インダクタンス行列のd-q-0変換　同期機

インダクタンス行列

関連記事のインダクタンス(制動巻線考慮なし)　同期機で求めたように同期機の各相の自己インダクタンスと相互インダクタンスは、以下のようになった。

自己インダクタンス

$$
\begin{align}
L_{\rm{aa}} &= L_{1}+L_{2}\cos \left(2\theta \right)\notag\\
L_{\rm{bb}} &= L_{1}+L_{2}\cos \left(2\theta + \frac{2}{3}\pi \right)\notag\\
L_{\rm{cc}} &= L_{1}+L_{2}\cos \left(2\theta - \frac{2}{3}\pi \right)\notag\\
\end{align}
$$

相互インダクタンス

$$
\begin{align}
L_{\rm{ab}}&= L_{\rm{ba}}= -L_{3} +L_{2}\cos\left(2\theta - \frac{2}{3}\pi \right)\notag\\
L_{\rm{bc}}&= L_{\rm{cb}}= -L_{3}+L_{2}\cos \left(2\theta \right)\notag\\
L_{\rm{ca}}&= L_{\rm{ac}}= -L_{3} +L_{2}\cos\left(2\theta + \frac{2}{3}\pi \right)\notag\\
\end{align}
$$

これを行列の形にすると、

$$
\begin{align}
L_{\rm{abc}} &=
\begin{pmatrix}
L_{\rm{aa}} & L_{\rm{ab}} & L_{\rm{ac}}\\
& & \\
L_{\rm{ba}} & L_{\rm{bb}} & L_{\rm{bc}}\\
& & \\
L_{\rm{ca}} & L_{\rm{cb}} & L_{\rm{cc}}\\
\end{pmatrix}\notag\\
&\notag\\
&= 
\left(
\begin{matrix}
L_{1}+L_{2}\cos \left(2\theta \right)&-L_{3} +L_{2}\cos\left(2\theta - \frac{2}{3}\pi \right) \\
& \\
-L_{3} +L_{2}\cos\left(2\theta - \frac{2}{3}\pi \right) &L_{1}+L_{2}\cos \left(2\theta + \frac{2}{3}\pi \right) \\
& \\
-L_{3} +L_{2}\cos\left(2\theta+ \frac{2}{3}\pi \right)&-L_{3}+L_{2}\cos \left(2\theta \right) \\
\end{matrix} \right.\notag\\
&\notag\\
& \qquad \qquad \qquad \qquad \qquad \qquad 
\left.
\begin{matrix}
 -L_{3} +L_{2}\cos\left(2\theta + \frac{2}{3}\pi \right)\\
 \\
 -L_{3}+L_{2}\cos \left(2\theta \right)\\
\\
L_{1}+L_{2}\cos \left(2\theta - \frac{2}{3}\pi \right)\\
\end{matrix} \right) \notag\\
\end{align}
$$

となる。

磁束鎖交数と電流の関係は、関連記事のd-q-0座標系の必要性　同期機で示したように、

$$
\begin{align}
\phi_{\rm{a}} &= L_{\rm{aa}}i_{\rm{a}}+L_{\rm{ab}}i_{\rm{b}}+L_{\rm{ac}}i_{\rm{c}}+L_{\rm{af}}i_{\rm{f}}\notag\\
\phi_{\rm{b}} &= L_{\rm{ba}}i_{\rm{a}}+L_{\rm{bb}}i_{\rm{b}}+L_{\rm{bc}}i_{\rm{c}}+L_{\rm{bf}}i_{\rm{f}}\notag\\
\phi_{\rm{c}} &= L_{\rm{ca}}i_{\rm{a}}+L_{\rm{cb}}i_{\rm{b}}+L_{\rm{cc}}i_{\rm{c}}+L_{\rm{cf}}i_{\rm{f}}\notag\\
\end{align}
$$

となる。
行列で表すと、

$$
\begin{align}
\left(
\begin{matrix}
\phi_{\rm{a}}\\
\\
\phi_{\rm{b}}\\
\\
\phi_{\rm{c}}
\end{matrix}
\right)
&=
\left(
\begin{matrix}
L_{\rm{aa}} & L_{\rm{ab}} & L_{\rm{ac}} \\
\\
L_{\rm{ba}} & L_{\rm{bb}} & L_{\rm{bc}}\\
\\
L_{\rm{ca}} & L_{\rm{cb}} & L_{\rm{cc}}\\
\end{matrix}
\right)
\left(
\begin{matrix}
i_{\rm{a}}\\
\\
i_{\rm{b}}\\
\\
i_{\rm{c}}\\
\end{matrix}
\right)
+
\left(
\begin{matrix}
L_{\rm{af}}\\
\\
L_{\rm{bf}}\\
\\
L_{\rm{cf}}\\
\end{matrix}
\right) i_{\rm{f}}\notag\\
\end{align}
$$

となる。
変換行列$${A}$$は、関連記事のd-q-0法　同期機より、

$$
\begin{align}
&\notag\\
A &=
\left(
\begin{matrix}
\frac{2}{3}\cos(\theta_{{\rm{a}}}) & \frac{2}{3}\cos(\theta_{{\rm{b}}})& \frac{2}{3}\cos(\theta_{{\rm{c}}})\\
&&\\
-\frac{2}{3}\sin(\theta_{{\rm{a}}}) & -\frac{2}{3}\sin(\theta_{{\rm{b}}})& -\frac{2}{3}\sin(\theta_{{\rm{c}}}) \\
&&\\
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\
\end{matrix}\right)\notag\\
\end{align}
$$

であり、

$$
\begin{align}
\theta_{{\rm{a}}} &= \varphi \notag\\
\theta_{{\rm{b}}} &= \varphi - \frac{2}{3}\pi\notag\\
\theta_{{\rm{c}}} &= \varphi+ \frac{2}{3}\pi\notag\\
\end{align}
$$

である。

左から変換行列をかけると、

$$
A\phi_{\rm{abc}} =AL_{\rm{abc}}i_{\rm{abc}}+ AL_{\rm{ff}}i_{\rm{f}}
$$

となる。
単位行列$${\bm{I}=A^{-1}A}$$の性質を用いれば、

$$
A\phi_{\rm{abc}} =AL_{\rm{abc}}A^{-1}Ai_{\rm{abc}}+ AL_{\rm{ff}}i_{\rm{f}}
$$

とすることができる。変換行列$${A}$$が左から掛けられているものは、d-q-0座標系に変換されるので、

$$
\phi_{\rm{dq0}} =AL_{\rm{abc}}A^{-1}i_{\rm{dq0}}+ L_{\rm{dq0f}}i_{\rm{f}}
$$

となる。

したがって、インダクタンス行列の変換式は、$${AL_{\rm{abc}}A^{-1}}$$を計算すれば良い。

行列計算

まず初めに$${AL_{\rm{abc}}}$$を計算する。

行列計算その1

($${1,1}$$)成分

$$
\begin{align}
&\notag\\
&\frac{2}{3}\cos(\theta_{{\rm{a}}})\left(L_{1}+L_{2}\cos \left(2\theta\right)\right) \notag\\
&\qquad \qquad + \frac{2}{3}\cos(\theta_{{\rm{b}}})\left(-L_{3} +L_{2}\cos\left(2\theta - \frac{2}{3}\pi \right)\right)\notag\\
&\qquad \qquad \qquad \qquad + \frac{2}{3}\cos(\theta_{{\rm{c}}})\left(-L_{3} +L_{2}\cos\left(2\theta+ \frac{2}{3}\pi \right)\right)\notag\\
&\notag\\
&= \frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{2}\cos(\varphi)\cos \left(2\theta\right) \notag\\
&\quad - \frac{2}{3}L_{3}\cos\left(\varphi - \frac{2}{3}\pi\right) + \frac{2}{3}L_{2}\cos\left(\varphi - \frac{2}{3}\pi\right)\cos\left(2\theta - \frac{2}{3}\pi \right)\notag\\
&\quad -\frac{2}{3}L_{3}\cos\left(\varphi+ \frac{2}{3}\pi\right) +\frac{2}{3}L_{2}\cos\left(\varphi+ \frac{2}{3}\pi\right)\cos\left(2\theta + \frac{2}{3}\pi \right)\notag\\
&\notag\\
&= \frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{2}\cos(\varphi)\cos \left(2\theta \right) \notag\\
&\quad - \frac{2}{3}L_{3}\left(-\frac{1}{2}\cos(\varphi)+\frac{\sqrt{3}}{2}\sin( \varphi)\right) \notag\\
&\quad + \frac{2}{3}L_{2}\left(-\frac{1}{2}\cos(\varphi)+\frac{\sqrt{3}}{2}\sin(\varphi)\right)\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right) \notag\\
& \quad -\frac{2}{3}L_{3}\left(-\frac{1}{2}\cos(\varphi)-\frac{\sqrt{3}}{2}\sin(\varphi)\right) \notag\\
&\quad +\frac{2}{3}L_{2}\left(-\frac{1}{2}\cos(\varphi)-\frac{\sqrt{3}}{2}\sin(\varphi)\right) \left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right) \notag\\
&\notag\\
&= \frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{2}\cos(\varphi)\cos \left(2\theta\right) \notag\\
&\quad +\frac{1}{3}L_{3}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) \notag\\
&\quad + \frac{2}{3}L_{2}\left(\frac{1}{4}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{4}\cos(\varphi)\sin(2\theta)\right.\notag\\
&\left. \qquad \qquad \qquad \qquad-\frac{\sqrt{3}}{4}\sin(\varphi)\cos(2\theta)+\frac{3}{4}\sin(\varphi)\sin(2\theta)\right) \notag\\
& \quad +\frac{1}{3}L_{3}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) \notag\\
&\quad +\frac{2}{3}L_{2}\left(\frac{1}{4}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{4}\cos(\varphi)\sin(2\theta) \right.\notag\\
&\left. \qquad \qquad \qquad \qquad+ \frac{\sqrt{3}}{4}\sin(\varphi)\cos(2\theta) + \frac{3}{4}\sin(\varphi)\sin(2\theta)\right) \notag\\
&\notag\\
&= \frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{2}\cos(\varphi)\cos \left(2\theta \right) \notag\\
&\quad +\frac{1}{3}L_{3}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) \notag\\
&\quad +\frac{1}{6}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{6}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
& \qquad \qquad \qquad \qquad-\frac{\sqrt{3}}{6}L_{2}\sin(\varphi)\cos(2\theta)+\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta) \notag\\
& \quad +\frac{1}{3}L_{3}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) \notag\\
&\quad +\frac{1}{6}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{6}L_{2}\cos(\varphi)\sin(2\theta) \notag\\
&\qquad \qquad \qquad \qquad+ \frac{\sqrt{3}}{6}L_{2}\sin(\varphi)\cos(2\theta) + \frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta) \notag\\
&\notag\\
&= \frac{2}{3}L_{1}\cos(\varphi)+L_{2}\cos(\varphi)\cos \left(2\theta \right) +\frac{2}{3}L_{3}\cos(\varphi) +L_{2}\sin(\varphi)\sin(2\theta) \notag\\
&\notag\\
&= \frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{3}\cos(\varphi) +L_{2}\cos(2\theta - \varphi) \notag\\
\end{align}
$$

($${1,2}$$)成分

$$
\begin{align}
&\frac{2}{3}\cos(\theta_{{\rm{a}}})\left(-L_{3} +L_{2}\cos\left(2\theta- \frac{2}{3}\pi \right)\right) \notag\\
&\qquad \qquad + \frac{2}{3}\cos(\theta_{{\rm{b}}})\left(L_{1}+L_{2}\cos \left(2\theta+ \frac{2}{3}\pi \right)\right)\notag\\
&\qquad \qquad \qquad \qquad + \frac{2}{3}\cos(\theta_{{\rm{c}}})\left(-L_{3}+L_{2}\cos \left(2\theta \right)\right)\notag\\
&\notag\\
&= -\frac{2}{3}L_{3}\cos(\varphi) +\frac{2}{3}L_{2}\cos(\varphi)\cos\left(2\theta- \frac{2}{3}\pi \right) \notag\\
&\quad +\frac{2}{3}L_{1}\cos\left(\varphi - \frac{2}{3}\pi\right)+\frac{2}{3}L_{2}\cos\left(\varphi - \frac{2}{3}\pi\right)\cos \left(2\theta + \frac{2}{3}\pi \right)\notag\\
&\quad -\frac{2}{3}L_{3}\cos\left(\varphi + \frac{2}{3}\pi\right)+\frac{2}{3}L_{2}\cos\left(\varphi+ \frac{2}{3}\pi\right)\cos \left(2\theta \right)\notag\\
&\notag\\
&= -\frac{2}{3}L_{3}\cos(\varphi) +\frac{2}{3}L_{2}\cos(\varphi)\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right)\notag\\
& \quad +\frac{2}{3}L_{1}\left(-\frac{1}{2}\cos(\varphi)+\frac{\sqrt{3}}{2}\sin( \varphi)\right)\notag\\
&\quad +\frac{2}{3}L_{2}\left(-\frac{1}{2}\cos(\varphi)+\frac{\sqrt{3}}{2}\sin(\varphi)\right)\left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right)\notag\\
&\quad -\frac{2}{3}L_{3}\left(-\frac{1}{2}\cos(\varphi)-\frac{\sqrt{3}}{2}\sin(\varphi)\right)\notag\\
&\quad +\frac{2}{3}L_{2}\left(-\frac{1}{2}\cos(\varphi)-\frac{\sqrt{3}}{2}\sin(\varphi)\right)\cos \left(2\theta \right)\notag\\
&\notag\\
&= -\frac{2}{3}L_{3}\cos(\varphi) -\frac{1}{3}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{3}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
& \quad -\frac{1}{3}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)\notag\\
&\quad +\frac{2}{3}L_{2}\left(\frac{1}{4}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{4}\cos(\varphi)\sin(2\theta)\right.\notag\\
&\qquad \qquad \qquad \qquad \left.-\frac{\sqrt{3}}{4}\cos(2\theta)\sin(\varphi)-\frac{3}{4}\sin(\varphi)\sin(2\theta)\right)\notag\\
& \quad +\frac{1}{3}L_{3}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\quad -\frac{1}{3}L_{2}\cos(\varphi)\cos (2\theta)-\frac{\sqrt{3}}{3}L_{2}\cos (2\theta)\sin(\varphi)\notag\\
&\notag\\
&= -\frac{2}{3}L_{3}\cos(\varphi) -\frac{1}{3}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{3}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
& \quad -\frac{1}{3}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)\notag\\
&\quad +\frac{1}{6}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{6}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\quad -\frac{\sqrt{3}}{6}L_{2}\cos(2\theta)\sin(\varphi)-\frac{3}{6}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
& \quad +\frac{1}{3}L_{3}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\quad -\frac{1}{3}L_{2}\cos(\varphi)\cos (2\theta)-\frac{\sqrt{3}}{3}L_{2}\cos (2\theta)\sin(\varphi)\notag\\
&\notag\\
&= -\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
& \quad -\frac{1}{3}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\quad -\frac{\sqrt{3}}{2}L_{2}\cos(2\theta)\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
\end{align}
$$

($${1,3}$$)成分

$$
\begin{align}
&\frac{2}{3}\cos(\theta_{{\rm{a}}})\left(-L_{3} +L_{2}\cos\left(2\theta + \frac{2}{3}\pi \right)\right) \notag\\
&\qquad \qquad + \frac{2}{3}\cos(\theta_{{\rm{b}}})\left(-L_{3}+L_{2}\cos \left(2\theta \right)\right)\notag\\
&\qquad \qquad \qquad \qquad + \frac{2}{3}\cos(\theta_{{\rm{c}}})\left(L_{1}+L_{2}\cos \left(2\theta - \frac{2}{3}\pi \right)\right)\notag\\
&\notag\\
&=\frac{2}{3}\cos(\varphi)\left(-L_{3} +L_{2}\cos\left(2\theta + \frac{2}{3}\pi \right)\right) \notag\\
&\qquad \qquad + \frac{2}{3}\cos\left(\varphi - \frac{2}{3}\pi\right)\left(-L_{3}+L_{2}\cos \left(2\theta \right)\right)\notag\\
&\qquad \qquad \qquad \qquad + \frac{2}{3}\cos\left(\varphi + \frac{2}{3}\pi\right)\left(L_{1}+L_{2}\cos \left(2\theta - \frac{2}{3}\pi \right)\right)\notag\\
&\notag\\
&=\frac{2}{3}\cos(\varphi)\left(-L_{3} +L_{2}\left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right)\right) \notag\\
&\quad + \frac{2}{3}\left(-\frac{1}{2}\cos(\varphi)+\frac{\sqrt{3}}{2}\sin(\varphi)\right)\left(-L_{3}+L_{2}\cos \left(2\theta \right)\right)\notag\\
&\quad + \frac{2}{3}\left(-\frac{1}{2}\cos(\varphi)-\frac{\sqrt{3}}{2}\sin(\varphi)\right)\notag\\
&\qquad \qquad \qquad \qquad \qquad \left(L_{1}+L_{2}\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right)\right)\notag\\
&\notag\\
&=-\frac{2}{3}L_{3}\cos(\varphi) -\frac{1}{3}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{3}L_{2}\cos(\varphi)\sin(2\theta) \notag\\
&\quad +\frac{1}{3}L_{3}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\quad -\frac{1}{3}L_{2}\cos(\varphi)\cos (2\theta )+\frac{\sqrt{3}}{3}L_{2}\cos (2\theta )\sin(\varphi)\notag\\
&\quad -\frac{1}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)\notag\\
&\quad+\frac{1}{6}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{6}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{2}\cos(\varphi)\sin(2\theta)-\frac{3}{6}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\notag\\
&=-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta) \notag\\
&\quad -\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{2}L_{2}\cos (2\theta )\sin(\varphi)\notag\\
&\quad -\frac{1}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
\end{align}
$$

($${2,1}$$)成分

$$
\begin{align}
&-\frac{2}{3}\sin(\theta_{{\rm{a}}})\left(L_{1}+L_{2}\cos \left(2\theta\right)\right) \notag\\
&\qquad \qquad -\frac{2}{3}\sin(\theta_{{\rm{b}}})\left(-L_{3} +L_{2}\cos\left(2\theta - \frac{2}{3}\pi \right)\right)\notag\\
&\qquad \qquad \qquad \qquad -\frac{2}{3}\sin(\theta_{{\rm{c}}})\left(-L_{3} +L_{2}\cos\left(2\theta+ \frac{2}{3}\pi \right)\right)\notag\\
&\notag\\
&=-\frac{2}{3}\sin(\varphi)\left(L_{1}+L_{2}\cos(2\theta)\right) \notag\\
&\quad  -\frac{2}{3}\sin\left(\varphi - \frac{2}{3}\pi\right)\left(-L_{3} +L_{2}\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right)\right)\notag\\
&\quad -\frac{2}{3}\sin\left(\varphi +\frac{2}{3}\pi\right)\left(-L_{3} +L_{2}\left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right)\right)\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\sin(\varphi)-\frac{2}{3}L_{2}\cos(2\theta)\sin(\varphi) \notag\\
&\quad  -\frac{2}{3}\left(-\frac{1}{2}\sin(\varphi)-\frac{\sqrt{3}}{2}\cos(\varphi)\right)\notag\\
&\qquad \qquad \qquad \qquad \left(-L_{3} +L_{2}\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right)\right)\notag\\
&\quad -\frac{2}{3}\left(-\frac{1}{2}\sin(\varphi)+\frac{\sqrt{3}}{2}\cos(\varphi)\right)\notag\\
&\qquad \qquad \qquad \qquad \left(-L_{3} +L_{2}\left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right)\right)\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\sin(\varphi)-\frac{2}{3}L_{2}\cos(2\theta)\sin(\varphi) \notag\\
&\quad -\frac{1}{3}L_{3}\sin(\varphi)-\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\quad-\frac{1}{6}L_{2}\cos(2\theta)\sin(\varphi)-\frac{\sqrt{3}}{6}L_{2}\cos(\varphi)\cos(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{2}\sin(\varphi)\sin(2\theta)+\frac{3}{6}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\quad -\frac{1}{3}L_{3}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\quad-\frac{1}{6}L_{2}\cos(2\theta)\sin(\varphi)+\frac{\sqrt{3}}{6}L_{2}\cos(\varphi)\cos(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{2}\sin(\varphi)\sin(2\theta)+\frac{3}{6}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\sin(\varphi)-\frac{2}{3}L_{3}\sin(\varphi)-L_{2}\cos(2\theta)\sin(\varphi)+L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\sin(\varphi)-\frac{2}{3}L_{3}\sin(\varphi)+L_{2}\sin(2\theta-\varphi)\notag\\
\end{align}
$$

($${2,2}$$)成分

$$
\begin{align}
&-\frac{2}{3}\sin(\theta_{{\rm{a}}})\left(-L_{3} +L_{2}\cos\left(2\theta- \frac{2}{3}\pi \right)\right) \notag\\
&\qquad \qquad -\frac{2}{3}\sin(\theta_{{\rm{b}}})\left(L_{1}+L_{2}\cos \left(2\theta+ \frac{2}{3}\pi \right)\right)\notag\\
&\qquad \qquad \qquad \qquad -\frac{2}{3}\sin(\theta_{{\rm{c}}})\left(-L_{3}+L_{2}\cos \left(2\theta \right)\right)\notag\\
&\notag\\
&=-\frac{2}{3}\sin(\varphi)\left(-L_{3} +L_{2}\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right)\right) \notag\\
&\quad -\frac{2}{3}\sin\left(\varphi - \frac{2}{3}\pi\right)\left(L_{1}+L_{2}\left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right)\right)\notag\\
&\quad -\frac{2}{3}\sin\left(\varphi + \frac{2}{3}\pi\right)\left(-L_{3}+L_{2}\cos (2\theta)\right)\notag\\
&\notag\\
&=-\frac{2}{3}\sin(\varphi)\left(-L_{3} -\frac{1}{2}L_{2}\cos(2\theta)+\frac{\sqrt{3}}{2}L_{2}\sin(2\theta)\right) \notag\\
&\quad -\frac{2}{3}\left(-\frac{1}{2}\sin(\varphi)-\frac{\sqrt{3}}{2}\cos(\varphi)\right)\left(L_{1}-\frac{1}{2}L_{2}\cos(2\theta)-\frac{\sqrt{3}}{2}L_{2}\sin(2\theta)\right)\notag\\
&\quad -\frac{2}{3}\left(-\frac{1}{2}\sin(\varphi)+\frac{\sqrt{3}}{2}\cos(\varphi)\right)\left(-L_{3}+L_{2}\cos (2\theta)\right)\notag\\
&\notag\\
&=\frac{2}{3}L_{3}\sin(\varphi) +\frac{1}{3}L_{2}\cos(2\theta)\sin(\varphi)-\frac{\sqrt{3}}{3}L_{2}\sin(\varphi)\sin(2\theta) \notag\\
&\quad -\frac{2}{3}\left(-\frac{1}{2}L_{1}\sin(\varphi)-\frac{\sqrt{3}}{2}L_{1}\cos(\varphi)\right.\notag\\
&\left. \qquad \qquad \qquad +\frac{1}{4}L_{2}\cos(2\theta)\sin(\varphi)+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\right. \notag\\
&\qquad \qquad \qquad \qquad \left. +\frac{\sqrt{3}}{4}L_{2}\sin(\varphi)\sin(2\theta)+\frac{3}{4}L_{2}\cos(\varphi)\sin(2\theta)\right)\notag\\
&\quad -\frac{2}{3}\left(\frac{1}{2}L_{3}\sin(\varphi)-\frac{\sqrt{3}}{2}L_{3}\cos(\varphi)\right.\notag\\
&\left.\qquad \qquad \qquad \qquad  -\frac{1}{2}L_{2}\cos (2\theta)\sin(\varphi)+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos (2\theta)\right)\notag\\
&\notag\\
&=\frac{2}{3}L_{3}\sin(\varphi) +\frac{1}{3}L_{2}\cos(2\theta)\sin(\varphi)-\frac{\sqrt{3}}{3}L_{2}\sin(\varphi)\sin(2\theta) \notag\\
&\quad +\frac{1}{3}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)\notag\\
& \quad -\frac{1}{6}L_{2}\cos(2\theta)\sin(\varphi)-\frac{\sqrt{3}}{6}L_{2}\cos(\varphi)\cos(2\theta) \notag\\
&\quad -\frac{\sqrt{3}}{6}L_{2}\sin(\varphi)\sin(2\theta)-\frac{3}{6}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\quad -\frac{1}{3}L_{3}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\quad +\frac{1}{3}L_{2}\cos (2\theta)\sin(\varphi)-\frac{\sqrt{3}}{3}L_{2}\cos(\varphi)\cos (2\theta)\notag\\
&\notag\\
&=\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{3}L_{3}\cos(\varphi) \notag\\
&\quad -\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta) +\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\quad -\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta) -\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
\end{align}
$$

($${2,3}$$)成分

$$
\begin{align}
&-\frac{2}{3}\sin(\theta_{{\rm{a}}})\left(-L_{3} +L_{2}\cos\left(2\theta + \frac{2}{3}\pi \right)\right) \notag\\
&\qquad \qquad -\frac{2}{3}\sin(\theta_{{\rm{b}}})\left(-L_{3}+L_{2}\cos \left(2\theta \right)\right)\notag\\
&\qquad \qquad \qquad \qquad -\frac{2}{3}\sin(\theta_{{\rm{c}}})\left(L_{1}+L_{2}\cos \left(2\theta - \frac{2}{3}\pi \right)\right)\notag\\
&\notag\\
&=-\frac{2}{3}\sin(\varphi)\left(-L_{3} +L_{2}\left(-\frac{1}{2}\cos(2\theta )-\frac{\sqrt{3}}{2}\sin(2\theta )\right)\right) \notag\\
&\quad -\frac{2}{3}\sin\left(\varphi - \frac{2}{3}\pi\right)\left(-L_{3}+L_{2}\cos(2\theta)\right)\notag\\
&\quad -\frac{2}{3}\sin\left(\varphi + \frac{2}{3}\pi\right)\left(L_{1}+L_{2}\left(-\frac{1}{2}\cos(2\theta )+\frac{\sqrt{3}}{2}\sin(2\theta )\right)\right)\notag\\
&\notag\\
&=-\frac{2}{3}\sin(\varphi)\left(-L_{3} -\frac{1}{2}L_{2}\cos(2\theta )-\frac{\sqrt{3}}{2}L_{2}\sin(2\theta )\right) \notag\\
& \quad -\frac{2}{3}\left(-\frac{1}{2}\sin(\varphi)-\frac{\sqrt{3}}{2}\cos(\varphi)\right)\left(-L_{3}+L_{2}\cos(2\theta)\right)\notag\\
& \quad -\frac{2}{3}\left(-\frac{1}{2}\sin(\varphi)+\frac{\sqrt{3}}{2}\cos(\varphi)\right)\left(L_{1}-\frac{1}{2}L_{2}\cos(2\theta )+\frac{\sqrt{3}}{2}L_{2}\sin(2\theta )\right)\notag\\
&\notag\\
&=\frac{2}{3}L_{3}\sin(\varphi) +\frac{1}{3}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{3}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
& \quad -\frac{2}{3}\left(\frac{1}{2}L_{3}\sin(\varphi)+\frac{\sqrt{3}}{2}L_{3}\cos(\varphi)\right.\notag\\
&\qquad \qquad \qquad \qquad \left. -\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi)-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)\right)\notag\\
& \quad -\frac{2}{3}\left(-\frac{1}{2}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{2}L_{1}\cos(\varphi)\right.\notag\\
&\qquad \qquad \qquad \left. +\frac{1}{4}L_{2}\cos(2\theta )\sin(\varphi)-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta )\right.\notag\\
&\qquad \qquad \qquad \qquad \left. -\frac{\sqrt{3}}{4}L_{2}\sin(\varphi)\sin(2\theta )+\frac{3}{4}L_{2}\cos(\varphi)\sin(2\theta )\right)\notag\\
&\notag\\
&=\frac{2}{3}L_{3}\sin(\varphi) +\frac{1}{3}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{3}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
& \quad -\frac{1}{3}L_{3}\sin(\varphi)-\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\quad +\frac{1}{3}L_{2}\cos(2\theta)\sin(\varphi)+\frac{\sqrt{3}}{3}L_{2}\cos(\varphi)\cos(2\theta)\notag\\
& \quad +\frac{1}{3}L_{1}\sin(\varphi)-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)\notag\\
&\quad -\frac{1}{6}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{6}L_{2}\cos(\varphi)\cos(2\theta )\notag\\
&\quad +\frac{\sqrt{3}}{6}L_{2}\sin(\varphi)\sin(2\theta )-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )\notag\\
&\notag\\
&=-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) -\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\quad +\frac{1}{2}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
&\quad+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )\notag\\
\end{align}
$$

($${3,1}$$)成分

$$
\begin{align}
&\frac{1}{3}\left(L_{1}+L_{2}\cos \left(2\theta\right)\right) \notag\\
&\qquad \qquad +\frac{1}{3}\left(-L_{3} +L_{2}\cos\left(2\theta - \frac{2}{3}\pi \right)\right)\notag\\
&\qquad \qquad \qquad \qquad + \frac{1}{3}\left(-L_{3} +L_{2}\cos\left(2\theta+ \frac{2}{3}\pi \right)\right)\notag\\
&\notag\\
&=\frac{1}{3}L_{1}+\frac{1}{3}L_{2}\cos (2\theta) \notag\\
&\qquad \qquad -\frac{1}{3}L_{3} +\frac{1}{3}L_{2}\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right)\notag\\
&\qquad \qquad \qquad \qquad -\frac{1}{3}L_{3} +\frac{1}{3}L_{2}\left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right)\notag\\
&\notag\\
&=\frac{1}{3}L_{1}+\frac{1}{3}L_{2}\cos (2\theta) \notag\\
&\quad -\frac{1}{3}L_{3}-\frac{1}{6}L_{2}\cos(2\theta)+\frac{\sqrt{3}}{6}L_{2}\sin(2\theta)\notag\\
&\quad -\frac{1}{3}L_{3} -\frac{1}{6}L_{2}\cos(2\theta)-\frac{\sqrt{3}}{6}L_{2}\sin(2\theta)\notag\\
&\notag\\
&=\frac{1}{3}L_{1}-\frac{2}{3}L_{3}\notag\\
\end{align}
$$

($${3,2}$$)成分

$$
\begin{align}
&\frac{1}{3}\left(-L_{3} +L_{2}\cos\left(2\theta- \frac{2}{3}\pi \right)\right) \notag\\
&\qquad \qquad + \frac{1}{3}\left(L_{1}+L_{2}\cos \left(2\theta+ \frac{2}{3}\pi \right)\right)\notag\\
&\qquad \qquad \qquad \qquad +\frac{1}{3}\left(-L_{3}+L_{2}\cos \left(2\theta \right)\right)\notag\\
&\notag\\
&=-\frac{1}{3}L_{3} +\frac{1}{3}L_{2}\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right) \notag\\
&\quad +\frac{1}{3}L_{1}+ \frac{1}{3}L_{2}\left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right)\notag\\
&\quad -\frac{1}{3}L_{3}+\frac{1}{3}L_{2}\cos(2\theta )\notag\\
&\notag\\
&=-\frac{1}{3}L_{3} -\frac{1}{6}L_{2}\cos(2\theta)+\frac{\sqrt{3}}{6}L_{2}\sin(2\theta) \notag\\
&\quad +\frac{1}{3}L_{1}-\frac{1}{6}L_{2}\cos(2\theta)-\frac{\sqrt{3}}{6}L_{2}\sin(2\theta)\notag\\
&\quad -\frac{1}{3}L_{3}+\frac{1}{3}L_{2}\cos(2\theta )\notag\\
&\notag\\
&=\frac{1}{3}L_{1} -\frac{2}{3}L_{3}\notag\\
\end{align}
$$

($${3,3}$$)成分

$$
\begin{align}
&\frac{1}{3}\left(-L_{3} +L_{2}\cos\left(2\theta + \frac{2}{3}\pi \right)\right) \notag\\
&\qquad \qquad +\frac{1}{3}\left(-L_{3}+L_{2}\cos \left(2\theta \right)\right)\notag\\
&\qquad \qquad \qquad \qquad +\frac{1}{3}\left(L_{1}+L_{2}\cos \left(2\theta - \frac{2}{3}\pi \right)\right)\notag\\
&\notag\\
&=-\frac{1}{3}L_{3} +\frac{1}{3}L_{2}\left(-\frac{1}{2}\cos(2\theta)-\frac{\sqrt{3}}{2}\sin(2\theta)\right) \notag\\
&\qquad \qquad -\frac{1}{3}L_{3}+\frac{1}{3}L_{2}\cos (2\theta )\notag\\
&\qquad \qquad \qquad \qquad +\frac{1}{3}L_{1}+\frac{1}{3}L_{2}\left(-\frac{1}{2}\cos(2\theta)+\frac{\sqrt{3}}{2}\sin(2\theta)\right) \notag\\
&\notag\\
&=-\frac{1}{3}L_{3} -\frac{1}{6}L_{2}\cos(2\theta)-\frac{\sqrt{3}}{6}L_{2}\sin(2\theta) \notag\\
&\quad -\frac{1}{3}L_{3}+\frac{1}{3}L_{2}\cos (2\theta )\notag\\
&\quad +\frac{1}{3}L_{1}-\frac{1}{6}L_{2}\cos(2\theta)+\frac{\sqrt{3}}{6}L_{2}\sin(2\theta) \notag\\
&\notag\\
&=\frac{1}{3}L_{1}-\frac{2}{3}L_{3} \notag\\
\end{align}
$$

$${AL_{\rm{abc}}}$$をまとめると、

$$
\begin{align}
AL_{\rm{abc}}&=\left (
\begin{matrix}
a_{11}&  a_{12}&a_{13} \\
& & & \\
a_{21}&  a_{22}&a_{23}\\
& & &\\
a_{31}&a_{32}&a_{33}\\
\end{matrix} \right )\notag\\
\end{align}
$$

$$
\begin{align}
&\notag\\
a_{11}&=\frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{3}\cos(\varphi) +L_{2}\cos(2\theta - \varphi)\notag\\
&\notag\\
a_{12} &= -\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
& \qquad -\frac{1}{3}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\qquad -\frac{\sqrt{3}}{2}L_{2}\cos(2\theta)\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\notag\\
a_{13}&=-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta) \notag\\
&\qquad -\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{2}L_{2}\cos (2\theta )\sin(\varphi)\notag\\
&\qquad -\frac{1}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\notag\\
a_{21}&=-\frac{2}{3}L_{1}\sin(\varphi)-\frac{2}{3}L_{3}\sin(\varphi)+L_{2}\sin(2\theta-\varphi)\notag\\
&\notag\\
a_{22} &=\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{3}L_{3}\cos(\varphi) \notag\\
&\qquad-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta) +\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\qquad -\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta) -\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\notag\\
a_{23}&=-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) -\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\qquad +\frac{1}{2}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
&\qquad+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )\notag\\
&\notag\\
a_{31} &= \frac{1}{3}L_{1}-\frac{2}{3}L_{3}\notag\\
&\notag\\
a_{32} &= \frac{1}{3}L_{1} -\frac{2}{3}L_{3}\notag\\
&\notag\\
a_{33} &=\frac{1}{3}L_{1}-\frac{2}{3}L_{3} \notag\\
\end{align}
$$

行列計算その2

変換行列の逆行列$${A^{-1}}$$は、関連記事のd-q-0法の逆行列　同期機より

$$
\begin{align}
A^{-1}&=\left (
\begin{matrix}
\cos(\theta_{{\rm{a}}})&  -\sin(\theta_{{\rm{a}}})&1  \\
& & & \\
\cos(\theta_{{\rm{b}}})&  -\sin(\theta_{{\rm{b}}}) &1\\
& & &\\
 \cos(\theta_{{\rm{c}}})&-\sin(\theta_{{\rm{c}}})&1\\
\end{matrix} \right )\notag\\
\end{align}
$$

である。

$${AL_{\rm{abc}}A^{-1}}$$を計算する。

($${1,1}$$)成分

$$
\begin{align}
&a_{11}\cos(\theta_{{\rm{a}}}) + a_{12}\cos(\theta_{{\rm{b}}})+a_{13} \cos(\theta_{{\rm{c}}})\notag\\
&\notag\\
&=\left(\frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{3}\cos(\varphi) +L_{2}\cos(2\theta - \varphi)\right)\cos(\varphi)\notag\\
&+\left(-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)\right.\notag\\
& \qquad -\frac{1}{3}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\qquad \left. -\frac{\sqrt{3}}{2}L_{2}\cos(2\theta)\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\right)\cos\left(\varphi - \frac{2}{3}\pi\right)\notag\\
&+\left(-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta) \right.\notag\\
&\qquad -\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{2}L_{2}\cos (2\theta )\sin(\varphi)\notag\\
&\qquad \left.-\frac{1}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\right)\cos\left(\varphi + \frac{2}{3}\pi\right)\notag\\
&\notag\\
&=\left(\frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{3}\cos(\varphi) +L_{2}\cos(2\theta - \varphi)\right)\cos(\varphi)\notag\\
&\qquad+\left(-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)\right.\notag\\
& \qquad -\frac{1}{3}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)-\frac{\sqrt{3}}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\quad \left. -\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\right)\left(-\frac{1}{2}\cos(\varphi)+\frac{\sqrt{3}}{2}\sin(\varphi)\right)\notag\\
&\qquad+\left(-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta) \right.\notag\\
&\qquad -\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{2}L_{2}\cos (2\theta )\sin(\varphi)-\frac{1}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)\notag\\
&\quad \left.-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\right)\left(-\frac{1}{2}\cos(\varphi)-\frac{\sqrt{3}}{2}\sin(\varphi)\right)\notag\\
&\notag\\
&=\frac{2}{3}L_{1}\left(\cos(\varphi)\right)^{2}+\frac{2}{3}L_{3}\left(\cos(\varphi)\right)^{2} +L_{2}\cos(\varphi)\cos(2\theta - \varphi)\notag\\
&\quad+\frac{1}{6}L_{3}\left(\cos(\varphi)\right)^{2} +\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)-\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
& \quad+\frac{1}{6}L_{1}\left(\cos(\varphi)\right)^{2}-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)-\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad+\frac{1}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)-\frac{\sqrt{3}}{6}L_{3}\cos(\varphi) \sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad+\frac{3}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta) -\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{1}{2}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{1}{2}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad-\frac{3}{4}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2} -\frac{\sqrt{3}}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{1}{6}L_{3}\left(\cos(\varphi)\right)^{2} +\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)+\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos (2\theta )\sin(\varphi)+\frac{1}{6}L_{1}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi) +\frac{1}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi) +\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad+\frac{3}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta) \notag\\
&\quad+\frac{1}{2}L_{3}\left(\sin(\varphi)\right)^{2}-\frac{3}{4}L_{2}\cos (2\theta )\left(\sin(\varphi)\right)^{2}+\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{1}{2}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{\sqrt{3}}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\notag\\
&=\frac{2}{3}L_{1}\left(\cos(\varphi)\right)^{2}+\frac{1}{6}L_{1}\left(\cos(\varphi)\right)^{2}+\frac{1}{6}L_{1}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)+\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{1}{2}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi) +\frac{1}{2}L_{1}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+L_{2}\cos(\varphi)\cos(2\theta - \varphi)+\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)\notag\\
&\quad+\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)-\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos (2\theta )\sin(\varphi)+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad+\frac{1}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)+\frac{3}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad+\frac{1}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)+\frac{3}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)+\frac{\sqrt{3}}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad-\frac{3}{4}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}-\frac{3}{4}L_{2}\cos (2\theta )\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+\frac{2}{3}L_{3}\left(\cos(\varphi)\right)^{2}+\frac{1}{6}L_{3}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad+\frac{1}{6}L_{3}\left(\cos(\varphi)\right)^{2}-\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi) +\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{3}\cos(\varphi) \sin(\varphi)+\frac{1}{2}L_{3}\left(\sin(\varphi)\right)^{2}+\frac{1}{2}L_{3}\left(\sin(\varphi)\right)^{2} \notag\\
&\notag\\
&=L_{1}\left(\cos(\varphi)\right)^{2}+L_{1}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+L_{2}\cos(\varphi)\left(\cos(2\theta)\cos(\varphi)+\sin(2\theta)\sin(\varphi)\right)\notag\\
&\quad+\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)\notag\\
&\quad+2L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)-\frac{3}{2}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+L_{3}\left(\cos(\varphi)\right)^{2}+L_{3}\left(\sin(\varphi)\right)^{2} \notag\\
&\notag\\
&=L_{1}+L_{3}+L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad+\frac{1}{2}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+2L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{3}{2}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}\notag\\
&\notag\\
&=L_{1}+L_{3}+\frac{3}{2}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)\notag\\
&\qquad \qquad \qquad+3L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)-\frac{3}{2}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}\notag\\
&\notag\\
&=L_{1}+L_{3}+\frac{3}{2}L_{2}\cos(2\theta)\left(\left(\cos(\varphi)\right)^{2}-\left(\sin(\varphi)\right)^{2}\right)\notag\\
&\qquad \qquad \qquad +\frac{3}{2}L_{2}\sin(2\varphi)\sin(2\theta)\notag\\
&\notag\\
&=L_{1}+L_{3}+\frac{3}{2}L_{2}\cos(2\theta)\left(\left(\cos(\varphi)\right)^{2}-\left(1-\left(\cos(\varphi)\right)^{2}\right)\right)\notag\\
&\qquad \qquad \qquad +\frac{3}{2}L_{2}\sin(2\varphi)\sin(2\theta)\notag\\
&\notag\\
&=L_{1}+L_{3}+\frac{3}{2}L_{2}\cos(2\theta)\left(2\left(\cos(\varphi)\right)^{2}-1\right)+\frac{3}{2}L_{2}\sin(2\varphi)\sin(2\theta)\notag\\
&\notag\\
&=L_{1}+L_{3}+\frac{3}{2}L_{2}\cos(2\theta)\cos(2\varphi)+\frac{3}{2}L_{2}\sin(2\theta)\sin(2\varphi)\notag\\
&\notag\\
&=L_{1}+L_{3}+\frac{3}{2}L_{2}\cos(2\theta-2\varphi)\notag\\
\end{align}
$$

($${1,2}$$)成分

$$
\begin{align}
&-a_{11}\sin(\theta_{{\rm{a}}}) - a_{12}\sin(\theta_{{\rm{b}}})-a_{13}\sin(\theta_{{\rm{c}}})\notag\\
&\notag\\
&=-\left(\frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{3}\cos(\varphi) +L_{2}\cos(2\theta - \varphi)\right)\sin(\varphi)\notag\\
&-\left(-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)\right.\notag\\
& \qquad -\frac{1}{3}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\qquad \left. -\frac{\sqrt{3}}{2}L_{2}\cos(2\theta)\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\right)\sin\left(\varphi - \frac{2}{3}\pi\right)\notag\\
&-\left(-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta) \right.\notag\\
&\qquad -\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{2}L_{2}\cos (2\theta )\sin(\varphi)\notag\\
&\qquad \left.-\frac{1}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\right)\sin\left(\varphi + \frac{2}{3}\pi\right)\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\cos(\varphi)\sin(\varphi)-\frac{2}{3}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad-L_{2}\left(\cos(2\theta)\cos(\varphi)+\sin(2\theta)\sin(\varphi)\right)\sin(\varphi)\notag\\
&\quad+\left(\frac{1}{3}L_{3}\cos(\varphi) +\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)\right.\notag\\
& \quad+\frac{1}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)-\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\left. +\frac{\sqrt{3}}{2}L_{2}\cos(2\theta)\sin(\varphi)+\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\right)\left(-\frac{1}{2}\sin(\varphi)-\frac{\sqrt{3}}{2}\cos(\varphi)\right)\notag\\
&\quad+\left(\frac{1}{3}L_{3}\cos(\varphi) +\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta) \right.\notag\\
&\quad+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) -\frac{\sqrt{3}}{2}L_{2}\cos (2\theta )\sin(\varphi)+\frac{1}{3}L_{1}\cos(\varphi)\notag\\
&\left.+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)+\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\right)\left(-\frac{1}{2}\sin(\varphi)+\frac{\sqrt{3}}{2}\cos(\varphi)\right)\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\cos(\varphi)\sin(\varphi)-\frac{2}{3}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad-L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)-L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad-\frac{1}{6}L_{3}\cos(\varphi)\sin(\varphi) -\frac{1}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
& \quad-\frac{1}{6}L_{1}\cos(\varphi)\sin(\varphi)+\frac{\sqrt{3}}{6}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{\sqrt{3}}{6}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
& \quad-\frac{\sqrt{3}}{4}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}-\frac{1}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{3}\left(\cos(\varphi)\right)^{2} -\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+\frac{3}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad -\frac{\sqrt{3}}{6}L_{1}\left(\cos(\varphi)\right)^{2}+\frac{1}{2}L_{1}\cos(\varphi)\sin(\varphi)+\frac{1}{2}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad -\frac{3}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{1}{6}L_{3}\cos(\varphi)\sin(\varphi) -\frac{1}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta) \notag\\
&\quad-\frac{\sqrt{3}}{4}L_{3}\left(\sin(\varphi)\right)^{2}+\frac{\sqrt{3}}{4}L_{2}\cos (2\theta )\left(\sin(\varphi)\right)^{2}-\frac{1}{6}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{1}\left(\sin(\varphi)\right)^{2}-\frac{1}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{3}\left(\cos(\varphi)\right)^{2} +\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+\frac{3}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta) \notag\\
&\quad+\frac{1}{2}L_{3}\cos(\varphi)\sin(\varphi) -\frac{3}{4}L_{2}\cos(\varphi)\cos (2\theta )\sin(\varphi)+\frac{\sqrt{3}}{6}L_{1}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad+\frac{1}{2}L_{1}\cos(\varphi)\sin(\varphi)+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\notag\\
&=-\frac{\sqrt{3}}{6}L_{1}\left(\cos(\varphi)\right)^{2}+\frac{\sqrt{3}}{6}L_{1}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad-\frac{2}{3}L_{1}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{1}{2}L_{1}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{1}{2}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1}\left(\sin(\varphi)\right)^{2}-\frac{\sqrt{3}}{6}L_{1}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+\frac{3}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+\frac{3}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta) \notag\\
&\quad-L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi) -\frac{1}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad-\frac{3}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)-\frac{1}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad-\frac{3}{4}L_{2}\cos(\varphi)\cos (2\theta )\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2} +\frac{\sqrt{3}}{4}L_{2}\cos (2\theta )\left(\sin(\varphi)\right)^{2}\notag\\
&\quad-\frac{1}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)-L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad-\frac{1}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{3}\left(\cos(\varphi)\right)^{2} +\frac{\sqrt{3}}{6}L_{3}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad-\frac{2}{3}L_{3}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad+\frac{1}{2}L_{3}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{1}{2}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{3}\left(\sin(\varphi)\right)^{2}-\frac{\sqrt{3}}{4}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)-3L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\qquad \qquad \qquad -\frac{3}{2}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\left(\left(\cos(\varphi)\right)^{2}-\left(\sin(\varphi)\right)^{2}\right)\sin(2\theta)\notag\\
&\qquad \qquad \qquad -\frac{3}{2}L_{2}\cos(2\theta)\sin(2\varphi)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\left(\left(\cos(\varphi)\right)^{2}-\left(1-\left(\cos(\varphi)\right)^{2}\right)\right)\sin(2\theta)\notag\\
&\qquad \qquad \qquad -\frac{3}{2}L_{2}\cos(2\theta)\sin(2\varphi)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\left(\left(2\cos(\varphi)\right)^{2}-1\right)\sin(2\theta) -\frac{3}{2}L_{2}\cos(2\theta)\sin(2\varphi)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\cos(2\varphi)\sin(2\theta) -\frac{3}{2}L_{2}\cos(2\theta)\sin(2\varphi)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\sin(2\theta-2\varphi)\notag\\
\end{align}
$$

($${1,3}$$)成分

$$
\begin{align}
&a_{11}+ a_{12}+a_{13}\notag\\
&\notag\\
&=\frac{2}{3}L_{1}\cos(\varphi)+\frac{2}{3}L_{3}\cos(\varphi) +L_{2}\cos(2\theta - \varphi)\notag\\
&\quad-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
& \quad -\frac{1}{3}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\quad  -\frac{\sqrt{3}}{2}L_{2}\cos(2\theta)\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{1}{3}L_{3}\cos(\varphi) -\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta) \notag\\
&\quad -\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{2}L_{2}\cos (2\theta )\sin(\varphi)\notag\\
&\quad -\frac{1}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\notag\\
&=\frac{2}{3}L_{1}\cos(\varphi)-\frac{1}{3}L_{1}\cos(\varphi)-\frac{1}{3}L_{1}\cos(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)-\frac{\sqrt{3}}{3}L_{1}\sin(\varphi)\notag\\
&\quad+L_{2}\cos(2\theta - \varphi)-\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)\notag\\
&\quad-\frac{1}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{\sqrt{3}}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{2}L_{2}\cos (2\theta )\sin(\varphi)+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\sin(2\theta)-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{1}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\quad+\frac{2}{3}L_{3}\cos(\varphi) -\frac{1}{3}L_{3}\cos(\varphi)-\frac{1}{3}L_{3}\cos(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)-\frac{\sqrt{3}}{3}L_{3}\sin(\varphi) \notag\\
&\notag\\
&=L_{2}\left(\cos(2\theta)\cos(\varphi)+\sin(2\theta)\sin(\varphi)\right)\notag\\
&\qquad \qquad \qquad -L_{2}\cos(\varphi)\cos(2\theta)-L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\notag\\
&=L_{2}\cos(2\theta)\cos(\varphi)+L_{2}\sin(2\theta)\sin(\varphi)\notag\\
&\qquad \qquad \qquad -L_{2}\cos(\varphi)\cos(2\theta)-L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\notag\\
&=0\notag\\
\end{align}
$$

($${2,1}$$)成分

$$
\begin{align}
&a_{21}\cos(\theta_{{\rm{a}}}) + a_{22}\cos(\theta_{{\rm{b}}})+a_{23} \cos(\theta_{{\rm{c}}})\notag\\
&\notag\\
&=\left(-\frac{2}{3}L_{1}\sin(\varphi)-\frac{2}{3}L_{3}\sin(\varphi)+L_{2}\sin(2\theta-\varphi)\right)\cos(\varphi)\notag\\
&\quad+\left(\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{3}L_{3}\cos(\varphi) \right.\notag\\
&\qquad-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta) +\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\qquad \left.-\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta) -\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta)\right)\cos\left(\varphi - \frac{2}{3}\pi\right)\notag\\
&+\left(-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) -\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\right.\notag\\
&\qquad +\frac{1}{2}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
&\qquad \left.+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )\right)\cos\left(\varphi + \frac{2}{3}\pi\right)\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\cos(\varphi)\sin(\varphi)-\frac{2}{3}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&+L_{2}\cos(\varphi)\left(\sin(2\theta)\cos(\varphi)-\cos(2\theta)\sin(\varphi)\right)\notag\\
&+\left(\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{3}L_{3}\cos(\varphi) \right.\notag\\
&\qquad-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta) +\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
& \left.-\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta) -\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta)\right)\left(-\frac{1}{2}\cos(\varphi)+\frac{\sqrt{3}}{2}\sin(\varphi)\right)\notag\\
&+\left(-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) -\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\right.\notag\\
&\qquad +\frac{1}{2}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
&\left.+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )\right)\left(-\frac{1}{2}\cos(\varphi)-\frac{\sqrt{3}}{2}\sin(\varphi)\right)\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\cos(\varphi)\sin(\varphi)-\frac{2}{3}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)-L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{1}\left(\cos(\varphi)\right)^{2}-\frac{1}{6}L_{1}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{3}\left(\cos(\varphi)\right)^{2} \notag\\
& \quad+\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta) -\frac{1}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad +\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta) +\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{1}{2}L_{1}\cos(\varphi)\sin(\varphi)+\frac{\sqrt{3}}{6}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{\sqrt{3}}{6}L_{3}\left(\sin(\varphi)\right)^{2} \notag\\
&\quad+\frac{1}{2}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad-\frac{3}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi) +\frac{\sqrt{3}}{4}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}\notag\\
&\quad -\frac{3}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta) -\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1}\left(\cos(\varphi)\right)^{2}-\frac{1}{6}L_{1}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad+\frac{\sqrt{3}}{6}L_{3}\left(\cos(\varphi)\right)^{2}\notag\\
& \quad-\frac{1}{4}L_{2}\cos(\varphi)\cos(2\theta )\sin(\varphi)-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta )\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{1}{2}L_{1}\cos(\varphi)\sin(\varphi)-\frac{\sqrt{3}}{6}L_{1}\left(\sin(\varphi)\right)^{2}-\frac{\sqrt{3}}{6}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+\frac{1}{2}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad -\frac{\sqrt{3}}{4}L_{2}\cos(2\theta )\left(\sin(\varphi)\right)^{2}-\frac{3}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta )\notag\\
&\quad-\frac{3}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta )\notag\\
&\notag\\
&=-\frac{\sqrt{3}}{6}L_{1}\left(\cos(\varphi)\right)^{2}+\frac{\sqrt{3}}{6}L_{1}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad-\frac{2}{3}L_{1}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{1}\cos(\varphi)\sin(\varphi)+\frac{1}{2}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad-\frac{1}{6}L_{1}\cos(\varphi)\sin(\varphi)+\frac{1}{2}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1}\left(\sin(\varphi)\right)^{2}-\frac{\sqrt{3}}{6}L_{1}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)-\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)\notag\\
&\quad+L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta) +\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{1}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)-L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi) \notag\\
&\quad-\frac{1}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)-\frac{3}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad-\frac{1}{4}L_{2}\cos(\varphi)\cos(2\theta )\sin(\varphi)-\frac{3}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2} -\frac{\sqrt{3}}{4}L_{2}\cos(2\theta )\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta) -\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta )+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta )\notag\\
&\quad-\frac{3}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)-\frac{3}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta )\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{3}\left(\cos(\varphi)\right)^{2}+\frac{\sqrt{3}}{6}L_{3}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad-\frac{2}{3}L_{3}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{1}{2}L_{3}\cos(\varphi)\sin(\varphi)-\frac{1}{6}L_{3}\cos(\varphi)\sin(\varphi)+\frac{1}{2}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{3}\left(\sin(\varphi)\right)^{2}-\frac{\sqrt{3}}{6}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)-3L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\qquad \qquad \qquad \qquad -\frac{3}{2}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\sin(2\theta)\left(\left(\cos(\varphi)\right)^{2}-\left(\sin(\varphi)\right)^{2}\right)\notag\\
&\qquad \qquad \qquad \qquad -3L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\sin(2\theta)\left(\left(\cos(\varphi)\right)^{2}-\left(1-\left(\cos(\varphi)\right)^{2}\right)\right)\notag\\
&\qquad \qquad \qquad \qquad -3L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\sin(2\theta)\left(2\left(\cos(\varphi)\right)^{2}-1\right)\notag\\
&\qquad \qquad \qquad \qquad -3L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\sin(2\theta)\cos(2\varphi)-\frac{3}{2}L_{2}\cos(2\theta)\sin(2\varphi)\notag\\
&\notag\\
&=\frac{3}{2}L_{2}\sin(2\theta-2\varphi)\notag\\
\end{align}
$$

($${2,2}$$)成分

$$
\begin{align}
&-a_{21}\sin(\theta_{{\rm{a}}}) - a_{22}\sin(\theta_{{\rm{b}}})-a_{23}\sin(\theta_{{\rm{c}}})\notag\\
&\notag\\
&=-\left(-\frac{2}{3}L_{1}\sin(\varphi)-\frac{2}{3}L_{3}\sin(\varphi)+L_{2}\sin(2\theta-\varphi)\right)\sin(\varphi)\notag\\
&-\left(\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\right. \notag\\
&\qquad-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta) +\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\qquad \left.-\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta) -\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta)\right)\sin\left(\varphi - \frac{2}{3}\pi\right)\notag\\
&-\left(-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) -\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\right.\notag\\
&\qquad +\frac{1}{2}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
&\qquad \left.+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )\right)\sin\left(\varphi + \frac{2}{3}\pi\right)\notag\\
&\notag\\
&=\frac{2}{3}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{2}{3}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&-L_{2}\sin(\varphi)\left(\sin(2\theta)\cos(\varphi)-\cos(2\theta)\sin(\varphi)\right)\notag\\
&+\left(-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)-\frac{1}{3}L_{1}\sin(\varphi)-\frac{1}{3}L_{3}\sin(\varphi) -\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\right. \notag\\
&\qquad+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta) -\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
& \left.+\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta) +\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta)\right)\left(-\frac{1}{2}\sin(\varphi)-\frac{\sqrt{3}}{2}\cos(\varphi)\right)\notag\\
&+\left(+\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)-\frac{1}{3}L_{1}\sin(\varphi)-\frac{1}{3}L_{3}\sin(\varphi) +\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\right.\notag\\
&\qquad -\frac{1}{2}L_{2}\cos(2\theta )\sin(\varphi)-\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
& \left.-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)+\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )\right)\left(-\frac{1}{2}\sin(\varphi)+\frac{\sqrt{3}}{2}\cos(\varphi)\right)\notag\\
&\notag\\
&=\frac{2}{3}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{2}{3}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad-L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)+L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)+\frac{1}{6}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{1}{6}L_{3}\left(\sin(\varphi)\right)^{2} \notag\\
&\quad+\frac{\sqrt{3}}{6}L_{3}\cos(\varphi) \sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi) +\frac{1}{4}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}\notag\\
&\quad -\frac{\sqrt{3}}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta) -\frac{1}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad+\frac{1}{2}L_{1}\left(\cos(\varphi)\right)^{2}+\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)+\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad+\frac{1}{2}L_{3}\left(\cos(\varphi)\right)^{2}\notag\\
& \quad-\frac{3}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta) +\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
& \quad-\frac{3}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta) -\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)+\frac{1}{6}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{1}{6}L_{3}\left(\sin(\varphi)\right)^{2} \notag\\
&\quad-\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
& \quad+\frac{1}{4}L_{2}\cos(2\theta )\left(\sin(\varphi)\right)^{2}+\frac{\sqrt{3}}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta )\notag\\
& \quad+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)-\frac{1}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta )\notag\\
&\quad+\frac{1}{2}L_{1}\left(\cos(\varphi)\right)^{2}-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)-\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad+\frac{1}{2}L_{3}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta )\sin(\varphi)-\frac{3}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta )\notag\\
&\quad-\frac{3}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)+\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\notag\\
&=\frac{1}{2}L_{1}\left(\cos(\varphi)\right)^{2}+\frac{1}{2}L_{1}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)+\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{2}{3}L_{1}\left(\sin(\varphi)\right)^{2}+\frac{1}{6}L_{1}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+\frac{1}{6}L_{1}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad-\frac{3}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta)-\frac{3}{4}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta) \notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)+\frac{\sqrt{3}}{4}L_{2}\left(\cos(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi) +\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta)\sin(\varphi)-\frac{\sqrt{3}}{4}L_{2}\cos(\varphi)\cos(2\theta )\sin(\varphi)\notag\\
&\quad+L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}+\frac{1}{4}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+\frac{1}{4}L_{2}\cos(2\theta )\left(\sin(\varphi)\right)^{2}-L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta) \notag\\
&\quad-\frac{1}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)-\frac{3}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{1}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta )-\frac{3}{4}L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta )\notag\\
&\quad-\frac{\sqrt{3}}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)+\frac{\sqrt{3}}{4}L_{2}\left(\sin(\varphi)\right)^{2}\sin(2\theta)\notag\\
&\quad+\frac{1}{2}L_{3}\left(\cos(\varphi)\right)^{2}+\frac{1}{2}L_{3}\left(\cos(\varphi)\right)^{2}\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{3}\cos(\varphi) \sin(\varphi)+\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi) \notag\\
&\quad-\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi)-\frac{\sqrt{3}}{6}L_{3}\cos(\varphi)\sin(\varphi)\notag\\
&\quad+\frac{2}{3}L_{3}\left(\sin(\varphi)\right)^{2}+\frac{1}{6}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&\quad+\frac{1}{6}L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&\notag\\
&=L_{1}\left(\cos(\varphi)\right)^{2}+L_{1}\left(\sin(\varphi)\right)^{2}-\frac{3}{2}L_{2}\left(\cos(\varphi)\right)^{2}\cos(2\theta) \notag\\
&\quad+\frac{3}{2}L_{2}\cos(2\theta)\left(\sin(\varphi)\right)^{2}-3L_{2}\cos(\varphi)\sin(\varphi)\sin(2\theta) \notag\\
&\quad+L_{3}\left(\cos(\varphi)\right)^{2}+L_{3}\left(\sin(\varphi)\right)^{2}\notag\\
&\notag\\
&=L_{1}\left(\left(\cos(\varphi)\right)^{2}+\left(\sin(\varphi)\right)^{2}\right)-\frac{3}{2}L_{2}\cos(2\theta)\left(\left(\cos(\varphi)\right)^{2}-\left(\sin(\varphi)\right)^{2}\right) \notag\\
&\qquad \qquad -\frac{3}{2}L_{2}\sin(2\varphi)\sin(2\theta) +L_{3}\left(\left(\cos(\varphi)\right)^{2}+\left(\sin(\varphi)\right)^{2}\right)\notag\\
&\notag\\
&=L_{1}-\frac{3}{2}L_{2}\cos(2\theta)\left(\left(\cos(\varphi)\right)^{2}-\left(1-\left(\cos(\varphi)\right)^{2}\right)\right) \notag\\
&\qquad \qquad -\frac{3}{2}L_{2}\sin(2\varphi)\sin(2\theta) +L_{3}\notag\\
&\notag\\
&=L_{1}-\frac{3}{2}L_{2}\cos(2\theta)\left(2\left(\cos(\varphi)\right)^{2}-1\right) -\frac{3}{2}L_{2}\sin(2\varphi)\sin(2\theta) +L_{3}\notag\\
&\notag\\
&=L_{1}-\frac{3}{2}L_{2}\cos(2\theta)\cos(2\varphi) -\frac{3}{2}L_{2}\sin(2\varphi)\sin(2\theta) +L_{3}\notag\\
&\notag\\
&=L_{1}-\frac{3}{2}L_{2}\left(\cos(2\theta)\cos(2\varphi) +\sin(2\varphi)\sin(2\theta)\right) +L_{3}\notag\\
&\notag\\
&=L_{1}-\frac{3}{2}L_{2}\cos(2\theta-2\varphi)+L_{3}\notag\\
\end{align}
$$

($${2,3}$$)成分

$$
\begin{align}
&a_{21}+ a_{22}+a_{23}\notag\\
&\notag\\
&=-\frac{2}{3}L_{1}\sin(\varphi)-\frac{2}{3}L_{3}\sin(\varphi)+L_{2}\sin(2\theta-\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) \notag\\
&\quad+\frac{\sqrt{3}}{3}L_{3}\cos(\varphi) -\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta) \notag\\
&\quad+\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\quad -\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta) -\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\quad-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi) -\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\quad +\frac{1}{2}L_{2}\cos(2\theta )\sin(\varphi)+\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
&\quad+\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )\notag\\
&\notag\\
&=\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{1}\cos(\varphi)\notag\\
&\quad-\frac{2}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{3}L_{1}\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta) +\frac{\sqrt{3}}{2}L_{2}\cos(\varphi)\cos(2\theta)\notag\\
&\quad+\frac{1}{2}L_{2}\cos(2\theta)\sin(\varphi) +\frac{1}{2}L_{2}\cos(2\theta )\sin(\varphi)\notag\\
&\quad-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta )-\frac{1}{2}L_{2}\cos(\varphi)\sin(2\theta)\notag\\
&\quad+L_{2}\sin(2\theta-\varphi)-\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta)\notag\\
&\quad+\frac{\sqrt{3}}{2}L_{2}\sin(\varphi)\sin(2\theta )\notag\\
&\quad+\frac{\sqrt{3}}{3}L_{3}\cos(\varphi) -\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\quad-\frac{2}{3}L_{3}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi)+\frac{1}{3}L_{3}\sin(\varphi)\notag\\
&\notag\\
&=L_{2}\cos(2\theta)\sin(\varphi) -L_{2}\cos(\varphi)\sin(2\theta )\notag\\
&\qquad \qquad \qquad \qquad +L_{2}\left(\sin(2\theta)\cos(\varphi)-\cos(2\theta)\sin(\varphi)\right)\notag\\
&\notag\\
&=L_{2}\cos(2\theta)\sin(\varphi) -L_{2}\cos(\varphi)\sin(2\theta )\notag\\
&\qquad \qquad \qquad \qquad +L_{2}\sin(2\theta)\cos(\varphi)-L_{2}\cos(2\theta)\sin(\varphi)\notag\\
&\notag\\
&=0\notag\\
\end{align}
$$

($${3,1}$$)成分

$$
\begin{align}
&a_{31}\cos(\theta_{{\rm{a}}}) + a_{32}\cos(\theta_{{\rm{b}}})+a_{33} \cos(\theta_{{\rm{c}}})\notag\\
&\notag\\
&=\left(\frac{1}{3}L_{1}-\frac{2}{3}L_{3}\right)\cos(\varphi)\notag\\
&+\left(\frac{1}{3}L_{1} -\frac{2}{3}L_{3}\right)\cos\left(\varphi - \frac{2}{3}\pi\right)\notag\\
&+\left(\frac{1}{3}L_{1}-\frac{2}{3}L_{3}\right)\cos\left(\varphi + \frac{2}{3}\pi\right)\notag\\
&\notag\\
&=\frac{1}{3}L_{1}\cos(\varphi)-\frac{2}{3}L_{3}\cos(\varphi)\notag\\
&\quad+\left(\frac{1}{3}L_{1} -\frac{2}{3}L_{3}\right)\left(-\frac{1}{2}\cos(\varphi)+\frac{\sqrt{3}}{2}\sin(\varphi)\right)\notag\\
&\quad+\left(\frac{1}{3}L_{1}-\frac{2}{3}L_{3}\right)\left(-\frac{1}{2}\cos(\varphi)-\frac{\sqrt{3}}{2}\sin(\varphi)\right)\notag\\
&\notag\\
&=\frac{1}{3}L_{1}\cos(\varphi)-\frac{2}{3}L_{3}\cos(\varphi)\notag\\
&\quad-\frac{1}{6}L_{1} \cos(\varphi)+\frac{1}{3}L_{3}\cos(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1} \sin(\varphi)-\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\quad-\frac{1}{6}L_{1}\cos(\varphi)+\frac{1}{3}L_{3}\cos(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{1}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\notag\\
&=\frac{1}{3}L_{1}\cos(\varphi)-\frac{1}{6}L_{1} \cos(\varphi)-\frac{1}{6}L_{1}\cos(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1} \sin(\varphi)-\frac{\sqrt{3}}{6}L_{1}\sin(\varphi)\notag\\
&\quad-\frac{2}{3}L_{3}\cos(\varphi)+\frac{1}{3}L_{3}\cos(\varphi)+\frac{1}{3}L_{3}\cos(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)+\frac{\sqrt{3}}{3}L_{3}\sin(\varphi)\notag\\
&\notag\\
&=0\notag\\
\end{align}
$$

($${3,2}$$)成分

$$
\begin{align}
&-a_{31}\sin(\theta_{{\rm{a}}}) - a_{32}\sin(\theta_{{\rm{b}}})-a_{33}\sin(\theta_{{\rm{c}}})\notag\\
&\notag\\
&=-\left(\frac{1}{3}L_{1}-\frac{2}{3}L_{3}\right)\sin(\varphi)\notag\\
&\quad-\left(\frac{1}{3}L_{1} -\frac{2}{3}L_{3}\right)\sin\left(\varphi - \frac{2}{3}\pi\right)\notag\\
&\quad-\left(\frac{1}{3}L_{1}-\frac{2}{3}L_{3}\right)\sin\left(\varphi + \frac{2}{3}\pi\right)\notag\\
&\notag\\
&= -\frac{1}{3}L_{1}\sin(\varphi)+\frac{2}{3}L_{3}\sin(\varphi)\notag\\
&\quad+\left(-\frac{1}{3}L_{1} +\frac{2}{3}L_{3}\right)\left(-\frac{1}{2}\sin(\varphi)-\frac{\sqrt{3}}{2}\cos(\varphi)\right)\notag\\
&\quad+\left(-\frac{1}{3}L_{1}+\frac{2}{3}L_{3}\right)\left(-\frac{1}{2}\sin(\varphi)+\frac{\sqrt{3}}{2}\cos(\varphi)\right)\notag\\
&\notag\\
&= -\frac{1}{3}L_{1}\sin(\varphi)+\frac{2}{3}L_{3}\sin(\varphi)\notag\\
&\quad+\frac{1}{6}L_{1}\sin(\varphi) -\frac{1}{3}L_{3}\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{6}L_{1} \cos(\varphi)-\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\quad+\frac{1}{6}L_{1}\sin(\varphi)-\frac{1}{3}L_{3}\sin(\varphi)\notag\\
&\quad-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)+\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)\notag\\
&\notag\\
&= \frac{\sqrt{3}}{6}L_{1} \cos(\varphi)-\frac{\sqrt{3}}{6}L_{1}\cos(\varphi)\notag\\
&\quad-\frac{1}{3}L_{1}\sin(\varphi)+\frac{1}{6}L_{1}\sin(\varphi)+\frac{1}{6}L_{1}\sin(\varphi)\notag\\
&\quad+\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)-\frac{\sqrt{3}}{3}L_{3}\cos(\varphi)+\frac{2}{3}L_{3}\sin(\varphi)\notag\\
& \quad-\frac{1}{3}L_{3}\sin(\varphi)-\frac{1}{3}L_{3}\sin(\varphi)\notag\\
&\notag\\
&=0\notag\\
\end{align}
$$

($${3,3}$$)成分

$$
\begin{align}
&a_{31}+ a_{32}+a_{33}\notag\\
&\notag\\
&=\frac{1}{3}L_{1}-\frac{2}{3}L_{3}+\frac{1}{3}L_{1} -\frac{2}{3}L_{3}+\frac{1}{3}L_{1}-\frac{2}{3}L_{3}\notag\\
&\notag\\
&= L_{1}-2L_{3}\notag\\
\end{align}
$$

変換後の行列

インダクタンス行列をd-q-0座標系に変換すると、

$$
\begin{align}
&L_{\rm{dq0}}=AL_{\rm{abc}}A^{-1}\notag\\
&\notag\\
&=\left (
\begin{matrix}
L_{1}+\frac{3}{2}L_{2}\cos(2\theta-2\varphi)+L_{3}&  \frac{3}{2}L_{2}\sin(2\theta-2\varphi)  \\
& \\
\frac{3}{2}L_{2}\sin(2\theta-2\varphi)&  L_{1}-\frac{3}{2}L_{2}\cos(2\theta-2\varphi)+L_{3}\\
&  \\
 0&0\\
\end{matrix} \right .\notag\\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\left.
\begin{matrix}
0  \\
 \\
0\\
 \\
 L_{1}-2L_{3}\\
\end{matrix} \right )\notag\\
\end{align}
$$

となる。

ここで、$${\theta = \varphi}$$とすれば、

$$
\begin{align}
L_{\rm{dq0}}&=AL_{\rm{abc}}A^{-1}\notag\\
&\notag\\
&=\left (
\begin{matrix}
L_{1}+\frac{3}{2}L_{2}+L_{3}&  0&0  \\
& &\\
0&  L_{1}-\frac{3}{2}L_{2}+L_{3}&0\\
& & \\
 0&0& L_{1}-2L_{3}\\
\end{matrix} \right )\notag\\
\end{align}
$$

となる。

これよりd軸巻線、q軸巻線、零軸巻線は自己インダクタンスのみを持ち、相互インダクタンスが存在しなくなる。また、$${\theta}$$にも無関係な定数として
扱える。
$${\theta = \varphi}$$と置くことは、回転子の位置$${\theta}$$が時間$${t}$$とともに変化するときに、d軸、q軸も変化していることを表す。
したがって、時刻$${t}$$のd軸、q軸巻線と時刻$${t+\Delta t}$$のd軸、q軸巻線は、異なったものとなるが、各時刻の$${\theta = \varphi}$$のd軸、q軸巻線の値を取ったものをd-q-0法では取り扱っている。

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