Heaviside condition
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The Heaviside condition, named for
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed ...
(1850–1925), is the condition an electrical
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
must meet in order for there to be no
distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signa ...
of a transmitted signal. Also known as the distortionless condition, it can be used to improve the performance of a transmission line by adding loading to the cable.


The condition

A transmission line can be represented as a
distributed-element model : ''This article is an example from the domain of electrical systems, which is a special case of the more general distributed-parameter systems.'' In electrical engineering, the distributed-element model or transmission-line model of electrical ...
of its
primary line constants The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevan ...
as shown in the figure. The primary constants are the electrical properties of the cable per unit length and are:
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
''C'' (in farads per meter), inductance ''L'' (in
henries The henry (symbol: H) is the unit of electrical inductance in the International System of Units (SI). If a current of 1 ampere flowing through a coil produces flux linkage of 1 weber turn, that coil has a self inductance of 1 henry.‌ The un ...
per meter), series resistance ''R'' (in ohms per meter), and shunt conductance ''G'' (in siemens per meter). The series resistance and shunt conductivity cause losses in the line; for an ideal transmission line, \scriptstyle R=G=0. The Heaviside condition is satisfied when :\frac = \frac. This condition is for no distortion, but not for no loss.


Background

A signal on a transmission line can become distorted even if the line constants, and the resulting transmission function, are all perfectly linear. There are two mechanisms: firstly, the attenuation of the line can vary with frequency which results in a change to the shape of a pulse transmitted down the line. Secondly, and usually more problematically, distortion is caused by a frequency dependence on phase velocity of the transmitted signal frequency components. If different frequency components of the signal are transmitted at different velocities the signal becomes "smeared out" in space and time, a form of distortion called
dispersion Dispersion may refer to: Economics and finance * Dispersion (finance), a measure for the statistical distribution of portfolio returns * Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variat ...
. This was a major problem on the first
transatlantic telegraph cable Transatlantic telegraph cables were undersea cables running under the Atlantic Ocean for telegraph communications. Telegraphy is now an obsolete form of communication, and the cables have long since been decommissioned, but telephone and data a ...
and led to the theory of the causes of dispersion being investigated first by
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
and then by Heaviside who discovered how it could be countered. Dispersion of
telegraph Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore is a method of telegraphy, whereas p ...
pulses, if severe enough, will cause them to overlap with adjacent pulses, causing what is now called
intersymbol interference In telecommunication, intersymbol interference (ISI) is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have a similar effect as noise, thus making ...
. To prevent intersymbol interference it was necessary to reduce the transmission speed of the transatlantic telegraph cable to the equivalent of baud. This is an exceptionally slow data transmission rate, even for human operators who had great difficulty operating a morse key that slowly. For voice circuits (telephone) the frequency response distortion is usually more important than dispersion whereas digital signals are highly susceptible to dispersion distortion. For any kind of analogue image transmission such as video or facsimile both kinds of distortion need to be eliminated.


Derivation

The transmission function of a transmission line is defined in terms of its input and output voltages when correctly terminated (that is, with no reflections) as :\frac = e^ where x represents distance from the transmitter in meters and :\gamma = \alpha +j \beta \, are the
secondary line constants The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
, ''α'' being the attenuation in nepers per metre and ''β'' being the phase change constant in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s per metre. For no distortion, ''α'' is required to be independent of the angular frequency ''ω'', while ''β'' must be proportional to ''ω''. This requirement for proportionality to frequency is due to the relationship between the velocity, ''v'', and
phase constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
, ''β'' being given by, :v = \frac and the requirement that phase velocity, ''v'', be constant at all frequencies. The relationship between the primary and secondary line constants is given by :\gamma^2 = (\alpha +j \beta)^2 = (R+j \omega L)(G + j \omega C)\, which has to be of the form \scriptstyle (A+j\omega B)^2 in order to meet the distortionless condition. The only way this can be so is if \scriptstyle (R+j \omega L) and \scriptstyle (G + j \omega C) differ by no more than a real constant factor. Since both have a real and imaginary part, the real and imaginary parts must independently be related by the same factor, so that; :\frac = \frac and the Heaviside condition is proved.


Line characteristics

The secondary constants of a line meeting the Heaviside condition are consequently, in terms of the primary constants: Attenuation, :\alpha = \sqrt   nepers/metre Phase change constant, :\beta = \omega \sqrt   radians/metre Phase velocity, :v = \frac   metres/second


Characteristic impedance

The characteristic impedance of a lossy transmission line is given by :Z_0=\sqrt In general, it is not possible to
impedance match In electronics, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or minimize sign ...
this transmission line at all frequencies with any finite network of discrete elements because such networks are rational functions of jω, but in general the expression for characteristic impedance is irrational due to the square root term.Schroeder, p. 226 However, for a line which meets the Heaviside condition, there is a common factor in the fraction which cancels out the frequency dependent terms leaving, :Z_0=\sqrt, which is a real number, and independent of frequency. The line can therefore be impedance-matched with just a resistor at either end. This expression for \scriptstyle Z_0 = \sqrt is the same as for a lossless line (\scriptstyle R = 0,\ G = 0) with the same ''L'' and ''C'', although the attenuation (due to ''R'' and ''G'') is of course still present.


Practical use

A real line, especially one using modern synthetic insulators, will have a ''G'' that is very low and will usually not come anywhere close to meeting the Heaviside condition. The normal situation is that :\frac \ll \frac. To make a line meet the Heaviside condition one of the four primary constants needs to be adjusted and the question is which one. ''G'' could be increased, but this is highly undesirable since increasing ''G'' will increase the loss. Decreasing ''R'' is sending the loss in the right direction, but this is still not usually a satisfactory solution. ''R'' must be decreased by a large fraction and to do this the conductor cross-sections must be increased dramatically. This not only makes the cable much more bulky but also adds significantly to the amount of copper (or other metal) being used and hence the cost. Decreasing the capacitance also makes the cable more bulky (since the insulation must now be thicker) but is not so costly as increasing the copper content. This leaves increasing ''L'' which is the usual solution adopted. The required increase in ''L'' is achieved by loading the cable with a metal with high magnetic permeability. It is also possible to load a cable of conventional construction by adding discrete loading coils at regular intervals. This is not identical to a distributed loading, the difference being that with loading coils there is distortionless transmission up to a definite
cut-off frequency In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
beyond which the attenuation increases rapidly. Loading cables to meet the Heaviside condition is no longer a common practice. Instead, regularly spaced digital repeaters are now placed in long lines to maintain the desired shape and duration of pulses for long-distance transmission.


See also

*
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
*
Telegrapher's equations The telegrapher's equations (or just telegraph equations) are a pair of coupled, linear partial differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver ...


References


Bibliography

* Nahin, Paul J, ''Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age'', JHU Press, 2002 . See especially pp. 231-232. * Schroeder, Manfred Robert, ''Fractals, Chaos, Power Laws'', Courier Corporation, 2012 . {{DEFAULTSORT:Heaviside Condition Transmission lines