User:Constant314/Telegrapher's equations frequency regimes

''' This is a work in progress. '''

It is intended to be a complementary article for Telegrapher's equations.

The telegrapher's equations are a set of two coupled, linear equations that predict the voltage and current distributions on a linear electrical transmission line. The equations are important because they allow transmission lines to be analyzed using circuit theory. The equations and their solutions are applicable from 0 Hz to frequencies at which the transmission line structure can support higher order waveguide modes. The equations can be expressed in both the time domain and the frequency domain. In the time domain approach the dynamical variables are functions of time and distance. The resulting time domain equations are partial differential equations of both time and distance. In the frequency domain approach the dynamical variables are functions of frequency, $$ \omega $$, or complex frequency, $$ s $$, and distance $$ x $$. The frequency domain variables can be taken as the Laplace transform or Fourier transform of the time domain variables or they can be taken to be phasors. The resulting frequency domain equations are ordinary differential equations of distance. An advantage of the frequency domain approach is that differential operators in the time domain become algebraic operations in frequency domain.

The Telegrapher's Equations are developed in similar forms in the following references: Kraus, Hayt, Marshall, Sadiku, Harrington, Karakash, Metzger.

Finite length


Johnson gives the following solution,



\frac {\mathbf{V}_L} {\mathbf{V}_S} = {[(\frac {\mathbf{H}^{-1} +\mathbf{H}} 2)(1+\frac {\mathbf{Z}_S}{\mathbf{Z}_L})+(\frac {\mathbf{H}^{-1} - \mathbf{H}} 2)(\frac {\mathbf{Z}_S}{\mathbf{\mathbf{Z}}_C} + \frac{\mathbf{Z}_C}{\mathbf{Z}_L}) ]}^{-1} = \frac {\mathbf{Z}_L \mathbf{Z}_C} {\mathbf{Z}_C(\mathbf{Z}_L+\mathbf{Z}_S)\cosh {\boldsymbol{\gamma} x}+( {\mathbf{Z}_L \mathbf{Z}_S} + {\mathbf{Z}_C}^2)\sinh {\boldsymbol{\gamma} x} }

$$

where


 * $$ \mathbf{H} = e^{-\boldsymbol{\gamma} x}, \ x = $$ length of the transmission line.

In the special case of $$ \mathbf{Z}_L=\mathbf{Z}_S=\mathbf{Z}_C $$ the solution reduces to $$ \frac {\mathbf{V}_L} {\mathbf{V}_S} = \frac 1 2 e^{-\boldsymbol{\gamma} x}$$


 * $$ \gamma = \alpha + j \beta = \sqrt{(R + s L)(G + s C)} $$.  $$  \alpha $$ is called the attenuation constant and $$  \beta $$ is called the phase constant.


 * $$ Z_c = \sqrt{\frac {(R + s L)} {(G + s C)}}   $$.  = the characteristic impedance.

Frequency regimes
The formulas of characteristic impedance and propagation constant can be reformulated into terms of simple parameter ratios by factoring.
 * $$ Z_c = \sqrt{\frac {(R_\omega + j \omega L_\omega)} {(G_\omega + j \omega C_\omega)}} = \sqrt { \frac {L_\omega} {C_\omega} } \sqrt{\frac {(1 - j r_\omega)} {(1 - j g_\omega)}}  $$



\gamma = \alpha + j \beta = \sqrt{(R_\omega + j \omega L_\omega)(G_\omega + j \omega C_\omega)} = j \omega \sqrt {L_\omega C_\omega} \sqrt{(\frac {R_\omega} { j \omega L_\omega} + 1)(\frac {G_\omega} { j \omega C_\omega} +1 )} = j \omega \sqrt {L_\omega C_\omega} \sqrt{(1 - j r_\omega)(1 - j g_\omega)} $$.


 * where $$ r_\omega=\frac {R_\omega} { \omega L_\omega}, \, g_\omega=\frac {G_\omega} {  \omega C_\omega}  \ $$  Note, $$ g_\omega $$ is also called dielectric loss tangent.

Where $$ \alpha $$ is called the attenuation constant and $$  \beta $$ is called the phase constant.

In conventional transmission lines, $$ C_\omega $$ and $$ L_\omega $$ are relatively constant compared to $$ r_\omega $$ and $$ g_\omega $$. Behavior of a transmission line over many orders of frequency is mainly determined by $$ r_\omega $$ and $$ g_\omega $$, each of which can be characterized as either being much less than unity, about equal to unity, much greater than unity, or infinite (at 0 Hz). Including 0 Hz, there are ten possible frequency regimes although in practice only six of them occur.

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