428 M. I. Pupin — Electrical Oscillations of 



% apCaa . , . 



xz=.^2, a ,. _ - — — : rSin (apt— CDa) 



i V(i-«ycL) ! + «yc 2 R 

 i - *y cl 



Where tan cp a = >=pft- 



apCK 



If we make 1— £> 2 CL=0, then the circuit is brought in 

 resonance with the fundamental harmonic and the current is 

 given by 



a, . * aa a . . 



x = =£ sin »£ + -2a — sin [apt— (Da). 



R 2 Vp s (l-a a ) 9 L a + a 8 R 9 



If the coefficient of self-induction is large then it is perfectly 

 evident that the amplitude of the fundamental harmonic cur- 

 rent is by far the largest especially when the frequency of the 

 fundamental harmonic of the impressed e. m. f. is high. 



For instance, let L = 2, R = 5, p = In x 100. 



I select these values so as to be near the conditions under which 

 the above experiment was performed. Under these conditions 

 we should have for the amplitude of the next harmonic, sup- 

 posing it to be an octave 



I -— _ (very nearly). 



The amplitude of the fundamental is therefore at least 360 

 times as large. In all probability this ratio is considerably 

 larger, considering that a 1 is generally several times larger 

 than a 2 . 



The higher harmonics have even much smaller amplitudes. 

 The rise of potential in the condenser is therefore just the same 

 as if a simple harmonic e. m.f of amplitude a x and pulsation 

 p, acted upon the circuit. 



The tuning of the circuit produces therefore two distinct 

 effects : 1st, It produces a rise of potential in the condenser, 

 and 2nd, It weeds out the upper harmonics. 



It may happen, however, that the circuit is tuned to one of 

 the upper harmonics, as for instance when a 2 j? 2 CL = 1. 



In this case the current is given by 



Ua . a (3 



x = ^pr sin a pt + 2 — • sin(6p — g>f>) 



R ^ VK-/S a )yL 2 +/3 8 R 2 xr ^ pt 



/? to take integral values from 1 to cc except the value a. 



It is evident that now the fundamental harmonic with all 

 the upper harmonics excepting the harmonic a is practically 

 weeded out on account of the strengthening of the harmonic 



