所看书籍：《Pulse Width Modulation for Power Converters Principles and Practice 》by D. Grahame Holmes, Thomas A. Lipo

先介绍下本人背景，电机控制新人一枚。也是在知乎搜索相关PWM相关书籍，看到有答主推荐上述书籍。

最近研究方向转到了逆变器调制，所以打算最近把这本书好好看一遍。这本书第一章1.6节，讲空间矢量的概念，个人感觉讲的很好。

看过一些其他资料，讲park，clark变换，都有各自不同的角度解释变换矩阵前面系数的问题，但是都不如这本书只从纯数学的角度推导坐标变换系数来的清楚（我感觉clark,park变换的系数先物理意义上去解释，容易让人犯糊涂，不如直接用清晰的数学公示，把坐标变换推导出来，后面再试着加一些物理意义进去）。

写这篇文章的目的，是希望分享一下本人这本书第一章，1.6节所推导出的公式1.37的详细补充（书中直接只给出了坐标的定义，以及公式结果，对于喜欢刨根问底的同学，不推导出来很痛苦）。

Figure 1.18

关于上图的解释和来源，我还是直接把原书英文贴上去吧，觉着看英文更好理解一点：

Consider, for instance, the stator currents of a three-phase induction machine load which is, in general, made up of three independent variables. These currents (phase variables) can be visualized as being components of a single three-dimensional vector (space vector) existing in a three-dimensional orthogonal space, i.e., the space defined by Figure 1.18.

通常情况下，电机三相负载中性点并没有引出连接线从而形成闭合回路，所以

(1.36）

The fact that Eq. (1.36) defines a particular plane is evident if it is recalled from analytic geometry that the general definition of a plane is ax + by + cz = d. This plane, the so-called d-q plane, is also illustrated in Figure 1.18. Components of the current and voltage vector in the plane are called the d-q components while the component in the axis normal to the plane (in the event that the currents do not sum to zero) is called the zero component.

By convention it is assumed that the projection ofthe phase a axis on the d-q plane forms the reference q axis for the case where the d-q axes are not rotating. A second axis on the plane is defined as being orthogonal to the q axis such that the cross product d x q yields a third axis, by necessity normal to the d-q plane, that produces a-third component of the vector having the conventional definition of the zero sequence quantity。

我个人理解：由于三相负载中性点并没有引出连接线从而形成闭合回路， Ia+Ib+Ic=0；所以电流空间 矢量就在上述平面内。这个 平面，定义为dq平面。根据 惯例，a轴在dq平面上的投 影是q轴。d轴以q轴为参考 ，顺时针90度。d叉乘q就产 生了零轴。

好了，现在abc正交坐标系有了，由于电机三相负载并没有引出连接线从而形成闭合回路，所以电流矢量一定在式1.36所定义的平面内，这个平面就是dq0平面（静止）。abc正交坐标系到dq0平面，dq坐标系的变换公式，书中直接给出了。如下

好了，铺垫到此已经全部做完，我所做的事情是，推导了一下上述这个公式是怎么来的。时间原因，我直接把手写的拍照上传。另外，今天是个很特别的日子，国庆节加中秋，祝祖国生日快乐！尤其感谢知乎大佬， @陈嘉豪 @Foxore ，一位是电机专业优秀的博士，一位在知乎上推荐了本文所引用的书籍。