Single phase and three phase underground cables

Most high-voltage underground electricity cables are three-phase.  That is, they have three conductors, carrying balanced currents.  But some are single-phase, with just two conductors, "out" and "back" or "go" and "return".  This can happen, for example, with railway supplies.

Note: on a three-phase system, you supply a single-phase load by connecting it across two of the three phases.  So in voltage terms, it is two-phase: each conductor has a voltage on it, and neither is at earth potential in the way that a neutral conductor at low voltages is.  In current terms, the current that flows out in one conductor flows back in the other, which means it is more naturally described as "single phase".  Single-phase or two-phase - it's the same thing in this context and the way you describe it depends which aspect you approach it from.

We calculate the exact field from a single-phase circuit numerically in just the same way as we do for a three-phase circuit (download a tutorial on how to do this).  But when you are a long way away from the cable compared to the separation of the conductors, there are simpler expressions for the field.  Consider first a single phase cable with a current I going out in one conductor and back in the other:diagram of single phase cable

We are interested in the field a distance r away.  When r is large compared to the separation of the conductors d, the field is given by

B = (μ0/2π) . I . (d/r2)

μ0/2π is just a constant.  So the field is this constant, times the current, times d/r2.

Now consider a three-phase cable:

diagram of three phase cable

Now the field is given by:

B = (μ0/2π) . I . (√3. d/r2)

The constant is the same, and so is the d/r2 term, so the only difference is that the field is the square-root-of-three (1.73) times bigger.  But that's with three conductors instead of two, taking up a total width of 2d instead of d.

In practical terms, the field from a single-phase and a three-phase cable at the same voltage are going to be similar, but given that the conductor spacings vary and the loads can be quite different, you can't really generalise about which is bigger, and need to do a specific calculation for each cable.