table of contents
- [Introduction to NXP Microcontrollers] [Motor Control] [Basics Part 3] Mechanism and Control Method ...
- The unsung hero! What are Clark Transform and Park Transform?
- It all began with the complex waves of three-phase AC
- Step 1: Clarke Transformation - Simplifying 3D into 2D
- Step 2: Park Transformation - "The Magic of the Merry-Go-Round" - Freeze the Moving World
- Why go through such a tedious conversion process? What are the huge benefits?
- Summary: The power of mathematics to manipulate complex waves at will
- The unsung hero! What are Clark Transform and Park Transform?
[Introduction to NXP Microcontrollers] [Motor Control] [Basics Part 3] Mechanism and Control Method of Permanent Magnet Synchronous Motors
The unsung hero! What are Clark Transform and Park Transform?
Hello!
When an electric vehicle (EV) takes off smoothly and a high-performance air conditioner operates surprisingly quietly, microcomputers perform complex calculations at ultra-high speeds and skillfully control the motor.
This time, let's unravel the mysteries of the "Clark Transformation" and the "Park Transformation" (transformations named after two great experts who work in " vector control ," the heart of control technology), with the GIF animation below!
It all began with the complex waves of three-phase AC
First, the top row. This is the world of three-phase AC , the basic energy required to run a motor.
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Graph on the right (Three-Phase Sine Waves) : Three waves, A, B, and C, are constantly changing in magnitude and direction as they flow. These three waves are interconnected, but as they are, it is extremely difficult to intuitively determine how much force should be applied to the motor at any given time. It's like trying to conduct three musicians who are each playing different pieces of music at the same time.
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Left graph (Rotating Vector) : However, when the forces of these three waves are combined, something interesting happens. A single force ( rotating vector ) is created that rotates smoothly and constantly, while maintaining a constant magnitude. Physically, this is the " rotating magnetic field " created by the stator coil. It is this rotating magnetic field that attracts the rotor magnet and is the source of the force that turns the motor.
Problem: How can we easily and accurately control these "three constantly changing waves" using a microcontroller?
Step 1: Clarke Transformation - Simplifying 3D into 2D
The first magic is the " Clark transformation ," which transforms a complex 3D world into a more understandable 2D world.
The calculation formula is as follows:
In normal motor control, the following simplified formula is used, assuming amplitude invariant transformation (K=2/3) and balanced three-phase (i_a+i_b+i_c=0).
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The graph on the right (Two-Phase Sine Waves α-β) : Look! The three waves have been consolidated into two waves, α (alpha) and β (beta). The wave shape (alternating current) is still there, but one variable has been removed, making it much easier to see.
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Graph on the left (Clarke Transformation α-β) : This shows the world as seen from two axes (α, β) that intersect at right angles to the rotation vectors seen from three axes (A, B, C). It's like turning a solid object seen from an angle into a flat view from directly above. The rotation vector itself continues to rotate in the same way without any change.
[Key points of Clarke conversion]
Without losing any information, we simplified the problem by converting from the somewhat difficult to handle three-phase coordinate system to Cartesian coordinates (α-β stationary coordinate system), which are familiar from mathematics.
Step 2: Park Transformation - "The Magic of the Merry-Go-Round" - Freeze the Moving World
The values of α and β are still changing like waves, and it is difficult to keep track of them. This is where the essence of vector control, the " Park transformation ," comes in!
The calculation formula is as follows:
This is a major shift in thinking : "Let's stop looking at the rotating vector from the stationary ground (α-β coordinates) and jump on a merry-go-round that rotates at the same speed as the rotating vector!" This new rotating coordinate system is called the " dq rotating coordinate system ."
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Right graph (Two-Phase Value dq) : What an amazing result! The two waves that had been changing so drastically have now turned into **almost constant values (direct current)** called d and q!
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Left graph (Park Transformation dq) : You can see that the dq coordinates rotate in perfect synchronization with the rotation vector. If you stand right next to a horse on a merry-go-round, the horse appears stationary to you, right? It's the exact same principle. From the perspective of the rotating object, it appears stationary.
[Points to note about park conversion]
By observing from a coordinate system (dq coordinates) that rotates at the same speed as the rotating vector, AC values can be treated as DC values.
Why go through such a tedious conversion process? What are the huge benefits?
This two-step transformation brings us a tremendous benefit: overwhelming simplification of control .
Controlling constantly changing AC values is difficult, but what about DC values? If it's higher than the target value, lower it; if it's lower, raise it. With this simple operation (PID control) that even an elementary school student can understand, you can achieve perfect control of a motor.
The DC values of d and q each have an important physical meaning.
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q-axis value (Quadrature-axis): This directly controls the motor's torque (rotational force). When you step on the accelerator of an EV, the car accelerates sharply because the microcomputer is raising the target value of this q-axis. It is truly a "power dial."
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d-axis value (Direct-axis): Controls the motor's magnetic flux (magnet strength). In the case of a permanent magnet motor, the strength of the rotor magnet is constant, so basically it is most efficient to control the d-axis current to zero. It is truly an "efficiency dial."
In other words, the Clarke and Park transformations are magical in that they separate the "power" and "efficiency" elements of a motor, which are normally mixed together, into two independent DC dials (d and q) .
Summary: The power of mathematics to manipulate complex waves at will
- The complex wave of three-phase AC creates a rotating force (rotation vector) inside the motor.
- Clarke transformation simplifies the problem by redrawing the three-dimensional world into two dimensions (α-β).
- The park transformation involves riding on a rotating merry-go-round (dq coordinates) and converting AC values into DC values.
- The DC torque (q) and magnetic flux (d) values can be easily and accurately controlled using a PID controller!
This series of elegant mathematical processes is the basis of "vector control," which supports modern high-performance motors. It is this magic that allows us to enjoy the full benefits of powerful, quiet, and energy-efficient motors.
Thank you for reading to the end!
If you would like to read the next basic chapter, click here ↓
[Basics Part 4] Practice! Let's see how vector control works using a block diagram!
Click here for an explanation of the specific setup method and how to run the sample code.
Here is a website that compiles articles about NXP motor control:
NXP Motor Control - Summary Page - (Japanese blog)
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This article will provide an easy-to-understand explanation of motor control using the NXP FRDM board "FRDM-MCXA156." It is divided into a basic section and a practical section, so we hope you will refer to the section that interests you.
・Basics ①～⑦, Practical ①～③
This time, as part 3 of the Basics, we will explain the mechanism and control method of permanent magnet synchronous motors using NXP microcontrollers.