Equations occurring in Wireless Telegraphy. 



215 



to two elements, (1) an oscillatory circuit containing self- 

 inductance, capacity, and small resistance, and (2) some type 

 of conductor possessing a non-linear voltage-current charac- 

 teristic. Sometimes the conductor is placed in series with the 

 capacity in the oscillatory circuit as in the Poulsen arc, but 

 more often it is included as a shunt across the condenser as 

 in the Dynatron* and the ordinary triode generator f. (See 

 accompanying figures where the conductor is shown as 

 possessing a characteristic current-voltage relation I = F(V).) 



I=F(V) 



i^F(v) 



i 



Arc 



Dynatron on Triode 



In either case the fundamental equation of the generator 

 representing the relation between the voltage Vj across the 

 condenser, and time t, may be reduced to 



dt 2 



+/(V; 



dt 



+ m a V! = 0, . 



(1) 



where /(Vx) and m 2 include the circuit constants C, r, and 

 L. For many purposes it is sufficient to consider the 

 characteristic I = F(V) as linear (e.g. in deducing the limiting- 

 conditions for instability), but it is clear that the inclusion 

 of the non-linear terms is essential in a discussion of the 

 maintained amplitude of the oscillations or the production 

 of harmonics. 



A certain amount of progress towards the solution of (1) 

 for the somewhat limited cases in which /(Vj) may be simply 

 expressed as a power series has been made by Dr. van der 

 Pol and the writer +, but no extension has been made to the 

 case where the right-hand side of the equation is not zero. 

 The latter case (dealt with in Part II. of Dr. Hobb's paper) 

 is of practical importance as representing the case of a 

 sinusoidal electromotive force impressed on a crystal or 

 triode receiver circuit. 



* Hull, Proc. Rad. Inst. Eng. vol. vi. p. 5 (1918). 



t Appleton and van der Pol, Phil. Mag. vol. xlii. p. 201 (Aug. 1921). 



J Phil. Mag. siqyra, p. 177. 



