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

Coaxial transmission line wih one source and one load

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 ${\frac {\mathbf {V} _{L}}{\mathbf {V} _{S}}}={\frac {1}{2}}e^{-{\boldsymbol {\gamma }}x}$

$\gamma =\alpha +j\beta ={\sqrt {(R+sL)(G+sC)}}$. $\alpha$ is called the attenuation constant and $\beta$ is called the phase constant.

$Z_{c}={\sqrt {\frac {(R+sL)}{(G+sC)}}}$. = the characteristic impedance.

Frequency regimes

The formulas of characteristic impedance and propagation constant can be reformulated into terms of simple parameter ratios by factoring.

$Z_{c}={\sqrt {\frac {(R_{\omega }+j\omega L_{\omega })}{(G_{\omega }+j\omega C_{\omega })}}}={\sqrt {\frac {L_{\omega }}{C_{\omega }}}}{\sqrt {\frac {(1-jr_{\omega })}{(1-jg_{\omega })}}}$

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

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.

Critical frequencies

Critical frequencies

Name	Definition	Notes
$\omega _{1}$	$g_{\omega }=10$	end of the RG regime
$\omega _{2}$	$g_{\omega }=1$	middle of the RGC regime, $\tau ={\tfrac {1}{\omega _{2}}}$ is also called the dielectric relaxation time constant]]
$\omega _{3}$	$g_{\omega }=0.1$	beginning of the RC regime
$\omega _{4}$	$r_{\omega }=10$	end of the RC regime
$\omega _{5}$	$r_{\omega }=1$	middle of the RLC regime
$\omega _{6}$	$r_{\omega }=0.1$	beginning of the LC regime
$\omega _{\theta }$	$r_{\omega }=g_{\omega }$	beginning of the dielectric loss dominated regime
$\omega _{\delta }$	Example	the frequency above which skin effect is significant
$\omega _{\mathrm {c} }$	Example	cutoff frequency of the lowest waveguide mode
Example	Example	Example
Example	Example	Example
Example	Example	Example
Example	Example	Example

Typical relationships

always true $\omega _{1}<\omega _{2}<\omega _{3}$, $\omega _{4}<\omega _{5}<\omega _{6}$ usually true $\omega _{1}<\omega _{2}<\omega _{3}<\omega _{4}<\omega _{5}<\omega _{6}<\omega _{\theta }$ $\omega _{1}<\omega _{2}<\omega _{3}<\omega _{4}<\omega _{5}<\omega _{6}<\omega _{\mathrm {c} }$ $\omega _{1}<\omega _{2}<\omega _{3}<\omega _{\delta }<\omega _{\theta }$ $\omega _{1}<\omega _{2}<\omega _{3}<\omega _{\delta }<\omega _{\mathrm {c} }$ usually true with exceptions $\omega _{\theta }<\omega _{\mathrm {c} }$ There are cases where $\omega _{\theta }>\omega _{\mathrm {c} }$ When the dielectric is very low loss, such as vacuum or dry nitrogen, then $\omega _{\theta }$ becomes very large (or even infinite in the case of ideal vacuum). When the separtion between conductors is large, then $\omega _{\mathrm {c} }$ becomes small, decreasing inversely with the separation.

Regimes

Regimes of the telegrapher's equations

Description	Dominant terms	lower frequency	upper frequency	$\gamma$	$Z_{c}$
DC	RG	0	0	Example	Example
Near DC	RG	${\text{0}}^{\text{+}}$	$\omega _{1}$	Example	Example
Very low frequency	RGC	$\omega _{1}$	$\omega _{3}$	Example	Example
Low frequency, voice frequency	RC	$\omega _{3}$	$\omega _{4}$	Example	Example
Intermediate frequency	RLC	$\omega _{4}$	$\omega _{6}$	Example	Example
High frequency	LC	$\omega _{6}$	$\infty$	Example	Example

Regimes of transmission lines

Regimes of transmission lines

Description	Dominant terms	lower frequency	upper frequency	$\gamma$	$Z_{c}$	Notes
Lumped (Pi model)	-	0	determined by $l^{2}(\alpha _{\omega }^{2}+\beta _{\omega }^{2})<0.0625$	Example	Example	less than 14.3° phase shift and .03 dB loss
RC	RC, RGC, RG	0	$\omega _{r=1}$	Example	Example
LC, Constant loss	LC	$\omega _{r=1}$	$\omega _{\delta }$	Example	Example	If $\omega _{\delta }<\omega _{r=1}$ then this regime does not exist
Skin effect	LC	$\omega _{\delta }$	$\omega _{\theta }$	Example	Example
Dielectric loss	LC	$\omega _{\theta }$	$\omega _{\mathrm {c} }$	Example	Example	If $\omega _{c}<\omega _{\theta }$ then this regime does not exist
Waveguide dispersion	LC	$\omega _{\mathrm {c} }$	$\infty$	Example	Example

Graphs

Typical Good Transmission Line Parameter Ratioes

Typical Good Transmission Line Velocity

Typical Good Transmission Line Characteristic Impedance

Typical Good Transmission Line Loss

Typical Good Transmission Line Characteristic Impedance Phase

Lengths of RG58 transmission lines at one fifth wavelength

Newfoundland-Azores 1928 Submarine Telegraph Cable Estimated Velocity vs Frequency