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

qq

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: