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Oliver Heaviside developed the transmission line model, also known as the telegrapher's equations, that describes how electrical voltage and current vary along a conductor.
The theory applies to high-frequency transmission lines (such as telegraph wires and radio frequency conductors) but is also important for designing high-voltage energy transmission lines. The equations consist of two linear differential equations in time and position: one for V(x,t) and the other one for I(x,t). The model demonstrates that the electrical current can be reflected on the wire, and that wave patterns can appear along the line.
The equations
The telegrapher's equations can be understood as a simplified case of Maxwell's equations. In a more practical approach, one assumes that the conductor is composed out of an infinite series of elementary components (four-pole model):
- The resistance of the conductor (expressed in ohms per unit length) is represented by a resistor.
- The electromagnetic behaviour of the wire (due to the magnetic field around it, as well inductance from twisting of the wires) is represented by a coil (inductance per unit length L).
- The capacitive behaviour of the insulation between the signal wire and the return wire is represented by a capacitor (capacitance per unit length C).
- The conductivity of the insulation material is accounted for by a resistor between the signal wire and the return wire (conductance per unit length).
As such we obtain a pair of first-order partial differential equations, one function describing the voltage V along the line and the other the current I, both function of position x and time t:
If we further consider electrical resistance, we obtain the equations:
where R represents resistance per unit length in the wire and G represents leakage conductance between the wire and ground. By differentiating the first equation with respect to x and the second with respect to t, and some algebraic manipulation, we obtain a pair of hyperbolic partial differential equations each involving only one unknown:
Note that these equations resemble the homogeneous wave equation with extra terms in V and I and their first derivatives. These extra terms cause the signal to decay and spread out with time and distance. If G = R = 0 (that is, no loss to resistance or leakage), both equations degenerate to the exact wave equation:
If the line has infinite length or when it is terminated with its characteristic impedance, these equations indicate the presence of a wave, traveling with a speed . Note that this propagation speed applies to the wave phenomenon on the line and has nothing to do with the electron drift velocity or the speed of light in a vacuum.
See also
- Heaviside condition
- Transverse electromagnetic wave
- Longitudinal electromagnetic wave
- Smith chart
External links and references
- "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) [Early Radio History.com]
- Avant! software, "Using Transmission Line Equations and Parameters". Star-Hspice Manual, June 2001.
- Boesch, "Basic Transmission Line Theory". (DOC format)
- 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.
- Han, Hsiu C., "Transmission-Line Equations". EE 313 Electromagnetic Fields and Waves. IAState.edu.
- Kupershmidt, Boris A., "Remarks on random evolutions in Hamiltonian representation". Math-ph/9810020. J. Nonlinear Math. Phys. 5 (1998), no. 4, 383-395.
- Naredo, J.L., A.C. Soudack, and J.R. Marti, "Simulation of transients on transmission lines with corona via the method of characteristics". Generation, telegraphers equations Transmission and Distribution, IEE Proceedings. Vol. 142.1, Inst. de Investigaciones Electr., Morelos, Jan 1995. ISSN 1350-2360
- "Transmission line matching". EIE403: High Frequency Circuit Design. Department of Electronic and Information Engineering, Hong Kong Polytechnic University. (PDF format)
- Wilson, Bill, "Telegrapher's Equations". Cnx.rice.edu.
- Wöhlbier, John Greaton, "Fundamental Equations". Modeling and Analysis of a Traveling Wave Tube Under Multitone Excitation.
- Wöhlbier, John Greaton, "Transforming the Telegrapher's Equation". Modeling and Analysis of a Traveling Wave Tube Under Multitone Excitation.
Electronics Topics
The field of electronics is the study and use of systems that operate by controlling the flow of electrons or other electrically charged particles in devices such as thermionic valves and semiconductors. The design and construction of electronic circuits to solve practical problems is part of the fields of electronic engineering, and the hardware design side of computer engineering. The study of new semiconductor devices and their technology is sometimes considered as a branch of physics.
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