Preliminaries - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 75

Hence, the rotation matrix is

cos

θ g sin θ g

T dq =

(2.18)

− sin θ g cos θ g

and

⎡ ⎤

1
2

1
2

⎡ ⎤

V d
V q

cos

1 − −

v a
v b
v c

⎣ ⎦

2
3

θ g sin θ g

⎣ ⎦

= √ 3

2

√ 3
2

− sin θ g cos θ g

0 −

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

cos( θ )

cos

1
2

1
2

2 3

cos 1 − −

θ −

2 V m
3

θ g sin θ g

= √ 3

2

√ 3
2

cos

− sin θ g cos θ g

2 3

0 −

θ +

cos cos

⎡
⎣

⎤
⎦

2 3 2 3

θ − + θ +

cos

cos θ −

2 V m
3

θ g sin θ g 2

=

cos cos

− θ g θ g

sin cos

2 3 2 3

θ − − θ +

√ 3

2

⎡
⎣

⎤
⎦

3
2 cos

cos

θ

2 V m
3

θ g sin θ g

=

− sin θ g cos θ g

3
2 sin

θ

V m cos

θ g cos θ + sin θ g sin θ

=

− sin θ g cos θ + cos θ g sin θ

V m cos(

θ g − θ

)

= .

− V m sin( θ g − θ )

The V d component remains unchanged but the V q component has changed its sign. In order
to minimise confusion, it is preferred to use the case shown in Figure 2.4 with the spatial
operator

e − j 2 3 α = for a system with the phase sequence abc . In this case, there is no

negative sign in V q .

2.3.4 The Case with Phase Sequence acb
For three-phase voltages with the phase sequence of acb described as

⎡ ⎤

θ

V m cos( )

V m cos

⎡ ⎤

3

⎣ 2 ⎦

v a
v b
v c

θ +

⎣ ⎦ =

,

V m cos

3

2

θ −