316 BELL SYSTEM TECHNICAL JOURNAL 



(11) with respect to 2?, L, G and C, in order to get a differential expression 

 da in terms of dR, dL, dC and dG. This may then be interpreted as a 

 change with temperature or a manufacturing variation, or a variation of 

 attenuation from pair to pair in the cable. This procedure applied to (10) 

 results in the formula 



4a. Ja = [g + RV{G' + J'C')/{R' + J'L')'] dR 



+ [r + GV{R' + cJ'L')/{G' + co'C')] dG 



-[co'C- </lV{G' + c/C')/{R' + </L')'] dL 



- Ic^'L - J'cViR' + c/L')/(G' + a,^C')] JC (13) 



Curves of Fig. 17 show the components of the temperature variation pro- 

 duced by changes in R, L, G and C for standard 19-gauge cable. 



A better formula from the point of view of equalizer design results from 

 applying Taylor's series expansion to equation (9) and taking the real part 

 of the resulting expressions. In this method, the variables are taken to be 

 LC, R/L and G/C which effectively reduces the number of variables by one. 

 There is a further advantage which appears in equation (14), below, namely, 

 that the coefficient of the per cent variation in LC is just 1/2 the attenuation 

 constant a and this means, therefore, only a slight addition to the basic 

 equalizer which matches the curve for a vs frequency. There are thus 

 added only two new types of temperature equalizers, one for R/L and one 

 for G/C correction. Since equation (10) is already in the real form, it is 

 more straightforward to expand it by Taylor's series and use the required 

 number of terms. The formula thus obtained is naturally the same as that 

 obtained from (9) and is as follows: 



Vf 



a A(LC) , 1 R / a2 + ^^^ZC A(R/L) 



2 LC ' 2 co£ y 1 + RycJ'U R/L 

 :2 + co^ZC A(G/C) 



1_^ /^ 



"^ 2 coC y 1 - 



+ GVaj2C2 G/C 



_ 1 ^ (R/oiLW^^^^f^JIC + Va^ - RG f MR/L) '? . . 



Sc^'L' V(l + R'/u^'L-^y I R/L ] ^ '" ^ 



Application of this formula gives slightly different values for the tempera- 

 ture-attenuation coefficient at different parts of the temperature range for 

 most frequencies. This means that the change of attenuation with tempera- 

 ture is not quite linear. The nonlinearity is so small below 100 kc. that it 

 has not been measured with any certainty on lengths of cable varying from 

 500 feet up to 10 miles, but on long cable carrier circuits corrections for it 

 may become necessary. 



The formula has another use, however, in determining the effects of small 

 manufacturing variations on the probable attenuation of cables made up 



