# Telegrapher's equations

Oliver Heaviside developed the transmission line theory known as the telegrapher's equations. The telegrapher's equations describe how electrical signals move along transmission lines such as telegraph wires. The equations embody coupled linear differential equations in time and position for V(x,t) and I(x,t). The resulting signal propagates along the line as an inhomogeneous wave.

## The equations

The telegrapher's equations are the result of applying Maxwell's equations to two-conductor transmission lines. Assuming that a piece of wire may be modeled as an inductor with inductance per unit length [itex]L[itex] and a capacitor with capacitance per unit length [itex]C[itex], we obtain two coupled first order partial differential equations describing the voltage [itex]V[itex] and current [itex]I[itex] as a function of position [itex]x[itex] and time [itex]t[itex]:

[itex]{\partial \over {\partial x}}V(x,t)=-{{L{\partial \over {\partial t}}I(x,t)}}[itex]
[itex]{\partial \over {\partial x}}I(x,t)=-{{C{\partial \over {\partial t}}V(x,t)}}[itex]

If we further consider electrical resistance, we obtain the equations:

[itex]{\partial \over {\partial x}}V(x,t)=-{{L{\partial \over {\partial t}}I(x,t)}}-RI(x,t)[itex]

[itex]{\partial \over {\partial x}}I(x,t)=-{{C{\partial \over {\partial t}}V(x,t)}}-GV(x,t)[itex]

where [itex]R[itex] represents resistance per unit length in the wire and [itex]G[itex] represents leakage conductance between the wire and ground. By differentiating the first equation with respect to [itex]x[itex] and the second with respect to [itex]t[itex], and some algebraic manipulation, we obtain a pair of hyperbolic partial differential equations each involving only one unknown:

[itex]{\partial^2 \over {\partial x}^2}V=CL{\partial^2 \over {\partial t}^2}V+(CR+GL){\partial \over {\partial t}}V+GRV[itex]
[itex]{\partial^2 \over {\partial x}^2}I=CL{\partial^2 \over {\partial t}^2}I+(CR+GL){\partial \over {\partial t}}I+GRI[itex]

Note that these equations resemble the homogeneous wave equation with extra terms in [itex]V[itex] and [itex]I[itex] and their first derivatives. These extra terms cause the signal to decay and spread out with time and distance. If [itex]G=R=0[itex] (that is, no loss to resistance or leakage), both equations degenerate to the exact wave equation:

[itex]{\partial^2 \over {\partial t}^2}V={1 \over {CL}}{\partial^2 \over {\partial x}^2}V[itex]
[itex]{\partial^2 \over {\partial t}^2}I={1 \over {CL}}{\partial^2 \over {\partial x}^2}I[itex]

In a line of infinite length, these equations each describe a wave traveling with speed [itex]c={1 \over \sqrt{CL}}[itex].

## External links and references

• Avant! software, "Using Transmission Line Equations and Parameters (http://www.ece.cmu.edu/~ee762/hspice-docs/html/hspice_and_qrg/hspice_2001_2-124.html)". Star-Hspice Manual, June 2001.
• Boesch, "Basic Transmission Line Theory (http://www.ece.stevens-tech.edu/people/Boesch/Boesch/10-part%201-Transmission%20line%20theory.doc)". (DOC format)
• Cornille, P, "On the propagation of inhomogeneous waves (http://www.iop.org/EJ/abstract/0022-3727/23/2/001)". 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 (http://www.ee.iastate.edu/~hsiu/ee313/notes8/16.html)". EE 313 Electromagnetic Fields and Waves. IAState.edu.
• Kupershmidt, Boris A., "Remarks on random evolutions in Hamiltonian representation (http://arxiv.org/abs/math-ph/9810020)". 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, Transmission and Distribution, IEE Proceedings. Vol. 142.1, Inst. de Investigaciones Electr., Morelos, Jan 1995. ISSN 1350-2360
• "Transmission line matching (http://cktse.eie.polyu.edu.hk/eie403/)". EIE403: High Frequency Circuit Design. Department of Electronic and Information Engineering, Hong Kong Polytechnic University. (PDF format)
• Wilson, Bill, "Telegrapher's Equations (http://cnx.rice.edu/content/m1044/latest/)". Cnx.rice.edu.
• Wöhlbier, John Greaton, "Fundamental Equations (http://www.wildwestwohlbiers.org/john/htmlmsthesis/node35.html)". Modeling and Analysis of a Traveling Wave Tube Under Multitone Excitation.
• Wöhlbier, John Greaton, "Transforming the Telegrapher's Equation (http://www.wildwestwohlbiers.org/john/htmlmsthesis/node41.html)". Modeling and Analysis of a Traveling Wave Tube Under Multitone Excitation.
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