派克变换

　　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。

定义

　　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]}{\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]}{\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{u}}_{dq0}={\mathbf{P}}{\mathbf{u}}_{abc}}，{\displaystyle{\mathbf{\Psi}}_{dq0}={\mathbf{P}}{\mathbf{\Psi}}_{abc}}{\displaystyle{\mathbf{\Psi}}_{dq0}={\mathbf{P}}{\mathbf{\Psi}}_{abc}}

几何解释

　　上图描绘了派克变换的几何意义，定子三相电流互成120度角，{\displaystyle\delta}\delta为定子电流落后于它们对应的相电压的角度。直轴与交轴电流分别等于定子三相电流在d轴与q轴上的投影。（图中的比例系数{\displaystyle{\sqrt{\frac{3}{2}}}}{\displaystyle{\sqrt{\frac{3}{2}}}}是由于图中所采用的是正交形式的派克变换）d-q坐标系在空间中以角速度{\displaystyle\omega}\omega逆时针旋转，故{\displaystyle\theta=\omega t}{\displaystyle\theta=\omega t}以d轴领先a相轴线的方向为正。当定子电流为三相对称的正弦交流电时，{\displaystyle i_{d}}{\displaystyle i_{d}},{\displaystyle i_{q}}{\displaystyle i_{q}}为直流电流，{\displaystyle i_{0}=0}{\displaystyle i_{0}=0}。

用派克变换化简同步发电机基本方程

变换后的磁链方程

　　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]}{\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}}}{\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]}{\displaystyle\left[{\begin{array}{*{20}c}{\mathbf{P}}&{}\\{}&{\mathbf{U}}\\\end{array}}\right]}，其中{\displaystyle{\mathbf{U}}}{\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{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]}{\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}}{\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}}{\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}}}{\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}}}{\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。

变换后的电压方程

　　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]}{\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]}{\displaystyle\left[{\begin{array}{*{20}c}{\mathbf{P}}&{}\\{}&{\mathbf{U}}\\\end{array}}\right]}，其中{\displaystyle{\mathbf{U}}}{\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]}{\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}}{\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}}{\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}}{\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]}{\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]}{\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]}{\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。

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