568 



BELL SYSTEM TECHNICAL JOURNAL 



an e.m.f. (£) at the sending end of circuit 1-2. Then at distance x 

 conductor 1 will be at the potential Eg— ^^/4 and carry the current 

 Ee~"^'J2k before the unbalances are introduced. The potential of 2 

 and the current carried by 2 will be the same with sign reversed. 

 Conductors 3, 4 and all others (S) in the system will be at potential 

 and carry no current. It follows at once that the impedances in 3 and 

 4 (/, m) and the admittances between these conductors and the con- 

 ductors (2) of the system (g, h) will contribute nothing towards the 

 crosstalk between 1-2 and the phantom. As equal impedances in- 

 serted in 1 and 2 will not unbalance the side circuit, the crosstalk 

 must depend upon the difference between r and s and in consequence 

 we may substitute {r — s)/2 for r in 1 and the negative of this in 2 

 without altering the crosstalk. Similarly, the effect of e and /depends 

 entirely upon their difference, and we may substitute ± (e — /)/2 for 

 e and /. The direct capacities (a, b, c, d) may be resolved into four 

 components,^ and of these only the third unbalances both 1-2 and 

 the phantom. The same applies to the mutual impedance unbalance. 

 It follows that the crosstalk depends solely upon 



X = {r - s), 



Y" = (a - b - c + d), 

 Z = {m — n — p + q), 



as indicated in Fig. 2. 



4 4 



X 



■AAAr 



X 

 2 



AAAr 



z_ z 



,4 4 



xrr 



-l—L 



Y" Y" 

 4 4 



X' 



Fig. 2. 



Substituting e.m.f. and currents for these in accordance with the 

 rules for small changes, we have Fig. 3. This system is perfectly 

 symmetrical with respect to the two wires in each side of the phantom 

 and we may now treat the two wires on each side as one conductor, 

 as in Fig. 4. The total e.m.f. around the phantom is Ee~'''{X + Z)/Ak 

 and the impedance of the two ends of the phantom in series is 2K_ 



^ See September 14 memorandum. 



