372 BELL SYSTEM TECHNICAL JOURNAL 



Proceeding in a similar way with the equation 



we get 



^-(^rM^+i^+f(i;+z-^)!- '^"°^ 



J 



In this way the steady-state series solution is built up term by 

 term, the component currents being poly-periodic as indicated in a 

 previous table. 



For sufficiently small impedance variations this method of solution 

 works very well, and leads to a rapidly convergent solution. In 

 other cases, however, the solution so obtained may be divergent, 

 even when the complete series solution from which it is derived is 

 absolutely convergent. The explanation of this lies in the fact that 

 the steady-state series so obtained is the sum of the limits (as / ap- 

 proaches infinity) of the terms of the complete series solution, whereas 

 the actual steady-state is the limit of the sum. These are not in 

 general equal; in particular the former may be and often is divergent 

 when the latter is convergent. 



In view of the foregoing considerations it is of importance to de- 

 velop another method of investigating the steady-state oscillations 

 which avoids the difficulties in the formal series solution. The fol- 

 lowing method has suggested itself to the writer and works very well 

 in cases where the previous form of solution fails. It should be stated 

 at the outset, however, that the absolute convergence of the solution 

 to be discussed, while reasonably certain in all physically possible 

 systems, has not been established by a rigorous mathematical in- 

 vestigation, which appears to present very considerably difficulties. 



We start with the problem just discussed and, in view of the results 

 of the formal series solution there obtained, assume a solution of the 

 form: 



X 



7 = ^ 2^^g.(n+-co)(^^^g-.(i2+ma,)/ (301) 



-N 



N 



= V^„,e'^«+"'")' (real part). (302) 



-N 



Here the series is supposed to extend from m = -{-N to m=—N^. 

 Ultimately, however, N will be put equal to infinity. As before, the 



