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。 派克正变换： i d q 0 = P i a b c = 2 3 [ cos ⁡ θ cos ⁡ ( θ − 120 ∘ ) cos ⁡ ( θ + 120 ∘ ) − sin ⁡ θ − sin ⁡ ( θ − 120 ∘ ) − sin ⁡ ( θ + 120 ∘ ) 1 2 1 2 1 2 ] [ i a i b i c ] {\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： i a b c = P − 1 i d q 0 = [ cos ⁡ θ − sin ⁡ θ 1 cos ⁡ ( θ − 120 ∘ ) − sin ⁡ ( θ − 120 ∘ ) 1 cos ⁡ ( θ + 120 ∘ ) − sin ⁡ ( θ + 120 ∘ ) 1 ] [ i d i q i 0 ] {\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： u d q 0 = P u a b c {\displaystyle {\mathbf {u} }_{dq0}={\mathbf {P} }{\mathbf {u} }_{abc}} ， Ψ d q 0 = P Ψ a b c {\displaystyle {\mathbf {\Psi } }_{dq0}={\mathbf {P} }{\mathbf {\Psi } }_{abc}} 磁链方程： [ Ψ a b c Ψ f D Q ] = [ L S S L S R L R S L R R ] [ − i a b c i f D Q ] {\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 L S S , L S R , L R S , L R R {\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，使得方程求解困难。 现对等式两边同时左乘 [ P U ] {\displaystyle \left[{\begin{array}{*{20}c}{\mathbf {P} }&{}\\{}&{\mathbf {U} }\\\end{array}}\right]} ，其中 U {\displaystyle {\mathbf {U} }} 为三阶单位矩阵。方程化为： [ Ψ d q 0 Ψ f D Q ] = [ P U ] [ L S S L S R L R S L R R ] [ P − 1 U ] [ − i a b c i f D Q ] {\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]} [ Ψ d q 0 Ψ f D Q ] = [ P L S S P − 1 P L S R L R S P − 1 L R R ] [ − i d q 0 i f D Q ] {\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 P L S S P − 1 = [ L d L q L 0 ] ≜ L d q 0 {\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 L S S {\displaystyle {\mathbf {L} }_{SS}} 起到了对角化的作用，并消去了其中的角度变量。 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 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。 电压方程： [ U a b c U f D Q ] = [ r S r R ] [ − i a b c i f D Q ] + [ Ψ ˙ a b c Ψ ˙ f D Q ] {\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 [ P U ] {\displaystyle \left[{\begin{array}{*{20}c}{\mathbf {P} }&{}\\{}&{\mathbf {U} }\\\end{array}}\right]} ，其中 U {\displaystyle {\mathbf {U} }} 为三阶单位矩阵。方程化为： [ U d q 0 U f D Q ] = [ r S r R ] [ − i d q 0 i f D Q ] + [ P Ψ ˙ a b c Ψ ˙ f D Q ] {\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 Ψ d q 0 = P Ψ a b c {\displaystyle {\mathbf {\Psi } }_{dq0}={\mathbf {P\Psi } }_{abc}} ， 对两边求导，得 Ψ ˙ d q 0 = P ˙ Ψ a b c + P Ψ ˙ a b c {\displaystyle {\mathbf {\dot {\Psi }} }_{dq0}={\mathbf {{\dot {P}}\Psi } }_{abc}+{\mathbf {P{\dot {\Psi }}} }_{abc}} ， 所以 P Ψ ˙ a b c = Ψ ˙ d q 0 − P ˙ Ψ a b c = Ψ ˙ d q 0 − P ˙ P − 1 Ψ d q 0 {\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 P ˙ P − 1 = [ ω − ω ] {\displaystyle {\mathbf {{\dot {P}}P} }^{-1}=\left[{\begin{array}{*{20}c}{}&\omega &{}\\{-\omega }&{}&{}\\{}&{}&{}\\\end{array}}\right]} ，令 S = P ˙ P − 1 Ψ d q 0 = [ ω − ω ] [ Φ d Φ q Φ 0 ] = [ ω Ψ q − ω Ψ d ] {\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 [ U d q 0 U f D Q ] = [ r S r R ] [ − i d q 0 i f D Q ] + [ Ψ ˙ d q 0 Ψ ˙ f D Q ] − [ S ] {\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。 定子电感矩阵 L S S = [ L a a M a b M a c M b a L b b M b c M c a M c b L c c ] {\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 L a a = l 0 + l 2 cos ⁡ ( 2 θ ) {\displaystyle L_{aa}=l_{0}+l_{2}\cos \left(2\theta \right)} L b b = l 0 + l 2 cos ⁡ 2 ( θ − 120 ∘ ) {\displaystyle L_{bb}=l_{0}+l_{2}\cos 2\left({\theta -120^{\circ }}\right)} L c c = l 0 + l 2 cos ⁡ 2 ( θ + 120 ∘ ) {\displaystyle L_{cc}=l_{0}+l_{2}\cos 2\left({\theta +120^{\circ }}\right)} M a b = M b a = − m 0 − m 2 cos ⁡ 2 ( θ + 30 ∘ ) {\displaystyle M_{ab}=M_{ba}=-m_{0}-m_{2}\cos 2\left({\theta +30^{\circ }}\right)} M b c = M c b = − m 0 − m 2 cos ⁡ 2 ( θ − 90 ∘ ) {\displaystyle M_{bc}=M_{cb}=-m_{0}-m_{2}\cos 2\left({\theta -90^{\circ }}\right)} M c a = M a c = − m 0 − m 2 cos ⁡ 2 ( θ + 150 ∘ ) {\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。