使用者:Gaoxuewei/sandbox

此頁面介紹的是電波傳輸線。關於電力傳輸線，請見「輸電系統」。

電波在無損耗傳輸線內流動原理圖。紅色代表高電壓，藍色代表低電壓。黑色圓點代表電子。傳輸線接於阻抗匹配的負載電阻（右邊的盒子）上，波形完全被吸收。

一種最常見的傳輸線——同軸電纜。

在通信工程和電子工程中，傳輸線是一種特殊的電纜或者其他結構，被設計用於承載電波中變化的電流，也就是說，電流的頻率高到一定程度時它們波的本質必須進行考慮。傳輸線一般用於連接發送器與接收器的天線，傳輸有線電視信號，中繼電信交換中心之間的路由呼叫，中繼計算機網絡鏈接以及中繼高速計算機總線。

在此僅討論雙導體傳輸線，包含平行線（梯線）、同軸電纜、帶狀線和微帶線。一些人認為波導管、介質波導甚至光纖也是傳輸線，然而這些線需要不同的分析技術，所以不在此進行討論；可參見電磁波導。

概述

普通電纜足以攜帶低頻交流電(例如家庭用電，每秒鐘變換100~120次方向）和聲音信號。然而，普通電纜不能用於承載電波頻率範圍的電流或更高頻率的電流 ，這種頻率的電流每秒鐘變更百萬次方向，能量易於從電纜中以電磁波的形式輻射出來，從而造成能量損耗。高頻電流也容易在電纜的連接處（如連接器和節點）反射回電源。 這些反射作為瓶頸，阻止了信號功率到達目的地。傳輸線使用了特殊的結構和阻抗匹配的方法，承載電磁信號以最小的反射和最小的功率損耗到達接收端。大多數傳輸線的顯着特點是它們具有沿其長度方向均勻的橫截面尺寸，使得傳輸線有着一致的阻抗，被稱為特性阻抗， 從而防止了反射的發生。傳輸線有多種形態，例如平行線(梯線、雙絞線)、同軸電纜、帶狀線以及微帶線。 電磁波的頻率與波長成反比。當線纜的長度與傳輸信號的波長相當時，就必須要使用傳輸線了。

傳輸微波頻率信號時，傳輸線的功率損失也會比較明顯，這時應當使用波導管替代傳輸線 ，波導管的功能是作為限制和引導電磁波的「管道」。 一些人將波導管視為一種傳輸線； 然而，這裡認為波導管和傳輸線是不同的。在更高的頻率上，例如太赫茲、紅外線、光的範圍，波導管也將對信號造成損失，這時需要使用光學方法（如稜鏡和鏡子）來引導電磁波。

聲波傳播的數學理論與電磁波的傳播數學理論是非常相似的，因此傳輸線的理論也被用來指導聲波的傳播；叫做聲學傳輸線。

歷史

電傳輸線的數學分析源於麥克斯韋、開爾文男爵和赫維賽德的工作。1855年開爾文男爵建立了一個關於海底電纜電流的微分模型。這個模型正確的預測了1858年穿越大西洋海底通信電纜的非良好性能。在1885年赫維賽德發表了第一篇關於描述他的電纜傳播分析和現代通信模式方程的論文。

適用範圍

在許多電子線路中，連接各器件的電線的長度是基本可以被忽略的。也就是說在電線各點同一時刻的電壓可以認為是相同的。但是，當電壓的變化和信號沿電線傳播下去的時間可以比擬時，電線的長度變得重要了，這時電線就必須被處理成傳輸線。換言之，當信號所包含的頻率分量的相應的波長較之電線長度小或二者可以比擬的時候，電線的長度是很重要的。

常見的經驗方法認為如果電纜或者電線的長度大於波長的1/10，則需被作為傳輸線處理。 在這個長度下相位延遲和線中的反射干擾非常顯著，那麼沒有用傳輸線理論仔細的研究設計過的系統就會出現一些不可預知行為。

四終端模型

傳輸線在電路圖中各種電路符號。

為了分析的需要，傳輸線可以用二端口網絡（四端網絡）進行建模，如下圖所：

在最簡單的情況，假設網絡是線性的（即任何端口之間的復電壓在沒有反射的情況下正比於復電流），且兩個端口可以互換。如果傳輸線在長度範圍內是均勻的，那麼其特性可以只用一個參數描述：特性阻抗， 符號是 Z₀ 。 特性阻抗是某一給定電波在傳輸線上任意一點復電壓與復電流的比值。常見電纜阻抗Z₀的典型數值：同軸電纜 - 50或75歐姆, 扭絞二股線 - 約100歐姆，廣播傳輸用的平行二股線 - 約300歐姆。

當在傳輸線上發送功率時， 最好的情況是儘可能多的功率被負載吸收，儘可能少的功率被反射回發送端。在負載阻抗等於特性阻抗Z₀時，這一點可以被保證，這時傳輸線被稱為阻抗匹配。

圖中兩條黑線代表傳輸線。在距離起點 x 處，每條線都流過了 I(x) 的電流，兩線之間的電壓差為 V(x) 。在單一信號沒有反射的情況下， V(x) / I(x) = Z₀， Z₀ 代表了傳輸線的 特性阻抗 。

由於傳輸線電阻的存在，一些被發送到傳輸線上的功率被損耗。這種現象叫做電阻損耗。在高頻處，另一種介電損耗變得非常明顯，加重了電阻引起的損耗。介電損耗是由於在傳輸線內的絕緣材料從電域吸收能量轉化為熱引起的。 傳輸線模型表現為電阻 (R) 與電感 (L) 的串聯以及電容 (C) 與電導 (G) 的並聯。電阻與電導引起了傳輸線的損耗。

傳輸線功率總損耗的單位是分貝每米 (dB/m)，並與信號頻率相關。生產廠家一般會提供一定範圍內以dB/m為單位的損耗圖。3dB代表大約損失一半的功率。

設計用於承載波長小於或可比於傳輸線長度電磁波的傳輸線稱為高頻傳輸線。在這種情況下，在低頻下的估值方法不再適用。高頻傳輸線常見於無線電，微波，光信號，金屬網濾光片和高速電子線路中的信號。

Telegrapher's equations

主條目：Telegrapher's equations

參見：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

主條目：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.

主條目：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

主條目：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

主條目：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

主條目：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

主條目：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

主條目：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

主條目：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

主條目：Twin-lead

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

Lecher lines

主條目：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

參見：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

主條目：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