# 派克变换

## 定义

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

${\mathbf{i}}_{abc} = {\mathbf{P}}^{ - 1} {\mathbf{i}}_{dq0} = \left[ {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } & 1 \\ {\cos \left( {\theta - 120^ \circ } \right)} & { - \sin \left( {\theta - 120^ \circ } \right)} & 1 \\ {\cos \left( {\theta + 120^ \circ } \right)} & { - \sin \left( {\theta + 120^ \circ } \right)} & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_d } \\ {i_q } \\ {i_0 } \\ \end{array} } \right]$

## 用派克变换化简同步发电机基本方程

### 变换后的磁链方程

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

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

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

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 ${\mathbf{L}}_{SS}$ 起到了对角化的作用，并消去了其中的角度变量。${L_d },{L_q},{L_0}$ 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

${L_d }$ 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

$\left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{abc} } \\ {{\mathbf{U}}_{fDQ} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{r}}_S } & {} \\ {} & {{\mathbf{r}}_R } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} { - {\mathbf{i}}_{abc} } \\ {{\mathbf{i}}_{fDQ} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {{\mathbf{\dot \Psi }}_{abc} } \\ {{\mathbf{\dot \Psi }}_{fDQ} } \\ \end{array} } \right]$

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

${\mathbf{\Psi }}_{dq0} = {\mathbf{P\Psi }}_{abc}$

## 注释

1. ^ 定子电感矩阵 ${\mathbf{L}}_{SS} = \left[ {\begin{array}{*{20}c} {L_{aa} } & {M_{ab} } & {M_{ac} } \\ {M_{ba} } & {L_{bb} } & {M_{bc} } \\ {M_{ca} } & {M_{cb} } & {L_{cc} } \\ \end{array} } \right]$
其中
$L_{aa} = l_0 + l_2 \cos \left( 2\theta \right)$
$L_{bb} = l_0 + l_2 \cos 2\left( {\theta - 120^ \circ } \right)$
$L_{cc} = l_0 + l_2 \cos 2\left( {\theta + 120^ \circ } \right)$
$M_{ab} = M_{ba} = - m_0 - m_2 \cos 2\left( {\theta + 30^ \circ } \right)$
$M_{bc} = M_{cb} = - m_0 - m_2 \cos 2\left( {\theta - 90^ \circ } \right)$
$M_{ca} = M_{ac} = - m_0 - m_2 \cos 2\left( {\theta + 150^ \circ } \right)$

## 参考书目

• 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。