Editing Transmission line (section)

==Telegrapher's equations==
{{Main|Telegrapher's equations}}
{{See also|Reflections on copper lines}}

The '''telegrapher's equations''' (or just '''telegraph equations''') are a pair of linear differential equations which describe the [[voltage]] (<math>V</math>) and [[Electric current|current]] (<math>I</math>) on an electrical transmission line with distance and time. They were developed by [[Oliver Heaviside]] who created the ''transmission line model'', and are based on [[Maxwell's equations]].

[[Image:Transmission line element.svg|thumb|right|250px|Schematic representation of the elementary component of a transmission line]]
The transmission line model is an example of the [[distributed-element model]]. It represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

* The distributed resistance <math>R</math> of the conductors is represented by a series resistor (expressed in [[ohm]]s per unit length).
* The distributed inductance <math>L</math> (due to the [[magnetic field]] around the wires, [[self-inductance]], etc.) is represented by a series [[inductor]] (in [[Henry (unit)|henries]] per unit length).
* The capacitance <math>C</math> between the two conductors is represented by a [[Shunt (electrical)|shunt]] [[capacitor]] (in [[farad]]s per unit length).
* The [[Electric conductance|conductance]] <math>G</math> of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (in [[Siemens (unit)|siemens]] per unit length).

The model consists of an ''infinite series'' of the elements shown in the figure, and the values of the components are specified ''per unit length'' so the picture of the component can be misleading. <math>R</math>, <math>L</math>, <math>C</math>, and <math>G</math> may also be functions of frequency. An alternative notation is to use <math>R'</math>, <math>L'</math>, <math>C'</math> and <math>G'</math> to emphasize that the values are derivatives with respect to length.  These quantities can also be known as the [[primary line constants]] to distinguish from the secondary line constants derived from them, these being the [[propagation constant]], [[attenuation constant]] and [[phase constant]].

The line voltage <math>V(x)</math> and the current <math>I(x)</math> can be expressed in the frequency domain as

:<math>\frac{\partial V(x)}{\partial x} = -(R + j\,\omega\,L)\,I(x)</math>

:<math>\frac{\partial I(x)}{\partial x} = -(G + j\,\omega\,C)\,V(x) ~\,.</math>

::(see [[differential equation]], angular frequency [[angular frequency|ω]] and imaginary unit {{mvar|[[Imaginary unit|j]]}})

===Special case of a lossless line===
When the elements <math>R</math> and <math>G</math> are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the <math>L</math> and <math>C</math> elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

:<math>\frac{\partial^2 V(x)}{\partial x^2}+ \omega^2 L\,C\,V(x) = 0</math>

:<math>\frac{\partial^2 I(x)}{\partial x^2} + \omega^2 L\,C\,I(x) = 0 ~\,.</math>

These are [[wave equation]]s which have [[plane wave]]s with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.

===General case of a line with losses===
In the general case the loss terms, <math>R</math> and <math>G</math>, are both included, and the full form of the Telegrapher's equations become:

:<math>\frac{\partial^2 V(x)}{\partial x^2} = \gamma^2 V(x)\,</math>

:<math>\frac{\partial^2 I(x)}{\partial x^2} = \gamma^2 I(x)\,</math>

where <math>\gamma</math> is the ([[complex number|complex]]) [[propagation constant]]. These equations are fundamental to transmission line theory. They are also [[wave equation]]s, and have solutions similar to the special case, but which are a mixture of sines and cosines with exponential decay factors. Solving for the propagation constant <math>\gamma</math> in terms of the primary parameters <math>R</math>, <math>L</math>, <math>G</math>, and <math>C</math> gives:

:<math>\gamma = \sqrt{(R + j\,\omega\,L)(G + j\,\omega\,C)\,}</math>

and the characteristic impedance can be expressed as

:<math>Z_0 = \sqrt{\frac{R + j\,\omega\,L}{G + j\,\omega\,C}\,} ~\,.</math>

The solutions for <math>V(x)</math> and <math>I(x)</math> are:

:<math>V(x) = V_{(+)} e^{-\gamma\,x} + V_{(-)} e^{+\gamma\,x} \,</math>

:<math>I(x) = \frac{1}{Z_0}\,\left( V_{(+)} e^{-\gamma\,x} - V_{(-)} e^{+\gamma\,x} \right) ~\,. </math>

The constants <math>V_{(\pm)}</math> must be determined from boundary conditions. For a voltage pulse <math>V_{\mathrm{in}}(t) \,</math>, starting at <math>x = 0</math> and moving in the positive <math>x</math>&nbsp;direction, then the transmitted pulse <math>V_{\mathrm{out}}(x,t) \,</math> at position <math>x</math> can be obtained by computing the Fourier Transform, <math>\tilde{V}(\omega)</math>, of <math>V_{\mathrm{in}}(t) \,</math>, attenuating each frequency component by <math>e^{-\operatorname{Re}(\gamma)\,x} \,</math>, advancing its phase by <math>-\operatorname{Im}(\gamma)\,x\,</math>, and taking the [[Fourier inversion theorem|inverse Fourier Transform]]. The real and imaginary parts of <math>\gamma</math> can be computed as

:<math>\operatorname{Re}(\gamma) = \alpha = (a^2 + b^2)^{1/4} \cos(\psi ) \,</math>

:<math>\operatorname{Im}(\gamma) = \beta = (a^2 + b^2)^{1/4} \sin(\psi) \,</math>

with

:<math>a ~ \equiv ~ R\, G\, - \omega^2 L\,C\ ~ = ~ \omega^2 L\,C\,\left[ \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right] </math>

:<math>b ~ \equiv ~ \omega\,C\,R + \omega\,L\,G ~ = ~ \omega^2 L\,C\,\left( \frac{R}{\omega\,L} + \frac{G}{\omega\,C} \right) </math>

the right-hand expressions holding when neither <math>L</math>, nor  <math>C</math>, nor <math>\omega</math> is zero, and with

:<math>\psi ~ \equiv ~ \tfrac{1}{2}\operatorname{atan2}(b,a)\,</math>

where [[atan2]] is the everywhere-defined form of two-parameter arctangent function, with arbitrary value zero when both arguments are zero.

Alternatively, the complex square root can be evaluated algebraically, to yield:

:<math> \alpha = \frac{\pm b}{\sqrt{2 \left( - a + \sqrt{a^2 + b^2} \right)~}},</math>
and
:<math> \beta = \pm { \sqrt{\tfrac{1}{2}\left( - a + \sqrt{a^2 + b^2} \right)~} },</math>

with the plus or minus signs chosen opposite to the direction of the wave's motion through the conducting medium. ({{mvar|a}}  is usually negative, since <math>G</math> and <math>R</math> are typically much smaller than <math>\omega C</math> and <math>\omega L</math>, respectively, so  {{mvar|&minus;a}}  is usually positive. {{mvar|b}} is always positive.)

===Special, low loss case===
For small losses and high frequencies, the general equations can be simplified: If <math>\tfrac{R}{\omega\,L} \ll 1 </math> and <math>\tfrac{G}{\omega\,C} \ll 1</math> then

:<math>\operatorname{Re}(\gamma) = \alpha \approx \tfrac{1}{2}\sqrt{L\,C\,}\,\left( \frac{R}{L} + \frac{G}{C} \right) \,</math>

:<math>\operatorname{Im}(\gamma) = \beta \approx \omega\,\sqrt{L\,C\,} ~.\,</math>

Since an advance in phase by <math>- \omega\,\delta</math> is equivalent to a time delay by <math>\delta</math>, <math>V_{out}(t)</math> can be simply computed as

:<math>V_{\mathrm{out}}(x,t) \approx V_{\mathrm{in}}(t - \sqrt{L\,C\,}\,x)\,e^{- \tfrac{1}{2}\sqrt{L\,C\,}\,\left( \frac{R}{L} + \frac{G}{C} \right)\,x }. \,</math>

===Heaviside condition===
{{Main|Heaviside condition}}

The [[Heaviside condition]] is <math> \frac {G}{C} = \frac {R}{L} </math>.  

If R, G, L, and C are constants that are ''not'' frequency dependent and the Heaviside condition is met, 
then waves travel down the transmission line without [[dispersion (optics)|dispersion]] distortion.