654 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



method to commercial designs of multipaired cable are included in what 

 follows. 



■ Low Frequency Inductance and Interaxial Spacing 



The self-inductance of a pair of straight parallel wires in free space or 

 within any non-magnetic shield is a function only of the ratio of conductor 

 diameter to the interaxial spacing if the frequency is so low that proximity, 

 shielding, and skin effects are negligible. The formula is as follows: 



L = 1.482 log -^ + 0.1609 (mh/mi) (1) 



a 



S = interaxial spacing 



d = conductor diameter 



The first term gives the self inductance due to net external flux-linkage 

 and the 0.1609 constant represents flux-linkage within the non-magnetic 

 conductors. 



Some textbooks give the formula with {2S-d)/d as the argument of the 

 logarithm. This form is not valid when d is not small compared to S, as is the 

 case with cable pairs. The derivation of formula (1) in Russell's "Alternating 

 Currents" shows that it is vaUd for any S, provided only that the current is 

 uniformly distributed across the conductor cross-section. 



While it was believed that formula (1) was valid for cable pairs at, say, 

 1000 cps, it was desirable to estabUsh experimentally whether or not the 

 twist in the pair, and magnetic coupling between pairs, affected the measured 

 inductance at a frequency in this range. The effect of coupling between pairs 

 is greatest when all of the cable pairs except the pair under test are shorted 

 together at both ends of the cable, thus providing a large number of closed 

 loops for any induced currents. In making laboratory inductance tests at 

 1000 cps on lengths of 19-gauge cable with 0.084 /xf/mi capacitance (Type 

 CNB) it was found that opening or shorting the surrounding pairs did not 

 affect the measured inductance. Also, grounding or floating the far end of the 

 test pair made no difference. The tests were repeated with the same results 

 on lengths ranging from 300' to 5000'. 



It should be noted that a correction term must be added to the measured 

 1000 cps inductance (L') to obtain the true distributed inductance (L) when 

 the cable length is such that propagation effects become appreciable. When 

 the correction term is included, the equation for L at 1000 cps becomes: 



L = V + l/3{R'yC 



Where L', R', and C are measured inductance, resistance, and mutual 

 capacitance, respectively. 



