A
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 transmis ...
which meets the Heaviside condition, named for
Oliver Heaviside
Oliver Heaviside ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, an ...
(1850–1925), and certain other conditions can transmit signals without dispersion and without distortion. The importance of the Heaviside condition is that it showed the possibility of dispersionless transmission of telegraph signals.
In some cases, the performance of a transmission line can be improved by adding inductive
loading to the cable.
The condition

A transmission line can be represented as a
distributed-element model of its
primary line constants as shown in the figure. The primary constants are the electrical properties of the cable per unit length and are:
capacitance
Capacitance is the ability of an object 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 are two closely related ...
''C'' (in
farad
The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units, International System of Units (SI), equivalent to 1 coulomb per volt (C/V). It is named afte ...
s per meter),
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The electric current produces a magnetic field around the conductor. The magnetic field strength depends on the magnitude of the ...
''L'' (in
henries per meter), series
resistance ''R'' (in
ohms per meter), and shunt
conductance ''G'' (in
siemens
Siemens AG ( ) is a German multinational technology conglomerate. It is focused on industrial automation, building automation, rail transport and health technology. Siemens is the largest engineering company in Europe, and holds the positi ...
per meter).
The Heaviside condition is satisfied when :
The series resistance and shunt conductivity cause losses in the line; for an ideal transmission line,
. An ideal line trivially meets the Heaviside condition.
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
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, t ...
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.
A transmission line is ''dispersionless'', if the velocity of signals is independent of frequency. Mathematically
.
A transmission line is ''distortionless'' if it is dispersionless and the
attenuation coefficient
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient val ...
is independent of frequency. Mathematically
.
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 a largely obsolete form of communication, and the cables have long since been decommissioned, but telephone and dat ...
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 physics, mathematical physicist and engineer. Born in Belfast, he was the Professor of Natural Philosophy (Glasgow), professor of Natur ...
and then by Heaviside who discovered in 1876 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 ...
pulses, if severe enough, will cause them to overlap with adjacent pulses, causing what is now called
intersymbol interference
In telecommunications, 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
In telecommunications and electronics, baud (; symbol: Bd) is a common unit of measurement of symbol rate, which is one of the components that determine the speed of communication over a data channel.
It is the unit for symbol rate or modulat ...
. 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 mitigated.
An analogous Heaviside condition for dispersionless propagation in left-handed transmission line
metamaterial
A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is a type of material engineered to have a property, typically rarely observed in naturally occu ...
s cannot be derived, since no combination of reactive and resistive elements would yield a constant group velocity.
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 constant in
neper
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. ...
s per metre and ''β'' being the
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 ...
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. It is defined such that one radian is the angle subtended at ...
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 ...
, ''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
:
If the Heaviside condition holds, then the square root function can be carried out explicitly as:
:
where
:
.
Hence
:
.
:
.
:
.
Velocity is independent of frequency if the product
is independent of frequency. Attenuation is independent of frequency if the product
is independent of frequency.
Characteristic impedance
The
characteristic impedance
The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a wave travelling in one direction along the line in the absence of reflections in th ...
of a lossy transmission line is given by
:
In general, it is not possible to
impedance match this transmission line at all frequencies with any finite network of discrete
elements because such networks are
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s of jω, but in general the expression for characteristic impedance is complex 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,
:
which is a real number, and independent of frequency if L/C is independent of frequency. The line can therefore be impedance-matched with just a resistor at either end. This expression for
is the same as for a lossless line (
) with the same ''L'' and ''C'', although the attenuation (due to ''R'' and ''G'') is of course still present.
Practical use

A real line 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
:
by several orders of magnitude.
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 number and to do this the conductor cross-sections must be increased dramatically. This not only makes the cable much bulkier, but also adds significantly to the amount of copper (or other metal) being used and hence the cost and weight. Decreasing the capacitance is difficult because it requires using a different dielectric with a lower permittivity.
Gutta-percha
Gutta-percha is a tree of the genus ''Palaquium'' in the family Sapotaceae, which is primarily used to create a high-quality latex of the same name. The material is rigid, naturally biologically Chemically inert, inert, resilient, electrically n ...
insulation used in the early trans-Atlantic cables has a dielectric constant of about 3, hence C could be decreased by a maximum factor or no more than 3. This leaves increasing ''L'' which is the usual solution adopted.
''L'' is increased by loading the cable with a metal with high
magnetic permeability
In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. It is the ratio of the magnetic ...
. 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 beyond which the attenuation increases rapidly.
Loading cables 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.
Frequency-dependent line parameters

When the line parameters are frequency dependent, there are additional considerations.
Achieving the Heaviside condition is more difficult when some or all of the line parameters depend on frequency. Typically, R (due to skin effect) and G (due to dielectric loss) are strong functions of frequency. If magnetic material is added to increase L, then L also becomes frequency dependent.
The chart on the left plots the ratios
for typical transmission lines made from non-magnetic materials. The Heaviside condition is satisfied where the blue curve touches or crosses a red curve.
The knee of the blue curve occurs at the frequency where
.
There are three red curves indicating typical low, medium, and high-quality dielectrics. Pulp insulation (used for telephone lines in the early 20th century),
gutta-percha
Gutta-percha is a tree of the genus ''Palaquium'' in the family Sapotaceae, which is primarily used to create a high-quality latex of the same name. The material is rigid, naturally biologically Chemically inert, inert, resilient, electrically n ...
, and modern foamed plastics are examples of low, medium, and high-quality dielectrics. The knee of each curve occurs at the frequency where
. The reciprocal of this frequency is known as the
dielectric relaxation time of the dielectric. Above this frequency, the value of G/(ωC) is the same as the
loss tangent
In electrical engineering, dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy (e.g. heat). It can be parameterized in terms of either the loss angle or the corresponding loss tangent . Both refer ...
of the dielectric material. The curve is depicted as flat on the figure, but loss tangent shows some frequency dependence. The value of G/(ωC) at all frequencies is determined entirely by properties of the dielectric and is independent of the transmission line cross-section.
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, Electrical network, electr ...
*
Telegrapher's equations
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