Preliminaries - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 73

Hence, the space vector (2.12) is

V m e − j θ . v s = ¯ V m cos θ − jV m sin θ =

When θ changes with time t , the length of the space vector ¯ v s does not change but the angle

changes. As a result, the vector ¯ increases with time t . The direction

of the rotation is the same as the spatial order abc shown in Figure 2.4(a). Note that the

v s rotates clockwise when θ

αβ
reference frame itself is stationary and does not rotate, hence the name stationary reference
frame. Here, the introduction of the space vector has converted translational movements on a
spatial diagram into rotational movements on the

reference frame.
For the case shown in Figure 2.5(a) with the spatial operator

αβ

e j 2 3 , the corresponding α =

transformation matrix is

⎡
⎣

⎤
⎦ =

1
2

1
2

1 − − 10

0

T αβ .

2
3

T αβ =

√ 3
2

√ 3
2

−

1

0 −

In this case, the real part v α remains unchanged but the imaginary part v β changes its sign. As

v s =

V m e j θ and it rotates counterclockwise when

θ

a result, the space vector is ¯ increases with

time t , as shown in Figure 2.5(b). Again, the direction of the rotation is the same as the spatial
order abc shown in Figure 2.5(a).

2.3.3 Synchronously Rotating Reference (dq) Frame
The voltages v α and v β in the stationary frame are still functions of time although the space

vector ¯ . If a reference frame that synchronously rotates

at the same speed of the space vector is introduced, then the space vector ¯

v s rotates with time t at the speed of ω

v s on this reference
frame is no longer a function of time and does not rotate. Such a reference frame is often called
the dq reference frame, or the synchronously rotating reference frame (Bose, 2001). Assume
that the new set of coordinates dq rotate in the same direction as the space vector with a phase
angle

θ g . Then, as shown in Figure 2.4(c), the angle of the space vector ¯ v s on the dq frame is

− θ − ( − θ g ) = θ g − θ and the space vector is

v β ) e j θ g V m e j ( θ g − θ ) v s = ¯ V d + jV q = ( v α + j = . (2.15)

Hence,

V d
V q

T dq v α

= (2.16)

v β