110 BELL SYSTEM TECHNICAL JOURNAL 



the grid circuit connected to an a.c. generator in series with a high 

 resistance of the order of a megohm, and with the plate circuit con- 

 nected to a resistance of the same order of magnitude as the internal 

 plate resistance. With a negative grid the input impedance is mainly 

 capacitive and at low frequencies nearly the entire applied e.m.f. 

 exists across the grid. At positive potentials however, when con- 

 ductive grid current flows and the grid resistance drops to something 

 like 10,000 ohms, by far the greater part of the drop is taken up in the 

 external resistance so that the positive lobe of the grid potential wave is 

 distorted. This distortion is equivalent to modulation since it implies 

 the presence of new frequencies. As a result of the varying reaction of 

 the tube then, modulation voltages are built up across the grid- 

 filament path. These potentials are amplified in the plate circuit 

 where the applied wave suffers further distortion due to the non-linear 

 relation between plate current and grid potential, which in a similar 

 way gives rise to plate modulation. Evidently plate modulation 

 would be alone effective if the external grid resistance were made small. 



General Relations 

 The production of modulated currents or potentials is characteristic 

 of any device in which the relation between instantaneous values of 

 current and voltage is not a linear one. Theoretically, such a relation 

 can be represented to any required degree of accuracy by an equation 

 of the form 



i = ao + ail' + a2v'^ + a^v^ + • • • , (1) 



in which i represents the current through the device and v the potential 

 drop across it. Now suppose a voltage wave which includes two 

 components of frequency pjlir and qjlir respectively to be impressed on 

 the non-linear element 



V = P cos pt -\- Q cos qt. (2) 



If we substitute this expression in eq. 1 for v, the current wave is found 

 to include components of the two original or fundamental frequencies, 

 together with new frequencies produced by the non-linear element: 



i = 



io -{- ip cos pt -\- iq cos qt 

 + to,, cos 2pt + 12 q cos 2qt 

 -f i+ cos ip + q)t -f i- cos (p — q)t 

 -\- izp cos 2>pt -+- iiq cos Zqt 

 -f t2p+5 COS {2p + q)t + iiv-1 cos {2p - q)t 

 -\- ip^2q cos {p + 2q)t -f ip_25 cos {p - 2q)t -f • • • , 



