Stationary Reference Frame (αβ)

From Rotating Vectors to Three-Phase Systems

In From Sinusoids to Rotating Vectors, a single sinusoidal quantity was interpreted as the projection of a rotating vector observed from a chosen reference frame. Oscillation arises from rotation relative to an observer.

A balanced three-phase system extends this idea. Instead of one sinusoid, there are three:

$$v_a = V_m \sin(\omega t)$$ $$v_b = V_m \sin(\omega t - 120°)$$ $$v_c = V_m \sin(\omega t - 240°)$$

Under balanced conditions, these quantities represent a structured rotating phenomenon. This structure allows them to be represented compactly as a single rotating space vector in a plane.

The stationary reference frame provides the geometric structure needed to describe this space vector.

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The Stationary Reference Frame

A stationary reference frame consists of:

• Two orthogonal axes, $\alpha$ and $\beta$.
• Zero angular velocity relative to the physical system.

The observer associated with this frame does not rotate. Electrical quantities rotate relative to these fixed axes.

The orientation of the $\alpha$ axis is a convention. In this article, it is aligned with phase $a$, but other alignments are possible without altering physical behaviour.

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Clarke Transformation – Standard Form

The Clarke transformation maps three-phase quantities $(v_a, v_b, v_c)$ into the stationary $\alpha\beta$ frame. In standard matrix form:

$$\begin{bmatrix} v_\alpha \\ v_\beta \\ v_0 \end{bmatrix} = \frac{2}{3} \begin{bmatrix} 1 & -\tfrac{1}{2} & -\tfrac{1}{2} \\ 0 & \tfrac{\sqrt{3}}{2} & -\tfrac{\sqrt{3}}{2} \\ \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \end{bmatrix} \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix}$$

The $v_0$ term represents the zero-sequence component. For balanced systems without zero-sequence content, $v_0 = 0$. The system is then fully described by $v_\alpha$ and $v_\beta$.

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Simplified Balanced-Case Form

Under balanced conditions and neglecting zero sequence, the transformation reduces to a two-dimensional mapping. A commonly used simplified form consistent with the above matrix is:

$$v_\alpha = v_a$$ $$v_\beta = \frac{1}{\sqrt{3}}(v_a + 2v_b)$$

Substituting balanced sinusoidal phase voltages yields:

$$v_\alpha = V_m \sin(\omega t)$$ $$v_\beta = V_m \cos(\omega t)$$

The pair $(v_\alpha, v_\beta)$ therefore defines a vector of constant magnitude rotating at angular speed $\omega$.

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Geometric Meaning of the Space Vector

The space vector is defined as:

$$\mathbf{v}_s = v_\alpha + j v_\beta$$

Geometrically, this space vector is the vector sum of the individual phase contributions projected into the $\alpha\beta$ plane.

Under balanced steady-state operation:

• The magnitude of $\mathbf{v}_s$ is constant.
• The angle increases linearly with time.
• The angular velocity equals the electrical frequency.

The three phase sinusoids are therefore projections of this rotating space vector onto axes separated by 120°.

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Connection to the Stationary Observer

In the stationary reference frame, the observer is fixed. The $\alpha$ and $\beta$ axes do not rotate.

Balanced three-phase voltages therefore appear as a circular trajectory in the $\alpha\beta$ plane. The space vector rotates at angular speed $\omega$ relative to the stationary observer.

If the observer were to rotate at angular speed $\omega$, the space vector would appear stationary. This observation prepares the transition to the rotating reference frame discussed in the next article.

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Geometric Meaning of Unbalance

If the three-phase system becomes unbalanced:

• The trajectory in the $\alpha\beta$ plane is no longer circular.
• The space vector magnitude may vary with time.
• The trajectory becomes elliptical or distorted.

This provides direct geometric insight into symmetry and disturbance behaviour.

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Summary

The stationary reference frame represents balanced three-phase signals as a single rotating space vector observed from fixed orthogonal axes. The Clarke transformation provides the formal mapping from phase quantities to this plane.

This representation preserves physical behaviour while changing its coordinate description. In the stationary frame, the space vector rotates at system frequency. If the observer rotates synchronously, this rotation disappears, forming the basis for the rotating reference frame.

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Reflective Questions

1. Why is only two-dimensional representation sufficient for balanced three-phase signals?
2. What determines the orientation of the $\alpha$ axis?
3. How would the space vector appear to an observer rotating at system frequency?