FOC控制之CLARK和PARK变换

CLARK变换

CLARK变换将三相系统（在 abc 坐标系中）的时域分量转换为正交静止坐标系 (αβ) 中的两个分量

转换公式

$\{\begin{array}{l}{U}_{\alpha }=U_{\alpha }-\cos (\pi /3)U_b-\cos (\pi /3)U_c\\ U_{\beta }=\sin (\pi /3)U_b-\sin (\pi /3)U_c\end{array}$

转换矩阵

∣ U α U β ∣ = K ∣ 1 − 1 / 2 − 1 / 2 0 ( 3 / 2 − ( 3 / 2 ∣ ∣ U a U b U c ∣ \begin{vmatrix} U_{α} \\ U_{β} \end{vmatrix}=K \begin{vmatrix} 1 & -1/2 & -1/2 \\ 0 & \sqrt{\mathstrut 3}/2 & -\sqrt{\mathstrut 3}/2 \end{vmatrix}\begin{vmatrix} U_{a} \\ U_{b} \\ U_{c} \end{vmatrix} ​Uα​Uβ​​ ​=K ​10​−1/2(3 ​/2​−1/2−(3 ​/2​ ​ ​Ua​Ub​Uc​​ ​

CLARK逆变换

2轴电流到3轴电流

变换公式

$\{\begin{array}{l}{U}_a=U_{\alpha }\\ U_b=\cos (2\pi /3)U_{\alpha }+\sin (\pi /3)U_{\beta }\\ U_c=\cos (2\pi /3)U_{\alpha }-\sin (\pi /3)U_{\beta }\end{array}$

变换矩阵

∣ U a U b U c ∣ = ∣ 1 0 − 1 / 2 ( 3 / 2 − 1 / 2 − ( 3 / 2 ∣ ∣ U α U β ∣ \begin{vmatrix} U_{a} \\ U_{b} \\ U_{c} \end{vmatrix}=\begin{vmatrix} 1 & 0 \\ -1/2 & \sqrt{\mathstrut 3}/2 \\ -1/2 & -\sqrt{\mathstrut 3}/2 \end{vmatrix}\begin{vmatrix} U_{\alpha} \\ U_{\beta} \end{vmatrix} ​Ua​Ub​Uc​​ ​= ​1−1/2−1/2​0(3 ​/2−(3 ​/2​ ​ ​Uα​Uβ​​ ​

PARK变换

PARK变换将 αβ 静止坐标系中的两个分量转换为一个正交旋转坐标系 (dq)

转换公式

$\{\begin{array}{l}{U}_d=U_{\alpha }\cos (\theta )+U_{\beta }\sin (\theta )\\ U_q=-U_{\alpha }\sin (\theta )+U_{\beta }\cos (\theta )\end{array}$

转换矩阵

$\left[\begin{array}{c}{U}_d\\ U_q\end{array}\right]=\left[\begin{array}{cc}\cos (\theta )& \sin (\theta )\\ -\sin (\theta )& \cos (\theta )\end{array}\right]\left[\begin{array}{c}{U}_{\alpha }\\ U_{\beta }\end{array}\right]$

PARK逆变换

旋转坐标系到静止坐标系

变换公式

$\{\begin{array}{l}{U}_{\alpha }=U_d\cos (\theta )-U_q\sin (\theta )\\ U_{\beta }=U_d\sin (\theta )+U_q\cos (\theta )\end{array}$

变换矩阵

$\left[\begin{array}{c}{U}_{\alpha }\\ U_{\beta }\end{array}\right]=\left[\begin{array}{cc}\cos (\theta )& -\sin (\theta )\\ \sin (\theta )& \cos (\theta )\end{array}\right]\left[\begin{array}{c}{U}_d\\ U_q\end{array}\right]$