# 传输线

## 电报员方程

• 导体的分布电阻 ${\displaystyle R}$ 表示为电阻串联（单位为欧姆每单位长度）。
• 分布电感 ${\displaystyle L}$（源于电线周围的磁场自感等）表示为电感串联（亨利每单位长度）。
• 两个导体之间的电容 ${\displaystyle C}$ 表示为并联电容 C（法拉每单位长度）。
• 分开两个导体的电介质材料的电导 ${\displaystyle G}$ 表示为信号线与回线间的并联电阻（西门子每单位长度）。

${\displaystyle {\frac {\partial V(x)}{\partial x}}=-(R+j\omega L)I(x)}$
${\displaystyle {\frac {\partial I(x)}{\partial x}}=-(G+j\omega C)V(x).}$

${\displaystyle {\frac {\partial ^{2}V(x)}{\partial x^{2}}}+\omega ^{2}LC\cdot V(x)=0}$
${\displaystyle {\frac {\partial ^{2}I(x)}{\partial x^{2}}}+\omega ^{2}LC\cdot I(x)=0.}$

${\displaystyle {\frac {\partial ^{2}V(x)}{\partial x^{2}}}=\gamma ^{2}V(x)}$
${\displaystyle {\frac {\partial ^{2}I(x)}{\partial x^{2}}}=\gamma ^{2}I(x)}$

${\displaystyle \gamma ={\sqrt {(R+j\omega L)(G+j\omega C)}}}$

${\displaystyle Z_{0}={\sqrt {\frac {R+j\omega L}{G+j\omega C}}}.}$

${\displaystyle V(x)}$${\displaystyle I(x)}$ 的解为：

${\displaystyle V(x)=V^{+}e^{-\gamma x}+V^{-}e^{\gamma x}\,}$
${\displaystyle I(x)={\frac {1}{Z_{0}}}(V^{+}e^{-\gamma x}-V^{-}e^{\gamma x}).\,}$

${\displaystyle \mathrm {Re} (\gamma )=(a^{2}+b^{2})^{1/4}\cos(\mathrm {atan2} (b,a)/2)\,}$
${\displaystyle \mathrm {Im} (\gamma )=(a^{2}+b^{2})^{1/4}\sin(\mathrm {atan2} (b,a)/2)\,}$

${\displaystyle a\equiv \omega ^{2}LC\left[\left({\frac {R}{\omega L}}\right)\left({\frac {G}{\omega C}}\right)-1\right]}$
${\displaystyle b\equiv \omega ^{2}LC\left({\frac {R}{\omega L}}+{\frac {G}{\omega C}}\right).}$

${\displaystyle \mathrm {Re} (\gamma )\approx {\frac {\sqrt {LC}}{2}}\left({\frac {R}{L}}+{\frac {G}{C}}\right)\,}$
${\displaystyle \mathrm {Im} (\gamma )\approx \omega {\sqrt {LC}}.\,}$

${\displaystyle V_{\mathrm {out} }(x,t)\approx V_{\mathrm {in} }(t-{\sqrt {LC}}x)e^{-{\frac {\sqrt {LC}}{2}}\left({\frac {R}{L}}+{\frac {G}{C}}\right)x}.\,}$

## 传输线的输入阻抗

${\displaystyle Z_{in}\left(l\right)={\frac {V(l)}{I(l)}}=Z_{0}{\frac {1+\Gamma _{L}e^{-2\gamma l}}{1-\Gamma _{L}e^{-2\gamma l}}}}$,

${\displaystyle Z_{in}\left(l\right)=Z_{0}{\frac {Z_{L}+Z_{0}\tanh \left(\gamma l\right)}{Z_{0}+Z_{L}\tanh \left(\gamma l\right)}}}$.

### 无损传输线的输入阻抗

${\displaystyle Z_{\mathrm {in} }(l)=Z_{0}{\frac {Z_{L}+jZ_{0}\tan(\beta l)}{Z_{0}+jZ_{L}\tan(\beta l)}}}$

## 参考书目

1. Jackman, Shawn M.; Matt Swartz; Marcus Burton; Thomas W. Head. CWDP Certified Wireless Design Professional Official Study Guide: Exam PW0-250. John Wiley & Sons. 2011: Ch. 7 [2020-12-16]. ISBN 1118041615. （原始内容存档于2020-04-14）.
2. Oklobdzija, Vojin G.; Ram K. Krishnamurthy. High-Performance Energy-Efficient Microprocessor Design. Springer. 2006: 297 [2014-10-10]. ISBN 0387340475. （原始内容存档于2016-12-24）.
3. ^ Guru, Bhag Singh; Hüseyin R. Hızıroğlu. Electromagnetic Field Theory Fundamentals, 2nd Ed.. Cambridge Univ. Press. 2004: 422–423 [2014-10-10]. ISBN 1139451928. （原始内容存档于2014-07-05）.
4. ^ Schmitt, Ron Schmitt. Electromagnetics Explained: A Handbook for Wireless/ RF, EMC, and High-Speed Electronics. Newnes. 2002: 153 [2014-10-10]. ISBN 0080505236. （原始内容存档于2016-12-24）.
5. ^ Carr, Joseph J. Microwave & Wireless Communications Technology. USA: Newnes. 1997: 46–47 [2014-10-10]. ISBN 0750697075. （原始内容存档于2016-12-24）.
6. Raisanen, Antti V.; Arto Lehto. Radio Engineering for Wireless Communication and Sensor Applications. Artech House. 2003: 35–37 [2014-10-10]. ISBN 1580536697. （原始内容存档于2020-11-04）.
7. ^ Ernst Weber and Frederik Nebeker, The Evolution of Electrical Engineering, IEEE Press, Piscataway, New Jersey USA, 1994 ISBN 978-0-7803-1066-7
• Steinmetz, Charles Proteus, "The Natural Period of a Transmission Line and the Frequency of lightning Discharge Therefrom". The Electrical world. August 27 1898. Pg. 203 - 205.
• Electromagnetism 2nd ed., Grant, I.S., and Phillips, W.R., pub John Wiley, ISBN 978-0-471-92712-9
• Fundamentals Of Applied Electromagnetics 2004 media edition., Ulaby, F.T., pub Prentice Hall, ISBN 978-0-13-185089-7
• Radiocommunication handbook, page 20, chaper 17, RSGB, ISBN 978-0-900612-58-9
• Naredo, J.L., A.C. Soudack, and J.R. Marti, Simulation of transients on transmission lines with corona via the method of characteristics. Generation, Transmission and Distribution, IEE Proceedings. Vol. 142.1, Inst. de Investigaciones Electr., Morelos, Jan 1995. ISSN 1350-2360