量值條件

量值條件（magnitude condition）是自動控制的根軌跡圖中，有關量值的限制條件，根軌跡圖中的點和閉迴路極點、零點組成向量的量值會滿足量值條件。量值條件和角度條件可以完全確定根軌跡圖。

根軌跡圖上的每一點和極點、零點組成向量的量值會滿足量值條件

令系統的特徵方程為${\displaystyle 1+{\textbf {G}}(s){\textbf {H}}(s)=0}$，而${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)={\frac {{\textbf {P}}(s)}{{\textbf {Q}}(s)}}}$，可改寫為以下各因式相乘的形式

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

則量值條件是指找到K值使下式成立：

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

也就是說

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

推導

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

將${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)}$改寫為各因式相乘的形式

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

量值條件即是

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

也就是

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

若將控制方程改為極坐標表示：

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

${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)=-1=e^{j(\pi +2k\pi )}}$其中 ${\displaystyle (k=0,1,2,...)}$，這些是方程式所有的解。

將每一個因子${\displaystyle (s-a_{p})}$及${\displaystyle (s-b_{q})}$都用等效的向量${\displaystyle A_{p}e^{j\theta _{p}}}$及${\displaystyle B_{q}e^{j\phi _{q}}}$來表示，因此 ${\displaystyle {\textbf {G}}(s){\textbf {H}}(s)}$可以再作整理。

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

簡化特徵方程式。

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

因此可以推導出另一個形式的量值條件：

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

也可以用類似的方式推導角度條件。