# 根軌跡圖

（重定向自根轨迹

## 定義

${\displaystyle T(s)={\frac {Y(s)}{X(s)}}={\frac {G(s)}{1+G(s)H(s)}}}$

${\displaystyle G(s)H(s)=K{\frac {(s+z_{1})(s+z_{2})\cdots (s+z_{m})}{(s+p_{1})(s+p_{2})\cdots (s+p_{n})}}}$

### 角度條件

${\displaystyle \angle (G(s)H(s))=\pi }$

${\displaystyle \sum _{i=1}^{m}\angle (s+z_{i})-\sum _{i=1}^{n}\angle (s+p_{i})=\pi }$

### 量值條件

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

${\displaystyle K{\frac {|s+z_{1}||s+z_{2}|\cdots |s+z_{m}|}{|s+p_{1}||s+p_{2}|\cdots |s+p_{n}|}}=1}$ .

## 繪製根軌跡圖

P為極點的個數，Z為零點的個數，兩者相減即為渐近线的數量：

${\displaystyle P-Z={\text{number of asymptotes}}\,}$

${\displaystyle \phi _{l}={\frac {180^{\circ }+(l-1)360^{\circ }}{P-Z}},l=1,2,\ldots ,P-Z}$
${\displaystyle \alpha ={\frac {\sum _{P}-\sum _{Z}}{P-Z}}}$

• 根據測試點的相位條件判斷其往外延伸的角度
• 計算分離點（breakaway/break-in points）

${\displaystyle {\frac {dG(s)H(s)}{ds}}=0{\text{ or }}{\frac {d{\overline {GH}}(z)}{dz}}=0}$

## 參考資料

1. ^ Kuo 1967，第331頁.
2. ^ Kuo 1967，第332頁.
3. ^ Evans, Walter R., Spirule Instructions, Whittier, CA: The Spirule Company, 1965
4. ^ Evans, W. R., Graphical Analysis of Control Systems, Trans. AIEE, January 1948, 67 (1): 547–551, ISSN 0096-3860, doi:10.1109/T-AIEE.1948.5059708
5. ^ Evans, W. R., Control Systems Synthesis by Root Locus Method, Trans. AIEE, January 1950, 69 (1): 66–69, ISSN 0096-3860, doi:10.1109/T-AIEE.1950.5060121

## 延伸閱讀

• Ash, R. H.; Ash, G. H., Numerical Computation of Root Loci Using the Newton-Raphson Technique, IEEE Trans. Automatic Control, October 1968, 13 (5), doi:10.1109/TAC.1968.1098980
• Williamson, S. E., Design Data to assist the Plotting of Root Loci (Part I), Control Magazine, May 1968, 12 (119): 404–407
• Williamson, S. E., Design Data to assist the Plotting of Root Loci (Part II), Control Magazine, June 1968, 12 (120): 556–559
• Williamson, S. E., Design Data to assist the Plotting of Root Loci (Part III), Control Magazine, July 1968, 12 (121): 645–647
• Williamson, S. E., Computer Program to Obtain the Time Response of Sampled Data Systems, IEE Electronics Letters, May 15, 1969, 5 (10): 209–210, doi:10.1049/el:19690159
• Williamson, S. E., Accurate Root Locus Plotting Including the Effects of Pure Time Delay (PDF), Proc. IEE, July 1969, 116 (7): 1269–1271, doi:10.1049/piee.1969.0235