364 BELL SYSTEM TECHNICAL JOURNAL 



formula reduces to the simple relation 



F = — ^°^" — = — ^'^^ 1 f2\ 



where Z44 = Z474 = series impedance per unit length of the circuit 

 composed of the outer conductors with sheath return.^ For a sheath 

 in contact this component is thus directly proportional to length since 

 interaction crosstalk from one elementary length into another has been 

 eliminated. 



Now, while we are actually concerned with the insulated sheath as 

 covered by (2) it is of considerable interest to study equations (1) 

 and (3) at this point. The total far-end crosstalk when the outer 

 conductors and sheath are in contact is the sum of the crosstalk com- 

 ponents 7^3 and Fi as given in equations (1) and (3), or 



F, + F, = Fi= ^ 



J /_ 



Z33 4Z4 



(4) 



This simple addition follows from the fact that the circuits for the two 

 modes of propagation covered by equations (1) and (3) are mutually 

 non-inductive because of symmetry so that there is no reaction be- 

 tween them. The recognition of this fact does away with the necessity 

 of complicated mathematics which would otherwise have to be used 

 in the general solution.* 



In formula (4) the second term in the bracket represents the contri- 

 bution of the tertiary circuit involving the sheath and is seen to be 

 opposite in sign to the first term which represents the crosstalk which 

 would exist in the absence of the sheath. The equation illustrates 

 mathematically the previous statement that conducting material in 

 contact with the coaxials acts to reduce the crosstalk. Since both 

 components are directly proportional to length, the total is also directly 

 proportional to length. 



It is apparent in formula (4) that the crosstalk would be zero if the 

 values of Z33 and 4Z44 were equal. In cables where steel tapes are 

 used on the outer surface of the coaxials this condition is approached. 

 For example, if we neglect external inductance and proximity effects, 

 Z33 would be equal to twice the surface self-impedance of a single outer 



^ It should be noted here that it is not really necessary to postulate a separate 

 sheath return in order to obtain expression (3) for 7^4 due to the longitudinal voltage 

 ei/2, since the return in continuous contact with the outer conductors will actually 

 tend to lose its identity. The device of introducing sheath return insulated from the 

 outer conductors and then shorting it to the conductors serves only to simplify the 

 concepts of Zn, Z4, 74. 



'This principle of symmetry can be extended to the case of four coaxial units 

 whether insulated or in contact. 



