Power law variations in the field from a power line

All fields fall with distance from the source. It is often convenient to express this as a power law: the field falls as the inverse "nth" power of distance.

For power lines we often use the following simplifications:

Single current:

  • n=1, inverse distance

Single circuit:

  • n=2, inverse distance squared

Double circuit powerline with untransposed phasing:

  • n=2, inverse distance squared

Double circuit powerline with transposed phasing:

  • n=3, inverse distance cubed

 

These are only simplifications and in the rest of this page we give more detail. Note that in every case the field still falls with distance - we are just discussing how rapidly it falls.

Consider first a double-circuit powerline with one circuit each side of the line, each in an exactly vertical array. When we are close to the conductors the power-law approximations break down. But when we are distant from the conductors an amount about equal to the spacing of the conductors, the field starts falling as a power law. With untransposed phasing we expect the fields to fall as the inverse square of distance - n=2. If we measure distance from the centre of the group of conductors, then this is indeed the case, as shown in this graph:

graph of power law variation 

In practice, we don't measure distance from the centre of the group of conductors, we usually measure it horizontally along the ground. This changes the power law slightly at closer distances but at larger distances it makes little difference and the power is still n=2:

graph of power law variation 

Next, UK double-circuit power lines often have transposed phasing. This makes n=3:

graph of power law variation 

This is still a little bit idealised. Firstly, most power lines don't have the conductors in exactly vertical arrays: they slope in slightly to the top (see picture here). This means the cancellation between the two circuits that transposed phasing produces is not perfect and n becomes a bit less than 3:

graph of power law variation 

In more detail: the field produced by each circuit is a horizontal dipole (n=2). With sloping circuits, the dipole can be resolved into two dipoles: the larger one still horizontal and a smaller vertical one expressing the small amount of slope. The horizontal components from the two circuits are exactly anti-parallel: this produces a quadrupole with n=3. The two small vertical components are parallel and produce a dipole with n=2. At smaller distances this dipole field is small compared to the quadrupole component and overall n is still close to 3. But at large distances, the n=2 dipole component, which falls more slowly with distance, ends up dominating the field and n tends towards 2.

Next, the currents in the two circuits may not be exactly equal, which again means the cancellation is not perfect. This graph shows what happens with a real power line (sloping in at the top) and unbalanced currents (in the ratio 3:2):

graph of power law variation 

In more detail: again, the actual currents resolve into a component which is equal in both circuits plus a component expressing the unbalance. The balanced component of current produces a quadrupole field with n=3 and the unbalanced component of current a dipole field with n=2. At smaller distances the balanced component is larger in this instance, but at larger distances the n=2 component dominates over the n=3 component.

Finally, in real circuits, the currents within the circuit may not be perfectly balanced either, usually expressed as there being a "zero sequence current". This graph shows the power law for the actual currents measured in one particular power line (the same one as here):

graph of power law variation 

In more detail: the sum of the zero-sequence current in each circuit plus the earth-wire current produce a small overall net current in the line which has n=1. But this net current induces an image current in the ground itself. The depth of that image current depends on the ground resistivity - the more resistive the ground, the deeper the current. At small distances from the line, all these out-of-balance currents produce fields which are small compared to the load current, and the power-law is determined by the load current. At very large distances - large compared to the depth of the image current in the ground, which can be many hundred metres - the net current in the line and the image current in the ground form a dipole with n=2. At intermediate distances, you can be close enough to the conductors for the n=1 of the net current to be important, but too far away from the image current in the ground for that to be important, and the overall n dips below 2 before rising back to 2 at larger distances. The following graph illustrates this for three different ground resistivities. The 10 Ωm curve - image depth 230 m - barely dips below n=2. The 100 Ωm curve - image depth 720 m - is just starting to rise. For infinite resistivity there is no image current and n tends towards 1.

graph of power law variation