Synchronous Reference Frame (dq)

• fundamentalsreference-framespark-transformdq-frame

From the Stationary Observer to the Rotating Observer

In Stationary Reference Frame (αβ), balanced three-phase voltages were represented as a rotating space vector in the stationary αβ reference frame. The α and β axes were fixed in space, and the observer remained stationary while the space vector rotated at electrical angular speed ω.

Balanced steady-state operation therefore appeared as a vector of constant magnitude rotating continuously in the αβ plane.

The next step is to examine how this picture changes when the observer itself rotates.

Rather than changing the electrical system, we change the coordinate system used to describe it. This change of viewpoint leads to the synchronous reference frame, commonly referred to as the dq frame.

The idea of choosing an observer's rotational speed was introduced in From Sinusoids to Rotating Vectors, where a single rotating vector was viewed from frames moving at different speeds. The same principle now extends to the three-phase space vector.

A Rotating Observer

Consider the rotating space vector introduced in the previous article.

The vector rotates at electrical angular speed:

ω\omega

Now introduce a new observer whose coordinate axes rotate at angular speed:

ωobs\omega_{obs}

The observer is no longer fixed in space. Instead, the observer carries a pair of orthogonal axes that rotate continuously.

The behaviour observed in this frame depends entirely on the relative motion between the space vector and the rotating axes.

Relative Motion Between Observer and Space Vector

The appearance of motion depends on the difference between the speed of the observer and the speed of the space vector.

If both rotate at different speeds, the observer continues to see motion. If both rotate at the same speed, the relative motion disappears.

Reference frames apply this same principle to rotating electrical quantities.

Case 1: Stationary Observer (ω_obs = 0)

When the observer is stationary, the coordinate axes do not rotate.

This corresponds to the αβ frame introduced previously.

The space vector rotates continuously at angular speed ω.

For balanced steady-state conditions:

  • The space vector traces a circular trajectory.
  • The vector angle increases continuously.
  • The α and β components vary sinusoidally.

The observer sees continuous rotation.

Case 2: Synchronous Observer (ω_obs = ω)

Now consider an observer rotating at exactly the same angular speed as the space vector.

Both the vector and the observer rotate together.

Because there is no relative motion between them, the vector appears stationary.

The vector magnitude remains unchanged and its angle relative to the rotating observer becomes constant.

Balanced steady-state quantities that appeared sinusoidal in the stationary frame now appear as constant values.

The reason these quantities become constant is that the relative angle between the observer and the space vector no longer changes.

In the stationary frame, the space vector continuously sweeps past the observer's axes, causing the measured components to vary with time. When the observer rotates at the same angular speed as the space vector, this relative motion disappears.

The projections onto the rotating axes therefore remain unchanged, producing constant d-axis and q-axis values under balanced steady-state conditions.

Case 3: Observer Speed Mismatch (ω_obs ≠ ω)

A third situation occurs when the observer rotates at a speed different from the space vector.

The apparent rotational speed becomes:

ωrel=ω−ωobs\omega_{rel} = \omega - \omega_{obs}

The greater the mismatch, the faster the apparent rotation.

The synchronous frame therefore only produces constant quantities when the observer rotates at the correct speed.

Introducing the d and q Axes

The rotating reference frame uses two orthogonal axes:

  • Direct axis (d)
  • Quadrature axis (q)

These axes rotate together at angular speed ωobs\omega_{obs}.

Like the αβ frame, the axes remain orthogonal at all times. The difference is that their orientation continuously changes.

Physical Meaning of the d-Axis

The d-axis is commonly chosen to align with the space vector.

When alignment occurs:

  • The entire vector lies on the d-axis.
  • The component along the d-axis equals the vector magnitude.
  • No component remains along the q-axis.

A useful visualization is to imagine the d-axis continuously rotating so that it remains aligned with the space vector at every instant.

When this alignment is maintained, the vector always lies along the d-axis. The d-component therefore represents the magnitude of the vector measured along its own direction, while the q-component measures any deviation from that alignment.

For a balanced steady-state voltage vector:

vd=∣V∣v_d = |V| vq=0v_q = 0

Physical Meaning of the q-Axis

The q-axis is orthogonal to the d-axis.

It measures the component of the vector perpendicular to the chosen alignment direction.

When the frame is perfectly aligned with the space vector, the q-component becomes zero.

If alignment is imperfect, a non-zero q-component appears.

What if the Observer Has the Correct Speed but the Wrong Angle?

Matching the angular speed of the space vector is not sufficient to achieve perfect alignment.

Consider an observer rotating at the correct speed:

ωobs=ω\omega_{obs} = \omega

but with an angular offset relative to the space vector.

In this situation, the observer maintains a constant phase difference with the vector. The vector therefore appears stationary, but it is not aligned with the d-axis.

As a result:

  • The d-component is reduced.
  • A non-zero q-component appears.
  • Both components remain constant because the angular error is constant.

The presence of a non-zero q-component indicates that the observer is rotating at the correct speed but is not oriented at the correct angle.

Determining and maintaining the correct angular position becomes an important practical requirement. This requirement motivates the angle-estimation techniques discussed later in the series.

Relationship Between the αβ and dq Frames

The αβ frame and the dq frame describe exactly the same physical quantity.

The difference lies only in the observer.

Only the coordinate representation differs.

Park Transformation

The conversion from the stationary αβ frame to the rotating dq frame is known as the Park transformation.

d=αcos⁡θ+βsin⁡θd = \alpha \cos\theta + \beta \sin\theta q=−αsin⁡θ+βcos⁡θq = -\alpha \sin\theta + \beta \cos\theta

where θ\theta is the angular position of the rotating observer.

What Happens When the Observer Rotates at the Wrong Speed?

Perfectly constant d and q quantities require the observer to rotate at exactly the same speed as the space vector.

If the observer rotates too slowly or too quickly:

  • The vector no longer appears stationary.
  • The d and q components begin to oscillate.
  • The oscillation depends on the speed mismatch.

This behaviour provides a direct indication that the observer is not properly synchronized with the electrical system.

Interactive Visualization

Presets:
%
0%ω (100%)150%
°
−180°0°+180°
abc Signals
vₐv_bv_c
vₐ = 0.000v_b = 0.000v_c = 0.000
αβ Stationary Frame
αβ
vα = 0.000vβ = 0.000θ = 0.0°
dq Rotating Frame
External view (lab frame)
αβdq
Observer view (rotating frame)
dq
v_d = 0.000v_q = 0.000θ_obs = -90.0°θ_err = 0.0°
αβ Components (time domain)
+10−1v (p.u.)tvαvβ
dq Components (time domain)
+10−1v (p.u.)tv_dv_q
ω_obs / ω1.00
|V|0.000
v_d0.000
v_q0.000
θ_err0.0°

Mode A (ω_obs = ω, θ_err = 0): The space vector appears stationary in the dq frame. v_d equals the vector magnitude and v_q = 0.

Mode B (ω_obs = ω, θ_err ≠ 0): The vector is stationary but misaligned. Both v_d and v_q are constant and non-zero. The q-axis value reveals the angular error.

Mode C (ω_obs ≠ ω): Speed mismatch causes continuous angular error growth. Both v_d and v_q oscillate at the difference frequency (ω − ω_obs).

The visualization above demonstrates the full abc → αβ → dq transformation chain. Key observations:

  • Panel 1 (abc): Three sinusoidal phase voltages with 120° separation. The instantaneous values feed directly into the Clarke transform.
  • Panel 2 (αβ): The space vector traces a circular locus under balanced conditions. Its angle θ advances continuously at electrical frequency ω.
  • Panel 3 (dq): The rotating observer axes (d in amber, q in green) follow the observer at speed ωobs. Use the presets or sliders to explore all three modes.
  • Waveform row: Time-domain αβ and dq signals show directly how sinusoidal quantities become constant (Mode A) or oscillating (Mode C).

Use Mode A to confirm that perfect alignment produces constant vd and zero vq. Use Mode B to see that speed-matched but misaligned observers produce a steady non-zero vq. Use Mode C to observe how speed mismatch drives continuous oscillation in both dq quantities.

Summary

The synchronous reference frame is created by allowing the observer to rotate.

When the observer rotates at the same angular speed as the space vector, balanced steady-state quantities appear as constant values. The rotating dq axes provide a convenient coordinate system for describing these quantities.

The effectiveness of this representation depends on choosing the correct angular position and rotational speed for the observer.

Determining that angle continuously is the remaining challenge. This naturally motivates the synchronization problem addressed later in the series through the Phase-Locked Loop (PLL).

Reflective Questions

  1. Why does the q-component become zero when the dq frame is perfectly aligned with the space vector?
  2. What is the physical interpretation of a non-zero q-component in an aligned dq frame?
  3. How does the Park transformation reduce to the Clarke transformation when θ=0\theta = 0?
  4. If the observer rotates at twice the electrical frequency, what does the space vector appear to do?