Impedances and transmission line effects...

[Gryphon]

2MHz corresponds to a wavelength of 150m in free space, so rather less than that in screened cable.  Someone upthread said that you need to start thinking transmission line theory if the line length approaches one-tenth of a wavelength (not the figure I would use myself, but never mind).  Mmm, 150/10 = 15.

Well if Trevor's tie-line gives him grief at 2MHz, I've no doubt that Trevor will know full well that he needs to terminate it. Certainly, we've established that it's absolutely fine for audio...

So if not 1/10, then what's your preference, and why? Some say 1/4. Interested in your thoughts...

If Murphy v310 does want to terminate his line correctly, it will help him enormously to know what the Zo is.  Which rather, I think, brings us back to where we started.

I'm going to give a politician's answer to your question, which means I'm going to duck it, but unlike most contemporary politicians, I'm going to have a stab at justifying the duck.

I, like, I imagine, many of my contemporaries, left University with the idea that lines of less than lambda/4 were easy, because you could treat them more-or-less like hook-up wire off the reel.  That little misconception didn't long survive encounters with real lines of all sorts, on behalf of my then employer.  So the answer I would now give is that such "rules-of-thumb" aren't helpful, because they can be misleading, and that the answer is what lawyers call "case-specific": in other words, it depends on what you are trying to do, how well you need to do it, and the unintended consequences of not doing it well enough.

As a generality, the problem is easier with low-level signals than with lots of power to transmit.  Power, especially r.f. power, is expensive and awkward to get rid of, so you want as much as possible going where it is intended.  Narrow-band applications are easier to solve than wideband: in the former case, tuned stubs and other resonant solutions are available. Nor is this an analogue-only problem: reflections on circuits carrying digital signals are a major cause of timing jitter.  This can be a serious problem in, for example, backplanes in equipment carrying high-speed digital signals, where the tracks act as a parallel line, spaced by a mixture of air and the backplane dielectric.

To show how easy it is to fall over in real life, if you don't think carefully enough about what's going on, consider for a moment the situation shown in the attached drawing. S and R are two centres heavily interconnected by lines. Some of these are co-axial tubes with interstitial pairs, which are routed directly between S and R.  We have tried to allocate the interstitial pairs to music circuits, but there are not enough of them, and so some musics are in multipair cables.  These do not run directly between S and R but via a third point which we will call L/SHE. Ignore for a moment the limb shown between L/SHE and E.

The cable we are interested in is the 50-pair shown in the drawing, with distances marked. Most of the pairs are used either for control lines or for voice telephony, but about 10 are equalised for use as music circuits. A user department complains that one of these musics is no longer serviceable as such, as it has a wideband notch in the frequency response, about 20dB deep and centred on 4.5kHz.  Investigation shows that all the other musics in this cable are like that as well.

Liaison with our friends at L/SHE reveals that, in preparation for some work on site at E, the run between L/SHE and E has been added to the line, by jumpering it on the frame at L/SHE, overnight. (S and E are in different buildings on the same site, about 200m apart)  The instructions to the frame technician were to jumper only the pairs in use for telephony, but he has misunderstood and jumpered all 50.  The line between L/SHE and E is not electrically terminated at E but the pair ends are left open-circuit on the frame there.

This is audio, right? No need to treat it as a transmission line over these distances, right? Lambda for 4.5kHz is about 67km in free space. 1km is lambda/67.  Yet the initial suspicion that the line between L/SHE and E is the cause of the problem is entirely confirmed by asking the Exchange Engineer at L/SHE to insert a plug in the break-jack which has been provided there.

What is going on?  Well, as described upthread, the characteristic impedance of the pairs is probably somewhere in the 110 - 140 ohm range.  They are, however, used with a sending chain at S which assumes that the Zo is 600 ohms, and terminated in 600 ohms at R.  So, even before we add the stub, there are probably standing waves on the line: however,  their effects  have been allowed for when the line equaliser was built.

Adding the stub does two things: it produces an impedance discontinuity at L/SHE, and it adds further standing waves due to the open circuit at E from which, clearly, reflection is total.  The saving grace is the line loss: the d.c. resistance is of the order of 170 ohms/km with the loss rising with frequency.