682 BELL SYSTEM TECHNICAL JOURNAL 



voltage divided by the discriminant and multiplied by the appropriate 

 minor of the determinant, in which p is replaced byjco. The character 

 of the response due to a slight disturbance and in the absence of any 

 periodic force is determined by the exponentials. In general, pk is a 

 complex quantity which may be written as ak -\- jwk- It is apparent 

 that in the critical case for which ak is zero, the corresponding term be- 

 comes e^"*', corresponding to an oscillation invariable in amplitude, 

 of frequency wkfix. If ak is negative, as is ordinarily the case when 

 the system is passive (containing no amplifier or negative impedance), 

 then the oscillation diminishes in course of time. When ak is positive, 

 however, the oscillation increases with time, and the system is said to 

 be unstable. Evidently the stability of a system is determined by the 

 signs of the a^'s. 



Several criteria which have previously been enunciated for the 

 maintenance of free oscillations are deducible from the above. One 

 states that the discriminant must vanish when p takes on the value jco. 

 Another states that the damping (a/;) must be zero at the frequency of 

 oscillation. These are clearly equivalent. Two derived criteria may 

 also be mentioned, based upon the properties of the system when the 

 circuit is broken. The first of these states that if the impedance is 

 measured looking into the two terminals provided by the break, the 

 impedance must be zero at the frequency of steady oscillation. 



The second criterion involving the transfer factor has become fairly 

 widespread, perhaps because it leads to a simple and plausible physical 

 picture. To determine the transfer factor around the feed-back loop, 

 the loop is broken at a convenient point, and the two sets of terminals 

 formed by the break are each terminated in a passive impedance equal 

 to that which is connected in the normal (unbroken) condition. Then 

 when a voltage of frequency co/27r is applied to one of the pairs of ter- 

 minals so provided — the input terminals ^ — and the corresponding 

 voltage is measured across the other pair, the transfer factor A (jco) is 

 obtained as the vector ratio of the output voltage to the input voltage. 



The manner in which the transfer factor enters into the problem may 

 be demonstrated directly by comparing the voltages at any point of the 

 main amplifier circuit under the two conditions in which the feed-back 

 path is opened and closed respectively. If with the feed-back path 

 open the voltage at any such point is Ee''\ then when the feed-back 

 path is closed the voltage will be changed ^ to 



£e^7[l -A{pn 



^ Input terminals are those across which an impressed potential leads to propaga- 

 tion in the normal direction of amplifier transmission. 

 ' Bell Sys. Tech. Jour., Vol. XI, p. 128. 



