32 



BELL SYSTEM TECHNICAL JOURNAL 



It is necessary, then, to calculate the ratio &2/61 and the correction c 

 for the different lengths of good and bad wires h and l\ in order to 

 determine the amount of error arising from the assumption that 

 &2/&1 = hlh- The values of capacitive reactance are determined from 

 the fact that 



& = Z sin </), 



where Z is the impedance and its angle as determined from 

 formula (1). 



The variation in h with variation in length of line is shown dia- 

 grammatically in Fig. 2. The total length of line is represented by 

 Oh. The reciprocal of h is plotted on the vertical axis for different 

 lengths of line. For an open at h the 

 location indicated by the ratio ^2/^1 is 

 at n whereas the true location is at m. 

 The correction mn = c must be sub- 

 tracted from the apparent location to 

 give the true location of the open. In 

 the lower curve, this error is plotted 

 against the total length of line I. 



It remains, then, to calculate b for 

 a large number of lengths from zero 

 to the maximum length of cable to be 

 encountered. These values of h are 

 used to calculate &2/&1 for different 

 total lengths of line / and different 

 fault locations /i: that is, h \s calcu- 

 lated for different lengths up to one 

 hundred miles; then a set of ratios of 

 bijbi and hlh can be determined using 

 the b of one hundred miles as b^ and b 

 for all the shorter lengths as bi. Simi- 

 larly, a set of ratios can be calculated 

 for 95, 90, 85, 80, 75, etc., miles as 

 total lengths. Since the interpolation 

 of hyperbolic functions is at best a 

 tedious calculation, even values of hyperbolics can be used in formula 

 (1) and the corresponding odd lengths in miles calculated. 



The ratio bojbi is then subtracted from the ratio hlh, in each instance 

 this procedure resulting in a family of correction curves expressed in 

 percentage such as that shown in Fig. 3. In this figure the correction 

 is plotted against apparent rather than actual percentage distance in 



Fig. 2 — Diagram showing the con- 

 struction of a correction curve. Ab- 

 scissas represent true linear distance; 

 ordinates of upper curve represent 

 measured capacitance; ordinates of 

 lower curve represent errors of com- 

 puted location. 



