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Technology / Technical guide
PMSG d–q Axes & the Rotating Reference Frame
A rotating reference frame can describe changing phase quantities using coordinates aligned with rotor motion. It changes the representation, not the physical winding. Understanding the angle, orientation and normalisation prevents a d–q equation from being mistaken for an unqualified control recipe.
Rotate the coordinates, not the hardware.
An apparently constant d–q quantity can represent a vector rotating in a stationary frame.
01 / Axis meaning
Define the reference before assigning components.
In a common permanent-magnet machine convention, the d axis aligns with the rotor magnetic reference and the q axis is 90 electrical degrees ahead of it.
In a common permanent-magnet machine convention, the d axis aligns with the rotor magnetic reference and the q axis is 90 electrical degrees ahead of it. The convention must state the positive rotation and current direction.
Electrical angle is related to mechanical angle through pole-pair count and a chosen origin. A number labelled rotor angle does not identify whether it is electrical or mechanical. A coordinate origin also affects which component is called zero.
02 / Rotation example
Use one explicit two-axis transformation.
For a stationary vector with components xα and xβ and a frame angle θ, define xd = xα cos θ + xβ sin θ and xq = −xα sin θ + xβ cos θ.
For a stationary vector with components xα and xβ and a frame angle θ, define xd = xα cos θ + xβ sin θ and xq = −xα sin θ + xβ cos θ. Here the q axis is ahead of d in the positive orientation.
This example rotates the coordinate basis while preserving the vector’s magnitude. It is the rotational part of a convention, not a complete phase-to-axis conversion or a command for any controller. The same formula can represent different physical quantities x.
03 / Constant components
See why synchronous quantities can look steady.
If the stationary vector is xα = A cos θ and xβ = A sin θ, the stated transform gives xd = A and xq = 0.
If the stationary vector is xα = A cos θ and xβ = A sin θ, the stated transform gives xd = A and xq = 0. The vector still rotates physically; its components are constant because the frame follows it.
A vector shifted by +90° under this convention gives xd = 0 and xq = A. These are mathematical identities. A frame-angle error mixes the components, and an unbalanced or harmonic waveform need not become constant in the rotating frame.
04 / Normalisation
Keep scaling with the phase transformation.
A full three-phase transform also specifies how phase values become stationary-axis quantities and whether zero sequence is included.
A full three-phase transform also specifies how phase values become stationary-axis quantities and whether zero sequence is included. Amplitude-invariant and power-invariant normalisations use different scale factors.
MathWorks documents an explicit Park transform and warns that an alternative implementation produces different numerical axis quantities. Mixing equations from different normalisations can change current, flux or torque factors even though the underlying physical machine is the same.
05 / Machine relations
Separate useful axes from implemented regulation.
Rotor-aligned axes can simplify how flux linkage, inductance and torque relationships are written.
Rotor-aligned axes can simplify how flux linkage, inductance and torque relationships are written. In a common ideal model, d and q inductances can differ; equal values are a separate simplifying assumption, not a consequence of using the transform.
Vector control uses a representation together with sensing or estimation, a model and an actual control implementation. The existence of a d–q diagram establishes none of those functions by itself. Representation and algorithm capability are different subjects.
06 / Sign and scope
Keep generator and motor conventions explicit.
Machine equations can adopt positive current into or out of the electrical terminals and different torque or angle signs.
Machine equations can adopt positive current into or out of the electrical terminals and different torque or angle signs. A minus sign that changes between references is not automatically a disagreement in physics; first compare definitions.
This page provides no current references, controller gains or FOC settings. FD14-50KW’s permanent-magnet direct-drive architecture is confirmed, but no MARSTON d–q model parameters, sensing arrangement or deployed vector-control function is asserted.
Continue the review with PMSG Magnetic Flux, Flux Linkage & Induced EMF.
Continue the review with Generator Inductance, Reactance & Impedance.
Continue the review with PI & PID Control: Proportional, Integral & Derivative Roles.
Continue the electromagnetic coordinate review with directional inductance and saliency and magnetic energy and coenergy in torque.
General background: MathWorks PMSM DQ0: explicit Park convention, electrical angle and scaling. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Frame angle | θ in electrical radians | Origin and positive direction are defined. |
| d component | Projection along the selected rotor reference | Not a physical extra winding. |
| q component | Projection 90° ahead in this convention | Sign depends on stated orientation. |
| Normalisation | Amplitude- or power-invariant definition | Scale factors cannot be mixed. |
| Implementation | Model, information and control functions | Not established by the transform alone. |
Engineering questions.
Does the transform physically reconnect windings?
No. It changes the mathematical coordinates.
Why can a rotating vector have constant d–q components?
The coordinate frame can rotate with the same vector.
Are d–q signs universal?
No. Angle, current, torque and axis conventions must be stated.
Does every balanced waveform become constant in d–q?
The synchronous fundamental can; harmonics and other components may still vary.
Does a d–q explanation prove installed FOC?
No. Actual control implementation is a separate capability.
Are current targets or gains provided?
No. Only a defined coordinate example and its interpretation are given.
Technical enquiry
Discuss the technical question.
A rotating reference frame can describe changing phase quantities using coordinates aligned with rotor motion. Share the model and applicable operating or design information for an engineering discussion.