角度条件

角度条件（angle condition）是自动控制的根轨迹图中，有关角度的限制条件，根轨迹图中的点和闭回路极点、零点组成向量的角度会满足角度条件。角度条件和量值条件可以完全确定根轨迹图。

根轨迹图上的每一点和极点、零点组成向量的量值会满足量值条件

令系统的特征方程为${\displaystyle 1+{\textbf {G}}(s){\textbf {H}}(s)=0}$，而${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)={\frac {{\textbf {P}}(s)}{{\textbf {Q}}(s)}}}$，可改写为以下各因式相乘的形式

${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)={\frac {{\textbf {P}}(s)}{{\textbf {Q}}(s)}}=K{\frac {(s-a_{1})(s-a_{2})\cdots (s-a_{n})}{(s-b_{1})(s-b_{2})\cdots (s-b_{m})}},}$

则角度条件是指

${\displaystyle \angle ({\textbf {G}}(s){\textbf {H}}(s))=\pi +2k\pi }$

其中${\displaystyle k}$为整数。

也就是说

${\displaystyle \sum _{i=1}^{m}\angle (s-b_{i})-\sum _{i=1}^{n}\angle (s-a_{i})=\pi +2k\pi }$

开回路零点到${\displaystyle s}$点角度的和，减去开回路极点到${\displaystyle s}$点角度的和，除${\displaystyle 2\pi }$后的余数需等于${\displaystyle \pi }$。

推导

假设。若将控制方程改为极坐标表示

${\displaystyle e^{j2\pi }+{\textbf {G}}(s){\textbf {H}}(s)=0}$

${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)=-1=e^{j(\pi +2k\pi )}}$

其中${\displaystyle k=0,1,2,\ldots }$为方程式中的解。将${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)}$改写为各因式相乘的形式

${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)={\frac {{\textbf {P}}(s)}{{\textbf {Q}}(s)}}=K{\frac {(s-a_{1})(s-a_{2})\cdots (s-a_{n})}{(s-b_{1})(s-b_{2})\cdots (s-b_{m})}},}$

再将因式${\displaystyle (s-a_{p})}$及${\displaystyle (s-b_{q})}$用向量的形式${\displaystyle A_{p}e^{j\theta _{p}}}$及${\displaystyle B_{q}e^{j\varphi _{q}}}$表示，因此可以改写${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)}$如下。

${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)=K{\frac {A_{1}A_{2}\cdots A_{n}e^{j(\theta _{1}+\theta _{2}+\cdots +\theta _{n})}}{B_{1}B_{2}\cdots B_{m}e^{j(\varphi _{1}+\varphi _{2}+\cdots +\varphi _{m})}}}}$

简化特征方程式，

${\displaystyle {\begin{aligned}e^{j(\pi +2k\pi )}&=K{\frac {A_{1}A_{2}\cdots A_{n}e^{j(\theta _{1}+\theta _{2}+\cdots +\theta _{n})}}{B_{1}B_{2}\cdots B_{m}e^{j(\varphi _{1}+\varphi _{2}+\cdots +\varphi _{m})}}}\\[6pt]&=K{\frac {A_{1}A_{2}\cdots A_{n}}{B_{1}B_{2}\cdots B_{m}}}e^{j(\theta _{1}+\theta _{2}+\cdots +\theta _{n}-(\varphi _{1}+\varphi _{2}+\cdots +\varphi _{m}))},\end{aligned}}}$

因此可以得到角度条件：

${\displaystyle \pi +2k\pi =\theta _{1}+\theta _{2}+\cdots +\theta _{n}-(\varphi _{1}+\varphi _{2}+\cdots +\varphi _{m})}$

其中${\displaystyle k=0,1,2,\ldots }$,

${\displaystyle \theta _{1},\theta _{2},\ldots ,\theta _{n}}$

为极点1至n的角度，而

${\displaystyle \varphi _{1},\varphi _{2},\ldots ,\varphi _{m}}$

为零点1至m的角度。

也可以用类似的方式推导量值条件。