Clarke Transform
First of all, there are many methods, which are equivalent in mathematics, such as 3+3+3+3=3*4=12. The same goes for the following.
For a more concise control, Ms. Clark proposed a simplified method of motor control, that is, to establish a coordinate system, the horizontal axis is α and the vertical axis is β, and the three-phase current is projected on this coordinate axis, using two orthogonally distributed projections replace. And describe the relationship between the two, so as to obtain iα and iβ at any time.
Its physical meaning is to use two orthogonally distributed coils (placed on the α-axis and β-axis respectively) to simulate three coils distributed at 120 degrees in a three-phase motor. The two are mathematically equivalent. This simplifies the control flow.
When actually driving the motor, only one inverse solution is required to drive the motor.
The projection method is as follows, coincide the magnetic field direction of phase A with iα, and draw the current vectors of phase B and phase C as shown below:
The green line segment is the projection of ib and ic on the iα axis, and the blue line segment is the projection of ib and ic on the iβ axis.
For the Iα axis, the final value is Iα = ia - bα - cα
For the Iβ axis, the final value is Iβ = bβ - cβ (there is no aβ because the projection of ia on the iβ axis is 0 at this time)
When the direction is not considered and only the length is considered,
bα = ib * sin30 = 0.5 * ib
cα = ic * sin30 = 0.5 * ic
bβ = ib * sin60 = (√3)/2 * ib
cβ = ic * sin60 = (√ 3)/2 * ic
Substitute into the previous formula to get the Clarke transformation formula
Iα = ia - 0.5 ib - 0.5 ic
Iβ = (√3)/2 * ib - (√3)/2 * ic
Written as a matrix is
Kirchhoff's laws
假设进入某节点的电流为正值，离开这节点的电流为负值，则所有涉及这节点的电流的代数和等于零
换个说法，即任意时刻，流入的电流等于流出的电流，这两个值大小相等，方向相反。
Assuming that the phase A current flows into ia=1A, then according to Kirchhoff’s law, the current flowing in phase B and phase C is ib = -0.5 and ic = -0.5, and substituting into the formula, Iα = 1 - 0.5*(-
0.5 ) - 0.5*(-0.5) = 1.5 = 1.5
Iβ = 0
Obviously, Iα and ia are not equal. For the convenience of subsequent calculations, we correspond Iα to ia, that is, multiply ia by a coefficient: 2/3, so The two values are then equal.
Iα = ia
Iβ = ((√3)/2 * ib - (√3)/2 * ic) * 2 / 3 = (ib - ic) / √3
Written as a matrix,
this is also called the constant-amplitude form of the Clarke transform.
The above formula can be further simplified, since ia + ib + ic = 0, then ic = - ia - ib, substituting it into Iβ = (ib - ic) / √3 to get Iα = ia Iβ = (
2
*
ib + ia) / √3
In this way, the current of phase C can be eliminated, and the motor can be controlled with only the current of phase A and B. For circuit design, there is no need to sample the current of phase C, which means that the cost is reduced.
Clark Equal Magnitude Inverse Transform
Substitute the first line in the formula into the second line, and solve ib, ic, which is the inverse Clarke transformation
ib = (√3 Iβ - Iα) / 2
Since ia + ib + ic = 0, then
ic = (- Iα - √3 Iβ) / 2
Finally, merge and list
ia = Iα
ib = (√3 * Iβ - Iα) / 2
ic = (- Iα - √3 * Iβ) / 2
ia, ib, ic are the currents we drive when we finally control the motor, then we only need to know Iα and Iβ to control the motor. These two quantities describe the vector of the magnetic field generated by the stator, because the magnetic field is equal to the current multiplied by the inductance, and the phase inductance of the motor is a value that can be known in advance, so controlling the phase current means controlling the magnetic field generated by the stator .
