Predictive Control of Three Phase AC/DC Converters

Appendix A
Coordinate transformations
Due to space vector theory it is possible to describe three phase circuits in various
rectangular coordinate systems. There are two main rectangular coordinate
systems:
• Stationary system (αβ)
• Rotating system (dq)
A.1 Stationary system
If we introduce stationary rectangular coordinate system in such way that α is
real axis and β imagine axis, space vector can be composed as:
k αβ = k α + jk β
(A.1)
Taking into account (2.2) transformation from natural abc to stationary αβ coordinate
system can be expressed as:
[ ] [ ] ⎡ ⎤
k
k α 1 0 0 a
= √ √
⎢ ⎥
k β 0 3
3
− 3 ⎣ k b ⎦
(A.2)
3 k c
where: [
1 0 0
is called matrix transformation.
The reversal transformation:
⎡ ⎤
k a
⎢ ⎥
⎣ k b ⎦ =
k c
0
√ √
3
3
− 3
3
⎡
⎢
⎣
]
1 0
− 1 2
= A abc2αβ (A.3)
√
3
2
− 1 2 − √
3
2
121
⎤
⎥
⎦
[ ]
k α
k β
(A.4)