Alpha–beta transformation

In electrical engineering, the alpha-beta (

α

β

γ

{\displaystyle \alpha \beta \gamma }

) transformation (also known as the Clarke transformation) is a mathematical transformation employed to simplify the analysis of three-phase circuits. Conceptually it is similar to the dq0 transformation. One very useful application of the

α

β

γ

{\displaystyle \alpha \beta \gamma }

transformation is the generation of the reference signal used for space vector modulation control of three-phase inverters.

History

In 1937 and 1938, Edith Clarke published papers with modified methods of calculations on unbalanced three-phase problems, that turned out to be particularly useful.

Definition

The

α

β

γ

{\displaystyle \alpha \beta \gamma }

transform applied to three-phase currents, as used by Edith Clarke, is

[

i

α

(

t

)

i

β

(

t

)

i

γ

(

t

)

]

=

i

α

β

γ

(

t

)

=

T

i

a

b

c

(

t

)

=

2

3

[

1

−

1

2

−

1

2

0

3

2

−

3

2

1

2

1

2

1

2

]

[

i

a

(

t

)

i

b

(

t

)

i

c

(

t

)

]

{\displaystyle {\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\\i_{\gamma }(t)\end{bmatrix}}=i_{\alpha \beta \gamma }(t)=Ti_{abc}(t)={\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\\\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}}}

where

i

a

b

c

(

t

)

{\displaystyle i_{abc}(t)}

is a generic three-phase current sequence and

i

α

β

γ

(

t

)

{\displaystyle i_{\alpha \beta \gamma }(t)}

is the corresponding current sequence given by the transformation

T

{\displaystyle T}

.

The inverse transform is:

[

i

a

(

t

)

i

b

(

t

)

i

c

(

t

)

]

=

i

a

b

c

(

t

)

=

T

−

1

i

α

β

γ

(

t

)

=

[

1

0

1

−

1

2

3

2

1

−

1

2

−

3

2

1

]

[

i

α

(

t

)

i

β

(

t

)

i

γ

(

t

)

]

.

{\displaystyle {\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}}=i_{abc}(t)=T^{-1}i_{\alpha \beta \gamma }(t)={\begin{bmatrix}1&0&1\\-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}&1\\-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}&1\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\\i_{\gamma }(t)\end{bmatrix}}.}

The above Clarke's transformation preserves the amplitude of the electrical variables which it is applied to. Indeed, consider a three-phase symmetric, direct, current sequence

i

a

(

t

)

=

2

I

cos

⁡

θ

(

t

)

,

i

b

(

t

)

=

2

I

cos

⁡

(

θ

(

t

)

−

2

3

π

)

,

i

c

(

t

)

=

2

I

cos

⁡

(

θ

(

t

)

+

2

3

π

)

,

{\displaystyle {\begin{aligned}i_{a}(t)=&{\sqrt {2}}I\cos \theta (t),\\i_{b}(t)=&{\sqrt {2}}I\cos \left(\theta (t)-{\frac {2}{3}}\pi \right),\\i_{c}(t)=&{\sqrt {2}}I\cos \left(\theta (t)+{\frac {2}{3}}\pi \right),\end{aligned}}}

where

I

{\displaystyle I}

is the RMS of

i

a

(

t

)

{\displaystyle i_{a}(t)}

,

i

b

(

t

)

{\displaystyle i_{b}(t)}

,

i

c

(

t

)

{\displaystyle i_{c}(t)}

and

θ

(

t

)

{\displaystyle \theta (t)}

is the generic time-varying angle that can also be set to

ω

t

{\displaystyle \omega t}

without loss of generality. Then, by applying

T

{\displaystyle T}

to the current sequence, it results

i

α

=

2

I

cos

⁡

θ

(

t

)

,

i

β

=

2

I

sin

⁡

θ

(

t

)

,

i

γ

=

0

,

{\displaystyle {\begin{aligned}i_{\alpha }=&{\sqrt {2}}I\cos \theta (t),\\i_{\beta }=&{\sqrt {2}}I\sin \theta (t),\\i_{\gamma }=&0,\end{aligned}}}

where the last equation holds since we have considered balanced currents. As it is shown in the above, the amplitudes of the currents in the

α

β

γ

{\displaystyle \alpha \beta \gamma }

reference frame are the same of that in the natural reference frame.