OPEN-WIRE CROSSTALK 47 



at point A for the second segment is equal to that for the first segment 

 times e"^'*"^ or NKde~^'^'^. The coupling at point A for the third seg- 

 ment is NKde'*^'^. The sum of the crosstalk couplings at point A at 

 all the segments is, therefore: 



NKd{l + e-^y^ + e-^y^ + e-^yi + etc.). 



This expression may be summed up for the number of segments 

 corresponding to the total length D. It is simpler, however, to let 

 d be an infinitesimal length and to integrate over the length D, i.e., 

 from point A to point B of Fig. 13. This gives for the total near-end 

 crosstalk for non-transposed circuits: 



1 _ ,-270 

 NK^—^ . 



In the special case when D is only the usual short segment between 

 transposition poles, the above expression is practically equal to NKD. 

 The near-end crosstalk between circuits having transposition poles 

 spaced a considerable distance D apart may now be computed. Figure 

 14 shows a length 2D in a parallel between two long circuits, there 

 being a transposition in one circuit at the center of 2D. The near-end 

 crosstalk for the length AB is given by the above expression. The 

 near-end crosstalk at point A for the length BC will be the same 

 expression multiplied by the propagation factor e~^y^ and reversed in 

 sign due to the effect of the transposition. The near-end crosstalk at 

 point A for the length 2D will, therefore, be the sum of the values for 

 lengths AB and BC. This sum is: 



1 _ .-270 

 NK—^ (1 - e-2^^). 



This quantity divided by NK is the type unbalance for the length 

 2D of Fig. 14. If D is only the length of a short segment the above 

 expression is about equal to NKD{2yD). 



Similarly the near-end crosstalk at point A for a length 2>D will be: 



NK- — ^ (1 - e-2^^ T 6-4^^°) 



27 



and the type unbalance is this quantity divided by NK. For a 

 length 4Z) the quantity in the parentheses becomes (1 — e"^''^^ T tr'^'^^ 

 1= fT^''^), etc. The sign of each term in the parentheses is determined 

 by the arrangement of "relative" transpositions, i.e., those at points 

 where only one of the two circuits is transposed. Each term corre- 



