Chapter 13 Transmission Lines

The first great digital information boom began when Samuel Morse patented the Telegraph in 1840.[1] His primary concern was the encoding of binary information and a method for automating the communication, since the electrical components of the transmitter and receiver were relatively simple. The automation process of encoded keys (like those that fit in a lock) was soon dispensed with in lieu of manually tapping a button; also called a key (like on a piano), and telegraph engineers became quite adept at sending Morse code manually. Telegraph technology spread quickly, telegraph lines became longer, and expectations arose for increased rates of data telemetry.

Transmission lines, which carry high frequency signals over large distances, were especially important to the development of electrodynamics, as the telegraph was the emerging communication technology in the nineteenth century. In Germany, Franz Neumann and Wilhelm Weber derived a wave equation for a signal in a wire, and predicted that it would travel at the same speed as light travels in a vacuum. In Britain, Michael Faraday, William Thomson, and Oliver Heaviside made significant gains in implementing telegraph cable technology.

Of particular interest to scientists and engineers of that era was the question of how to string a telegraph cable under the ocean. Among the many issues that dogged early efforts in laying the trans-Atlantic cable, for example, was signal attenuation and distortion due to the conductivity of salt water.[2] At the behest of the Electric Telegraph Company, Faraday, in 1854, witnessed an experimental demonstration of electrical signaling retardation in a long insulated cable submerged in the River Thames. He reasoned that the insulation acted as a dielectric placed between the copper wire and the slightly conducting water. Transmission of the electric current would therefore be limited because the cable acted like a capacitor that had to charge and discharge before the signal could be received at the other end.[3]

William Thomson soon developed a partial differential equation, identical in form to Fourier’s equation describing the diffusion of heat, which modeled the behavior of electric current as it passed through a long cable of finite capacitance and resistance.[4] The solutions Thomson obtained for his equation allowed him to correctly predict that current pulses along the line would retard at a rate proportional to the square of the cable’s length, thus limiting signal speed and causing poor performance. His expressed concerns about the first trans-Atlantic cable went unheeded, and the cable deteriorated only months after Queen Victoria exchanged the first telegraph messages with U.S. President James Buchanan in 1858. Thomson’s early work was not in vain, however. He was subsequently appointed chief engineer for the company that successfully laid the transatlantic cable in 1867. His technical contributions—both theoretical and practical—were so crucial to the success of the project that Queen Victoria knighted him within the year.[5]

Gustav Kirchhoff, in his 1857 series of papers on the propagation of electric signals through telegraph lines, demonstrated “a very remarkable analogy between the propagation of electricity in a wire and the propagation of a wave in a [tense string].” He also found the propagation velocity to be “very nearly equal to the velocity of light in vacuo,” although he failed to comment on the possible implications of this result.[6] Independently of Kirchhoff, Weber also performed a similar investigation, but his work wasn’t published until shortly after Kirchhoff’s. The remarkable implication of their analyses was that in a circuit of negligible resistivity, oscillating currents could be propagated along the wire with a constant velocity numerically equal to the velocity of light. Furthermore, this velocity was found to be independent of the nature of the conductors, of the cross section of the wire, and of the electric current density. This result of Kirchhoff and Weber is all the more remarkable as it came before Maxwell derived the wave equation from his introduction of the displacement current in Ampère’s law in 1865.

Oliver Heaviside, a nephew of British inventor Charles Wheatstone (from whom the “Wheatstone bridge” gets its name), quit his first and only employment, as a telegraph operator, to move back to his parents’ house and pursue research. Having chanced upon a copy of Maxwell’s Treatise, Heaviside was inspired to devote the rest of his life to clarifying and applying Maxwell’s theory. Heaviside was acquainted with the practical side of telegraphy through his job, and even published several early papers on elementary circuits and telegraph technology during that period. Despite, or perhaps because of, the fact that he was mostly self-taught, he made outstanding and profound contributions to electrodynamics, electrical engineering, and vector calculus in the 1870s and 1880s. Heaviside improved upon William Thomson’s work by including inductance and modeling transmission cables as a long chain of coupled LC circuits.[7] And, although they had long been used in undersea telegraphy, Heaviside first explained, and then patented, the idea of a coaxial cable.[8]

We use the results of Heaviside’s research to calculate the speed of transmission, the flux of energy, and the pressure exerted by electromagnetic fields along a coaxial cable. We will derive the telegrapher’s equations for lossless transmission in Thought Experiment 13.4 (p. 533), for example, and show how these yield a wave equation for signal propagation along the cable. It is interesting to note that John Henry Poynting actually used the examples of a long straight wire and, later, of an underwater telegraph cable in discussing the energy flow by electromagnetic fields.[9]

[1] S.F.B. Morse, “Improvement in the mode of communicating by signals by the application of electromagnetism,” U.S. Patent No. 1647, awarded on June 20, 1840.

[2] The plate on the opposite page is from C.G. Abott, Great Inventions, ©1932 by Smithsonian Institution Series Inc. (

[3] M. Faraday, “On electric induction-Associated causes of current and static effects,” Philosophical Magazine Series 4, 7:44, (1854), 197-208.

[4] W. Thomson, “On the theory of the electric telegraph,” Proceedings of the Royal Society 2, (1855), 382.

[5] The Queen had also tried to knight Faraday, but he refused the honor.

[6] G. Kirchhoff, “On the motion of electricity in wires,” Philosophical Magazine, 13 (1857), 393-412.

[7] O. Heaviside, “Electromagnetic induction and its propagation” in Electrical Papers by Oliver Heaviside in 2 vols. (Bronx, New York: Chelsea Publishing Company 1970), I: 492-56, II: 39-146.

[8] H. Griffiths, “Oliver Heaviside” in History of Wireless, ed. Tapan K. Sarkar et al. (Hoboken, New Jersey: J. Wiley & Sons, 2006), pp. 239-40; David Kraueter, British Radio and Television Pioneers: A Patent Bibliography (USA: Scarecrow Press), p. 66.

[9] J.H. Poynting, “Molecular electricity,” Collected Scientific Papers (Cambridge: Cambridge University Press, 1920), pp. 269-298.