# 电报员方程

## 无耗传输

${\displaystyle V=V(x,t)}$
${\displaystyle I=I(x,t)}$

### 方程

${\displaystyle {\frac {\partial V}{\partial x}}=-L{\frac {\partial I}{\partial t}}}$
${\displaystyle {\frac {\partial I}{\partial x}}=-C{\frac {\partial V}{\partial t}}}$

${\displaystyle {\frac {\partial ^{2}V}{{\partial t}^{2}}}-u^{2}{\frac {\partial ^{2}V}{{\partial x}^{2}}}=0}$
${\displaystyle {\frac {\partial ^{2}I}{{\partial t}^{2}}}-u^{2}{\frac {\partial ^{2}I}{{\partial x}^{2}}}=0}$

${\displaystyle u={\frac {1}{\sqrt {LC}}}}$

### 正弦稳态

${\displaystyle V(x,t)=\mathrm {Re} \{V(x)\cdot e^{j\omega t}\}}$
${\displaystyle I(x,t)=\mathrm {Re} \{I(x)\cdot e^{j\omega t}\}}$,

${\displaystyle {\frac {dV}{dx}}=-j\omega LI}$
${\displaystyle {\frac {dI}{dx}}=-j\omega CV}$

${\displaystyle {\frac {d^{2}V}{dx^{2}}}+k^{2}V=0}$
${\displaystyle {\frac {d^{2}I}{dx^{2}}}+k^{2}I=0}$

${\displaystyle k=\omega {\sqrt {LC}}={\omega \over u}.}$

${\displaystyle V(x)=V_{1}e^{-jkx}+V_{2}e^{+jkx}}$

${\displaystyle I(x)={V_{1} \over Z_{0}}e^{-jkx}-{V_{2} \over Z_{0}}e^{+jkx}}$

${\displaystyle Z_{0}={\sqrt {L \over C}}}$

${\displaystyle V_{1}}$${\displaystyle V_{2}}$ 是积分的两个任意常量，由两个边界条件（传输线的两端各一个）确定。

### 通解

${\displaystyle V(x,t)\ =\ f_{1}(x-ut)+f_{2}(x+ut)}$

• ${\displaystyle f_{1}}$${\displaystyle f_{2}}$ 可以是任意函数，并且
• ${\displaystyle u={\frac {1}{\sqrt {LC}}}}$ 是波形的传播速度（也被称为相速度）。

f1 表示波从左到右向 x 轴正方向传播的波 f2 表示从右到左传播的波。可以看出传输线上任意一点 x 的瞬时电压为两个波的电压之和。

${\displaystyle I(x,t)\ =\ {\frac {f_{1}(x-ut)}{Z_{0}}}-{\frac {f_{2}(x+ut)}{Z_{0}}}}$

## 有耗传输线

${\displaystyle {\frac {\partial }{\partial x}}V(x,t)=-L{\frac {\partial }{\partial t}}I(x,t)-RI(x,t)}$
${\displaystyle {\frac {\partial }{\partial x}}I(x,t)=-C{\frac {\partial }{\partial t}}V(x,t)-GV(x,t)}$

${\displaystyle {\frac {\partial ^{2}}{{\partial x}^{2}}}V=LC{\frac {\partial ^{2}}{{\partial t}^{2}}}V+(RC+GL){\frac {\partial }{\partial t}}V+GRV}$
${\displaystyle {\frac {\partial ^{2}}{{\partial x}^{2}}}I=LC{\frac {\partial ^{2}}{{\partial t}^{2}}}I+(RC+GL){\frac {\partial }{\partial t}}I+GRI}$

## 电报员方程的解作为电路元件

${\displaystyle V_{1}=V_{2}\cosh(\gamma x)+I_{2}Z\sinh(\gamma x)\,}$
${\displaystyle I_{1}=V_{2}{\frac {1}{Z}}\sinh(\gamma x)+I_{2}\cosh(\gamma x).\,}$

ABCD型的二端口给出了 ${\displaystyle V_{1}\,}$${\displaystyle I_{1}\,}$ 表示为 ${\displaystyle V_{2}\,}$${\displaystyle I_{2}\,}$ 的函数的形式。上面的方程当对 ${\displaystyle V_{1}\,}$${\displaystyle I_{1}\,}$ 作为 ${\displaystyle V_{2}\,}$${\displaystyle I_{2}\,}$ 的函数求解时，都会得到相同的方程。

## 注释

1. ^ Kraus (1989， pp. 380–419)
2. ^ Hayt (1989， pp. 382–392)
3. ^ Marshall (1987， pp. 359–378)
4. ^ Sadiku (1989， pp. 497–505)
5. ^ Harrington (1961， pp. 61–65)
6. ^ Karakash (1950， pp. 5–14)
7. ^ Metzger (1969， pp. 1–10)
8. ^ Sadiku (1989， pp. 501–503)
9. ^ Marshall (1987， pp. 369–372)
10. ^ McCammon, Roy, SPICE Simulation of Transmission Lines by the Telegrapher's Method (PDF), [22 Oct 2010]
11. ^ William H. Hayt. Engineering Circuit Analysis second. New York, NY: McGraw-Hill. 1971. ISBN 0070273820., pp. 73-77
12. ^ John J. Karakash. Transmission Lines and Filter Networks First. New York, NY: Macmillan. 1950., p. 44

## 参考文献

• Chen, Walter Y., Home Networking Basics, Prentice Hall, 2004, ISBN 0-13-016511-5
• Harrington, Roger F., Time-Harmonic Electromagnetic Fields, McGraw-Hill, 1961
• Hayt, William, Engineering Electromagnetics 5th, McGraw-Hill, 1989, ISBN 0-07-027406-1
• Karakash, John J., Transmission Lines and Filter Networks 1st, Macmillan, 1950
• Kraus, John D., Electromagnetics 3rd, McGraw-Hill, 1984, ISBN 0-07-035423-5
• Marshall, Stanley V., Electromagnetic Concepts & Applications 1st, Prentice-Hall, 1987, ISBN 0-13-249004-8
• Metzger, Georges; Vabre, Jean-Paul, Transmission Lines with Pulse Excitation, Academic Press, 1969
• Reeve, Whitman D., Subscriber Loop Signaling and Transmission Handbook, IEEE Press, 1995, ISBN 0-7803-0440-3
• Sadiku, Matthew N. O., Elements of Electromagnetics 1st, Saunders College Publishing, 1989, ISBN 0030134846
• Terman, Frederick Emmons, Radio Engineers' Handbook 1st, McGraw-Hill, 1943