634 BELL SYSTEM TECHNICAL JOURNAL 



frequencies, and the fact that the actual gain curv^e Hes 3 db from an iso- 

 lated simple asymptotic corner, the gain curve can usually be drawn in 

 without further computation.^'' The phase curve also is easily constructed 

 by adding up the elementary phase curves associated with the various cor- 

 ners. As may be seen from (7.3), these component phase curves all will 

 have the same shape on a logarithmic frequency plot, merely being shifted 

 along the frequency scale. The phase contributed by each simple corner 

 will be ± 45 degrees at the corner frequency, the sign depending upon 

 whether the associated root appears in the numerator or in the denominator. 

 It is an extremely important fact that the very requirement of stability 

 imposes an unambiguous interrelationship between the gain and the phase 

 shift of most types of transfer characteristic! By the general mathema- 

 tical methods leading to the previously discussed stability criteria, Bode^^ 

 has shown that this is true for the broad class of network structures com- 

 monly used in feedback loop design. That is, if either the transfer gain 

 or phase shift is specified at all frequencies, the attendant phase or gain can 

 be computed without further information. This class of networks is called 

 minimum phase. Any stable structure composed of lumped circuit ele- 

 ments will have a transfer characteristic of the minimum phase type, pro- 

 vided it does not include an all-pass section. ^^ All-pass characteristics are 

 seldom used in the design of feedback loops, since their inclusion in the loop 

 always reduces the stability margins achievable with a given high-frequency 

 cut-off. Thus the unique interrelationship between phase and gain may 

 be assumed for the loop characteristic — /Xj8 in single-loop feedback systems. 

 The nature of this relationship is discussed in detail by Bode. Briefly, the 

 phase shift at any frequency coc is proportional to a weighted average of 

 the gain slope in db/octave, over the entire logarithmic frequency scale. 

 The weighting factor sharply emphasizes gain slopes in the immediate vicin- 

 ity of 03c , while the contributions of gain slopes at remote frequencies are 

 reduced in proportion to the logarithmic frequency span from the par- 

 ticular frequency coc -^^ For transfer characteristics of the form w , 

 having a constant gain slope of ±6^ db/octave,^^ the associated phase shift 

 is also constant and equal to ±90yfe degrees. For transfer functions which 

 behave approximately as co^'' over a finite frequency span, the phase shift 



"The corner frequency concept is less useful if the roots are complex. However a 

 great many servo systems are so constructed that i^P has only real roots. 



^^Loc. cit. Also see "Relations between attenuation and phase in feedback amphfier 

 design," by H. W. Bode, 5. 5. r. /., July 1940, p. 421. 



'^ An all-pass section is one which has constant gain but varying phase shift versus 

 frequency, and is usually composed of a lattice, bridged T, or other bridge type circuit. 



*^ About 60% of the area under tliis weighting function lies between co = 0.5 wc and 

 u = 2 uc , 80% between 0.25 Wc and 4 oic ■ 



^* That is, for transfer characteristics whose absolute magnitude is given by w'^^ ■ • ■■ 



