A cross section of a submarine communications cable.
1 - Polyethylene
2 - Mylar tape
3 - Stranded steel wires
4 - Aluminium water barrier
5 - Polycarbonate
6 - Copper or aluminium tube
7 - Petroleum jelly
8 - Optical fibers

Submarine cables are laid using special cable layer ships, such as the modern René Descartes, operated by France Telecom Marine.

A submarine communications cable is a cable laid on the sea bed between land-based stations to carry telecommunication signals across stretches of ocean.

The first submarine communications cables carried telegraphy traffic. Subsequent generations of cables carried telephony traffic, then data communications traffic. Modern cables use only optical fiber technology to carry digital payloads, which carry telephone, Internet and private data traffic.

Modern cables are typically 69 millimetres (2.7 in) in diameter and weigh around 10 kilograms per metre (7 lb/ft), although thinner and lighter cables are used for deep-water sections.^[1]

As of 2010, submarine cables link all the world's continents except Antarctica.

Early history: telegraph and coaxial cables[link]

Trials[link]

After William Cooke and Charles Wheatstone had introduced their working telegraph in 1839, the idea of a submarine line across the Atlantic Ocean began to be thought of as a possible triumph of the future. Samuel Morse proclaimed his faith in it as early as the year 1840, and in 1842 he submerged a wire, insulated with tarred hemp and India rubber,^[2][3] in the water of New York Harbor, and telegraphed through it. The following autumn Wheatstone performed a similar experiment in Swansea Bay. A good insulator to cover the wire and prevent the electric current from leaking into the water was necessary for the success of a long submarine line. India rubber had been tried by Moritz von Jacobi, the Prussian electrical engineer, as far back as the early 19th century.

Another insulating gum which could be melted by heat and readily applied to wire made its appearance in 1842. Gutta-percha, the adhesive juice of the Palaquium gutta tree, was introduced to Europe by William Montgomerie, a Scottish surgeon in the service of the British East India Company. Twenty years earlier he had seen whips made of it in Singapore, and he believed that it would be useful in the fabrication of surgical apparatuses. Michael Faraday and Wheatstone soon discovered the merits of gutta-percha as an insulator, and in 1845 the latter suggested that it should be employed to cover the wire which was proposed to be laid from Dover to Calais. It was tried on a wire laid across the Rhine between Deutz and Cologne. In 1849 C.V. Walker, electrician to the South Eastern Railway, submerged a wire coated with gutta-percha along the coast off Dover.^{[citation needed]}

The first commercial cables[link]

In August 1850, John Watkins Brett's Anglo-French Telegraph Company laid the first line across the English Channel. It was simply a copper wire coated with gutta-percha, without any other protection. The experiment served to keep alive the concession, and the next year, on November 13, 1851, a protected core, or true cable, was laid from a government hulk, the Blazer, which was towed across the Channel. The next year, Great Britain and Ireland were linked together. In 1852, a cable laid by the Submarine Telegraph Company linked London to Paris. In May, 1853, England was joined to the Netherlands by a cable across the North Sea, from Orford Ness to The Hague. It was laid by the Monarch, a paddle steamer which had been fitted for the work.

Transatlantic telegraph cable[link]

The first attempt at laying a transatlantic telegraph cable was promoted by Cyrus West Field, who persuaded British industrialists to fund and lay one in 1858. However, the technology of the day was not capable of supporting the project; it was plagued with problems from the outset, and was in operation for only a month. Subsequent attempts in 1865 and 1866 with the world's largest steamship, the SS Great Eastern, used a more advanced technology and produced the first successful transatlantic cable. The Great Eastern later went on to lay the first cable reaching to India from Aden, Yemen, in 1870.

British dominance of early cable[link]

From the 1850s until 1911, British submarine cable systems dominated the most important market, the North Atlantic Ocean. The British had both supply side and demand side advantages. In terms of supply, Britain had entrepreneurs willing to put forth enormous amounts of capital necessary to build, lay and maintain these cables. In terms of demand, Britain's vast colonial empire led to business for the cable companies from news agencies, trading and shipping companies, and the British government. Many of Britain’s colonies had significant populations of European settlers, making news about them of interest to the general public in the home country.

British officials believed that depending on telegraph lines that passed through non-British territory posed a security risk, as lines could be cut and messages could be interrupted during wartime. They sought the creation of a worldwide network within the empire, which became known as the All Red Line, and conversely prepared strategies to quickly interrupt enemy communications.^[4]

The submarine cables were an economic boon to trade companies because owners of ships could communicate with captains when they reached their destination on the other side of the ocean and even give directions as to where to go next to pick up more cargo based on reported pricing and supply information. The British government had obvious uses for the cables in maintaining administrative communications with governors throughout its empire as well as in engaging other nations diplomatically and communicating with its military units in wartime. Location of Britain’s territory was also an advantage as it possessed both Ireland and Newfoundland, making for the shortest route across the Atlantic Ocean (reducing cost significantly).

A few facts put this dominance of the industry in perspective. In 1896, there were thirty cable laying ships in the world and twenty-four of them were owned by British companies. In 1892, British companies owned and operated two-thirds of the world’s cables and by 1923 their share was still 42.7 percent.^[5] During World War I Britain's telegraph communications were almost completely uninterrupted, while it was able to quickly cut Germany's cables worldwide.^[4]

Cable to India, Singapore, Far East and Australasia[link]

Eastern Telegraph Company network in 1901

An 1863 cable to Mumbai, India (then known as Bombay) provided a crucial link to Saudi Arabia.^[6] In 1870 Bombay was linked to London via submarine cable in a combined operation by four cable companies, at the behest of the British Government. In 1872 these four companies were combined to form the mammoth globespanning Eastern Telegraph Company, owned by John Pender. A spin-off from Eastern Telegraph Company was a second sister company, the Eastern Extension, China and Australasia Telegraph Company, commonly known simply as "the Extension". In 1872, Australia was linked by cable to Bombay via Singapore and China and in 1876 the cable linked the British Empire from London to New Zealand.^[7]

Submarine cable across the Pacific[link]

This was completed in 1902–03, linking the US mainland to Hawaii in 1902 and Guam to the Philippines in 1903.^[8] Canada, Australia, New Zealand and Fiji were also linked in 1902.^[9]

Decades later, the North Pacific Cable system was the first regenerative (repeatered) system to completely cross the Pacific from the US mainland to Japan. The US portion of NPC was manufactured in Portland, Oregon, from 1989 to 1991 at STC Submarine Systems, and later Alcatel Submarine Networks. The system was laid by Cable & Wireless Marine on the CS Cable Venture in 1991.

Construction[link]

Transatlantic cables of the 19th century consisted of an outer layer of iron and later steel wire, wrapping India rubber, wrapping gutta-percha, which surrounded a multi-stranded copper wire at the core. The portions closest to each shore landing had additional protective armor wires. Gutta-percha, a natural polymer similar to rubber, had nearly ideal properties for insulating submarine cables, with the exception of a rather high dielectric constant which made cable capacitance high. Gutta-percha was not replaced as a cable insulation until polyethylene was introduced in the 1930s. In the 1920s, the American military experimented with rubber-insulated cables as an alternative to gutta-percha, since American interests controlled significant supplies of rubber but no gutta-percha manufacturers.

Bandwidth problems[link]

Early long-distance submarine telegraph cables exhibited formidable electrical problems. Unlike modern cables, the technology of the 19th century did not allow for in-line repeater amplifiers in the cable. Large voltages were used to attempt to overcome the electrical resistance of their tremendous length but the cables' distributed capacitance and inductance combined to distort the telegraph pulses in the line, reducing the cable's bandwidth, severely limiting the data rate for telegraph operation to 10-12 words per minute.

As early as 1823,^{[citation needed]} Francis Ronalds had observed that electric signals were retarded in passing through an insulated wire or core laid underground, and the same effect was noticed by Latimer Clark (1853) on cores immersed in water, and particularly on the lengthy cable between England and The Hague. Michael Faraday showed that the effect was caused by capacitance between the wire and the earth (or water) surrounding it. Faraday had noted that when a wire is charged from a battery (for example when pressing a telegraph key), the electric charge in the wire induces an opposite charge in the water as it travels along. In 1831, Faraday described this effect in what is now referred to as Faraday's law of induction. As the two charges attract each other, the exciting charge is retarded. The core acts as a capacitor distributed along the length of the cable which, coupled with the resistance and inductance of the cable limits the speed at which a signal travels through the conductor of the cable.

Early cable designs failed to analyze these effects correctly. Famously, E.O.W. Whitehouse had dismissed the problems and insisted that a transatlantic cable was feasible. When he subsequently became electrician of the Atlantic Telegraph Company he became involved in a public dispute with William Thomson. Whitehouse believed that, with enough voltage, any cable could be driven. Because of the excessive voltages recommended by Whitehouse, Cyrus West Field's first transatlantic cable never worked reliably, and eventually short circuited to the ocean when Whitehouse increased the voltage beyond the cable design limit.

Thomson designed a complex electric-field generator that minimized current by resonating the cable, and a sensitive light-beam mirror galvanometer for detecting the faint telegraph signals. Thomson became wealthy on the royalties of these, and several related inventions. Thomson was elevated to Lord Kelvin for his contributions in this area, chiefly an accurate mathematical model of the cable, which permitted design of the equipment for accurate telegraphy. The effects of atmospheric electricity and the geomagnetic field on submarine cables also motivated many of the early polar expeditions.

Thomson had produced a mathematical analysis of propagation of electrical signals into telegraph cables based on their capacitance and resistance, but since long submarine cables operated at slow rates, he did not include the effects of inductance. By the 1890s, Oliver Heaviside had produced the modern general form of the telegrapher's equations which included the effects of inductance and which were essential to extending the theory of transmission lines to higher frequencies required for high-speed data and voice.

Transatlantic telephony[link]

Five submarine communication cables crossing the Scottish shore at Scad Head on Hoy, Orkney.

While laying a transatlantic telephone cable was seriously considered from the 1920s, a number of technological advances were required for cost-efficient telecommunications that did not arrive until the 1940s. A first attempt to lay a pupinized telephone cable failed in the early 1930s due to the Great Depression.

In 1942, Siemens Brothers of New Charlton, London in conjunction with the United Kingdom National Physical Laboratory, adapted submarine communications cable technology to create the world's first submarine oil pipeline in Operation Pluto during World War II.

TAT-1 (Transatlantic No. 1) was the first transatlantic telephone cable system. Between 1955 and 1956, cable was laid between Gallanach Bay, near Oban, Scotland and Clarenville, Newfoundland and Labrador. It was inaugurated on September 25, 1956, initially carrying 36 telephone channels.

In the 1960s, transoceanic cables were coaxial cables that transmitted frequency-multiplexed voiceband signals. A high voltage direct current on the inner conductor powered the repeaters. The first-generation repeaters are among the most reliable vacuum tube amplifiers ever designed.^[10] Later ones were transistorized. Many of these cables are still usable, but abandoned because their capacity is too small to be commercially viable. Some have been used as scientific instruments to measure earthquake waves and other geomagnetic events.^[11]

Modern history[link]

Optical telephone cables[link]

Diagram of an optical submarine cable repeater.

In the 1980s, fiber optic cables were developed. The first transatlantic telephone cable to use optical fiber was TAT-8, which went into operation in 1988. A fiber-optic cable comprises multiple pairs of fibers. Each pair has one fiber in each direction. TAT-8 had two operational pairs and one backup pair.

Modern optical fiber repeaters use a solid-state optical amplifier, usually an Erbium-doped fiber amplifier. Each repeater contains separate equipment for each fiber. These comprise signal reforming, error measurement and controls. A solid-state laser dispatches the signal into the next length of fiber. The solid-state laser excites a short length of doped fiber that itself acts as a laser amplifier. As the light passes through the fiber, it is amplified. This system also permits wavelength-division multiplexing, which dramatically increases the capacity of the fiber.

Repeaters are powered by a constant direct current passed down the conductor near the center of the cable, so all repeaters in a cable are in series. Power feed equipment is installed at the terminal stations. Typically both ends share the current generation with one end providing a positive voltage and the other a negative voltage. A virtual earth point exists roughly half way along the cable under normal operation. The amplifiers or repeaters derive their power from the potential difference across them.

The optic fiber used in undersea cables is chosen for its exceptional clarity, permitting runs of more than 100 kilometers between repeaters to minimize the number of amplifiers and the distortion they cause.

The rising demand for these fiber-optic cables outpaced the capacity of providers such as AT&T. Having to shift traffic to satellites resulted in poorer quality signals. To address this issue, AT&T had to improve its cable laying abilities. It invested $100 million in producing two specialized fiber-optic cable laying vessels. These included laboratories in the ships for splicing cable and testing its electrical properties. Such field monitoring is important because the glass of fiber-optic cable is less malleable than the copper cable that had been formerly used. The ships are equipped with additional propellers that increase maneuverability. This capability is important because fiber-optic cable must be laid straight from the stern (another factor copper cable laying ships did not have to contend with).^[12]

Originally, submarine cables were simple point-to-point connections. With the development of submarine branching units (SBUs), more than one destination could be served by a single cable system. Modern cable systems now usually have their fibers arranged in a self-healing ring to increase their redundancy, with the submarine sections following different paths on the ocean floor. One driver for this development was that the capacity of cable systems had become so large that it was not possible to completely back-up a cable system with satellite capacity, so it became necessary to provide sufficient terrestrial back-up capability. Not all telecommunications organizations wish to take advantage of this capability, so modern cable systems may have dual landing points in some countries (where back-up capability is required) and only single landing points in other countries where back-up capability is either not required, the capacity to the country is small enough to be backed up by other means, or having back-up is regarded as too expensive.

A further redundant-path development over and above the self-healing rings approach is the "Mesh Network" whereby fast switching equipment is used to transfer services between network paths with little to no effect on higher-level protocols if a path becomes inoperable. As more paths become available to use between two points, the less likely it is that one or two simultaneous failures will prevent end-to-end service.

As of 2012 operators had "successfully demonstrated long-term, error-free transmission at 100Gbps across Atlantic Ocean" routes of up to 6000km [5], meaning a typical cable can move tens of terabits per second overseas. Speeds improved rapidly in the last few years, with 40Gbit/s having been offered on that route only three years earlier in August 2009 [6].

Switching and all-by-sea routing commonly increases the distance and thus the round trip latency by more than 50%. For example, the round trip delay (RTD) or latency of the fastest transatlantic connections is under 60ms, close to the theoretical maximum for an all-sea route. While in theory a great circle route between London and New York City is only 5600km [7] this requires several land masses (Ireland, Newfoundland, Prince Edward Island and the isthmus connecting New Brunswick to Nova Scotia) to be traversed, as well as the extremely tidal Bay of Fundy and a land route along Massachusetts' north shore from Gloucester to Boston and through fairly built up areas to Manhattan itself. In theory using this partly land route could result in round trip times below 40ms, not counting switching. Along routes with less land in the way, speeds can approach speed of light minimums in the long term.

Importance of submarine cables[link]

As of 2006, overseas satellite links accounted for only 1 percent of international traffic, while the remainder was carried by undersea cable. The reliability of submarine cables is high, especially when (as noted above) multiple paths are available in the event of a cable break. Also, the total carrying capacity of submarine cables is in the terabits per second while satellites typically offer only megabits per second and display higher latency. However, a typical multi-terabit, transoceanic submarine cable system costs several hundred million dollars to construct.^[13]

As a result of these cables' cost and usefulness they are highly valued not only by the corporations building and operating them for profit, but also by national governments. For instance, the Australian government considers its submarine cable systems to be “vital to the national economy.” Accordingly, the Australian Communications and Media Authority (ACMA) has created protection zones that restrict activities that could potentially damage cables linking Australia to the rest of the world. The ACMA also regulates all projects to install new submarine cables.^[14]

Investment in and financing of submarine cables[link]

Almost all fiber optic cables from TAT-8 in 1988 until approximately 1997 were constructed by "consortia" of operators. For example, TAT-8 counted 35 participants including most major international carriers at the time such as AT&T.^[15] Two privately financed, non-consortium cables were constructed in the late-1990s, which preceded a massive, speculative rush to construct privately financed cables that peaked in more than $22 billion worth of investment between 1999 and 2001. This was followed by the bankruptcy and reorganization of cable operators such as Global Crossing, 360networks, FLAG, Worldcom, and Asia Global Crossing.

There has been an increasing tendency in recent years to expand submarine cable in the Pacific Ocean (the previous bias always having been to lay communications cable across the Atlantic Ocean which separates the United States and Europe). For example, between 1998 and 2003, approximately 70% of undersea fiber-optic cable was laid in the Pacific. This is in part a response to the emerging significance of Asian markets in the global economy.^[16]

A map of active and anticipated submarine communications cables servicing the African continent

Although much of the investment in submarine cables has been directed toward developed markets such as the transatlantic and transpacific routes, in recent years there has been an increased effort to expand the submarine cable network to serve the developing world. For instance, in July 2009 an underwater fiber optic cable line plugged East Africa into the broader Internet. The company that provided this new cable was SEACOM, which is 75% owned by Africans.^[17] The project is based on the hope that new information technology will reduce the cost of doing business in Africa and between Africa and other international parties. Still, the project was delayed by a month due to increased piracy along the coast, a reminder that the developing world faces other struggles that new technologies such as this may not necessarily solve.^[18]

Out of range[link]

Antarctica is the only continent yet to be reached by a submarine telecommunications cable. All phone, video, and e-mail traffic must be relayed to the rest of the world via satellite, which is still quite unreliable. Bases on the continent itself are able to communicate with one another via radio, but this is only a local network. To be a viable alternative, a fiber-optic cable would have to be able to withstand temperatures of −80˚ C as well as massive strain from ice flowing up to 10 meters per year. Thus, plugging into the larger Internet backbone with the high bandwidth afforded by fiber-optic cable is still an as yet infeasible economic and technical challenge in the Antarctic.^[19]

Cable repair[link]

Cables can be broken by fishing trawlers, anchors, earthquakes, turbidity currents and even shark bites.^[20] Based on surveying breaks in the Atlantic Ocean and the Caribbean Sea, it was found that between 1959 and 1996 fewer than 9% were due to natural events. In response to this threat to the communications network, the practice of cable burial has developed. The average incidence of cable faults was 3.7 per 1,000 km per year from 1959 to 1979. That rate was reduced to 0.44 faults per 1,000 km per year after 1985, due to widespread burial of cable starting in 1980.^[21] Still, cable breaks are by no means a thing of the past, with more than 50 repairs a year in the Atlantic alone,^[22] and significant breaks in 2006, 2008 and 2009.

The propensity for fishing trawler nets to cause cable faults may well have been exploited during the Cold War. For example, in February 1959 a series of 12 breaks occurred in five American trans-Atlantic communications cables. In response, a United States naval vessel, the U.S.S. Roy O. Hale detained and investigated the Soviet trawler Novorosiysk. A review of the ship’s log indicated it had been in the region of each of the cables when they broke. Broken sections of cable were also found on the deck of the Novorosiysk. It appeared that the cables had been dragged along by the ship’s nets, and then cut once they were pulled up onto the deck in order to release the nets. The Soviet Union’s stance on the investigation was that it was unjustified, but the United States cited the Convention for the Protection of Submarine Telegraph Cables of 1884 to which Russia had been committed (prior to the formation of the Soviet Union) as evidence of violation of international protocol.^[23]

Shore stations can locate a break in a cable by electrical measurements, such as through spread-spectrum time-domain reflectometry (SSTDR). SSTDR is a type of time-domain reflectometry that can be used in live environments very quickly. Presently SSTDR can collect a complete data set in 20ms.^[24] Spread spectrum signals are sent down the wire and then the reflected signal is observed. It is then correlated with the copy of the sent signal and mathematical algorithms are applied to the shape and timing of the signals to locate the break.

A cable repair ship will be sent to the location to drop a marker buoy near the break. Several types of grapples are used depending on the situation. If the sea bed in question is sandy, a grapple with rigid prongs is used to plough under the surface and catch the cable. If the cable is on a rocky sea surface, the grapple is more flexible, with hooks along its length so that it can adjust to the changing surface.^[25] In especially deep water, the cable may not be strong enough to lift as a single unit, so a special grapple that cuts the cable soon after it has been hooked is used and only one length of cable is brought to the surface at a time, whereupon a new section is spliced in.^[26] The repaired cable is longer than the original, so the excess is deliberately laid in a 'U' shape on the seabed. A submersible can be used to repair cables that lie in shallower waters.

A number of ports near important cable routes became homes to specialised cable repair ships. Halifax, Nova Scotia was home to a half dozen such vessels for most of the 20th century including long-lived vessels such as the CS Cyrus West Field, CS Minia and CS Mackay-Bennett. The latter two were contracted to recover victims from the sinking of the RMS Titanic. The crews of these vessels developed many new techniques and devices to repair and improve cable laying, such as the "plough".

Intelligence gathering[link]

Underwater cables, which cannot be kept under constant surveillance, have tempted intelligence-gathering organizations since the late 19th century. Frequently at the beginning of wars nations have cut the cables of the other sides in order to shape the information flows into cables that were being monitored. The most ambitious efforts occurred in World War I, when British and German forces systematically attempted to destroy the others' worldwide communications systems by cutting their cables with surface ships or submarines.^[27] During the Cold War the United States Navy and National Security Agency (NSA) succeeded in placing wire taps on Soviet underwater communication lines in Operation Ivy Bells.

Environmental impact[link]

The main point of interaction of cables with marine life is in the benthic zone of the oceans where the majority of cable lies. Recent studies (in 2003 and 2006) have indicated that cables pose minimal impacts on life in these environments. In sampling sediment cores around cables and in areas removed from cables, there were few statistically significant differences in organism diversity or abundance. The main difference was that the cables provided an attachment point for anemones that typically could not grow in soft sediment areas. Data from 1877 to 1955 showed a total of 16 cable faults caused by the entanglement of various whales, but such deadly entanglements have entirely ceased after the transition from telegraph cables to coaxial cables and then fiber-optic cables (the new cables are better designed in terms of torsional balance so that they have less of a tendency to coil).^[28]

Notable events[link]

The Newfoundland earthquake of 1929 broke a series of trans-Atlantic cables by triggering a massive undersea mudslide. The sequence of breaks helped scientists chart the progress of the mudslide.

In July 2005, a portion of the SEA-ME-WE 3 submarine cable located 35 kilometres (22 mi) south of Karachi that provided Pakistan's major outer communications became defective, disrupting almost all of Pakistan's communications with the rest of the world, and affecting approximately 10 million Internet users.^[29][30][31]

The 2006 Hengchun earthquake on December 26, 2006 rendered numerous cables between Taiwan and Philippines inoperable.

In March, 2007, pirates stole an 11-kilometre (6.8 mi) section of the T-V-H submarine cable that connected Thailand, Vietnam, and Hong Kong, affecting Vietnam's Internet users with far slower speeds. The thieves attempted to sell the 100 tons of cable as scrap.^[32]

The 2008 submarine cable disruption was a series of cable outages, two of the three Suez Canal cables, two disruptions in the Persian Gulf, and one in Malaysia. It caused massive communications disruptions to India and the Middle East.^[33][34]

In April 2010 the undersea cable SEA-ME-WE 4 was under an outage the South East Asia–Middle East–Western Europe 4 (SEA-ME-WE 4) submarine communications cable system, which connects South East Asia and Europe, was reportedly cut in three places, off Palermo, Italy.

The 2011 Tōhoku earthquake and tsunami damaged a number of undersea cables that make landings in Japan, including^[35]:

• APCN-2, an intra-Asian cable that forms a ring linking China, Hong Kong, Japan, the Republic of Korea, Malaysia, the Philippines, Singapore and Taiwan.
• Pacific Crossing West and Pacific Crossing North
• Segments of the East Asia Crossing network (reported by PacNet)
• A segment of the Japan-U.S. Cable Network (reported by Korea Telecom)
• PC-1 submarine cable system (reported by NTT)

In February 2012, breaks in the EASSy and TEAMS cables disconnected about half of the networks in Kenya and Uganda from the global Internet.^[36]

See also[link]

• List of domestic submarine communications cables
• List of international submarine communications cables

References[link]

External links[link]

Wikimedia Commons has media related to: Undersea telecommunications

• The International Cable Protection Committee -- includes a register of submarine cables worldwide (though not always updated as often as one might hope)
• Timeline of Submarine Communications Cables, 1850-2010
• Kingfisher Information Service - Cable Awareness; UK Fisherman's Submarine Cable Awareness site
• France Telecom's Fishermen's/Submarine Cable Information
• Oregon Fisherman's Cable Committee
• Website with comprehensive list of cable landing sites and suppliers globally (contains many duplicates and incomplete data)
• SubOptic -- the industry's major conference, held every 3 years
• Submarine Telecoms Forum -- a trade magazine dedicated to the submarine cable industry

Articles[link]

• History of the Atlantic Cable & Submarine Telegraphy - Wire Rope and the Submarine Cable Industry
• Mother Earth Mother Board - Wired article by Neal Stephenson about submarine cables
• Nature article - Geomagnetic induction on a transatlantic communications cable
• Hunt, Bruce J. Lord Cable. Europhysics News (2004), Vol. 35 No 6.
• Winkler, Jonathan Reed. Nexus: Strategic Communications and American Security in World War I. (Cambridge, MA: Harvard University Press, 2008) Account of how U.S. government discovered strategic significance of communications lines, including submarine cables, during World War I.
• Animations from Alcatel showing how submarine cables are installed and repaired
• Work begins to repair severed net

Maps[link]

Wikimedia Commons has media related to: Maps of submarine communication cables

• TeleGeography Interactive Cable Map Comprehensive online submarine cable map displaying cable routes, landing points, owners, length, ready-for-service (RFS) date, and website.
• Greg's Cable Map Google map display of submarine cables with downloadable KML data.
• Map and Satellite views of US landing sites for transatlantic cables
• Map and Satellite views of US landing sites for transpacific cables
• World map of submarine cables from the Guardian
• Positions and Route information of Submarine Cables in the Seas Around the UK

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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 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, and computer network connections.

Explanation[link]

Ordinary electrical cables suffice to carry low frequency AC, such as mains power, which reverses direction 100 to 120 times per second (cycling 50 to 60 times per second). However, they cannot be used to carry currents in the radio frequency range or higher, which reverse direction millions to billions of times per second, because the energy tends to radiate off the cable as radio waves, causing power losses. Radio frequency currents also tend to reflect from discontinuities in the cable such as connectors, and travel back down the cable toward the source. These reflections act as bottlenecks, preventing the power from reaching the destination. Transmission lines use specialized construction such as precise conductor dimensions and spacing, and impedance matching, to carry electromagnetic signals with minimal reflections and power losses. Types of transmission line include ladder line, coaxial cable, dielectric slabs, stripline, optical fiber, and waveguides. The higher the frequency, the shorter are the waves in a transmission medium. Transmission lines must be used when the frequency is high enough that the wavelength of the waves begins to approach the length of the cable used. To conduct energy at frequencies above the radio range, such as millimeter waves, infrared, and light, the waves become much smaller than the dimensions of the structures used to guide them, so transmission line techniques become inadequate and the methods of optics are used.

The theory of sound wave propagation is very similar mathematically to that of electromagnetic waves, so techniques from transmission line theory are also used to build structures to conduct acoustic waves; and these are also called transmission lines.

History[link]

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. The model 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.^[1]

Applicability[link]

In many electric circuits, the length of the wires connecting the components can for the most part be ignored. That is, the voltage on the wire at a given time can be assumed to be the same at all points. However, when the voltage changes in a time interval comparable to the time it takes for the signal to travel down the wire, the length becomes important and the wire must be treated as a transmission line. Stated another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to or less than the length of the wire.

A common rule of thumb is that the cable or wire should be treated as a transmission line if the length is greater than 1/10 of the wavelength. At this length the phase delay and the interference of any reflections on the line become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.

The four terminal model[link]

Variations on the schematic electronic symbol for a transmission line.

For the purposes of analysis, an electrical transmission line can be modelled as a two-port network (also called a quadrupole network), as follows:

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 of a given wave to the complex current of the same wave 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 as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the load impedance 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 (see ohmic heating). 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 (see dielectric heating). The transmission line is modeled with a resistance (R) and inductance (L) in series with a capacitance (C) and conductance (G) in parallel. The resistance and conductance contribute to the loss in a transmission line.

The total loss of power in a transmission line is often specified in decibels per metre (dB/m), 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 those designed to carry electromagnetic waves whose wavelengths are shorter than or 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, metal mesh optical filters, and with the signals found in high-speed digital circuits.

Telegrapher's equations[link]

The Telegrapher's Equations (or just Telegraph Equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's Equations.

Schematic representation of the elementary component of a transmission line.

The transmission line model represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

• The distributed resistance Failed to parse (Missing texvc executable; please see math/README to configure.): R

of the conductors is represented by a series resistor (expressed in ohms per unit length).

• The distributed inductance Failed to parse (Missing texvc executable; please see math/README to configure.): L

(due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).

• The capacitance Failed to parse (Missing texvc executable; please see math/README to configure.): C

between the two conductors is represented by a shunt capacitor C (farads per unit length).

• The conductance Failed to parse (Missing texvc executable; please see math/README to configure.): G

of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (siemens per unit length).

The model consists of an infinite series of the elements shown in the figure, and that the values of the components are specified per unit length so the picture of the component can be misleading. Failed to parse (Missing texvc executable; please see math/README to configure.): R , Failed to parse (Missing texvc executable; please see math/README to configure.): L , Failed to parse (Missing texvc executable; please see math/README to configure.): C , and Failed to parse (Missing texvc executable; please see math/README to configure.): G

may also be functions of frequency. An alternative notation is to use Failed to parse (Missing texvc executable; please see math/README to configure.): R'

, Failed to parse (Missing texvc executable; please see math/README to configure.): L' , Failed to parse (Missing texvc executable; please see math/README to configure.): C'

and Failed to parse (Missing texvc executable; please see math/README to configure.): G'
to emphasize that the values are derivatives with respect to length.  These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the propagation constant, attenuation constant and phase constant.

The line voltage Failed to parse (Missing texvc executable; please see math/README to configure.): V(x)

and the current Failed to parse (Missing texvc executable; please see math/README to configure.): I(x)
can be expressed in the frequency domain as

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial V(x)}{\partial x} = -(R + j \omega L)I(x)

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial I(x)}{\partial x} = -(G + j \omega C)V(x).

When the elements Failed to parse (Missing texvc executable; please see math/README to configure.): R

and Failed to parse (Missing texvc executable; please see math/README to configure.): G
are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the Failed to parse (Missing texvc executable; please see math/README to configure.): L
and Failed to parse (Missing texvc executable; please see math/README to configure.): C
elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial^2V(x)}{\partial x^2}+ \omega^2 LC\cdot V(x)=0

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial^2I(x)}{\partial x^2} + \omega^2 LC\cdot I(x)=0.

These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.

If Failed to parse (Missing texvc executable; please see math/README to configure.): R

and Failed to parse (Missing texvc executable; please see math/README to configure.): G
are not neglected, the Telegrapher's equations become:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial^2V(x)}{\partial x^2} = \gamma^2 V(x)

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial^2I(x)}{\partial x^2} = \gamma^2 I(x)

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma = \sqrt{(R + j \omega L)(G + j \omega C)}

and the characteristic impedance is:

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_0 = \sqrt{\frac{R + j \omega L}{G + j \omega C}}.

The solutions for Failed to parse (Missing texvc executable; please see math/README to configure.): V(x)

and Failed to parse (Missing texvc executable; please see math/README to configure.): I(x)
are:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(x) = V^+ e^{-\gamma x} + V^- e^{\gamma x} \,

Failed to parse (Missing texvc executable; please see math/README to configure.): I(x) = \frac{1}{Z_0}(V^+ e^{-\gamma x} - V^- e^{\gamma x}). \,

The constants Failed to parse (Missing texvc executable; please see math/README to configure.): V^\pm

and Failed to parse (Missing texvc executable; please see math/README to configure.): I^\pm
must be determined from boundary conditions. For a voltage pulse Failed to parse (Missing texvc executable; please see math/README to configure.): V_{\mathrm{in}}(t) \,

, starting at Failed to parse (Missing texvc executable; please see math/README to configure.): x=0

and moving in the positive Failed to parse (Missing texvc executable; please see math/README to configure.): x

-direction, then the transmitted pulse Failed to parse (Missing texvc executable; please see math/README to configure.): V_{\mathrm{out}}(x,t) \,

at position Failed to parse (Missing texvc executable; please see math/README to configure.): x
can be obtained by computing the Fourier Transform, Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{V}(\omega)

, of Failed to parse (Missing texvc executable; please see math/README to configure.): V_{\mathrm{in}}(t) \, , attenuating each frequency component by Failed to parse (Missing texvc executable; please see math/README to configure.): e^{\mathrm{-Re}(\gamma) x} \, , advancing its phase by Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{-Im}(\gamma)x \, , and taking the inverse Fourier Transform. The real and imaginary parts of Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma

can be computed as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{Re}(\gamma) = (a^2 + b^2)^{1/4} \cos(\mathrm{atan2}(b,a)/2) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{Im}(\gamma) = (a^2 + b^2)^{1/4} \sin(\mathrm{atan2}(b,a)/2) \,

where atan2 is the two-parameter arctangent, and

Failed to parse (Missing texvc executable; please see math/README to configure.): a \equiv \omega^2 LC \left[ \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right]

Failed to parse (Missing texvc executable; please see math/README to configure.): b \equiv \omega^2 LC \left( \frac{R}{\omega L} + \frac{G}{\omega C} \right).

For small losses and high frequencies, to first order in Failed to parse (Missing texvc executable; please see math/README to configure.): R / \omega L

and Failed to parse (Missing texvc executable; please see math/README to configure.): G / \omega C
one obtains

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{Re}(\gamma) \approx \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{Im}(\gamma) \approx \omega \sqrt{LC}. \,

Noting that an advance in phase by Failed to parse (Missing texvc executable; please see math/README to configure.): - \omega \delta

is equivalent to a time delay by Failed to parse (Missing texvc executable; please see math/README to configure.): \delta

, Failed to parse (Missing texvc executable; please see math/README to configure.): V_{out}(t)

can be simply computed as

Failed to parse (Missing texvc executable; please see math/README to configure.): V_{\mathrm{out}}(x,t) \approx V_{\mathrm{in}}(t - \sqrt{LC}x) e^{- \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) x }. \,

Input impedance of lossless transmission line[link]

The characteristic impedance Failed to parse (Missing texvc executable; please see math/README to configure.): Z_0

of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.

For a lossless transmission line, it can be shown that the impedance measured at a given position Failed to parse (Missing texvc executable; please see math/README to configure.): l

from the load impedance Failed to parse (Missing texvc executable; please see math/README to configure.): Z_L
is

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_\mathrm{in} (l)=Z_0 \frac{Z_L + jZ_0\tan(\beta l)}{Z_0 + jZ_L\tan(\beta l)}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \beta=\frac{2\pi}{\lambda}

is the wavenumber.

In calculating Failed to parse (Missing texvc executable; please see math/README to configure.): \beta , the wavelength is generally different inside the transmission line to what it would be in free-space and the velocity constant of the material the transmission line is made of needs to be taken into account when doing such a calculation.

Special cases[link]

Half wave length[link]

For the special case where Failed to parse (Missing texvc executable; please see math/README to configure.): \beta l= n\pi

where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_\mathrm{in}=Z_L \,

for all Failed to parse (Missing texvc executable; please see math/README to configure.): n . This includes the case when Failed to parse (Missing texvc executable; please see math/README to configure.): n=0 , meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.

Quarter wave length[link]

For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_\mathrm{in}=\frac{{Z_0}^2}{Z_L}. \,

Matched load[link]

Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_\mathrm{in}=Z_L=Z_0 \,

for all Failed to parse (Missing texvc executable; please see math/README to configure.): l

and all Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda

.

Short[link]

For the case of a shorted load (i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): Z_L=0 ), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_\mathrm{in} (l)=j Z_0 \tan(\beta l). \,

Open[link]

For the case of an open load (i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): Z_L=\infty ), the input impedance is once again imaginary and periodic

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_\mathrm{in} (l)=-j Z_0 \cot(\beta l). \,

Stepped transmission line[link]

A simple example of stepped transmission line consisting of three segments.

A stepped transmission line^[2] is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be Z_0,i. The input impedance can be obtained from the successive application of the chain relation

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_\mathrm{i+1}=Z_\mathrm{0,i} \frac{Z_i + jZ_\mathrm{0,i}\tan(\beta_i l_i)}{Z_\mathrm{0,i} + jZ_i\tan(\beta_i l_i)}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_i

is the wave number of the ith transmission line segment and l_i is the length of this segment, and Z_i is the front-end impedance that loads the ith segment.

The impedance transformation circle along a transmission line whose characteristic impedance Z_0,i is smaller than that of the input cable Z₀. And as a result, the impedance curve is off-centered towards the -x axis. Conversely, if Z_0,i > Z₀, the impedance curve should be off-centered towards the +x axis.

Because the characteristic impedance of each transmission line segment Z_0,i is often different from that of the input cable Z₀, the impedance transformation circle is off centered along the x axis of the Smith Chart whose impedance representation is usually normalized against Z₀.

Practical types[link]

Coaxial cable[link]

Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. 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 and magnetic mode (TEM) only, which means that the electric and magnetic fields are both perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential). However, at frequencies for which the wavelength (in the dielectric) is significantly shorter than the circumference of the cable, transverse electric (TE) and transverse magnetic (TM) waveguide modes can also propagate. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another.

The most common use for coaxial cables is for television and other signals with bandwidth of multiple megahertz. In the middle 20th century they carried long distance telephone connections.

A type of transmission line called a cage line, used for high power, low frequency applications. It functions similarly to a large coaxial cable. This example is the antenna feedline for a longwave radio transmitter in Poland, which operates at a frequency of 225 kHz and a power of 1200 kW.

Microstrip[link]

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. Microstrip is an open structure whereas coaxial cable is a closed structure.

Stripline[link]

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.

Balanced lines[link]

A balanced line is a transmission line consisting of two conductors of the same type, and equal impedance to ground and other circuits. There are many formats of balanced lines, amongst the most common are twisted pair, star quad and twin-lead.

Twisted pair[link]

Twisted pairs are commonly used for terrestrial telephone communications. In such cables, many pairs are grouped together in a single cable, from two to several thousand.^[3] The format is also used for data network distribution inside buildings, but the cable is more expensive because the transmission line parameters are tightly controlled.

Quad, star quad[link]

Quad is four-conductor cable used sometimes for two circuits, as in 4-wire telephony, and other times for a single circuit, called star quad, a balanced circuit for audio signals. All four conductors are twisted together around the cable axis. In the quad format, each pair uses non-adjacent conductors. For star quad, two non-adjacent conductors are terminated together at both ends of the cable, and the other two conductors are also terminated together.

Interference picked up by the cable arrives as a virtually perfect common mode signal, which is easily removed by coupling transformers. Because the conductors are always the same distance from each other, cross talk is reduced relative to cables with two separate twisted pairs.

The combined benefits of twisting, differential signalling, and quadrupole pattern give outstanding noise immunity, especially advantageous for low signal level applications such as long microphone cables, even when installed very close to a power cable. The disadvantage is that star quad, in combining two conductors, typically has double the capacitance of similar two-conductor twisted and shielded audio cable. High capacitance causes increasing distortion and greater loss of high frequencies as distance increases.^[4][5]

Twin-lead[link]

Twin-lead consists of a pair of conductors held apart by a continuous insulator.

Lecher lines[link]

Lecher lines are a form of parallel conductor that can be used at UHF for creating resonant circuits. They are a convenient practical format that fills the gap between lumped components (used at HF/VHF) and resonant cavities (used at UHF/SHF).

Single-wire line[link]

Unbalanced lines were formerly much used for telegraph transmission, but this form of communication has now fallen into disuse. Cables are similar to twisted pair in that many cores are bundled into the same cable but only one conductor is provided per circuit and there is no twisting. All the circuits on the same route use a common path for the return current (earth return). There is a power transmission version of single-wire earth return in use in many locations.

Waveguide[link]

Waveguides are rectangular or circular metallic tubes inside which an electromagnetic wave is propagated and is confined by the tube. Waveguides are not capable of transmitting the transverse electromagnetic mode found in copper lines and must use some other mode. Consequently, they cannot be directly connected to cable and a mechanism for launching the waveguide mode must be provided at the interface.

Optical fiber[link]

Optical fiber is a solid transparent fiber of glass or polymer that carries an optical signal. Optical fiber is a variety of waveguide. Optical fiber transmission lines form the backbone of modern terrestrial communications networks due to their low cost, low loss, and high signal bandwidth (high data rate).

General applications[link]

Signal transfer[link]

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.

Pulse generation[link]

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, although with half the voltage. A Blumlein transmission line is a related pulse forming device that overcomes this limitation. These are sometimes used as the pulsed energy sources for radar transmitters and other devices.

Stub filters[link]

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.

Acoustic transmission lines[link]

An acoustic transmission line is the acoustic analog of the electrical transmission line, typically thought of as a rigid-walled tube that is long and thin relative to the wavelength of sound present in it.

Solutions of the Telegrapher's Equations as Circuit Components[link]

Equivalent circuit of a transmission line described by the Telegrapher's equations.

Solutions of the Telegrapher's Equations as Components in the Equivalent Circuit of a Balanced Transmission Line Two-Port Implementation.

The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the left figure implements the solutions of the telegrapher's equations.^[6]

The right hand circuit is derived from the left hand circuit by source transformations.^[7] It also implements the solutions of the telegrapher's equations.

The solution of the telegrapher's equations can be expressed as an ABCD type Two-port network with the following defining equations^[8]

Failed to parse (Missing texvc executable; please see math/README to configure.): V_1 = V_2 cosh ( \gamma x) + I_2 Z sinh (\gamma x) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): I_1 = V_2 \frac{1}{Z} sinh (\gamma x) + I_2 cosh(\gamma x). \,

The symbols: Failed to parse (Missing texvc executable; please see math/README to configure.): E_s, E_L, I_s, I_L, l \,

 in the source book have been replaced by the symbols :Failed to parse (Missing texvc executable; please see math/README to configure.):  V_1, V_2, I_1, I_2, x  \,
in the preceding two equations.

The ABCD type two-port gives Failed to parse (Missing texvc executable; please see math/README to configure.): V_1 \,

 and  Failed to parse (Missing texvc executable; please see math/README to configure.): I_1 \, 
as functions of  Failed to parse (Missing texvc executable; please see math/README to configure.): V_2 \, 
and  Failed to parse (Missing texvc executable; please see math/README to configure.): I_2 \, 

. Both of the circuits above, when solved for Failed to parse (Missing texvc executable; please see math/README to configure.): V_1 \,

 and  Failed to parse (Missing texvc executable; please see math/README to configure.): I_1 \, 
as functions of  Failed to parse (Missing texvc executable; please see math/README to configure.): V_2 \, 
and  Failed to parse (Missing texvc executable; please see math/README to configure.): I_2 \, 
yield exactly the same equations.

In the right hand circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedances Z(s), the voltage dependent current sources (VDCSs) and the difference amplifiers (the triangle with the number "1") account for the interaction of the transmission line with the external circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the T(s) blocks carries the forward wave and the other carries the backward wave. The circuit, as depicted, is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from Failed to parse (Missing texvc executable; please see math/README to configure.): V_1 \,

to Failed to parse (Missing texvc executable; please see math/README to configure.): V_2 \, 
in the sense that Failed to parse (Missing texvc executable; please see math/README to configure.): V_1 \, 

, Failed to parse (Missing texvc executable; please see math/README to configure.): V_2 \, , Failed to parse (Missing texvc executable; please see math/README to configure.): I_1 \,

and Failed to parse (Missing texvc executable; please see math/README to configure.): I_2 \, 
would be same whether this circuit or an actual transmission line was connected between Failed to parse (Missing texvc executable; please see math/README to configure.): V_1 \, 
and Failed to parse (Missing texvc executable; please see math/README to configure.): V_2 \, 

. There is no implication that there are actually amplifiers inside the transmission line.

Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which may be called shield, sheath, common, Earth or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential and common modes. The circuit shown on the right only models the differential mode.

In the left hand circuit, the voltage doublers, the difference amplifiers and impedances Z(s) account for the interaction of the transmission line with the external circuit. This circuit, as depicted, is also fully symmetric, and also not drawn that way. This circuit is a useful equivalent for an unbalanced transmission line like a coaxial cable or a micro strip line.

These are not the only possible equivalent circuits.

See also[link]

References[link]

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

• Steinmetz, Charles Proteus (August 27, 1898), "The Natural Period of a Transmission Line and the Frequency of lightning Discharge Therefrom", The Electrical World: 203–205 
• Grant, I. S.; Phillips, W. R., Electromagnetism (2nd ed.), John Wiley, ISBN 0-471-92712-0 
• Ulaby, F. T., Fundamentals of Applied Electromagnetics (2004 media ed.), Prentice Hall, ISBN 0-13-185089-X 
• "Chapter 17", Radio communication handbook, Radio Society of Great Britain, 1982, pp. 20, ISBN 0-900612-58-4 
• Naredo, J. L.; Soudack, A. C.; Marti, J. R. (Jan 1995), "Simulation of transients on transmission lines with corona via the method of characteristics", IEE Proceedings. Generation, Transmission and Distribution. (Morelos: Institution of Electrical Engineers) 142 (1), ISSN 1350-2360 

External articles and further reading[link]

Wikimedia Commons has media related to: Transmission lines

• Annual Dinner of the Institute at the Waldorf-Astoria. Transactions of the American Institute of Electrical Engineers, New York, January 13, 1902. (Honoring of Guglielmo Marconi, January 13, 1902)
• Avant! software, Using Transmission Line Equations and Parameters. Star-Hspice Manual, June 2001.
• Cornille, P, On the propagation of inhomogeneous waves. J. Phys. D: Appl. Phys. 23, February 14, 1990. (Concept of inhomogeneous waves propagation — Show the importance of the telegrapher's equation with Heaviside's condition.)
• Farlow, S.J., Partial differential equations for scientists and engineers. J. Wiley and Sons, 1982, p. 126. ISBN 0-471-08639-8.
• Kupershmidt, Boris A., Remarks on random evolutions in Hamiltonian representation. Math-ph/9810020. J. Nonlinear Math. Phys. 5 (1998), no. 4, 383-395.
• Pupin, M., U.S. Patent 1,541,845, Electrical wave transmission.
• Transmission line matching. EIE403: High Frequency Circuit Design. Department of Electronic and Information Engineering, Hong Kong Polytechnic University. (PDF format)
• Wilson, B. (2005, October 19). Telegrapher's Equations. Connexions.
• John Greaton Wöhlbier, ""Fundamental Equation" and "Transforming the Telegrapher's Equations". Modeling and Analysis of a Traveling Wave Under Multitone Excitation.
• Agilent Technologies. Educational Resources. Wave Propagation along a Transmission Line. Edutactional Java Applet.
• Qian, C., Impedance matching with adjustable segmented transmission line. J. Mag. Reson. 199 (2009), 104-110.

External links[link]

• Transmission Line Parameter Calculator
• Interactive applets on transmission lines
• SPICE Simulation of Transmission Lines

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