, current or flux and does not necessarily has to be sinusoidal
Space Vector

2
x  x a ( t )  ax b ( t )  a 2 x c ( t )
3

Space Vector


2
vx  vx a ( t )  av
ax b ( t )  a 2 vx c ( t )
3
Let’s consider 3-phase sinusoidal voltage:
va(t) = Vmsin(t)
vb(t) = Vmsin(t - 120o)
vc(t) = Vmsin(t + 120o)
Space Vector

2
v  v a ( t )  av b ( t )  a 2 v c ( t )
3
Let’s consider 3-phase sinusoidal voltage:
At t=t1, t = (3/5) (= 108o)
va = 0.9511(Vm)
vb = -0.208(Vm)
vc = -0.743(Vm)
t=t1

Space Vector

2
v  v a ( t )  av b ( t )  a 2 v c ( t )
3

Let’s consider 3-phase sinusoidal voltage:
b
At t=t1, t = (3/5) (= 108o)
va = 0.9511(Vm)
a
vb = -0.208(Vm)
vc = -0.743(Vm)
c
Space Vector

2
v  v a ( t )  av b ( t )  a 2 v c ( t )
3

Let’s consider 3-phase sinusoidal voltage:
At t=t1, t = (3/5) (= 108o)
va = 0.9511(Vm)
vb = -0.208(Vm)
vc = -0.743(Vm)
va (t) = 0.9511Vm
avb (t) = ( cos120 o + j sin120 o ) (-0.208Vm )
= 0.1040 - j0.1801
a 2 vc (t) = ( cos 240o + j sin 240o ) (-0.743Vm )
= 0.3715 - j0.6435
v=
2
(1.4266 + j0.4634) Vm
3
Þ v = Vm
Space Vector
Space vector can also be represented in its d-q axis:
v = vd + jvq
é2
ù 2æ
1
1 ö
2
vd = Re [ v ] = Reê ( va + avb + a vc )ú = ç va - vb - vc ÷
ë3
û 3è
2
2 ø
é2
ù 1
vq = Im [ v ] = Im ê ( va + avb + a 2 vc )ú =
vb - vc )
(
ë3
û
3
q
v = v e jq
θ
d
Space Vector
v = vd + jvq
If
v
rotates, and vd and vq will oscillate on the stationary d and q axes
qe
q
v
vq
v
de
vde
e
q
vd
d
If we define a rotating axes de and qe that rotates synchronously
with , then we can write
e
e
d
q
v
v = v + jv
vde
and
vqe
will appear as DC on this rotating frame
Space Vector
a is the angle between the stationary and rotating frames
qe
q
v
vq
v
e
q
α
θ
de
vde
d
vd
In rotating reference frame,
v= ve
This is
j (q -a )
jq
= v e ×e
- ja
v expressed in stationary reference frame
Space Vector
v = v e j (q -a ) = v e jq × e- ja
e- ja
is the rotator vector - transforms stationary frame to rotating frame.
The transformation can also be written in matrix form:
é e ù é
ê vd ú = ê cos a sin a
ê vqe ú ë - sin a cos a
ë
û
ùé vd ù
ú
úê
ûêë vq úû