The Relation between Risetime of Pulse and Length of Cable

The figure 1 shows typical step-responces of a cable, in other words, voltage waveform at the load end of the cable when voltage is suddenly added at the another end of cable. This is the basic of pulse transmission waveform, because, with step-responces, you can calculate output of any waveforms.

[Step response of the cable]
Fig. 1 step response of relatively short cable

Please be noted that this is a very general type of waveform which forms with good quality cables under any conditions. Some people may think that they have seen sharper waveforms than this. Once enlarged, though, they are the same waveform as this.

Now, I assume almost everybody knows that, the longer the cable is, the more time the transmission of pulse waveform takes to rise due to waveform weakening. For example, the figure 2 shows how it is with slightly longer cable of the same kind as the above;

[Step response of long cable]
Fig. 2 step response of more longer cable

Risetime, which is defined as the time to reach from 10% to 90% in ordinary circuits, is often used as an index to see how pulse waveform weakens. In case of cables, however, the waveform rises sharply in the beginning of rise and slowly at the end. So, let's define "risetime" of pulse waveform of cable as the time to reach to 50%, here. In the above figure, "t0" is the risetime.

If you do not like this definition, please use the following relation which is formed with typical cables so that you can convert it to risetime of ordinary circuits;

 10-90% risetime = 28.6 * 50% risetime
Here, "=" means "approximately" equal. Exactly speaking, it is 28.64.....

By the way, it must be easy to understand by the linear circuit theory that waveform weakening in cable transmission is caused by attenuation distortion and phase distortion of cables. These distortions are proportionate to the length of the cable. For example, if the cable is doubled, so is attenuation.

If so,

  Is risetime proportionate to length of cable?
Or, is the relation between risetime and length of cable more complicated and unpredictable?

This is the question, this time.

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