KenD Wrote:eg, A line has a civil line speed of 160km/h
I used the standard braking distance of 1189m,
What do you mean by "standard braking distance"? The IRSE exam generally gives the braking rate of 0.5m/s/s; 1189m seems a little short for the maximum permissible line speed so it looks like you are assuming higher brake rate as I make it 1998m (so round to 2000m).
Whatever the value is, this is the closest spacing of 3 aspect signals and therefore lowest possible headway distance is then derived by:
double signal spacing + sighting distance + train length + overlap length. Convert this to headway time by DIVIDING [thanks Ken!] by the timetabled speed of the train (i.e. could be the maximum train speed but may be less). Then, as you have said, you need to ensure that there is an appropriate % contingency to arrive at the timetabled headway.
Obviously other constraints dictate where signals can be placed so that in reality won't be able to put every signal at true minimum spacing, so unless your desired headway can still be achieved with signals spaced at 115% braking then it's not really a tenable solution and thus 4 aspects are needed.
Similar calculation for 4 aspects except instead of the braking from the Green aspect to the Red occuring over 2 signal sections, it occurs over 3 sections; hence multiply the spacing by 3 rather than 2, BUT don't forget that in the 4 aspect case the minimum signal spacing is 0.5 braking distance!
You need to do the sums for stopping headways (and that is where a graph can be helpful), but looking at the figures it seems that the non-stop is probably the more onerous. The reason that I say this: 120km/h = 75 mph = approx 34 m/s and thus will take just about 68s to stop at the usual brake rate and similar to accelerate, so compared to the non-stop train lose say 80sec (perhaps signals not a minimum spacing). If dwell time only 30s then total loss = 120s but there is 180s extra; seems highly likely that the signalling that gives 3 min non-stop headway can also give 6 min stopping.
PJW