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User:Constant314/Telegrapher's equations frequency regimes

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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.[1]: 381–392  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, , or complex frequency, , and distance . 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,[2]: 380–419  Hayt,[1]: 381–392  Marshall,[3]: 359–378  Sadiku,[4]: 497–505  Harrington,[5]: 61–65  Karakash,[6]: 5–14  Metzger.[7]: 1–10 

Finite length

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Coaxial transmission line wih one source and one load

Johnson gives the following solution,[8]: 739–741 

where

length of the transmission line.

In the special case of the solution reduces to

.[1]: 385  is called the attenuation constant and is called the phase constant.
.[1]: 385  = the characteristic impedance.

Frequency regimes

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

.
where Note, is also called dielectric loss tangent.

Where is called the attenuation constant and is called the phase constant.

In conventional transmission lines, and are relatively constant compared to and . Behavior of a transmission line over many orders of frequency is mainly determined by and , 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

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Critical frequencies
Name Definition Notes
end of the RG regime
middle of the RGC regime, is also called the dielectric relaxation time constant]]
beginning of the RC regime
end of the RC regime
middle of the RLC regime
beginning of the LC regime
beginning of the dielectric loss dominated regime[8]: 200 
Example the frequency above which skin effect is significant[8]: 185 
Example cutoff frequency of the lowest waveguide mode[8]: 217 
Example Example Example
Example Example Example
Example Example Example
Example Example Example

Typical relationships

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always true

,

usually true

usually true with exceptions

There are cases where

  • When the dielectric is very low loss, such as vacuum or dry nitrogen, then becomes very large (or even infinite in the case of ideal vacuum).
  • When the separtion between conductors is large, then becomes small, decreasing inversely with the separation.

Regimes

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Regimes of the telegrapher's equations
Description Dominant terms lower frequency upper frequency
DC RG 0 0 Example Example
Near DC RG Example Example
Very low frequency RGC Example Example
Low frequency, voice frequency RC Example Example
Intermediate frequency RLC Example Example
High frequency LC Example Example

Regimes of transmission lines

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Regimes of transmission lines[8]: 121–236 
Description Dominant terms lower frequency upper frequency Notes
Lumped (Pi model) - 0 determined by Example Example less than 14.3° phase shift and .03 dB loss
RC RC, RGC, RG 0 Example Example
LC, Constant loss LC Example Example If then this regime does not exist
Skin effect LC Example Example
Dielectric loss LC Example Example If then this regime does not exist
Waveguide dispersion LC Example Example

Graphs

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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

References

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  1. ^ a b c d Hayt, William H. (1989), Engineering Electromagnetics (5th ed.), McGraw-Hill, ISBN 0070274061
  2. ^ Kraus, John D. (1984), Electromagnetics (3rd ed.), McGraw-Hill, ISBN 0-07-035423-5
  3. ^ Marshall, Stanley V.; Skitek, Gabriel G. (1987), Electromagnetic Concepts and Applications (2nd ed.), Prentice-Hall, ISBN 0-13-249004-8
  4. ^ Sadiku, Matthew N.O. (1989), Elements of Electromagnetics (1st ed.), Saunders College Publishing, ISBN 0-03-013484-6
  5. ^ Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields (1st ed.), McGraw-Hill, ISBN 0-07-026745-6
  6. ^ Karakash, John J. (1950), Transmission lines and Filter Networks (1st ed.), Macmillan
  7. ^ Metzger, Georges; Vabre, Jean-Paul (1969), Transmission Lines with Pulse Excitation (1st ed.), Academic Press, LCCN 69-18342
  8. ^ a b c d e Johnson, Howard; Graham, Martin (2003), High Speed Signal Propagation (1st ed.), Prentice-Hall, ISBN 0-13-084408-X