Post on 31-Dec-2015
description
ELECTRIC DRIVES
SPACE VECTORS
Dr. Nik Rumzi Nik IdrisDept. of Energy Conversion, UTM
2013
WHY space vectors?
Representation of 3-phase equations (for 3-phase AC motor) is more compact: only one equations is needed
Space Vector Space Vector
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.
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:
x – can be a voltage, current or flux and does not necessarily has to be sinusoidal
a = ej2/3 = cos(2/3) + jsin(2/3) a2 = ej4/3 = cos(4/3) + jsin(4/3)
)t(xa)t(ax)t(x32
x c2
ba
Space Vector Space Vector
)t(xa)t(ax)t(x32
x c2
ba
Space Vector Space Vector
)t(xa)t(ax)t(x32
x c2
ba
Let’s consider 3-phase sinusoidal voltage:
va(t) = Vmsin(t)
vb(t) = Vmsin(t - 120o)
vc(t) = Vmsin(t + 120o)
)t(va)t(av)t(v32
v c2
ba
Space Vector Space Vector
)t(va)t(av)t(v32
v c2
ba
Let’s consider 3-phase sinusoidal voltage:
t=t1
At t=t1, t = (3/5) (= 108o)
va = 0.9511(Vm)
vb = -0.208(Vm)
vc = -0.743(Vm)
Space Vector Space Vector
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)
b
c
a
)t(va)t(av)t(v32
v c2
ba
Space Vector Space Vector
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)
Space Vector Space Vector
)t(va)t(av)t(v32
v c2
ba
Space vector can also be represented in its d-q axis:
Space Vector Space Vector
dd θθ
Space Vector Space Vector
If rotates, and vd and vq will oscillate on the stationary d and q axes
If we define a rotating axes de and qe that rotates synchronously with , then we can write
and will appear as DC on this rotating frame
ddee
qqee qq
dd
Space Vector Space Vector
ddee
qqee qq
dd θθ αα
In rotating reference frame,
This is expressed in stationary reference frame
is the angle between the stationary and rotating frames
Space Vector Space Vector
is the rotator vector - transforms stationary frame to rotating frame.
The transformation can also be written in matrix form: