872 BELL SYSTEM TECHNICAL JOURNAL 



Bode points out,^ however, that the building up of the complete im- 

 aginary characteristic from a single primitive curve, namely a semi-infinite 

 real slope, suffers from the disadvantage that the phase contributions of the 

 individual slopes may be rather large positive and negative quantities, 

 even though the net phase shift is fairly small. In order to avoid this dis- 

 advantage, Bode recommends that the individual finite line segments which 

 constitute the straight line approximation to the real characteristic be 

 regarded as the elementary characteristics used in the summation of the 

 total phase. He then gives a series of charts, plotted as a function of ///"o, 

 of the phase associated with a finite line segment having a 1 db change in 

 attenuation and with a ratio of the geometric mean frequency (/o) of the 

 two termmal frequencies of the finite line segment to the lower terminal 

 frequency as a parameter (ratio designated a). 



However, problems have arisen where, even with the finite line segment 

 phase charts, the phase contributions of the various elements were suth- 

 ciently large and nearly equal positive and negative quantities that diffi- 

 culties in interpolation between the curves for the various values of a, given 

 on the charts, resulted in a sufficient lack of precision that the quantity 

 being sought was lost. 



Because of the usefuhiess of the method in question, and with its applica- 

 tion to a wider variety of problems, means of increasing its over-all precision 

 and simplification of computation have constantly been sought. It had 

 occurred to several engineers independently that a table of phase versus 

 frequency for a semi-infinite unit slope of attenuation would prove extremely 

 useful. The phase in radians at frequency /c, associated with a semi- 

 infinite unit slope of attenuation commencing at frequency /o, is given by 

 Bode as® 



£W=?(.., + | + |+...) (1) 



where: 



7o Wo 



The computation time required to determine the phase at a given frequency 

 by summation of the above series is such, that the work required to get the 

 phase at a sufficient number of points and to a sufficient number of sig- 

 nificant figures to prepare an adequate table proved to be sufficient to dis- 

 courage this procedure. 



5 Ibid: Chap. XV page 338. 

 "Ibid: Chap. XV, page 343. 



