Transmission line

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. Components of transmission lines include wires, coaxial cables, dielectric slabs, optical fibers, electric power lines, and waveguides.

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. ^[1]

Transmission line vs wire

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 (justified in the input impedance section) 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

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

Image:Transmission line 4 port.svg

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 source and load impedances equal to Z₀, in which case the transmission line is said to be matched.

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, 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 transmission lines that are 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

Main article: Telegraphers 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 Failed to parse (Missing texvc executable; please see math/README to configure.): R

of the conductors is represented by a series resistor (expressed in ohms per unit length).

• The distributed inductance Failed to parse (Missing texvc executable; please see math/README to configure.): L

(due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).

• The capacitance Failed to parse (Missing texvc executable; please see math/README to configure.): C

between the two conductors is represented by a shunt capacitor C (farads per unit length).

• The conductance Failed to parse (Missing texvc executable; please see math/README to configure.): G

of the dielectric material separating the two conductors is represented by a conductance G shunted 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. Failed to parse (Missing texvc executable; please see math/README to configure.): R , Failed to parse (Missing texvc executable; please see math/README to configure.): L , Failed to parse (Missing texvc executable; please see math/README to configure.): C , and Failed to parse (Missing texvc executable; please see math/README to configure.): G

may also be functions of frequency. An alternative notation is to use Failed to parse (Missing texvc executable; please see math/README to configure.): R'

, Failed to parse (Missing texvc executable; please see math/README to configure.): L' , Failed to parse (Missing texvc executable; please see math/README to configure.): C'

and Failed to parse (Missing texvc executable; please see math/README to configure.): G'
to emphasize that the values are derivatives with respect to length.

The line voltage Failed to parse (Missing texvc executable; please see math/README to configure.): V(x)

and the current Failed to parse (Missing texvc executable; please see math/README to configure.): I(x)
can be expressed in the frequency domain as

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial V(x)}{\partial x} = -(R + j \omega L)I(x)

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial I(x)}{\partial x} = -(G + j \omega C)V(x)

When the elements Failed to parse (Missing texvc executable; please see math/README to configure.): R

and Failed to parse (Missing texvc executable; please see math/README to configure.): G
are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the Failed to parse (Missing texvc executable; please see math/README to configure.): L
and Failed to parse (Missing texvc executable; please see math/README to configure.): C
elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial^2V(x)}{\partial x^2}+ \omega^2 LC\cdot V(x)=0

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): R

and Failed to parse (Missing texvc executable; please see math/README to configure.): G
are not neglected, the Telegrapher's equations become:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial^2V(x)}{\partial x^2} = \Gamma^2 V(x)

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial^2I(x)}{\partial x^2} = \Gamma^2 I(x)

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma = \sqrt{(R + j \omega L)(G + j \omega C)}

and the characteristic impedance is:

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_0 = \sqrt{\frac{R + j \omega L}{G + j \omega C}}

The solutions for Failed to parse (Missing texvc executable; please see math/README to configure.): V(x)

and Failed to parse (Missing texvc executable; please see math/README to configure.): I(x)
are:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(x) = V_- e^{-\Gamma x} + V_+ e^{\Gamma x} \,

Failed to parse (Missing texvc executable; please see math/README to configure.): I(x) = I_- e^{-\Gamma x} + I_+ e^{\Gamma x} \,

The constants Failed to parse (Missing texvc executable; please see math/README to configure.): V_\pm

and Failed to parse (Missing texvc executable; please see math/README to configure.): I_\pm
must be determined from boundary conditions. For a voltage pulse Failed to parse (Missing texvc executable; please see math/README to configure.): V_{in}(t) \,

, starting at Failed to parse (Missing texvc executable; please see math/README to configure.): x=0

and moving in the positive Failed to parse (Missing texvc executable; please see math/README to configure.): x

-direction, then the transmitted pulse Failed to parse (Missing texvc executable; please see math/README to configure.): V_{out}(x,t) \,

at position Failed to parse (Missing texvc executable; please see math/README to configure.): x
can be obtained by computing the Fourier Transform, Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{V}(\omega)

, of Failed to parse (Missing texvc executable; please see math/README to configure.): V_{in}(t) \, , attenuating each frequency component by Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-Re(\Gamma) x} \, , advancing its phase by Failed to parse (Missing texvc executable; please see math/README to configure.): -Im(\Gamma)x \, , and taking the inverse Fourier Transform. The real and imaginary parts of Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma

can be computed as

Failed to parse (Missing texvc executable; please see math/README to configure.): Re(\Gamma) = (a^2 + b^2)^{1/4} \cos(\mathrm{atan2}(b,a)/2) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): Im(\Gamma) = (a^2 + b^2)^{1/4} \sin(\mathrm{atan2}(b,a)/2) \,

where atan2 is the two-parameter arctangent, and

Failed to parse (Missing texvc executable; please see math/README to configure.): a \equiv \omega^2 LC \left[ \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right]

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): R / \omega L

and Failed to parse (Missing texvc executable; please see math/README to configure.): G / \omega C
one obtains

Failed to parse (Missing texvc executable; please see math/README to configure.): Re(\Gamma) \approx \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): Im(\Gamma) \approx \omega \sqrt{LC} \,

Noting that an advance in phase by Failed to parse (Missing texvc executable; please see math/README to configure.): - \omega \delta

is equivalent to a time delay by Failed to parse (Missing texvc executable; please see math/README to configure.): \delta

, Failed to parse (Missing texvc executable; please see math/README to configure.): V_{out}(t)

can be simply computed as

Failed to parse (Missing texvc executable; please see math/README to configure.): V_{out}(x,t) \approx V_{in}(t - \sqrt{LC}x) e^{- \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) x } \,

Input impedance of a transmission line

The characteristic impedance Failed to parse (Missing texvc executable; please see math/README to configure.): Z_0

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 Failed to parse (Missing texvc executable; please see math/README to configure.): l

from the load impedance Failed to parse (Missing texvc executable; please see math/README to configure.): Z_L
is

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_{in} (l)=Z_0 \frac{Z_L\cos(\beta l) + Z_0j\sin(\beta l)}{Z_0\cos(\beta l) + Z_Lj\sin(\beta l)}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \beta=\frac{2\pi}{\lambda}

is the wavenumber.

For the special case where Failed to parse (Missing texvc executable; please see math/README to configure.): \beta l\approx n\pi

where n is an integer (meaning that the length of the line is a very close to a multiple of half a wavelength), the expression reduces to the load impedance so that Failed to parse (Missing texvc executable; please see math/README to configure.): Z_{in}=Z_L
for all Failed to parse (Missing texvc executable; please see math/README to configure.): l

. This includes the case when Failed to parse (Missing texvc executable; please see math/README to configure.): n=0 , meaning that the length of the transmission line is less than about 1/100 of the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.

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 Failed to parse (Missing texvc executable; please see math/README to configure.): Z_{in}=Z_0

In calculating Failed to parse (Missing texvc executable; please see math/README to configure.): \beta , 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.

Practical types of electrical transmission line

Coaxial cable

Main article: 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), which means that the electric and magnetic fields are both perpendicular to the direction of propagation. However, above a certain frequency called the cutoff frequency, the cable behaves as a waveguide, and propagation switches to either a transverse electric (TE) or a transverse magnetic (TM) mode or a mixture of modes. This effect enables coaxial cables to be used at microwave frequencies, although they are not as efficient as the more expensive, purpose-built waveguides.

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

Main article: 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.

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

Twin-lead

Main article: Twin-lead

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

Lecher lines

Main article: Lecher lines

Lecher lines are a form of parallel conductor that can be used at UHF for creating resonant circuits. They are used at frequencies between HF/VHF where lumped components are used, and UHF/SHF where resonant cavities are more practical.

General applications of transmission lines

Transferring signals from one point to another

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

Main article: Acoustic transmission lines

See also

• Heaviside condition
• Smith chart a graphical method to solve transmission line equations
• Transverse electromagnetic wave
• Longitudinal electromagnetic wave
• Electric power transmission
• Radio Frequency Power Transmission
• Standing wave

References

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

1. ^ 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

External articles and 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.
• Han, Hsiu C., Transmission-Line Equations. EE 313 Electromagnetic Fields and Waves.
• Kupershmidt, Boris A., Remarks on random evolutions in Hamiltonian representation. Math-ph/9810020. J. Nonlinear Math. Phys. 5 (1998), no. 4, 383-395.
• Pupin, M., U.S. Patent 1,541,845 , Electrical wave transmission.
• 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.de:Elektrische Leitung

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