Breakdown voltage in HV transmission lines (was: this was..)

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Antonio Carlos M. de Queiroz" <acmdq-at-uol-dot-com.br> 

Tesla list wrote:
 >
 > Original poster: "Dr. Resonance" <resonance-at-jvlnet-dot-com>
 >
 > Ed:
 >
 > I have always wondered how these long distance EHV lines avoid producing
 > excessive corona.
 >
 > The cables do not appear to be very large in diameter --- perhaps an inch or
 > two at best.  With that small radius and diameter, why don't they emit
 > tremendous corona?
 >
 > With corona inception potential around 67 kV per inch (30 kV/cm) they should
 > be glowing a lot with their small diameters.  Especially in rainy weather.

The fundamental reason is because the wires are essentially straight.
Cylindric conductors donīt follow the same rule of spheres, where 30 kV
per cm of radius is enough to create corona.

If you try to calculate the electric field at the surface of a long
wire that is at a given potential, the result is that that if the wire
has infinite length and is really straight, the breakdown voltage is
infinite. But real wires are always somewhat curved by gravity.

I can use the field of a toroid implemented in the Inca program to have
an idea of what happens. Consider a ring of wire, with wire diameter of
1 cm and varying major diameter. I list below the breakdown voltages
calculated by the program:

Major diameter:
1 cm:  15.0 kV (a ball with 1 cm of diameter)
2 cm:  22.6 kV
4 cm:  32.9 kV
10 cm: 50.0 kV
1 m:   95.2 kV
10 m: 133.8 kV
100 m: 169.8 kV
1000 m: 203.9 kV

The breakdown goes slowly to infinity as the radius of curvature
(half of the major diameter of the toroid) of the wire decreases.

A bunch of wires results in a larger effective diameter of the wire,
and in greater breakdown voltage. I can still use the program to
evaluate this case.
Consider 4 wires with 1 cm of diameter disposed as a square with
distance between the centers of the wires of 4 cm.
A ring of this "wire" with distance from the center to the center
of the inner wires of 10 cm is described in the program as:

* Toroidal conductor with 4 wires
C1 wire 50 0.1 0.02 0.005 0 360
C2 wire 50 0.1 -0.02 0.005 0 360
C3 wire 50 0.14 0.02 0.005 0 360
C4 wire 50 0.14 -0.02 0.005 0 360

The breakdown voltage for this case results as 112.4 kV
For larger rings (also changing the distance from the center to the
center of the inner wires, that is approximately the radius of
curvature of the composite wire):
0.5 m: 181 kV (compare with 95.2 kV with 1 wire)
5 m:   288 kV (compare with 133.8 kV)
50 m:  385 kV (compare with 169.8 kV)
500 m: 478 kV (compare with 203.9 kV)

So, it's not difficult to keep transmission lines at very high
potentials
without excessive corona at the wires. The most problematic areas are
the middle points between towers, where the radius of curvature is
maximum, and the suspension devices in the towers, where it's common to
see corona rings.

Antonio Carlos M. de Queiroz