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BELL SYSTEM TECHNICAL JOURNAL 



The equation is the same for the three types of circuits considered and is 

 derived for the current ii , in Figs. 12.4 and 12.5, although it may be set up 

 in terms of any of the currents or voltages existing in the circuit. It is 

 of the form 



d ii d ii d ii dii . /n i\ 



The P coefficients are functions of the circuit elements and are defined for 

 each type of circuit in the following sections. 



The solution of (12.1) normally represents a doubly periodic function 

 arising from the two coupled antiresonant meshes (a) and (b). The normal 



Fig. 12.5 — Equivalent circuit of oscillator with crystal connected between 

 grid and cathode 



modes of oscillation consist of two currents in each mesh with frequency 

 and damping factors fi\ and ai , ^2 and a2 respectively. 



The conditions for undamped oscillation as derived from the general 

 equation (12.1) are expressed in terms of the coefficients by 



Pg ^ Pa ± VpI - 4P4 

 Pi 2 



(12.2) 



and the angular frequencies are 



^ _ P2 ± Vpl - 4P4 



(12.3) 



where the plus sign gives the condition for one damping factor to be zero 

 and the minus sign that for the other to be zero. 



