214 BELL SYSTEM TECHNICAL JOURNAL 



CALCULATED 



5 

 MEGACYCLES 



Fig. 12.33 — Calculated and measured values of PI for BT crystals. The discrepancy 

 is a measure of mounting loss 



12.87 Measurement of PI and M 



In all the discussions so far regarding the performance of crystals in 

 oscillator circuits, the crystal has been represented by the equivalent circuit 

 of Fig. 12.3 in which all the elements were considered constant. It is 

 possible to obtain crystals in which this is essentially the case, but in general 

 there are three secondary effects which complicate the picture. These are, 

 first, the effect of other modes of vibration of the crystal, second, variations 

 in the crystal constants resulting from variations in the amplitude of vibra- 

 tion, and third, the leakage or dielectric loss in the crystal holder. These 

 factors will be considered in the order named. 



Secondary modes of vibration affect the crystal for oscillator purposes 

 only when the frequencies of these modes are sufficiently close to the princi- 

 pal one to alter its impedance characteristic in the frequency range of 

 oscillation; that is, to alter the reactance as shown in Fig. 12.27 between the 

 frequency /i and/o and the corresponding effective resistance between these 

 two frequencies. With interfering modes present, the equivalent crystal 

 circuit is so complicated as to make it impractical to compute PI or M from 

 such measurable quantities as resonant resistance Ri , series resonant 

 frequency /i , anti-resonant frequency /2 , etc. For this reason it is 

 necessary to measure the reactance and effective resistance of the crystal at 

 the operating frequency in order to obtain a measure of crystal quality 

 which will correlate with the crystal performance. For the same reason 

 it is important when comparing oscillator circuits that the crystal should be 

 operated at the same frequency in each case. 



It is believed that the non-linear effect noticed in crystals when used as 

 oscillators is produced by the changes in the mounting as the amplitude of 



