Tables of Phase Associated with a Semi-Infinite Unit 

 Slope of Attenuation 



By D. E. THOMAS 



This paper presents tables of the phase associated with a semi-infinite unit slope 

 of attenuation. The phase is given in degrees to .001 degree with an accuracy of 

 ± .001 degree and in radians to .00001 radian with an accuracy of d= .000015 

 radian. The method of constructing the tables and a brief analysis of the errors 

 are given. An appendix, which gives a detailed explanation with specific exam- 

 ples of the use of the tables in determining the phase associated with a given 

 attenuation characteristic or the reactance associated with a given resistance 

 characteristic by means of the straight line approximation method given in Bode's 

 "Network Analysis and Feedback Amplifier Design," is included for the benefit of 

 those who are not already acquainted with this method. The Appendix also 

 presents an example of a non-minimum phase network^ in which the minimum 

 phase determined from the attenuation characteristic fails to predict the true 

 phase of the network. 



THE method described by Bode^ for the determination of the phase 

 associated with a given attenuation characteristic or the reactance 

 associated with a given resistance characteristic has proved to be an ex- 

 tremely useful laboratory and design tool. In this method the attenuation 

 (or real) characteristic, plotted versus the log of frequency, is approximated 

 by a series of straight lines. The phase (or imaginary component) is then 

 determined by summing up the individual contributions of each elementary 

 straight line segment to the total phase (or imaginary component). 



The most elementary straight line characteristic which can be used to 

 construct a given straight line approximation is that in which the attenua- 

 tion plotted against the log of frequency is constant on one side of a 

 prescribed frequency, /o, and has a constant slope thereafter. Such a 

 characteristic has been called by Bode a "semi-infinite constant slope" 

 characteristic.^ A semi-infinite unit slope of attenuation or one in which j 

 the attenuation changes 6 dh per octave, or 20 dh per decade is shown in 

 Fig. 1. The phase associated with this attenuation characteristic is plotted 

 in Fig. 2} The independent variable was chosen as///o for values of/ less 

 than /o and /o// for values of / greater than /o to keep it finite for all values 

 of/ and in order to show the phase plotted exactly as it is given in the tables 

 to follow. The phase associated with a semi-infinite constant slope of 



' For a complete discussion of minimum phase see Hendrik W. Bode, "Network Analysis 

 and Feedback Amplifier Design," D. Van Nostrand Company, Inc., New York, N. Y., 

 1945. 



2 Ibid: Chap. XV, page 344. 



■VIbid:Chap. XIV, page 316. 



•Il)id: Chap. XIV page 317. 



870 



