.Spot.UsersGuide.Introduction.Transforms

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Coordinate Transforms

We consider the current and voltage signals i and v of the three-phase system as abstract signal vectors that can be represented in different reference frames.
In the following s is used for either i or v.
We distinguish between an 'absolute' system (s_abc) and a manifold of transformed systems (s_abc_theta and s_dqo_theta) depending on a transformation angle theta.

  s_abc           signals of phase a, b, c (R, S, T) in inertial frame 'abc'
  s_abc_theta     signals relative to a transformed frame 'abc'
  s_dqo_theta     signals relative to a transformed frame 'dqo'

  s_abc_theta = R_abc'*s_abc,  s_abc = R_abc*s_abc_theta
  s_dqo_theta = P*s_abc,       s_abc = P'*s_dqo_theta

where ' denotes 'transposed'. R_abc and P obey the orthogonality condition

  R_abc' = inverse(R_abc)
  P' = inverse(P)

Both R_abc and P depend on an angle theta.P can be factorised into a constant, angle independent matrix P0 and an angle-dependent rotation R_dqo

  P(theta) = R_dqo'(theta)*P0

As the choice of theta is arbitrary, R_abc and P do not define one specific but a whole manifold of abc- and dqo-systems.

Particular choices of practical importance are

• theta is the rotor angle of a machine, synchronous machine (Park) or asynchronous machine
• theta is defined through

  der(theta) = omega

  by the frequency omega of a source or a system-frequency
• theta = 0
  The resulting 'abc'-system is identical to the inertial (absolute) 'abc'-system.
  (In the present context this is interpreted as abc(theta=0))
  The resulting 'dqo'-system is traditionally called 'alpha-beta-gamma' system
  (In the present context this is interpreted as dqo(theta=0))

As a consequence of the orthogonality of R and P we obtain the invariance of power under all transforms

  p = v_abc*i_abc = v_abc_theta*i_abc_theta = v_dqo_theta*i_dqo_theta

The main difference between the transforms R_abc and P is the following:
Whereas R_abc leaves impedance matrices of symmetrical systems invariant, P0 and P diagonalise these matrices. We have

         [xs, xm, xm]         [xs, xm, xm]
  R_abc'*[xm, xs, xm]*R_abc = [xm, xs, xm]
         [xm, xm, xs]         [xm, xm, xs]

         [xs, xm, xm]         [x, 0, 0 ]
       P*[xm, xs, xm]*P'    = [0, x, 0 ]     with
         [xm, xm, xs]         [0, 0, x0]

       x  = xs - xm       stray reactance (dq-components)
       x0 = xs + 2*xm     zero-reactance (o-component)

The 1:2 components of s_dqo_theta signals can be interpreted as phasors with respect to the frequency

  omega = der(theta).

Historically P was introduced by Park in the context of synchronous machines.
Here it is used in a generalised sense. Nevertheless we call it 'Park-transform'.

Note: packages ACabc and ACdqo use v and i for the transformed variables.

  ACabc: v = v_abc_theta, i = i_abc_theta
  ACdqo  v = v_dqo_theta, i = i_dqo_theta

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