10 



BELL SYSTEM TECHNICAL JOURNAL 



tion, as indicated in 5/, the first five terms of the series are accurately 

 reproduced. 



The degree of approximation represented by dropping ofif various 

 terms of this series is indicated for three typical cases in Figures 2, 3, 

 and 4. Fig. 2 represents a typical 11,000 volt distribution line. It is 

 to be noted that even neglecting the third term, which is the first in 

 which the capacity of the line appears, results in an error of only 0.4 

 per cent, as indicated by consulting the values of ratios between suc- 

 cessive terms of the series which are given with the diagram. The 

 first ratio shown is that between the second and first term, and the 



20 Miles, 11 Kv., Three Phase Distribution Line 



200 Kw. at 90% Power Factor, 60 ~ 

 Ratio of Terms: 



I 2 



Fig. 2 



Zl_ 



Zr 



.0540/2°[l i •••] 

 YlZr = .0668/1 16°[^ i •••] 



second ratio when divided by two is that between the third and second 

 term. The ratios between successive terms in the series are equal 

 alternately to these two values divided by coefficients increasing in 

 simple arithmetical progression as indicated. This fact would seem 

 to make the series very convenient for computations. 



To take the opposite extreme of power transmission, Fig. 3 has been 

 prepared showing the degree of approximation resulting from the use 

 of different numbers of terms of the series for a 220 kv. line, 250 miles 



250 Mile, 220 Kv., Three Phase Power Line 

 100,000 Kw. at 90% Power Factor, 60 ~ 

 Ratio of Terms: 



Zr 



.464/60°[l 



•] 



Fig. 3 



YlZr = .588/1 16°[^ 



•] 



long, transmitting 100,000 kilowatts. Here it is seen that in order to 

 get a precision of a fraction of one per cent, four terms are necessary. 

 The results obtained by a simple Pi network are indicated by the 

 results of the first three terms. It is seen that even for this somewhat 

 extreme case the computation by the series expansion is not at all 

 laborious. This is, of course, due to the fact that even for 250 miles 

 at 60 cycles the power line represents only about one-fifteenth of a 

 wave-length. 



Series expansions of the type discussed are, of course, not novel. 

 I ha\'e dwelt on these somewhat, however, because of the value which 



