OSCILLATIONS IN ELECTROMECHANICAL SYSTEM 445 



the new frcc|iicncy roniponents suddenly appear and rapidly build up 

 to large amplitudes as the voltage is increased. The current of the 

 impressed frequency, co^, remains practically constant and independent 

 of voltage above the threshold. 



There remains to be described the method by which the mechanical 

 amplitude was obtained. The current voltage relation for a con- 

 denser is 



F = ' I /■ dt. 



=;-/ 



The current through the condenser involves only three frequency 

 components in this case, so it can be written in the form 



i = Ag cos (uyt + ify) -\- Ad cos [(co„ — w„,)/ + <pd)^ 



+ ^, cos [(cOff — 2C0„,)/ + (pe)2 



and the capacity of a condenser, in e.s.u., is 



where 



C = e-^ = —^ — j— ^; farads 



6 (Oo + Oto cos OOmt) 



€ = 8.85 X 10-1" = permittivity, 



A = plate area in cm.^. 



So = constant, or average spacing cm., 



Sm — amplitude of mechanical displacement. 



From these equations the amplitude of the mechanical vibration 

 in terms of the electrical amplitudes can be determined. Neglecting 

 phase angles the relation is 



SoAe 



Ve- 



eA(cOg — 2cOm) 



A, 



2€A(cOg — U>m) 



Ve = amplitude voltage component of frequency w^ — 2com. 



The neglect of the phase angles will make some inaccuracy in the 

 results. They would be exact, in the above formula, were Ag negligibly 

 small. The term involving Ae'is a correction term necessitated by the 

 incomplete suppression of that frequency component. 



