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此页面介紹的是电波传输线。关于电力传输线，请见「輸電系統」。

电波在无损耗传输线内流动原理图。红色代表高电压，蓝色代表低电压。黑色圆点代表电子。传输线接于阻抗匹配的负载电阻（右边的盒子）上，波形完全被吸收。

一种最常见的传输线——同轴电缆。

在通信工程和电子工程中，传输线是一种特殊的电缆或者其他结构，被设计用于承载电波中变化的电流，也就是说，电流的频率高到一定程度时它们波的本质必须进行考虑。传输线一般用于连接发送器与接收器的天线，传输有线电视信号，中继电信交换中心之间的路由呼叫，中继计算机网络链接以及中继高速计算机总线。

在此仅讨论双导体传输线，包含平行线（梯线）、同轴电缆、带状线和微带线。一些人认为波导管、介质波导甚至光纤也是传输线，然而这些线需要不同的分析技术，所以不在此进行讨论；可参见电磁波导。

概述

普通电缆足以携带低频交流电(例如家庭用电，每秒钟变换100~120次方向）和声音信号。然而，普通电缆不能用于承载电波频率范围的电流或更高频率的电流 ，这种频率的电流每秒钟变更百万次方向，能量易于从电缆中以电磁波的形式辐射出来，从而造成能量损耗。高频电流也容易在电缆的连接处（如连接器和节点）反射回电源。 这些反射作为瓶颈，阻止了信号功率到达目的地。传输线使用了特殊的结构和阻抗匹配的方法，承载电磁信号以最小的反射和最小的功率损耗到达接收端。大多数传输线的显着特点是它们具有沿其长度方向均匀的横截面尺寸，使得传输线有着一致的阻抗，被称为特性阻抗， 从而防止了反射的发生。传输线有多种形态，例如平行线(梯线、双绞线)、同轴电缆、带状线以及微带线。 电磁波的频率与波长成反比。当线缆的长度与传输信号的波长相当时，就必须要使用传输线了。

传输微波频率信号时，传输线的功率损失也会比较明显，这时应当使用波导管替代传输线 ，波导管的功能是作为限制和引导电磁波的“管道”。 一些人将波导管视为一种传输线； 然而，这里认为波导管和传输线是不同的。在更高的频率上，例如太赫兹、红外线、光的范围，波导管也将对信号造成损失，这时需要使用光学方法（如棱镜和镜子）来引导电磁波。

声波传播的数学理论与电磁波的传播数学理论是非常相似的，因此传输线的理论也被用来指导声波的传播；叫做声学传输线。

历史

电传输线的数学分析源于麦克斯韦、开尔文男爵和赫维赛德的工作。1855年开尔文男爵建立了一个关于海底电缆电流的微分模型。这个模型正确的预测了1858年穿越大西洋海底通信电缆的非良好性能。在1885年赫维赛德发表了第一篇关于描述他的电缆传播分析和现代通信模式方程的论文。

适用范围

在许多电子线路中，连接各器件的电线的长度是基本可以被忽略的。也就是说在电线各点同一时刻的电压可以认为是相同的。但是，当电压的变化和信号沿电线传播下去的时间可以比拟时，电线的长度变得重要了，这時电线就必须被处理成传输线。换言之，当信号所包含的频率分量的相应的波长较之电线长度小或二者可以比拟的时候，电线的长度是很重要的。

常见的经验方法认为如果电缆或者电线的长度大于波长的1/10，则需被作为传输线处理。 在这个长度下相位延迟和线中的反射干扰非常显著，那么没有用传输线理论仔细的研究设计过的系统就会出现一些不可预知行为。

四终端模型

传输线在电路图中各种电路符号。

为了分析的需要，传输线可以用二端口网络（四端网络）进行建模，如下图所：

在最简单的情况，假设网络是线性的（即任何端口之间的复电压在没有反射的情况下正比于复电流），且两个端口可以互换。如果传输线在长度范围内是均匀的，那么其特性可以只用一个参数描述：特性阻抗， 符号是 Z₀ 。 特性阻抗是某一给定电波在传输线上任意一点复电压与复电流的比值。常見电缆阻抗Z₀的典型數值：同軸電纜 - 50或75欧姆, 扭絞二股线 - 约100欧姆，广播传输用的平行二股线 - 约300欧姆。

当在传输线上发送功率时， 最好的情况是尽可能多的功率被负载吸收，尽可能少的功率被反射回发送端。在负载阻抗等于特性阻抗Z₀时，这一点可以被保证，这时传输线被称为阻抗匹配。

图中两条黑线代表传输线。在距离起点 x 处，每条线都流过了 I(x) 的电流，两线之间的电压差为 V(x) 。在单一信号没有反射的情况下， V(x) / I(x) = Z₀， Z₀ 代表了传输线的 特性阻抗 。

由于传输线电阻的存在，一些被发送到传输线上的功率被损耗。这种现象叫做电阻损耗。在高频处，另一种介电损耗变得非常明显，加重了电阻引起的损耗。介电损耗是由于在传输线内的绝缘材料从电域吸收能量转化为热引起的。 传输线模型表现为电阻 (R) 与电感 (L) 的串联以及电容 (C) 与电导 (G) 的并联。电阻与电导引起了传输线的损耗。

传输线功率总损耗的单位是分贝每米 (dB/m)，并与信号频率相关。生产厂家一般会提供一定范围内以dB/m为单位的损耗图。3dB代表大约损失一半的功率。

设计用于承载波长小于或可比于传输线长度电磁波的传输线称为高频传输线。在这种情况下，在低频下的估值方法不再适用。高频传输线常见于无线电，微波，光信号，金属网滤光片和高速电子线路中的信号。

Telegrapher's equations

主条目：Telegrapher's equations

参见：Reflections on copper lines

The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's Equations.

Schematic representation of the elementary component of a transmission line.

The transmission line model represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

• The distributed resistance $R$ of the conductors is represented by a series resistor (expressed in ohms per unit length).
• The distributed inductance $L$ (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
• The capacitance $C$ between the two conductors is represented by a shunt capacitor C (farads per unit length).
• The conductance $G$ of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (siemens per unit length).

The model consists of an infinite series of the elements shown in the figure, and that the values of the components are specified per unit length so the picture of the component can be misleading. $R$, $L$, $C$, and $G$ may also be functions of frequency. An alternative notation is to use $R'$, $L'$, $C'$ and $G'$ to emphasize that the values are derivatives with respect to length. These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the propagation constant, attenuation constant and phase constant.

The line voltage $V(x)$ and the current $I(x)$ can be expressed in the frequency domain as

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

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

When the elements $R$ and $G$ are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the $L$ and $C$ elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

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

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

These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.

If $R$ and $G$ are not neglected, the Telegrapher's equations become:

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

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

where γ is the propagation constant

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

and the characteristic impedance can be expressed as

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

The solutions for $V(x)$ and $I(x)$ are:

$V(x)=V^{+}e^{-\gamma x}+V^{-}e^{\gamma x}\,$

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

The constants $V^{\pm }$ and $I^{\pm }$ must be determined from boundary conditions. For a voltage pulse $V_{\mathrm {in} }(t)\,$, starting at $x=0$ and moving in the positive $x$-direction, then the transmitted pulse $V_{\mathrm {out} }(x,t)\,$ at position $x$ can be obtained by computing the Fourier Transform, ${\tilde {V}}(\omega )$, of $V_{\mathrm {in} }(t)\,$, attenuating each frequency component by $e^{\mathrm {-Re} (\gamma )x}\,$, advancing its phase by $\mathrm {-Im} (\gamma )x\,$, and taking the inverse Fourier Transform. The real and imaginary parts of $\gamma$ can be computed as

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

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

where atan2 is the two-parameter arctangent, and

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

$b\equiv \omega ^{2}LC\left({\frac {R}{\omega L}}+{\frac {G}{\omega C}}\right).$

For small losses and high frequencies, to first order in $R/\omega L$ and $G/\omega C$ one obtains

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

$\mathrm {Im} (\gamma )\approx \omega {\sqrt {LC}}.\,$

Noting that an advance in phase by $-\omega \delta$ is equivalent to a time delay by $\delta$, $V_{out}(t)$ can be simply computed as

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

Input impedance of transmission line

Looking towards a load through a length l of lossless transmission line, the impedance changes as l increases, following the blue circle on this impedance Smith chart. (This impedance is characterized by its reflection coefficient V_reflected / V_incident.) The blue circle, centered within the chart, is sometimes called an SWR circle (short for constant standing wave ratio).

The characteristic impedance Z₀ of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.

The impedance measured at a given distance, l, from the load impedance Z_L may be expressed as,

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

where γ is the propagation constant and $\Gamma _{L}=\left(Z_{L}-Z_{0}\right)/\left(Z_{L}+Z_{0}\right)$ is the voltage reflection coefficient at the load end of the transmission line. Alternatively, the above formula can be rearranged to express the input impedance in terms of the load impedance rather than the load voltage reflection coefficient:

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

Input impedance of lossless transmission line

For a lossless transmission line, the propagation constant is purely imaginary, γ=jβ, so the above formulas can be rewritten as,

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

where $\beta ={\frac {2\pi }{\lambda }}$ is the wavenumber.

In calculating β, the wavelength is generally different inside the transmission line to what it would be in free-space and the velocity constant of the material the transmission line is made of needs to be taken into account when doing such a calculation.

Special cases of lossless transmission lines

Half wave length

For the special case where $\beta l=n\pi$ where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that

$Z_{\mathrm {in} }=Z_{L}\,$

for all n. This includes the case when n=0, meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.

Quarter wave length

主条目：quarter wave impedance transformer

For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes

$Z_{\mathrm {in} }={\frac {{Z_{0}}^{2}}{Z_{L}}}.\,$

Matched load

Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that

$Z_{\mathrm {in} }=Z_{L}=Z_{0}\,$

for all $l$ and all $\lambda$.

Short

Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Colors represent voltages, and black dots represent electrons.

主条目：stub

For the case of a shorted load (i.e. $Z_{L}=0$), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)

$Z_{\mathrm {in} }(l)=jZ_{0}\tan(\beta l).\,$

Open

主条目：stub

For the case of an open load (i.e. $Z_{L}=\infty$), the input impedance is once again imaginary and periodic

$Z_{\mathrm {in} }(l)=-jZ_{0}\cot(\beta l).\,$

Stepped transmission line

A simple example of stepped transmission line consisting of three segments.

A stepped transmission line is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be Z_0,i. The input impedance can be obtained from the successive application of the chain relation

$Z_{\mathrm {i+1} }=Z_{\mathrm {0,i} }{\frac {Z_{i}+jZ_{\mathrm {0,i} }\tan(\beta _{i}l_{i})}{Z_{\mathrm {0,i} }+jZ_{i}\tan(\beta _{i}l_{i})}}$

where $\beta _{i}$ is the wave number of the ith transmission line segment and l_i is the length of this segment, and Z_i is the front-end impedance that loads the ith segment.

The impedance transformation circle along a transmission line whose characteristic impedance Z_0,i is smaller than that of the input cable Z₀. And as a result, the impedance curve is off-centered towards the -x axis. Conversely, if Z_0,i > Z₀, the impedance curve should be off-centered towards the +x axis.

Because the characteristic impedance of each transmission line segment Z_0,i is often different from that of the input cable Z₀, the impedance transformation circle is off centered along the x axis of the Smith Chart whose impedance representation is usually normalized against Z₀.

Practical types

Coaxial cable

主条目：coaxial cable

Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them. In radio-frequency applications up to a few gigahertz, the wave propagates in the transverse electric and magnetic mode (TEM) only, which means that the electric and magnetic fields are both perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential). However, at frequencies for which the wavelength (in the dielectric) is significantly shorter than the circumference of the cable, transverse electric (TE) and transverse magnetic (TM) waveguide modes can also propagate. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another.

The most common use for coaxial cables is for television and other signals with bandwidth of multiple megahertz. In the middle 20th century they carried long distance telephone connections.

A type of transmission line called a cage line, used for high power, low frequency applications. It functions similarly to a large coaxial cable. This example is the antenna feedline for a longwave radio transmitter in Poland, which operates at a frequency of 225 kHz and a power of 1200 kW.

Microstrip

主条目：microstrip

A microstrip circuit uses a thin flat conductor which is parallel to a ground plane. Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating layer determine the characteristic impedance. Microstrip is an open structure whereas coaxial cable is a closed structure.

Stripline

主条目：Stripline

A stripline circuit uses a flat strip of metal which is sandwiched between two parallel ground planes. The insulating material of the substrate forms a dielectric. The width of the strip, the thickness of the substrate and the relative permittivity of the substrate determine the characteristic impedance of the strip which is a transmission line.

Balanced lines

主条目：Balanced line

A balanced line is a transmission line consisting of two conductors of the same type, and equal impedance to ground and other circuits. There are many formats of balanced lines, amongst the most common are twisted pair, star quad and twin-lead.

Twisted pair

主条目：Twisted pair

Twisted pairs are commonly used for terrestrial telephone communications. In such cables, many pairs are grouped together in a single cable, from two to several thousand. The format is also used for data network distribution inside buildings, but the cable is more expensive because the transmission line parameters are tightly controlled.

Star quad

Star quad is a four-conductor cable in which all four conductors are twisted together around the cable axis. It is sometimes used for two circuits, such as 4-wire telephony and other telecommunications applications. In this configuration each pair uses two non-adjacent conductors. Other times it is used for a single, balanced circuit, such as audio applications and 2-wire telephony. In this configuration two non-adjacent conductors are terminated together at both ends of the cable, and the other two conductors are also terminated together.

Interference picked up by the cable arrives as a virtually perfect common mode signal, which is easily removed by coupling transformers. Because the conductors are always the same distance from each other, cross talk is reduced relative to cables with two separate twisted pairs.

The combined benefits of twisting, differential signalling, and quadrupole pattern give outstanding noise immunity, especially advantageous for low signal level applications such as long microphone cables, even when installed very close to a power cable. The disadvantage is that star quad, in combining two conductors, typically has double the capacitance of similar two-conductor twisted and shielded audio cable. High capacitance causes increasing distortion and greater loss of high frequencies as distance increases.

Twin-lead

主条目：Twin-lead

Twin-lead consists of a pair of conductors held apart by a continuous insulator.

Lecher lines

主条目：Lecher lines

Lecher lines are a form of parallel conductor that can be used at UHF for creating resonant circuits. They are a convenient practical format that fills the gap between lumped components (used at HF/VHF) and resonant cavities (used at UHF/SHF).

Single-wire line

Unbalanced lines were formerly much used for telegraph transmission, but this form of communication has now fallen into disuse. Cables are similar to twisted pair in that many cores are bundled into the same cable but only one conductor is provided per circuit and there is no twisting. All the circuits on the same route use a common path for the return current (earth return). There is a power transmission version of single-wire earth return in use in many locations.

General applications

Signal transfer

Electrical transmission lines are very widely used to transmit high frequency signals over long or short distances with minimum power loss. One familiar example is the down lead from a TV or radio aerial to the receiver.

Pulse generation

Transmission lines are also used as pulse generators. By charging the transmission line and then discharging it into a resistive load, a rectangular pulse equal in length to twice the electrical length of the line can be obtained, although with half the voltage. A Blumlein transmission line is a related pulse forming device that overcomes this limitation. These are sometimes used as the pulsed power sources for radar transmitters and other devices.

Stub filters

参见：Distributed element filter § Stub band-pass filters

If a short-circuited or open-circuited transmission line is wired in parallel with a line used to transfer signals from point A to point B, then it will function as a filter. The method for making stubs is similar to the method for using Lecher lines for crude frequency measurement, but it is 'working backwards'. One method recommended in the RSGB's radiocommunication handbook is to take an open-circuited length of transmission line wired in parallel with the feeder delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in the strength of the signal observed at a receiver can be found. At this stage the stub filter will reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the stub will become a filter rejecting the even harmonics.

Acoustic transmission lines

主条目：Acoustic transmission line

An acoustic transmission line is the acoustic analog of the electrical transmission line, typically thought of as a rigid-walled tube that is long and thin relative to the wavelength of sound present in it.

Solutions of the telegrapher's equations as circuit components

Equivalent circuit of a transmission line described by the Telegrapher's equations.

Solutions of the Telegrapher's Equations as Components in the Equivalent Circuit of a Balanced Transmission Line Two-Port Implementation.

The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the top figure implements the solutions of the telegrapher's equations.

The bottom circuit is derived from the top circuit by source transformations. It also implements the solutions of the telegrapher's equations.

The solution of the telegrapher's equations can be expressed as an ABCD type Two-port network with the following defining equations

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

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

The symbols: $E_{s},E_{L},I_{s},I_{L},l\,$ in the source book have been replaced by the symbols :$V_{1},V_{2},I_{1},I_{2},x\,$ in the preceding two equations.

The ABCD type two-port gives $V_{1}\,$ and $I_{1}\,$ as functions of $V_{2}\,$ and $I_{2}\,$. Both of the circuits above, when solved for $V_{1}\,$ and $I_{1}\,$ as functions of $V_{2}\,$ and $I_{2}\,$ yield exactly the same equations.

In the bottom circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedances Z(s), the voltage dependent current sources (VDCSs) and the difference amplifiers (the triangle with the number "1") account for the interaction of the transmission line with the external circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the T(s) blocks carries the forward wave and the other carries the backward wave. The circuit, as depicted, is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from $V_{1}\,$ to $V_{2}\,$ in the sense that $V_{1}\,$, $V_{2}\,$, $I_{1}\,$ and $I_{2}\,$ would be same whether this circuit or an actual transmission line was connected between $V_{1}\,$ and $V_{2}\,$. There is no implication that there are actually amplifiers inside the transmission line.

Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which may be called shield, sheath, common, Earth or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential and common modes. The circuit shown on the bottom only models the differential mode.

In the top circuit, the voltage doublers, the difference amplifiers and impedances Z(s) account for the interaction of the transmission line with the external circuit. This circuit, as depicted, is also fully symmetric, and also not drawn that way. This circuit is a useful equivalent for an unbalanced transmission line like a coaxial cable or a micro strip line.

These are not the only possible equivalent circuits.

See also

• Distributed element model
• Electric power transmission
• Heaviside condition
• Longitudinal electromagnetic wave
• Lumped components
• Propagation velocity
• Radio frequency power transmission
• Smith chart, a graphical method to solve transmission line equations
• Standing wave
• Time domain reflectometer
• Transverse electromagnetic wave

References

Part of this article was derived from Federal Standard 1037C.

• Steinmetz, Charles Proteus, The Natural Period of a Transmission Line and the Frequency of lightning Discharge Therefrom, The Electrical World, August 27, 1898: 203–205 
• Grant, I. S.; Phillips, W. R., Electromagnetism 2nd, John Wiley, ISBN 0-471-92712-0 
• Ulaby, F. T., Fundamentals of Applied Electromagnetics 2004 media, Prentice Hall, ISBN 0-13-185089-X 
• Chapter 17, Radio communication handbook, Radio Society of Great Britain: 20, 1982, ISBN 0-900612-58-4 
• Naredo, J. L.; Soudack, A. C.; Marti, J. R., Simulation of transients on transmission lines with corona via the method of characteristics, IEE Proceedings. Generation, Transmission and Distribution. (Morelos: Institution of Electrical Engineers), Jan 1995, 142 (1), ISSN 1350-2360 

Further reading

• Annual Dinner of the Institute at the Waldorf-Astoria. Transactions of the American Institute of Electrical Engineers, New York, January 13, 1902. (Honoring of Guglielmo Marconi, January 13, 1902)
• Avant! software, Using Transmission Line Equations and Parameters. Star-Hspice Manual, June 2001.
• Cornille, P, On the propagation of inhomogeneous waves. J. Phys. D: Appl. Phys. 23, February 14, 1990. (Concept of inhomogeneous waves propagation — Show the importance of the telegrapher's equation with Heaviside's condition.)
• Farlow, S.J., Partial differential equations for scientists and engineers. J. Wiley and Sons, 1982, p. 126. ISBN 0-471-08639-8.
• Kupershmidt, Boris A., Remarks on random evolutions in Hamiltonian representation. Math-ph/9810020. J. Nonlinear Math. Phys. 5 (1998), no. 4, 383-395.
• Transmission line matching. EIE403: High Frequency Circuit Design. Department of Electronic and Information Engineering, Hong Kong Polytechnic University. (PDF format)
• Wilson, B. (2005, October 19). Telegrapher's Equations. Connexions.
• John Greaton Wöhlbier, ""Fundamental Equation" and "Transforming the Telegrapher's Equations". Modeling and Analysis of a Traveling Wave Under Multitone Excitation.
• Agilent Technologies. Educational Resources. Wave Propagation along a Transmission Line. Edutactional Java Applet.
• Qian, C., Impedance matching with adjustable segmented transmission line. J. Mag. Reson. 199 (2009), 104–110.

External links

• Transmission Line Parameter Calculator
• Interactive applets on transmission lines
• SPICE Simulation of Transmission Lines