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 L and a capacitor with capacitance per unit length C, we obtain two coupled first order partial differential equations describing the voltage V and current I as a function of position x and time t:
{\partial \over {\partial x}}V(x,t)=-I(x,t)}}
{\partial \over {\partial x}}I(x,t)=-V(x,t)}}
If we further consider electrical resistance, we obtain the equations:
{\partial \over {\partial x}}V(x,t)=-I(x,t)}}-RI(x,t)
{\partial \over {\partial x}}I(x,t)=-V(x,t)}}-GV(x,t)
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:
{\partial^2 \over {\partial x}^2}V=CL{\partial^2 \over {\partial t}^2}V+(CR+GL){\partial \over {\partial t}}V+GRV
{\partial^2 \over {\partial x}^2}I=CL{\partial^2 \over {\partial t}^2}I+(CR+GL){\partial \over {\partial t}}I+GRI
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:
{\partial^2 \over {\partial t}^2}V={1 \over {CL}}{\partial^2 \over {\partial x}^2}V
{\partial^2 \over {\partial t}^2}I={1 \over {CL}}{\partial^2 \over {\partial x}^2}I
In an line of infinite length, these equations each describe a wave traveling with speed c={1 \over \sqrt{CL}}.

See also

External links and references

  • Whlbier, John Greaton, "Fundamental Equations". Modeling and Analysis of a Traveling Wave Tube Under Multitone Excitation.
* Whlbier, John Greaton, "Transforming the Telegrapher's Equation". Modeling and Analysis of a Traveling Wave Tube Under Multitone Excitation.

 

<< PreviousWord BrowserNext >>
carbachol
l'assomption, quebec
a story timeline of another world
bundesliga (ice hockey)
dekasegi
plva county
valga county
vru county
ayra (fire emblem)
ibm cpc
claro m. recto
ddr oberliga
sherlock holmes in the 22nd century
diego luna
dolbeau mistassini, quebec
sigurd (fire emblem)
gael garca bernal
agent 212
arthur avenue (the bronx)
magog, quebec
council for british research in the levant
film colorization
austrian champions
zhang daoling
keiko fukuda
hurricane fabian
kamelopedia
villa madama
method & red
rosie perez
rajneeshee cult
bushwick
list of telephony terminology
anzac cove
new spring
robert shankland
agami heron
list of religious populations
cuacuatzin
serart
masahiko kimura
thermal printer
zero install
hydrotherapy
Copyright 2005-2009 OnPedia.com. All Rights Reserved