546 BELL SYSTEM TECHNICAL JOURNAL 



In other words, to an increasing oscillation reactive elements have a "loss" 

 component of admittance or impedance. This "loss" component corre- 

 sponds not to dissipation but to the increasing storage of electric or magnetic 

 energy in the reactive elements as the oscillation increases in amplitude. 



The admittance curves plotted in Figs. 41-46 may be regarded as contours 

 in the admittance plane for a = 0. If such a contour is known either by 

 calculation or experiment, and it is divided into equal frequency increments, 

 a simple construction will give a neighboring curve for w = w — jAa where 

 Aa is a small constant. Suppose that the change in F for a frequency 

 Acoi is AFi . Then for a change —jAa 



AY = -j — Aa. (12 .4) 



•^ Awi ^ 



Thus, to construct from a constant amplitude admittance curve an admit- 

 tance curve for an increasing oscillation, one takes a constant fraction of 

 each admittance increment between constant frequency increment points 

 (a constant fraction of each space between circles in Figs. 41-46), rotates it 

 90 degrees clockwise, and thus establishes a point on the new curve. 



This construction holds equally well for any conformal representation of 

 the admittance plane (for instance, for the reflection coefficient plane repre- 

 sented on the Smith chart). 



The general appearance of these curves for increasing oscillations in terms 

 of the curve for real frequency can be appreciated at once. The increasing 

 amplitude curve will lie to the right of the real frequency curve where the 

 latter is rising and to the left where the latter is falling. Thus the loops 

 will be diminished or eliminated altogether for increasing amplitude oscilla- 

 tions, and the low conductance portions w^ill move to the right, to regions 

 of higher conductance. This is consistent with the idea that for an increas- 

 ing oscillation a "loss" component is added to each reactance, thus degrading 

 the "Q", increasing the conductance, and smoothing out the admittance 

 curve. 



The oscillation starts from a very small amplitude, presumably that due 

 to shot noise of the electron stream. For an appreciable fraction of the 

 build-up period the oscillation will remain so small that nonlinearities are 

 unimportant. The exponential build-up during this period is determined 

 by the electronic admittance for very small signals. 



As an example, consider a case in which the electronic admittance for 

 small signals is a pure conductance with a value of — ye . Here the fact that 

 that the quantity is negative is recognized by prefixing a minus sign. 



Assume also that the circuit admittance including the load may b'^ ex- 

 pressed as in (a-22) of Appendix I, which holds very nearly in case there 

 is only one resonance in resonator and load. Then for a complex frequency 

 Wo — jao the circuit admittance will be 



