Transmission line

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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 fibres, electric power lines, and waveguides.

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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. This law 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. Template:Ref

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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:Transmissionline4port.png

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 to the complex current 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 all the power is absorbed by the load and none of it is 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. 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.

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 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.

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Telegrapher's equations

Oliver Heaviside developed the transmission line model, also known as the telegrapher's equations, that describes how electrical voltage and current vary along a conductor.

The theory applies to high-frequency transmission lines (such as telegraph wires and radio frequency conductors) but is also important for designing high-voltage energy transmission lines. The equations consist of two linear differential equations in time and position: one for V(x, t) and the other one for I(x, t). The model demonstrates that the electrical current can be reflected on the wire, and that wave patterns can appear along the line.

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The equations

Image:Transmission line element.svgThe telegrapher's equations can be understood as a simplified case of Maxwell's equations. In a more practical approach, one assumes that the conductor is composed out of an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

  • The resistivity of the conductor (expressed in ohms per unit length) is represented by a series resistor R.
  • The distributed inductance (due to the magnetic field around the wire, self-inductance, etc.) is represented by a series coil (inductance per unit length L).
  • The capacitive behaviour of the insulation between the signal wire and the return wire is represented by a shunt capacitor C (capacitance per unit length).
  • The conductivity of the insulation material is accounted for by a resistor G shunted between the signal wire and the return wire (conductance per unit length).

When the elements R and G are very small, their effects can be neglected, and the transmission line is considered as an idealized, lossless, structure. In this case, the model depends only on the L and C elements, and we obtain a pair of first-order partial differential equations, one function describing the voltage V along the line and the other the current I, both function of position x and time t:

<math>

\frac{\partial}{\partial x} V(x,t) = -L \frac{\partial}{\partial t} I(x,t) </math>

<math>

\frac{\partial}{\partial x} I(x,t) = -C \frac{\partial}{\partial t} V(x,t) </math>

These equations may be combined to form either of two exact wave equations:

<math>

\frac{\partial^2}{{\partial t}^2} V = \frac{1}{LC} \frac{\partial^2}{{\partial x}^2} V </math>

<math>

\frac{\partial^2}{{\partial t}^2} I = \frac{1}{LC} \frac{\partial^2}{{\partial x}^2} I </math>

If the line has infinite length or when it is terminated with its characteristic impedance, these equations indicate the presence of a wave, travelling with a speed <math>c = \frac{1}{\sqrt{LC}}</math>.

(Note that this propagation speed applies to the wave phenomenon on the line and has nothing to do with the electron drift velocity.) For a coaxial transmission line, made of perfect conductors with vacuum between them, it can be shown that this speed is equal to the speed of light.

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Lossy transmission line

When the loss elements R and G are not negligible, the original differential equations describing the elementary segment of line become

<math>

\frac{\partial}{\partial x} V(x,t) = -L \frac{\partial}{\partial t} I(x,t) - R I(x,t) </math>

<math>

\frac{\partial}{\partial x} I(x,t) = -C \frac{\partial}{\partial t} V(x,t) - G V(x,t) </math>

By differentiating the first equation with respect to x and the second with respect to t, and some algebraic manipulation, we obtain a pair of hyperbolic partial differential equations each involving only one unknown:

<math>

\frac{\partial^2}{{\partial x}^2} V = L C \frac{\partial^2}{{\partial t}^2} V + (R C + G L) \frac{\partial}{\partial t} V + G R V </math>

<math>

\frac{\partial^2}{{\partial x}^2} I = L C \frac{\partial^2}{{\partial t}^2} I + (R C + G L) \frac{\partial}{\partial t} I + G R I </math>

Note that these equations resemble the homogeneous wave equation with extra terms in V and I and their first derivatives. These extra terms cause the signal to decay and spread out with time and distance.

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Direction of signal propagation

The wave equations above indicate that there are two solutions for the travelling wave: one forward and one reverse.

<math>V(x,t) \ = \ f_1(\omega t-kx) + f_2(\omega t+kx)</math>

where:

<math> k = \omega \sqrt{LC} = {\omega \over v}</math>
is called the wave number and has units of radians per meter,
ω is the angular frequency (in radians per second),
<math>f_1</math> and <math>f_2</math> can be any functions whatsoever, and
<math>v = { 1 \over \sqrt{LC} } </math>
is the waveform's speed of propagation.

<math>f_1</math> represents a wave travelling from left to right in a positive x direction whilst <math>f_2</math> represents a wave travelling from right to left. It can be seen that the instantaneous voltage at any point x on the line is the sum of the voltages due to both waves.

Since the current I is related to the voltage V by the telegrapher's equations, we can write

<math>I(x,t) \ = \ { f_1(\omega t-kx) \over Z } - { f_2(\omega t+kx) \over Z }</math>

where

<math> Z = \sqrt { {L \over C} } </math>

is the characteristic impedance (in ohms) of the transmission line.

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Cable impedance variation with frequency

The impedance of normal transmission lines is not constant, but varies with frequency. At low frequencies, when

<math>\omega L \ll R</math> and <math>\omega C \ll G</math>,

the characteristic impedance of a transmission line is

<math>Z_0 = \sqrt{R/G}</math>.

At high frequencies where

<math>\omega L \gg R</math>, and <math>\omega C \gg G</math>,

then the characterstic impedance is

<math>Z_0 = \sqrt{L/C}</math>.

So there are two distinct characteristic impedances for every line. Usually G is very small so the low-frequency impedance is high, whereas the high-frequency impedance is low. The break points in the impedance frequency graph are at <math>\omega_1 = G/C</math> and <math>\omega_2 = R/L</math> (where <math>\omega =2 \pi f</math>). If <math>R/G \gg L/C</math>, it is obvious that <math>\omega_2 \gg \omega_1</math>. Between these two break frequencies the cable impedance decreases smoothly.

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Example

Take the case of a 50Ω coax with polyethylene dielectric. R is about 100 mΩ/m and G < 20 pS/m (based on measurements of leakage resistance in a 1 m length). Using <math>L=CZ^2</math>, L can be calculated at about 250 nH/m. So,

ω2 = R/L = 200 krad/s (f2 = 30 kHz)

and

ω1 = G/C = 0.2 rad/s (f1 = 30 millihertz)

At 100 Hz the 50 ohm coax will have an impedance of about 900 ohms, only reaching 50 ohms at about 30 or 40 kHz. The phase angle of the impedance between the two break frequencies is leading (the cable looks capacitive).

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Practical types of electrical transmission line

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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 magnetic (TEM) mode, 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.

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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.

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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.

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Balanced lines

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Lecher lines

Template:Main 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.

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General applications of transmission lines

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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.

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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. These are sometimes used as the pulsed energy sources for radar transmitters and other devices.

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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.


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Acoustic transmission lines

A duct for sound propagation also behaves like a transmission line (e.g. air conditioning duct, car muffler, ...). The duct contains some medium, such as air, that supports sound propagation. Its length is normally of a similar order than the wavelengths of the sound it will be used with, but the dimensions of its cross-section are normally smaller than one quarter of a wavelength. Sound is introduced at one end of the tube by forcing the pressure across the whole cross-section to vary with time. A plane wave will travel down the line at the speed of sound. When the wave reaches the end of the transmission line, behaviour depends on what is present at the end of the line. There are three possible scenarios:

  • A low impedance load (e.g. leaving the end open in free air) will cause a reflected wave in which the sign of the pressure variation reverses, but the direction of air movement remains the same.
  • A load that matches the characteristic impedance (defined below) will completely absorb the wave and the energy associated with it. No reflection will occur.
  • A high impedance load (e.g. by plugging the end of the line) will cause a reflected wave in which the direction of air movement is reversed but the sign of the pressure remains the same.

Since a transmission line behaves like a four terminal model, one cannot really define or measure the impedance of a transmission line component. One can however measure its input or output impedance. It depends on the cross-sectional area and length of the line, the sound frequency, as well as the characteristic impedance of the sound propagating medium within the duct. Only in the exceptional case of a closed end tube (to be compared with electrical short circuit), the input impedance could be regarded as a component impedance.

Where a transmission line of finite length is mismatched at both ends, there is the potential for a wave to bounce back and forth many times until it is absorbed. This phenomenon is a kind of resonance and will tend to attenuate any signal fed into the line.

When this resonance effect is combined with some sort of active feedback mechanism and power input, it is possible to set up an oscillation which can be used to generate periodic acoustic signals such as musical notes (e.g. in an organ pipe).

The application of transmission line theory is however seldom used in acoustics. An equivalent four terminal model which splits the downstream and upstream waves is used. This eases the introduction of physically measurable acoustic characteristics, reflection coefficients, material constants of insulation material, the influence of air velocity on wavelength (Mach number), etc. This approach also circumvents unpractical theoretical concepts, such as acoustic impedance of a tube, which is not measurable because of its inherent interaction with the sound source and the load of the acoustic component.

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See also

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References

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

  1. 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.
  2. Template:Note Ernst Weber and Frederik Nebeker, The Evolution of Electrical Engineering, IEEE Press, Piscataway, New Jersey USA, 1994 ISBN 0780310557
  3. Template:Note G.R. Jessop, VHF UHF manual, RSGB, Potters Bar, 1983, ISBN 0900612924
  • Electromagnetism 2nd ed., Grant, I.S., and Phillips, W.R., pub John Wiley, ISBN 0-471-92712-0
  • Radiocommunication handbook, page 20, chaper 17, RSGB, ISBN 0900612584
  • 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
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External articles and further reading

nl:Transmissielijn fi:Siirtolinja

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