470 BELL SYSTEM TECHNICAL JOURNAL 



circuit shunted by the capacitances. At the frequency for which 

 {Xx + X2 + ^3) is zero the three-element combination has, as induc- 

 tive reactance, 7X20, which can be computed. The inductance Lq 

 should then be such that 



i. = ^-^. (33) 



Evidently the three-element combination in the Z2 branch is such that 

 it might be replaced by a piezoelectric crystal. To keep the inductance 

 Lo small, the capacitances Ci and C3 should be fairly large so that the 

 resonance of {Xi + X2 + Xz) lies close to the crystal resonance. 



The foregoing examples are based on the constant-^ low-pass filter 

 as a prototype. Evidently high-pass or band-pass filters of the various 

 known kinds might also be used as prototypes and diversified in similar 

 manner. Additional forms may also be found by increasing the number 

 of meshes in the network, but in such cases, the simplest circuits 

 providing frequency stability appear to be the homogeneous single 

 pass-band filter networks. The stable oscillation frequency is the 

 frequency within the pass-band for which the phase constant is equal 

 to 180 degrees. If the network is equivalent to six ladder- type half- 

 sections or more, there will be two or more frequencies for which the 

 phase constant is 180 degrees. Such networks are generally not well- 

 suited for oscillator circuits. 



Certain simple configurations which do not admit of complete stabili- 

 zation in the above manner may be partially stabilized and in actual 

 use may exhibit a very high degree of constancy. The common quartz 

 crystal controlled oscillator with the crystal connected between the 

 grid and cathode of the tube is an example of a partially stabilized 

 circuit. The impedance characteristic of the crystal itself is primarily 

 responsible for the stabilization. Usually the circuit is such as to re- 

 quire the crystal to exhibit an inductive reactance at the oscillation 

 frequency and the impedance characteristic is such that this occurs only 

 in an extremely small frequency interval. The main determinant D for 

 this circuit has a single zero at the resonance of the crystal and com- 

 plete stabilization would require that the minor 1^11,22 have its zero 

 at this frequency also. However, oscillation under this condition 

 would be impossible since the crystal resonance would short-circuit 

 the feedback path and reduce the magnitude of /i/3 to zero. Actually 

 the zero of Dn, 22 lies somewhere in the inductive interval of the crystal 

 at a point fairly close to the resonance frequency. The range in which 

 oscillation is possible is therefore only a fraction of the inductive inter- 

 val of the crystal and a high degree of stability results. 



