Talk:Heaviside condition

The article says that "the condition is satisfied when the ratio of conductance G (in siemens) to capacitance C (in farads) equals the ratio of resistance R (in ohms) to inductance L (in henries)"

     G/C = R/L

But, G=1/R, therefore

G/C = R/L <=> 1/RC = R/L <=> C = L

So, it appears that resistance is actually irrelevant. Am I missing something? Nikola 18:22, 15 November 2005 (UTC)[reply]

Of course I am. It's hard to work with slashes, so actually L/C=R^2. Resistance is relevant. Nikola 05:05, 22 January 2006 (UTC)[reply]

Also, G is not 1/R. R represents the series resistance per unit length; it's (approximately, ignoring G) what you'd get if you short-circuited one end of a length of cable, and measured the other end with an ohmmeter. G represents the shunt resistance; it's (approximately, ignoring R) what you'd get if you open-circuited the cable and repeated your measurement. They just happen to be expressible in the same units (ohms, or 1/ohms). It makes the math work out nicer to express the series resistance in ohms, and the shunt resistance in 1/ohms. —The preceding unsigned comment was added by 66.30.10.35 (talk • contribs) 17:01, 3 December 2006 (UTC).[reply]

R is the series resistance component in the equivalent circuit model of a non-ideal transmission line, and G is the shunt conductance in the same model. Physically, the R and the G arise due to completely different reasons: R is due to the small, yet finite, resistivity in the non-ideal metal conductor(s) of the line, whereas G is due to the small, yet again finite, conductivity of the non-ideal dielectric material between the metal conductor(s) of the line. Therefore, in a lossless transmission line, where the metal is a perfect conductor and the dielectric is a perfect insulator, the metal exhibits no resistivity and the dielectric exhibits no conductivity. Hence, the characteristic impedance of the lossless line reduces to Z0 = sqrt(L/C).

To look at it another way, the non-ideal metals and dielectrics that are used in transmission lines (which is practically the case) are the root cause of signal distortion, but by implementing Heaviside's condition distortionless transmission lines can be designed even with lossy materials.

NOTE: I keep using the terms "lossy" and "lossless." "Lossy" simply means the transmission line has some R and/or G associated with it, and since any resistance when none is wanted (or conversely any conductance when non is wanted) results in loss of some of the energy of a signal, a signal propagating through a "lossy" transmission line will lose some energy. On the other hand a "lossless," i.e., perfect transmission line will cause no loss of signal energy.

Idunno271828 (talk) 09:25, 21 May 2008 (UTC)[reply]

Was there ever any transmission line designed that actually achieved the Heaviside condition

I have been looking at the early transatlantic cables. The dielectric is gutta-percha which is a good, low conductivity dielectric. In other words, G, in the Heaviside condition G/C=R/L is very close to zero. It seems just about impossible to meet the Heaviside condition by just increasing L. There are plenty of examples where L is increased, so I am not questioning that. However, I haven’t found any where L was increased enough to meet the Heaviside condition. I’m asking if anybody has a reference or a link describing an actual application of the Heaviside condition. By the way, I have looked in detail at the 1928 Newfoundland-Azores cable. It was at least a factor of 10000 away from meeting the Heaviside condition. Constant314 (talk) 18:09, 2 November 2020 (UTC)[reply]

I would be very surprised if anyone tried to do that, except maybe for very short lines, like on a board. It would mean very large attenuation across the frequency spectrum in order to get a flat spectrum; not a good trade. Dicklyon (talk) 18:14, 2 November 2020 (UTC)[reply]

I don't have any sources, but leaky feeder might be a candidate. The line losses are not caused by high G (they are due to radiation), nor is it designed to meet the Heaviside condition, but the losses can be modelled as a high-G line. It might very well be closer to the ideal Heaviside line than any line using high-mu materials. SpinningSpark 15:19, 21 November 2020 (UTC)[reply]

Thanks for the suggestion. I'm not a radiating systems expert by a long margin, but I suspect that the losses in a leaky feeder show up as radiation resistance and thus increase the R term. Constant314 (talk) 15:35, 21 November 2020 (UTC)[reply]

Is the Heaviside condition the Zeno's paradox of transmission lines?

Greetings. I hope you like the title of this section. I am trying to pull in some editors to discuss the issue of whether the Heaviside condition has or ever had any practical significance, or whether it is like Zeno's paradox: much talked about but without effect on anything.

My answer to that question is a resounding No!.

The appeal of the Heaviside condition is simple, if you could achieve the condition G/C = R/L, then the phase velocity, $v={\frac {1}{\sqrt {LC}}}$ and the impedance, $Z_{0}={\sqrt {\frac {L}{C}}},$ would be independent of frequency. Well, not quite. You must also have L and C independent of frequency.

There is a problem: there are no known materials that you can build a transmission line out of that can meet the Heaviside condition over a useful frequency range. The problems are two-fold. First, R and G are strong functions of frequency. If you want to increase L to achieve the Heaviside condition, then you must add magnetic materials. That would cause L to also be a strong function of frequency. If you could achieve the Heaviside condition at a single frequency, the bandwidth would be extremely narrow. The other problem is the that R/L is about $10^{8}$ times larger than G/C. You simply cannot increase L enough to achieve the Heaviside condition.

Here is a chart that gives more insight into the problem:

The Heaviside condition is met only where the blue line crosses a red line! So, it is not impossible, but it is at only one frequency. I gave three representative red lines for three representative dielectrics. The line labeled "high" is typical of modern dielectrics. The line labeled "med" is based on gutta-percha which is the dielectric used for the early trans-ocean telegraph cables. If we were to try to force the blue line to coincide with the medium red line, we would have to increase L by 8 orders of magnitude and the matching would only work up to 0.01 Hz. That would not be enough bandwidth.

Note that the scales are logarithmic. Variations of the transmission line cross section only cause minor variations in the curves and would not change the conclusions.

I used some very simple equations to plot this graph. They were

$C$ = a constant

$G=G_{dc}+\omega C\tan \delta$

$L=L_{\infty }+L_{\Delta ,\omega }$

$L_{\Delta ,\omega }$ and $R$ computed from a formula by Weeks as desribed in Skin effect#Impedance of round wire.

I realize that this is synthesis so that I cannot write it into the article without a strong consensus. Constant314 (talk) 15:51, 14 April 2023 (UTC)[reply]

I came across [[1]] which has data on the 1928 submarine cable. Particularly look at figure 4, which shows the loading profile. The inductance ranges from 5.6 mH/nm (nm is nautical mile) in the unloaded end sections to 205 mH/nm in the maximally loaded middle section. That is a factor of about 37, or 1.5 orders of magnitude. That is about a factor of a million short of the Heaviside condition. But notice that they increased the inductance toward the middle of the cable. These guys knew what they were doing. They were optimizing the overall dispersion rather than trying to meet the Heaviside condition on particular sections. I don't know what method they used, but I strongly suspect that they used filter theory and treated the sections as lumped elements. Credit is due to the telegrapher's equations for telling them that they can treat the sections as lumped elements.

Here is a plot of the propagation velocity.

.

As can be seen, the velocity curve is pretty flat from 3 to 100 Hz which is about where cable was operating. Constant314 (talk) 21:21, 14 April 2023 (UTC)[reply]

Working on the article

Please give me a little while before making changes. Constant314 (talk) 23:11, 18 April 2023 (UTC)[reply]

I am done. The article no longer makes excessive claims about the importance or application of the Heaviside condition. I would have like to have said more, but don't have the sources. Constant314 (talk) 01:39, 21 April 2023 (UTC)[reply]