量值条件

量值条件（magnitude condition）是自动控制的根轨迹图中，有关量值的限制条件，根轨迹图中的点和闭回路极点、零点组成向量的量值会满足量值条件。量值条件和角度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

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

则量值条件是指找到K值使下式成立：

${\displaystyle |G(s)H(s)|=1}$

也就是说

${\displaystyle K{\frac {|s-a_{1}||s-a_{2}|\cdots |s-a_{m}|}{|s-b_{1}||s-b_{2}|\cdots |s-b_{n}|}}=1}$.

推导

令${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)={\frac {{\textbf {P}}(s)}{{\textbf {Q}}(s)}}}$。

将${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)}$改写为各因式相乘的形式

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

量值条件即是

${\displaystyle {\frac {|{\textbf {P}}(s)|}{|{\textbf {Q}}(s)|}}=1}$

也就是

${\displaystyle K{\frac {|s-a_{1}||s-a_{2}|\cdots |s-a_{n}|}{|s-b_{1}||s-b_{2}|\cdots |s-b_{m}|}}=1,}$

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 ${\displaystyle (k=0,1,2,...)}$，这些是方程式所有的解。

将每一个因子${\displaystyle (s-a_{p})}$及${\displaystyle (s-b_{q})}$都用等效的向量${\displaystyle A_{p}e^{j\theta _{p}}}$及${\displaystyle B_{q}e^{j\phi _{q}}}$来表示，因此 ${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)}$可以再作整理。

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

简化特征方程式。

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

因此可以推导出另一个形式的量值条件：

${\displaystyle 1=K{\frac {A_{1}A_{2}\cdots A_{n}}{B_{1}B_{2}\cdots B_{m}}}.}$

也可以用类似的方式推导角度条件。