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BELL SYSTEM TECHNICAL JOURNAL 



measure the response with a sinusoidal impressed wave, but it would 

 involve great difficulties of experiment as well as of interpretation to 

 determine the response with negatively damped waves corresponding 

 to values of p in the right-hand half of the plane. However, these re- 

 sults may be brought within the field of practical experience by a pro- 

 cedure widely used for the purpose. 



To include all roots in the right-hand half of the ^-plane, the con- 

 tour must be taken of infinite extent. The path ordinarily followed for 

 this purpose extends from the value -\- R to — Ron the imaginary axis, 

 and is closed by a semicircle of radius R, where R is assumed to expand 

 without limit. It may be noted that in actual amplifier circuits the 

 transfer factor becomes zero when \p\ becomes infinite, so that A{p) 

 is zero along the semicircular part of the closed contour. Conse- 

 quently, the only values of A (p) which dififer from zero are those corre- 

 sponding to finite values of p, along the imaginary axis. In other 

 words, the plot of A(p) under these conditions comes down to the plot 

 of A (jo) where w is finite. Hence, if we plot A (jco) for all values of co 

 from minus to plus infinity, there will be no roots with positive real 

 parts and the system will be stable when the vector from (1, 0) to the 

 curve sweeps through a net angle of zero. The system will be unstable 

 when the vector sweeps through 360 degrees, or an integral multiple 

 thereof. 



Two types of transfer factor curves may be considered as illustra- 

 tions. The first of these shown in Fig. 3 corresponds to that for a re- 



X 



3-M-C 



Fig. 3 — Schematic of a reversed feed-back oscillator circuit at the left. At the 

 right plot of the transfer factor A (jco) around the feed-back loop of Fig. 3a over the 

 frequency range from zero to very high frequencies. The imaginary part of the 

 transfer factor is plotted as ordinate against the real part as abscissa for the three 

 curves a, b, c, which correspond to increasing gains around the loop. Condition a 

 is stable, while b and c are unstable. 



