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The **Heaviside condition**, named for Oliver Heaviside (1850–1925), is the condition an electrical transmission line must meet in order for there to be no distortion 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.

**Oliver Heaviside** FRS was an English self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations, reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science for years to come.

In radio-frequency engineering, a **transmission line** is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

**Distortion** is the alteration of the original shape of something. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal representing sound or a video signal representing images, in an electronic device or communication channel.

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 *C* (in farads per meter), inductance *L* (in henries 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, .

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. This is in contrast to the more common lumped-element model, which assumes that these values are lumped into electrical components that are joined by perfectly conducting wires. In the distributed-element model, each circuit element is infinitesimally small, and the wires connecting elements are not assumed to be perfect conductors; that is, they have impedance. Unlike the lumped-element model, it assumes nonuniform current along each branch and nonuniform voltage along each wire. The distributed model is used at high frequencies where the wavelength becomes comparable to the physical dimensions of the circuit, making the lumped model inaccurate.

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 relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

**Capacitance** is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. There are two closely related notions of capacitance: *self capacitance* and *mutual capacitance*. Any object that can be electrically charged exhibits *self capacitance*. A material with a large self capacitance holds more electric charge at a given voltage than one with low capacitance. The notion of *mutual capacitance* is particularly important for understanding the operations of the capacitor, one of the three elementary linear electronic components.

The Heaviside condition is satisfied when

This condition is for no distortion, but not for no loss.

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.

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 circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, **propagation constant **measures the change undergone by the source quantity as it propagates from one port to the next.

The **phase velocity** of a wave is the rate at which the phase of the wave propagates in space. 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 will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

In optics, **dispersion** is the phenomenon in which the phase velocity of a wave depends on its frequency.

This was a major problem on the first transatlantic telegraph cable 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. To prevent intersymbol interference it was necessary to reduce the transmission speed of the transatlantic telegraph cable to the equivalent of ^{1}⁄_{15} baud. This is an exceptionally slow data transmission rate, even for human operators who had great difficulty operating a morse key that slowly.

A **transatlantic telegraph cable** is an undersea cable running under the Atlantic Ocean used for telegraph communications. The first was laid across the floor of the Atlantic from Telegraph Field, Foilhommerum Bay, Valentia Island in western Ireland to Heart's Content in eastern Newfoundland. The first communications occurred August 16, 1858, reducing the communication time between North America and Europe from ten days—the time it took to deliver a message by ship—to a matter of minutes. Transatlantic telegraph cables have been replaced by transatlantic telecommunications cables.

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 similar effect as noise, thus making the communication less reliable. The spreading of the pulse beyond its allotted time interval causes it to interfere with neighboring pulses. ISI is usually caused by multipath propagation or the inherent linear or non-linear frequency response of a communication channel causing successive symbols to "blur" together.

In telecommunication and electronics, **baud** is a common measure of symbol rate, one of the components that determine the speed of communication over a data channel.

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.

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 nepers per metre and *β* being the phase change constant in radians 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, *β* being given by,

The **neper** 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. As is the case for the decibel and bel, the neper is a unit defined in the international standard ISO 80000. It is not part of the International System of Units (SI), but is accepted for use alongside the SI.

The **radian** is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees. The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.

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 in order to meet the distortionless condition. The only way this can be so is if and 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;

and the Heaviside condition is proved.

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

The characteristic impedance of a lossy transmission line is given by

The **characteristic impedance** or **surge impedance** (usually written Z_{0}) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. Alternatively and equivalently it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

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 functions of jω, but in general the expression for characteristic impedance is irrational due to the square root term.^{ [1] } 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 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.

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. 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 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.

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 rather than passing through.

A **loading coil** or **load coil** is an inductor that is inserted into an electronic circuit to increase its inductance. The term originated in the 19th century for inductors used to prevent signal distortion in long-distance telegraph transmission cables. The term is also used for inductors in radio antennas, or between the antenna and its feedline, to make an electrically short antenna resonant at its operating frequency.

**Coaxial cable**, or **coax** is a type of electrical cable that has an inner conductor surrounded by a tubular insulating layer, surrounded by a tubular conducting shield. Many coaxial cables also have an insulating outer sheath or jacket. The term coaxial comes from the inner conductor and the outer shield sharing a geometric axis. Coaxial cable was invented by English physicist, engineer, and mathematician Oliver Heaviside, who patented the design in 1880.

A **resistor–capacitor circuit**, or **RC filter** or **RC network**, is an electric circuit composed of resistors and capacitors driven by a voltage or current source. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

The **Smith chart**, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

The **old quantum theory** is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics.

In microwave and radio-frequency engineering, a **stub** or **resonant stub** is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit or short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus function as capacitors, inductors and resonant circuits at radio frequencies.

The **electromagnetic wave equation** is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field **E** or the magnetic field **B**, takes the form:

The **telegrapher's 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 Heaviside who developed the *transmission line model* in the 1880s. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line.

**Zobel networks** are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.

The **Π pad** is a specific type of attenuator circuit in electronics whereby the topology of the circuit is formed in the shape of the Greek letter "Π".

**Constant k filters**, also **k-type filters**, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive components. Historically, they are the first filters that could approach the ideal filter frequency response to within any prescribed limit with the addition of a sufficient number of sections. However, they are rarely considered for a modern design, the principles behind them having been superseded by other methodologies which are more accurate in their prediction of filter response.

**m-derived filters** or **m-type filters** are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was an improvement on the existing constant k type filter. The main problem being addressed was the need to achieve a better match of the filter into the terminating impedances. In general, all filters designed by the image method fail to give an exact match, but the m-type filter is a big improvement with suitable choice of the parameter m. The m-type filter section has a further advantage in that there is a rapid transition from the cut-off frequency of the pass band to a pole of attenuation just inside the stop band. Despite these advantages, there is a drawback with m-type filters; at frequencies past the pole of attenuation, the response starts to rise again, and m-types have poor stop band rejection. For this reason, filters designed using m-type sections are often designed as composite filters with a mixture of k-type and m-type sections and different values of m at different points to get the optimum performance from both types.

In the fields of nonlinear optics and fluid dynamics, **modulational instability** or **sideband instability** is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.

**Characteristic admittance** is the mathematical inverse of the characteristic impedance. The general expression for the characteristic admittance of a transmission line is:

An **RLC circuit** is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

- ↑ Schroeder, p. 226

- Nahin, Paul J,
*Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age*, JHU Press, 2002 ISBN 0801869099. See especially pp. 231-232. - Schroeder, Manfred Robert,
*Fractals, Chaos, Power Laws*, Courier Corporation, 2012 ISBN 0486134784.

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