Regeneration Theory 



By H. NYQUIST 



Regeneration or feed-back is of considerable importance in many appli- 

 cations of vacuum tubes. The most obvious example is that of vacuum tube 

 oscillators, where the feed-back is carried beyond the singing point. Another 

 application is the 21-circuit test of balance, in which the current due to the 

 unbalance between two impedances is fed back, the gain being increased 

 until singing occurs. Still other applications are cases where portions of 

 the output current of amplifiers are fed back to the input either unin- 

 tentionally or by design. For the purpose of investigating the stability of 

 such devices they may be looked on as amplifiers whose output is connected 

 to the input through a transducer. This paper deals with the theory of 

 stability of such systems. 



Preliminary Discussion 



WHEN theoutput of an amplifier is connected to the input through 

 a transducer the resulting combination may be either stable or 

 unstable. The circuit will be said to be stable when an impressed small 

 disturbance, which itself dies out, results in a response which dies out. 

 It will be said to be unstable when such a disturbance results in a 

 response which goes on indefinitely, either staying at a relatively small 

 value or increasing until it is limited by the non-linearity of the 

 amplifier. When thus limited, the disturbance does not grow further. 

 The net gain of the round trip circuit is then zero. Otherwise stated, 

 the more the response increases the more does the non-linearity decrease 

 the gain until at the point of operation the gain of the amplifier is just 

 equal to the loss in the feed-back admittance. An oscillator under 

 these conditions would ordinarily be called stable but it will simplify 

 the present paper to use the definitions above and call it unstable. 

 Now, this fact as to equality of gain and loss appears to be an accident 

 connected with the non-linearity of the circuit and far from throwing 

 light on the conditions for stability actually diverts attention from the 

 essential facts. In the present discussion this difficulty will be avoided 

 by the use of a strictly linear amplifier, which implies an amplifier of 

 unlimited power carrying capacity. The attention will then be 

 centered on whether an initial impulse dies out or results in a runaway 

 condition. If a runaway condition takes place in such an amplifier, it 

 follows that a non-linear amplifier having the same gain for small 

 current and decreasing gain with increasing current will be unstable as 

 well. 



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