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{{Unreferenced|date=December 2009}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
The '''Heaviside condition''', named for [[Oliver Heaviside]] (1850–1925), is the condition an electrical [[transmission line]] must meet in order for there to be no [[distortion]] of a transmitted signal.  Also known as the '''distortionless condition''', it can be used to improve the performance of a transmission line by adding [[loading coil|loading]] to the cable.


==The condition==
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[[File:Line model Heaviside.svg|right|thumb|350px|Heaviside's model of a transmission line.]]
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A transmission line can be represented as a [[distributed element model]] of its [[primary line constants]] as shown in the figure.  The primary constants are the electrical properties of the cable per unit length and are: [[capacitance]] ''C'' (in [[farad]]s per meter), [[inductance]] ''L'' (in [[Henry (unit)|henries]] per meter), series [[Electrical resistance|resistance]] ''R'' (in [[Ohm (unit)|ohms]] per meter), and shunt [[Electrical conductance|conductance]] ''G'' (in [[Siemens (unit)|siemens]] per meter). The series resistance and shunt conductivity cause losses in the line; for an ideal transmission line, <math style="vertical-align:0%;">\scriptstyle R=G=0</math>.  
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The Heaviside condition is satisfied when
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:<math>\frac{G}{C} = \frac{R}{L}.</math>


This condition is for no distortion, but not for no loss.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==Background==
<!--'''PNG''' (currently default in production)
A signal on a transmission line can become distorted even if the line constants, and the resulting [[transmission function]], are all perfectly linear. This happens in two ways: firstly, the attenuation of the line can vary with frequency which results in a change to the shape of a pulse transmitted down the line.  Secondly, and usually more problematically, distortion is caused by a frequency dependence on [[phase velocity]] of the transmitted signal frequency components.  If different frequency components of the signal are transmitted at different velocities the signal becomes "smeared out" in space and time, a form of distortion called [[dispersion (optics)|dispersion]].
:<math forcemathmode="png">E=mc^2</math>


This was a major problem on the first [[transatlantic telegraph cable]] and led to the theory of the causes of dispersion being investigated first by [[Lord Kelvin]] and then by Heaviside who discovered how it could be countered.  Dispersion of [[electric telegraph|telegraph]] pulses, if severe enough, will cause them to overlap with adjacent pulses, causing what is now called [[intersymbol interference]].  To prevent intersymbol interference it was necessary to reduce the transmission speed of the transatlantic telegraph cable to the equivalent of {{Frac|1|15}} [[baud]].  This is an exceptionally slow data transmission rate, even for human operators who had great difficulty operating a morse key that slowly.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


For voice circuits (telephone) the frequency response distortion is usually more important than dispersion whereas digital signals are highly susceptible to dispersion distortion. For any kind of analogue image transmission such as video or facsimile both kinds of distortion need to be eliminated.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==Derivation==
==Demos==
The transmission function of a transmission line is defined in terms of its input and output voltages when correctly terminated (that is, with no reflections) as


:<math>\frac{V_\mathrm{in}}{V_\mathrm{out}} = e^{\gamma x}</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


where <math>x</math> represents distance from the transmitter in meters and


:<math>\gamma = \alpha +j \beta \,</math>
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
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are the [[secondary line constants]], ''α'' being the attenuation in [[neper]]s per metre and ''β'' being the phase change constant in [[radian]]s per metre.  For no distortion, ''α'' is required to be constant with angular frequency ''ω'', while ''β'' must be proportional to ''ω''.  This requirement for proportionality to frequency is due to the relationship between the velocity, ''v'', and [[phase constant]], ''β'' being given by,
==Test pages ==


:<math>v = \frac{\omega}{\beta}</math>
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and the requirement that phase velocity, ''v'', be constant at all frequencies.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
The relationship between the primary and secondary line constants is given by
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
:<math>\gamma^2 = (\alpha +j \beta)^2 = (R+j \omega L)(G + j \omega C)\,</math>
 
which has to be of the form <math style="vertical-align:-10%;">\scriptstyle (A+j\omega B)^2</math> in order to meet the distortionless condition.  The only way this can be so is if <math style="vertical-align:-10%;">\scriptstyle (R+j \omega L)</math> and <math style="vertical-align:-10%;">\scriptstyle (G + j \omega C)</math> differ by no more than a constant factor.  Since both have a real and imaginary part, the real and imaginary parts must independently be related by the same factor, so that;
 
:<math>\frac {R}{G} = \frac {j \omega L}{j \omega C} </math>
 
and the Heaviside condition is proved.
 
===Line characteristics===
The secondary constants of a line meeting the Heaviside condition are consequently, in terms of the primary constants:
 
Attenuation,
 
:<math>\alpha = \sqrt {RG} </math>&nbsp; nepers/metre
 
Phase change constant,
 
:<math>\beta = \omega \sqrt {LC} </math>&nbsp; radians/metre
 
Phase velocity,
 
:<math>v = \frac {1}{\sqrt {LC}}</math>&nbsp; metres/second
 
===Characteristic impedance===
The [[characteristic impedance]] of a lossy transmission line is given by
:<math>Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}}</math>
In general, it is not possible to match this transmission line at all frequencies because the square root causes the expression to be [[Rational function|irrational]] and it consequently cannot be realised exactly with a network of discrete elements.  However, for a line which meets the Heaviside condition, there is a common factor in the fraction which cancels out the frequency dependent terms leaving,
:<math>Z_0=\sqrt{\frac{L}{C}},</math>
which is a real number, and independent of frequency. The line can therefore be matched with just a resistor at either end. This expression for <math>\scriptstyle Z_0 = \sqrt{L/C}</math> is the same as for a lossless line (<math style="vertical-align:-15%;">\scriptstyle R = 0,\  G = 0</math>) with the same ''L'' and ''C'', although the attenuation (due to ''R'' and ''G'') is of course still present.
 
==Practical use==
[[File:Mu-metal cable.svg|thumb|400px|An example of loaded cable]]
A real line, especially one using modern synthetic insulators, will have a ''G'' that is very low and will usually not come anywhere close to meeting the Heaviside condition.  The normal situation is that
 
:<math>\frac{G}{C} \ll \frac{R}{L}.</math>
 
To make a line meet the Heaviside condition one of the four primary constants needs to be adjusted and the question is which one.  ''G'' could be increased, but this is highly undesirable since increasing ''G'' will increase the loss.  Decreasing ''R'' is sending the loss in the right direction, but this is still not usually a satisfactory solution.  ''R'' must be decreased by a large fraction and to do this the conductor cross-sections must be increased dramatically.  This not only makes the cable much more bulky but also adds significantly to the amount of copper (or other metal) being used and hence the cost.  Decreasing the capacitance also makes the cable more bulky (since the insulation must now be thicker) but is not so costly as increasing the copper content. This leaves increasing ''L'' which is the usual solution adopted.
 
The required increase in ''L'' is achieved by loading the cable with a metal with high [[permeability (electromagnetism)|magnetic permeability]].  It is also possible to load a cable of conventional construction by adding discrete [[loading coil]]s at regular intervals.  This is not identical to a distributed loading, the difference being that with loading coils there is distortionless transmission up to a definite [[cut-off frequency]] beyond which the attenuation increases rapidly.
 
Loading cables to meet the Heaviside condition is no longer a common practice. Instead, regularly spaced digital [[repeaters]] are now placed in long lines to maintain the desired shape and duration of pulses for long-distance transmission.
 
==See also==
* [[Telegrapher's equations]]
* [[Maxwell's equations]]
 
{{DEFAULTSORT:Heaviside Condition}}
[[Category:Electrical engineering]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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