派克变换（也译作帕克变换，英语：Park's Transformation），是目前分析同步电动机运行最常用的一种坐标变换，由美国工程师派克（R.H.Park）在1929年提出。派克变换将定子的a,b,c三相电流投影到随着转子旋转的直轴（d轴），交轴（q轴）与垂直于dq平面的零轴（0轴）上去，从而实现了对定子电感矩阵的对角化，对同步电动机的运行分析起到了简化作用。

派克正变换：

${\displaystyle {\mathbf {i} }_{dq0}={\mathbf {P} }{\mathbf {i} }_{abc}={\frac {2}{3))\left[{\begin{array}{*{20}c}{\cos \theta }&{\cos \left({\theta -120^{\circ ))\right)}&{\cos \left({\theta +120^{\circ ))\right)}\\{-\sin \theta }&{-\sin \left({\theta -120^{\circ ))\right)}&{-\sin \left({\theta +120^{\circ ))\right)}\\{\frac {1}{2))&{\frac {1}{2))&{\frac {1}{2))\\\end{array))\right]\left[{\begin{array}{*{20}c}{i_{a))\\{i_{b))\\{i_{c))\\\end{array))\right]}$

逆变换：

${\displaystyle {\mathbf {i} }_{abc}={\mathbf {P} }^{-1}{\mathbf {i} }_{dq0}=\left[{\begin{array}{*{20}c}{\cos \theta }&{-\sin \theta }&1\\{\cos \left({\theta -120^{\circ ))\right)}&{-\sin \left({\theta -120^{\circ ))\right)}&1\\{\cos \left({\theta +120^{\circ ))\right)}&{-\sin \left({\theta +120^{\circ ))\right)}&1\\\end{array))\right]\left[{\begin{array}{*{20}c}{i_{d))\\{i_{q))\\{i_{0))\\\end{array))\right]}$

派克变换也作用在定子电压与定子绕组磁链上： ${\displaystyle {\mathbf {u} }_{dq0}={\mathbf {P} }{\mathbf {u} }_{abc))$，${\displaystyle {\mathbf {\Psi } }_{dq0}={\mathbf {P} }{\mathbf {\Psi } }_{abc))$

磁链方程：

${\displaystyle \left[{\begin{array}{*{20}c}((\mathbf {\Psi } }_{abc))\\((\mathbf {\Psi } }_{fDQ))\\\end{array))\right]=\left[{\begin{array}{*{20}c}((\mathbf {L} }_{SS))&((\mathbf {L} }_{SR))\\((\mathbf {L} }_{RS))&((\mathbf {L} }_{RR))\\\end{array))\right]\left[{\begin{array}{*{20}c}{-{\mathbf {i} }_{abc))\\((\mathbf {i} }_{fDQ))\\\end{array))\right]}$

上式中的电感系数矩阵 ${\displaystyle ((\mathbf {L} }_{SS)),((\mathbf {L} }_{SR)),((\mathbf {L} }_{RS)),((\mathbf {L} }_{RR))}$ 事实上都含有随时间变化的角度参数^[1]，使得方程求解困难。

现对等式两边同时左乘 ${\displaystyle \left[{\begin{array}{*{20}c}{\mathbf {P} }&{}\\{}&{\mathbf {U} }\\\end{array))\right]}$，其中${\displaystyle {\mathbf {U} ))$为三阶单位矩阵。方程化为：

${\displaystyle \left[{\begin{array}{*{20}c}((\mathbf {\Psi } }_{dq0))\\((\mathbf {\Psi } }_{fDQ))\\\end{array))\right]=\left[{\begin{array}{*{20}c}{\mathbf {P} }&{}\\{}&{\mathbf {U} }\\\end{array))\right]\left[{\begin{array}{*{20}c}((\mathbf {L} }_{SS))&((\mathbf {L} }_{SR))\\((\mathbf {L} }_{RS))&((\mathbf {L} }_{RR))\\\end{array))\right]\left[{\begin{array}{*{20}c}((\mathbf {P} }^{-1))&{}\\{}&{\mathbf {U} }\\\end{array))\right]\left[{\begin{array}{*{20}c}{-{\mathbf {i} }_{abc))\\((\mathbf {i} }_{fDQ))\\\end{array))\right]}$

${\displaystyle \left[{\begin{array}{*{20}c}((\mathbf {\Psi } }_{dq0))\\((\mathbf {\Psi } }_{fDQ))\\\end{array))\right]=\left[{\begin{array}{*{20}c}((\mathbf {PL} }_{SS}{\mathbf {P} }^{-1))&((\mathbf {PL} }_{SR))\\((\mathbf {L} }_{RS}{\mathbf {P} }^{-1))&((\mathbf {L} }_{RR))\\\end{array))\right]\left[{\begin{array}{*{20}c}{-{\mathbf {i} }_{dq0))\\((\mathbf {i} }_{fDQ))\\\end{array))\right]}$

其中 ${\displaystyle {\mathbf {PL} }_{SS}{\mathbf {P} }^{-1}=\left[{\begin{array}{*{20}c}{L_{d))&{}&{}\\{}&{L_{q))&{}\\{}&{}&{L_{0))\\\end{array))\right]\triangleq {\mathbf {L} }_{dq0))$。

① 变换后的电感系数都变为常数，可以假想dd绕组，qq绕组是固定在转子上的，相对转子静止。

② 派克变换阵对定子自感矩阵 ${\displaystyle {\mathbf {L} }_{SS))$ 起到了对角化的作用，并消去了其中的角度变量。${\displaystyle {L_{d)),{L_{q)),{L_{0))}$ 为其特征根。

③ 变换后定子和转子间的互感系数不对称，这是由于派克变换的矩阵不是正交矩阵。

④ ${\displaystyle {L_{d))}$ 为直轴同步电感系数，其值相当于当励磁绕组开路，定子合成磁势产生单纯直轴磁场时，任意一相定子绕组的自感系数。

电压方程：

${\displaystyle \left[{\begin{array}{*{20}c}((\mathbf {U} }_{abc))\\((\mathbf {U} }_{fDQ))\\\end{array))\right]=\left[{\begin{array}{*{20}c}((\mathbf {r} }_{S))&{}\\{}&((\mathbf {r} }_{R))\\\end{array))\right]\left[{\begin{array}{*{20}c}{-{\mathbf {i} }_{abc))\\((\mathbf {i} }_{fDQ))\\\end{array))\right]+\left[{\begin{array}{*{20}c}((\mathbf {\dot {\Psi )) }_{abc))\\((\mathbf {\dot {\Psi )) }_{fDQ))\\\end{array))\right]}$

现对等式两边同时左乘 ${\displaystyle \left[{\begin{array}{*{20}c}{\mathbf {P} }&{}\\{}&{\mathbf {U} }\\\end{array))\right]}$，其中${\displaystyle {\mathbf {U} ))$为三阶单位矩阵。方程化为：

${\displaystyle \left[{\begin{array}{*{20}c}((\mathbf {U} }_{dq0))\\((\mathbf {U} }_{fDQ))\\\end{array))\right]=\left[{\begin{array}{*{20}c}((\mathbf {r} }_{S))&{}\\{}&((\mathbf {r} }_{R))\\\end{array))\right]\left[{\begin{array}{*{20}c}{-{\mathbf {i} }_{dq0))\\((\mathbf {i} }_{fDQ))\\\end{array))\right]+\left[{\begin{array}{*{20}c}((\mathbf {P{\dot {\Psi ))} }_{abc))\\((\mathbf {\dot {\Psi )) }_{fDQ))\\\end{array))\right]}$

由 ${\displaystyle {\mathbf {\Psi } }_{dq0}={\mathbf {P\Psi } }_{abc))$ ，

对两边求导，得 ${\displaystyle {\mathbf {\dot {\Psi )) }_{dq0}={\mathbf ((\dot {P))\Psi } }_{abc}+{\mathbf {P{\dot {\Psi ))} }_{abc))$ ，

所以 ${\displaystyle {\mathbf {P{\dot {\Psi ))} }_{abc}={\mathbf {\dot {\Psi )) }_{dq0}-{\mathbf ((\dot {P))\Psi } }_{abc}={\mathbf {\dot {\Psi )) }_{dq0}-{\mathbf ((\dot {P))P} }^{-1}{\mathbf {\Psi } }_{dq0))$

其中 ${\displaystyle {\mathbf ((\dot {P))P} }^{-1}=\left[{\begin{array}{*{20}c}{}&\omega &{}\\{-\omega }&{}&{}\\{}&{}&{}\\\end{array))\right]}$ ，令 ${\displaystyle {\mathbf {S} }={\mathbf ((\dot {P))P} }^{-1}{\mathbf {\Psi } }_{dq0}=\left[{\begin{array}{*{20}c}{}&\omega &{}\\{-\omega }&{}&{}\\{}&{}&{}\\\end{array))\right]\left[{\begin{array}{*{20}c}{\Phi _{d))\\{\Phi _{q))\\{\Phi _{0))\\\end{array))\right]=\left[{\begin{array}{*{20}c}{\omega \Psi _{q))\\{-\omega \Psi _{d))\\{}\\\end{array))\right]}$

于是有 ${\displaystyle \left[{\begin{array}{*{20}c}((\mathbf {U} }_{dq0))\\((\mathbf {U} }_{fDQ))\\\end{array))\right]=\left[{\begin{array}{*{20}c}((\mathbf {r} }_{S))&{}\\{}&((\mathbf {r} }_{R))\\\end{array))\right]\left[{\begin{array}{*{20}c}{-{\mathbf {i} }_{dq0))\\((\mathbf {i} }_{fDQ))\\\end{array))\right]+\left[{\begin{array}{*{20}c}((\mathbf {\dot {\Psi )) }_{dq0))\\((\mathbf {\dot {\Psi )) }_{fDQ))\\\end{array))\right]-\left[{\begin{array}{*{20}c}{\mathbf {S} }\\{}\\\end{array))\right]}$

上式右边第一项为绕组电阻的压降，第二项为变压器电势，第三项为发电机电势或旋转电势。

1. ^ 定子电感矩阵 ${\displaystyle {\mathbf {L} }_{SS}=\left[{\begin{array}{*{20}c}{L_{aa))&{M_{ab))&{M_{ac))\\{M_{ba))&{L_{bb))&{M_{bc))\\{M_{ca))&{M_{cb))&{L_{cc))\\\end{array))\right]}$，
   其中
   

   ${\displaystyle L_{aa}=l_{0}+l_{2}\cos \left(2\theta \right)}$

   ${\displaystyle L_{bb}=l_{0}+l_{2}\cos 2\left({\theta -120^{\circ ))\right)}$

   ${\displaystyle L_{cc}=l_{0}+l_{2}\cos 2\left({\theta +120^{\circ ))\right)}$

   ${\displaystyle M_{ab}=M_{ba}=-m_{0}-m_{2}\cos 2\left({\theta +30^{\circ ))\right)}$

   ${\displaystyle M_{bc}=M_{cb}=-m_{0}-m_{2}\cos 2\left({\theta -90^{\circ ))\right)}$

   ${\displaystyle M_{ca}=M_{ac}=-m_{0}-m_{2}\cos 2\left({\theta +150^{\circ ))\right)}$

• 电机电子类科《电力系统暂态分析》，ISBN 978-7-5083-4825-4，作者：李光琦，中国电力出版社。