Clarke and Park transforms are used to convert three‑phase electrical quantities into simpler coordinate systems for control.
Clarke maps the three‑phase signals into a fixed two‑axis (αβ) plane, while Park rotates that plane into a synchronous dq frame where the signals become almost DC.
- Clarke Functions
Classical Clarke Transform (αβ)
The classical Clarke transform converts a three‑phase system (abc) into a two‑axis system (αβ)by projecting the phase variables onto a stationary reference frame.
It is mainly used to obtain a bidimensional representation of phase quantities in vector‑control applications.
Direct transform:
- Converts (abc) into α and β.
- Preserves the physical magnitude of the vector and separates the quadrature component.
- Allows the behavior of the three‑phase system to be represented on a 2D plane.
Inverse transform:
- Reconstructs the phase quantities from α and β.
- Used when αβ‑domain processing must be converted back to phase variables (e.g., PWM generation or simulation tasks).
Full Clarke Transform (αβ0)
The full version also preserves the zero‑sequence component , which is useful when the neutral point is not perfectly constrained or when analyzing unbalanced conditions.
Full direct transform:
- Converts (abc) into (αβ0).
- The zero‑sequence component highlights possible asymmetries or non‑sinusoidal conditions.
Full inverse transform:
- Reconstructs (abc) from (αβ0).
- Required when the zero‑sequence must be maintained for accurate reconstruction.
Simplified Clarke Transform
Used when the system is assumed to be balanced, meaning the zero‑sequence component is zero and can be omitted.
Simplified direct transform:
- Computes α, β using a reduced matrix.
- Decreases computational load and is suitable for high‑performance control.
Simplified inverse transform:
- Rebuilds (abc) from α, β assuming 0 = 0.
- Ideal for symmetric systems such as induction and synchronous motors.
- Park Functions
Classical Park Transform (dq)
The classical Park transform converts the stationary (αβ)system into a rotating (dq) system synchronized to an angle .
It transforms time‑varying quantities into quasi‑DC values, enabling simpler control of motors and power converters.
Direct transform:
- Converts αβ into dq through a rotational transformation.
- The d component aligns with the rotating reference frame.
- The q component is in quadrature and is essential for torque or reactive power control.
Inverse transform:
- Converts dq back into the stationary αβ frame.
- Used after dq ‑domain regulation to return the values to a form suitable for PWM or other control stages.
Modified Park Transform
The modified version applies alternative scaling factors or conventions to:
- preserve physical amplitude consistency,
- simplify controller tuning,
- comply with specific standards or modeling choices.
Modified direct transform:
- Apply the rotation together with different scaling factors compared to the classical form.
- Sometimes normalizes amplitudes to ensure identical gains across channels.
Modified inverse transform:
- Clarke Functions
Classical Clarke Transform (αβ)
The classical Clarke transform converts a three‑phase system (abc) into a two‑axis system (αβ)by projecting the phase variables onto a stationary reference frame.
It is mainly used to obtain a bidimensional representation of phase quantities in vector‑control applications.
Direct transform:
- Converts (abc) into α and β.
- Preserves the physical magnitude of the vector and separates the quadrature component.
- Allows the behavior of the three‑phase system to be represented on a 2D plane.
Inverse transform:
- Reconstructs the phase quantities from α and β.
- Used when αβ‑domain processing must be converted back to phase variables (e.g., PWM generation or simulation tasks).
Full Clarke Transform (αβ0)
The full version also preserves the zero‑sequence component , which is useful when the neutral point is not perfectly constrained or when analyzing unbalanced conditions.
Full direct transform:
- Converts (abc) into (αβ0).
- The zero‑sequence component highlights possible asymmetries or non‑sinusoidal conditions.
Full inverse transform:
- Reconstructs (abc) from (αβ0).
- Required when the zero‑sequence must be maintained for accurate reconstruction.
Simplified Clarke Transform
Used when the system is assumed to be balanced, meaning the zero‑sequence component is zero and can be omitted.
Simplified direct transform:
- Computes α, β using a reduced matrix.
- Decreases computational load and is suitable for high‑performance control.
Simplified inverse transform:
- Rebuilds (abc) from α, β assuming 0 = 0.
- Ideal for symmetric systems such as induction and synchronous motors.
- Park Functions
Classical Park Transform (dq)
The classical Park transform converts the stationary (αβ)system into a rotating (dq) system synchronized to an angle .
It transforms time‑varying quantities into quasi‑DC values, enabling simpler control of motors and power converters.
Direct transform:
- Converts αβ into dq through a rotational transformation.
- The d component aligns with the rotating reference frame.
- The q component is in quadrature and is essential for torque or reactive power control.
Inverse transform:
- Converts dq back into the stationary αβ frame.
- Used after dq ‑domain regulation to return the values to a form suitable for PWM or other control stages.
Modified Park Transform
The modified version applies alternative scaling factors or conventions to:
- preserve physical amplitude consistency,
- simplify controller tuning,
- comply with specific standards or modeling choices.
Modified direct transform:
- Apply the rotation together with different scaling factors compared to the classical form.
- Sometimes normalizes amplitudes to ensure identical gains across channels.
Modified inverse transform:
- Converts the values back to the αβ frame while maintaining the same modified scaling.
- Ensures compatibility with controllers or models that require these conventions.