Why is a railway's third line or overhead line DC and not AC?

Railway infrastructure is expensive. It is relatively rare to create totally new tracks and when you do, they most often conform to the engineering norms of the existing tracks (gauge etc) to allow for flexibility in rolling stock usage etc.

Therefore decisions about electrification were made in the 19th century (e.g. 1890 in London). At that time, speed control of large motors was probably easier for DC than for AC where the speed is linked to AC frequency.

Also at that time DC distribution had advantages over AC.

Subsequent technological revolutions are generally hampered by the need to maximise return on very long term investment in large-scale infrastructure.

An interesting case is London's Thameslink which has trains that operate on overhead 25KV AC for the northern part of the journey and on third-rail 750V DC tracks for the portion of the journey south of Farringdon station. The costs of introducing incompatible infrastructure can be considerable.

This is actually a very interesting power electronics question; none of the answers have hit all the major points:

Drive Side Perspective

Regardless, we need DC to drive motors

• Before transistors enabling inverters (Thyristors and IGBTs) were available, the only effective way to achieve highly variable speed was with DC, since AC motor speed is fixed to frequency. Likewise, mercury arc rectifiers were too heavy to be transported on trains so putting the AC => DC conversion there was infeasible.
• The efficiency of AC motors along with superior mechanical characteristics of induction/brushless motors makes AC at the drive side attractive. However, this requires a Variable Frequency Drive Inverter which must be fed from DC as there is no easy way to change frequency or use power electronics for direct AC-AC conversion.

Therefore, the question is: Where do you put the rectifier?

1. AC Transmission

Rectify AC to DC on the train and use HVAC at 25 kV to get the power to the trains

Pros:

• Higher transmission line efficiency due to lower current.

Cons:

• Rectifier must be weight optimized; probably has a lower power factor and efficiency.
• Single phase rectifier means voltage nulls require power storage elements and efficiency reduction.
• Rectifier must be transported by the train. Rectifiers for high power use are heavy.

2. DC Transmission

Rectify track-side and use 600V-3kV DC to transmit to trains

Pros:

• Trains are lighter
• Rectifieris more efficient, better power factor
• Three phase rectifier

Cons:

• Higher transmission line currents mean higher losses

I recall reading about a Russian experiment in the 80s that compared solutions 1 and 2 above and found that despite the losses in the transmission line that the overall system was more efficient with DC transmission due to the power electronics required. Nevertheless, many regional trains and high speed rail do use HVAC.

There are some other considerations:

• High Voltage AC is not used on third rail systems; requires overhead wire. Safety considerations limit voltages of third rails to ~750V, which also limits the effective power, air conditioning, etc. (Not that you couldn't fry yourself pretty well at that voltage.)
• It's only practical to transmit one phase (though there are a few examples of three phase trains). DC systems can use three phase rectifiers track-side increasing efficiency.
• Skin depth limits effectiveness of large diameter AC wires; this is not a problem for DC systems which can use thicker gauge wires to transmit higher currents.

Note that power is not usually transmitted long distance along the track (especially for medium voltage DC): the lines are fed along the way, not just from one end.

The whole of the south eastern end of the UK use the third rail system - it was never used anywhere else in the UK and I believe a major reason was that a lot of this area is urban with low bridges hence a third rail system. DC overhead lines (5kV) were used along an old stretch from Manchester to Sheffield.

DC control is one aspect but there is another and that is induction to track control and telephony systems. An AC third rail would represent a big source of magnetic interference for track signalling and track telephone systems. Originally the signalling and track control was done mechanically so AC wouldn't be a threat so this "reason" is more a 20th century explanation rather than a 19th century one.

However, track-side telephone systems would have been affected by AC from the onset and, because the voltage is lower than overhead AC power feeds, the current would be higher and induction greater. A third rail is much closer to the track-side telephone wires as well making things worse.

As an example, when the UK's east coast mainline was electrified (overhead), engineers reported that telephony problems were occurring on lengths of cable about 1500m to 1700m (1 mile) or greater. For a third rail where the current is probably going to be at least ten times higher than overhead 25kV systems and about one-third the distance from cables you can guess AC just wouldn't work even on short distances.