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# Telegrapher's Equations

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**New for September 2018!** The term "telegrapher" brings to mind an old-school telegraph office with a clerk wearing a green eye shade furiously clicking a key pad. Surely, this is wishful thinking by historians, modern microwave engineering, home of massive MIMO and engineered bandgaps could not be related to telegraphy?

While we are on the subect of telegraphy, here's an answer to a September 2018 New York Times crossword puzzle that none of us should get wrong: verb in first telegraphed message? "Hath", as in "what hath God wrought?" This message was sent by Samuel Morse himself, on the occasion of the opening of the first commercial telegraph which operated between Washington DC and Baltimore, Maryland.

Just as modern physics is built on the foundation of Newton's Laws, microwave transmission lines are perfectly described by Microwave Hall-of-Famer Oliver Heaviside's work in trouble-shooting a trans-Atlantic cable, in the 1880s, which came to be called the Telegrapher's Equations. A related topic, Maxwell's Equations, predate the Telegrapher's Equations and are equally important in microwave theory. Just remember, telegrapher's equations are to transmission lines as Maxwell's equations are to electromagnetic radiation. Just as Newton's laws break down at relativistic velocities, you can find that Maxwell and Heaviside could not anticipate quantum mechanics....

The Telegraphers' Equations come from a transmission line model, answering the question, "if I impose a time-varying voltage on one side of the transmission line (the input), what happens on the other side (the output)?" The lumped element model represents an infinitesimally small section of a transmission line. The "prime" notation is an abbreviation, these are derivatives with respect to length. Not shown on our graphic are the generator, its internal impedance, and the load impedance, which set up boundary conditions for the solution.

Kirchoff's Laws were in place in 1845, prior to Heaviside's work. Kirchoff stated that the sum of voltages around a loop muct equal zero, and the sum of currents into a node must also equal zero.

Noting that voltages and currents in inductors and capacitors are functions of time for sinusoidal steady-state signals, the solution to the aformentioned question using Kirchoff's Laws is two partial differential equations, in time and distance, for current and voltage. The solution is an exponential function which can be re-written with trigonometric functions, with sinusoidal waves in both forward and backward directions. Note that the solution always varies over frequency, which is what keeps microwave engineers employed.

Below, Dr. Bart Smolders describes how the transmission line model is solved into usable equations, and shows some of the simple solutions to different loading such as short circuit, open circuit and matched load. Bravo!

Professor Bart Smolders' lecture on Transmission line theory

In addition to the wave solutions described in the video, transmission line behavior can be predicted based on propagation constant, attenuation constant, and characteristic impedance. Our page on transmission line model describes this.

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