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BELL SYSTEM TECHNICAL JOURNAL 



into a single impedance as shown in Fig. 12.22A. Here Zt represents the 

 impedance looking into the oscillator from the crystal terminals. 



The requirements for sustained oscillations are that the sum of the 

 reactances around the loop equal zero and the sum of thfe resistances equal 

 zero as previously stated in section 12.21. These conditions are obtained 

 when Zt is a negative resistance p in parallel with (or in series with) a 

 capacitance Ct as shown in Fig. 12.22C. The crystal is considered to be 

 operating as an inductance Lc and resistance Re as determined in the pre- 



A B C 



Fig. 12.22 — Equivalent fepresentations of crystal and oscillator circuit 



vious sections. The frequency equation has been derived by Reich 

 from the differential equation for the current in the loop. It is 



/' 



+ Ro 1 

 P LcCt 



and the condition for oscillation is shown to be 



\Ctp\ S 



(12.43) 



(12.44) 



We shall consider the crystal connected between the grid and cathode of 



the tube, in which case Zt is the input impedance of the vacuum tube. 



1 17 ... . 



The expression for ^ was developed by Chaffee from which it is possible 

 Zt 



to determine the circuit conditions necessary for the input resistance and 

 reactance to be negative. The effect of the circuit variables upon the abso- 

 lute values of p and Ci determines their effect upon the frequency and activity 

 according to equations (12.43) and (12.44). 



12.41 Input Admittance or the Vacuum Tube 

 With the assumption that the grid current is negligible and the static 

 tube capacitances Cp and Cg are part of the external circuit, Chaffee's 

 equation for input conductance becomes 



Clo,{K + Gi) + Czi^ixKiCzoi - Bi) 



{K + G^Y + (C3C0 - B^y 



(12.45) 



