Talk:Alpha–beta transformation

Untitled

The arrow of the angular velocity shows in the wrong direction.

The alpha/beta/gamma transformation has no effect on the Rotation of the components so the systems rotates mathematically positive.

Should talk about the various reference frames (alpha leading beta, beta leading alpha, alpha pointing down, alpha pointing up, etc). Also should show the reason about the "2/3" factor before the transformation matrix.

Shouldn't the scalar value be sqrt(2/3) and the third row in the Clarke transform (corresponding to the RMS) be 1/sqrt(2) ? — Preceding unsigned comment added by 194.132.104.253 (talk) 09:08, 30 April 2012 (UTC)

αß Transform Derivation

The α-ß transform, aka [Edith] Clarke transform, projects the (dependent/non-orthogonal) three vector (ABC) basis at 0°, 120°, & 240° onto the (independent/orthogonal) two vector (αß) basis at 0° & 90°.
Since the ABC reference-frame is not orthogonal it is not possible to vary one axis without affecting another axis - this complicates (introduces instability into) applications such as 3-phase motor-control.
Subject to an additional constraint, e.g. all currents sum to zero or all currents sum to one, it is possible to create a transform and companion inverse transform to & from an orthogonal x-y reference called α-ß in motor-control jargon.


[ 1 -1/2 -1/2 ]
[ 0 √3/2 -√3/2 ]

That's cos(2*PI/3), cos(4*PI/3) & sin(2*PI/3), sin(4*PI/3) (120° -> 2PI/3 rad)
If you have more legs/phases, or if they are aligned differently, adjust accordingly.
Motors are generally 3-phase & balanced, other applications might not be (e.g. boosting or charging).
If you want to tack-on the homogeneous row it should be [ 1/3 1/3 1/3 ] (There's three phases and each counts for 1/3 if you want it balanced.)
[ 1 -1/2 -1/2 ]
[ 0 √3/2 -√3/2 ]
[ 1/3 1/3 1/3 ]


The 2/3 factor is added to the transform tacked on to maintain unity so that 1Apu in ABC-reference is 1Apu in αß-reference.
(Apu -> normalized Amps / Amp per-unit) This is because 3 vectors are collapsed into 2.
If you have more legs/phases this factor will change accordingly.
The 2/3 factor only applies to the projection, the first two rows, not the homogeneous (balancing) row. Which is why the Clarke transform has the peculiar [ 1/2 1/2 1/2 ] for the homogeneous row.

3phase to 2phase conversion by rotation of the coordinate system

For conversion of three phase voltages 120 degrees apart to two phase voltages 90 degrees apart a transformation like the following might be a better choice, as it provides a pure rotation of the coordinate system.

${\begin{bmatrix}U_{\alpha }\\U_{\beta }\\U_{\gamma }\end{bmatrix}}={\begin{bmatrix}{\frac {2}{\sqrt {6}}}&-{\frac {1}{\sqrt {6}}}&-{\frac {1}{\sqrt {6}}}\\0&{\frac {\sqrt {2}}{2}}&-{\frac {\sqrt {2}}{2}}\\{\frac {1}{\sqrt {3}}}&{\frac {1}{\sqrt {3}}}&{\frac {1}{\sqrt {3}}}\\\end{bmatrix}}{\begin{bmatrix}U_{a}\\U_{b}\\U_{c}\end{bmatrix}}$

This is similar to the Clarke transformation, and corresponds to a Dqo transformation with an angle of zero.

The mapping corresponds to a rotation by the euler angles α=45°, β=atan(1/sqrt(2))≈35.26439° and γ=30°

Then directional vectors (rows) are proper unit vectors, the inverse matrix is equal to the transposed matrix and the determinant is one. As energy is independent of the chosen coordinate system, power calculations can be done directly on the transformed variables, that is: this transformation is power invariant.

Actual values will be

$A={\begin{bmatrix}0.8164966&-0.4082483&-0.4082483\\0.0000000&0.7071068&-0.7071068\\0.5773503&0.5773503&0.5773503\\\end{bmatrix}}$

Concordia

In France, the Power Invariant Version of the Clark is known as the Concordia transform (see http://fr.wikipedia.org/wiki/Transform%C3%A9e_de_Concordia) Also, the comment that αβ transform is of little use is rubbish — Preceding unsigned comment added by Wiredrabbit (talk • contribs) 12:33, 29 January 2015 (UTC)

dpo is dp0

It's d-q-zero not d-q-o.