Heaviside condition
Encyclopedia
The Heaviside condition, due to Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...

 (1850–1925), is the condition an electrical transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...

 must meet in order for there to be no distortion
Distortion
A distortion is the alteration of the original shape of an object, image, sound, waveform or other form of information or representation. Distortion is usually unwanted, and often many methods are employed to minimize it in practice...

 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
Loading coil
In electronics, a loading coil or load coil is a coil that does not provide coupling to any other circuit, but is inserted in a circuit to increase its inductance. The need was discovered by Oliver Heaviside in studying the disappointing slow speed of the Transatlantic telegraph cable...

 to the cable.

The condition

A transmission line can be represented as a distributed element model
Distributed element model
In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit...

 of its primary line constants
Primary line constants
The primary line constants are parameters that describe the characteristics of copper transmission lines in terms of the physical electrical properties of the line. The primary line constants are only relevant to copper lines and are to be contrasted with the secondary line constants, which can...

 as shown in the figure. The primary constants are the electrical properties of the cable per unit length and are: capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

 C (in farad
Farad
The farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...

s per meter), inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

 L (in henries per meter), series resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

 R (in ohms per meter), and shunt conductance G (in siemens
Siemens (unit)
The siemens is the SI derived unit of electric conductance and electric admittance. Conductance and admittance are the reciprocals of resistance and impedance respectively, hence one siemens is equal to the reciprocal of one ohm, and is sometimes referred to as the mho. In English, the term...

 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

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
Transmission function
Transmission function can refer to*transfer function*propagation constant...

, are all perfectly linear. This happens in two ways: 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
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the 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 (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

.

This was a major problem on the first transatlantic telegraph cable
Transatlantic telegraph cable
The transatlantic telegraph cable was the first cable used for telegraph communications laid across the floor of the Atlantic Ocean. It crossed from , Foilhommerum Bay, Valentia Island, in western Ireland to Heart's Content in eastern Newfoundland. The transatlantic cable connected North America...

 and led to the theory of the causes of dispersion being investigated first by Lord Kelvin and then by Heaviside who discovered how it could be countered. Dispersion of telegraph pulses, if severe enough, will cause them to overlap with adjacent pulses, causing what is now called intersymbol interference
Intersymbol interference
In telecommunication, intersymbol interference 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 similar effect as noise, thus making the communication less reliable...

. To prevent intersymbol interference it was necessary to reduce the transmission speed of the transatlantic telegraph cable to the equivalent of baud
Baud
In telecommunications and electronics, baud is synonymous to symbols per second or pulses per second. It is the unit of symbol rate, also known as baud rate or modulation rate; the number of distinct symbol changes made to the transmission medium per second in a digitally modulated signal or a...

. 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


where represents distance from the transmitter in meters and


are the secondary line constants, α being the attenuation in neper
Neper
The neper is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. It has the unit symbol Np. The unit's name is derived from the name of John Napier, the inventor of logarithms...

s per metre and β being the phase change constant in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s per metre. For no distortion, α is required to be constant with 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, β being given by,


and the requirement that phase velocity, v, be constant at all frequencies.

The relationship between the primary and secondary line constants is given by


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 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;


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,
  nepers/metre

Phase change constant,
  radians/metre

Phase velocity,
  metres/second

Characteristic impedance

The characteristic impedance
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...

 of a lossy transmission line is given by
In general, it is not possible to match this transmission line at all frequencies because the square root causes the expression to be irrational
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

 and it consequently cannot be realised exactly with a network of discrete elements. 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,
which is a real number, and independent of frequency. The line can therefore be matched with just a resistor at either end. This expression for 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


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
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

. It is also possible to load a cable of conventional construction by adding discrete loading coil
Loading coil
In electronics, a loading coil or load coil is a coil that does not provide coupling to any other circuit, but is inserted in a circuit to increase its inductance. The need was discovered by Oliver Heaviside in studying the disappointing slow speed of the Transatlantic telegraph cable...

s 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 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

  • Telegrapher's equations
  • Maxwell's equations
    Maxwell's equations
    Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

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