72 Scientific Proceedings, Royal Dtihlin Society. 



regulation, and corona loss, and also by mechanical considerations, such as the 

 size of wire or cable convenient for manufacture, and its mechanical strength 

 necessitated by its employment as an overhead line exposed to wind and weather. 

 In long spans the question of mechanical strength becomes of great importance. 



The annual cost involved in the use of a particular size of conductor 

 (assuming that the electrical and mechanical conditions are satisfied) depends 

 mainly on two items. The first of these is' the annual value of the power lost in 

 the line, which is greater the smaller the size of the conductor, and which also 

 depends on the current to be transmitted, and its fluctuations in value at different 

 times throughout the year. 



The second item is the interest and depreciation on the cost of the conductor ; 

 and this is greater the larger the size of conductor employed. In the case of 

 overhead lines the supporting poles or towers will in general increase in size to a 

 slight extent with increase in weight of the line ; but this can probably be 

 sufficiently allowed for by a slight increase in the basic rate for cost of conductor, 

 thus regarding the fluctuating part of the cost of poles as a small percentage added 

 to the cost of the conductor proper. 



The best size of conductor then is that for which the sum of these two items 

 of annual cost is as small as possible ; and this consideration leads to a mathe- 

 matical determination of the best area of cross-section. 



4. When the size of conductor is known, it is necessary to determine its 

 electrical performance. The line has resistance, inductance, capacity, and leak- 

 ance. In long distance lines it is necessary to take all these factors into account. 

 In short lines, however, it is usually permissible to neglect the effects of capacity 

 and leakance, with the result that the calculations are greatly simplified. 



The solution of the long distance high tension transmission problem commences 

 with Heaviside's general differential equations for the line. Their solution leads 

 to hyperbolic functions of vector quantities. The method to be described below 

 deals directly with these functions, and manipulates them by the laws of complex 

 quantities. The process is illustrated by numerical examples, which indicate how 

 the work can be systematically arranged. 



As a result of the calculation of the electrical performance of the line, it may 

 be found, for example, that the voltage regulation is so bad that a change of size 

 of the conductor must be made from that which otherwise gives theoretically the 

 most economic performance. In other cases, the use of auxiliary apparatus, such 

 as synchronous condensers, may be necessary. When the voltage is high and the 

 size of conductor small, the corona loss may become relatively so great that a 

 modification of the design is essential. 



5. It is necessary now to set out symbols for the several quantities involved 

 in the calculations. The following notation will be adopted : — 



Let X be the distance in miles measured from the sending end to any point in 

 the line where, at a particular instant t seconds from some zero of time, the 

 instantaneous electrical pressure to neutral is v, and the current is c. 



Let the length of the line between stations be I miles, and the line be of over- 

 head type. We will suppose that the system is three-phase or single-phase, with a 

 frequency of / cycles per second. 



Let the circular conductor (solid or stranded) be of radius a inches, and of 

 cross-sectional area A square inches. 



Let the conductors be equally spaced at h inches mutually apart. For each 

 conductor let the resistance be R ohms per mile ; the inductance be L henrys per 



