# Transmission line: Wikis

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# Encyclopedia

A transmission line is the material medium or structure that forms all or part of a path from one place to another for directing the transmission of energy, such as electromagnetic waves or acoustic waves, as well as electric power transmission. Types of transmission line include wires, coaxial cables, dielectric slabs, striplines, optical fibers, electric power lines, and waveguides.[1]

## History

Mathematical analysis of the behaviour of electrical transmission lines grew out of the work of James Clerk Maxwell, Lord Kelvin and Oliver Heaviside. In 1855 Lord Kelvin formulated a diffusion model of the current in a submarine cable. The model correctly predicted the poor performance of the 1858 trans-Atlantic submarine telegraph cable. In 1885 Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the telegrapher's equations.[2]

## Applicability

In many electric circuits, the length of the wires connecting the components can for the most part be ignored. That is, the voltage on the wire at a given time can be assumed to be the same at all points. However, when the voltage changes in a time interval comparable to the time it takes for the signal to travel down the wire, the length becomes important and the wire must be treated as a transmission line. Stated another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to or less than the length of the wire.

A common rule of thumb is that the cable or wire should be treated as a transmission line if the length is greater than 1/10 of the wavelength. At this length the phase delay and the interference of any reflections on the line become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.

## The four terminal model

Variations on the schematic electronic symbol for a transmission line.

For the purposes of analysis, an electrical transmission line can be modelled as a two-port network (also called a quadrupole network), as follows:

In the simplest case, the network is assumed to be linear (i.e. the complex voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable. If the transmission line is uniform along its length, then its behaviour is largely described by a single parameter called the characteristic impedance, symbol Z0. This is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z0 are 50 or 75 ohms for a coaxial cable, about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission.

When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the load impedance equal to Z0, in which case the transmission line is said to be matched. Ensuring the source impedance matches Z0 will maximize power transfer from the source to the transmission line, but has no other effect on the behavior of the line.

Some of the power that is fed into a transmission line is lost because of its resistance. This effect is called ohmic or resistive loss (see ohmic heating). At high frequencies, another effect called dielectric loss becomes significant, adding to the losses caused by resistance. Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to heat (see dielectric heating).

The total loss of power in a transmission line is often specified in decibels per metre (dB/m), and usually depends on the frequency of the signal. The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies. A loss of 3 dB corresponds approximately to a halving of the power.

High-frequency transmission lines can be defined as those designed to carry electromagnetic waves whose wavelengths are shorter than or comparable to the length of the line. Under these conditions, the approximations useful for calculations at lower frequencies are no longer accurate. This often occurs with radio, microwave and optical signals, and with the signals found in high-speed digital circuits.

## Telegrapher's equations

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:

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

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:

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

where

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

and the characteristic impedance is:

$Z_0 = \sqrt{\frac{R + j \omega L}{G + j \omega C}}$

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 Γ 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 / ωL and G / ωC 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 − ωδ is equivalent to a time delay by δ, Vout( 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 lossless transmission line

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.

For a lossless transmission line, it can be shown that the impedance measured at a given position l from the load impedance ZL is

$Z_\mathrm{in} (l)=Z_0 \frac{Z_L + jZ_0\tan(\beta l)}{Z_0 + jZ_L\tan(\beta l)}$

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

#### Half wave length

For the special case where βl = nπ 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

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} \$

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 λ.

#### Short

Main article: stub

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

$Z_\mathrm{in} (l)=j Z_0 \tan(\beta l) \,$

#### Open

Main article: stub

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

$Z_\mathrm{in} (l)=-j Z_0 \cot(\beta l) \,$

### Stepped transmission line

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

Stepped transmission line is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in serial, with the characteristic impedance of each individual element to be, Z0,i. And 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 βi is the wave number of the ith transmission line segment and li is the length of this segment, and Zi is the front-end impedance that loads the ith segment.

The impedance transformation circle along a transmission line whose characteristic impedance Z0,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 Z0,i > Z0, the impedance curve should be off-centered towards the +x axis.

Because the characteristic impedance of each transmission line segment Z0,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

Coaxial lines confine the electromagnetic wave to the area inside the cable, between the center conductor and the shield. The transmission of energy in the line occurs totally through the dielectric inside the cable between the conductors. 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.

### 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

Main article : 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

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

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 in this case the cable used is more expensive with much tighter controlled parameters and either two or four pairs per cable.

Star quad is another balanced format used at low frequencies. Applications include 4-wire telephony and microphone circuits.

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

#### 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.

### Waveguide

Waveguides are rectangular or circular metallic tubes inside which an electromagnetic wave is propagated and is confined by the tube. Waveguides are not capable of transmitting the transverse electromagnetic mode found in copper lines and must use some other mode. Consequently, they cannot be directly connected to cable and a mechanism for launching the waveguide mode must be provided at the interface.

### Optical fiber

Optical fibers are a solid transparent fiber of glass or polymer which transmits a signal at optical, or near infrared, wavelengths. They form the backbone of all modern terrestrial communications networks due to the very high bandwidths that can be achieved. Optical fiber are another variety of waveguide.

## 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 energy sources for radar transmitters and other devices.

### Stub 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

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.

## References

1. ^ Federal Standard 1037C
2. ^ Ernst Weber and Frederik Nebeker, The Evolution of Electrical Engineering, IEEE Press, Piscataway, New Jersey USA, 1994 ISBN 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 0-471-92712-0
• Fundamentals Of Applied Electromagnetics 2004 media edition., Ulaby, F.T., pub Prentice Hall, ISBN 0-13-185089-X
• Radiocommunication handbook, page 20, chaper 17, RSGB, ISBN 0-900612-58-4
• 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