If any given power is to be transmitted with a given percentage of maximum loss in line conductors, the weight of these conductors will increase as the square of their length, and decrease as the square of the full voltage of operation in every case.
Thus, if the length of this transmission is doubled, the weight of the conductors must be multiplied by four, the voltage remaining the same; but if the voltage is doubled and the line length remains unchanged, the weight of conductors must be divided by four. With the length of line and the voltage of transmission either lowered or raised together, the weight of the conductors remains fixed, for constant power and loss.
An illustration of this last rule may be drawn from the case of lines designed to transmit any given power a distance of ten miles at 10,000 volts, and a distance of fifty miles at 50,000 volts, in which the total weight of conductors would be the same for each line if the percentage of loss was constant.
This statement of the rule as to proportionate increase of voltage and distance presents the advantages of high voltages in their most favorable light. Though a uniform ratio between the voltage of operation and the length of line allows a constant weight of conductors to be employed for the transmission of a given power with unchanging efficiency of conductors, yet other considerations soon limit the advantage thus obtained.
Important among these considerations may be mentioned the mechanical strength of line conductors, difficulties of line insulation, losses between conductors through the air, limits of generator voltages, and the cost of transformers.
If the ten-mile transmission at 10,000 volts, above mentioned, requires a circuit of two No. 1/0 copper wires, the total weight of these wires will be represented by (5,500 × 10 × 2 × 320) ÷ 1,000 = 35,200 pounds, allowing 5,500 feet of wire per mile of single conductor to provide something for sag between poles, and 320 pounds being the weight of bare No. 1/0 copper wire per 1,000 feet.
When the length of line is raised to 50 miles, the two-wire circuit will contain 5,500 × 50 × 2 = 550,000 feet of single conductor, and since the voltage is raised to 50,000 at the same time, the total weight of conductors will be 35,200 pounds as before. The weight of single conductor per 1,000 feet is therefore only 64 pounds in the 50-mile line.
A No. 7 copper wire, B. & S. gauge, has a weight of 63 pounds per 1,000 feet, and is the nearest regular size to that required for the 50-mile line as just found. It would be poor policy to string a wire of this size for a transmission line, because it is so weak mechanically that breaks would probably be frequent in stormy weather. The element of unreliability introduced by the use of this small wire on a 50-mile line would cost far more in the end than a larger conductor.
As a rule, No. 4 B. & S. gauge wire is the smallest that should be used on a long transmission line in order to give fair mechanical strength, and this size has just twice the weight of a No. 7 wire of equal length. Here, then, is one of the practical limits to the advantages that may be gained by increasing the voltage with the length of line.
As line voltage goes up, the strain on line insulation increases rapidly, and the insulators for a circuit operated at 50,000 volts must be larger and of a much more expensive character than those for a 10,000-volt circuit. In this way a part of the saving in conductors effected by the use of very high voltages on long lines is offset by the increased cost of insulation.