190 BELL SYSTEM TECHNICAL JOURNAL 



where Fo is the value of Young's modulus along the bar, p the density, and 

 t the length of the bar. 



A significant feature of the equivalent circuit is that there is always a 

 definite ratio between Co and Ci for a given crystal cut. This is really a 

 measure of the ratio of electrical to mechanical energy stored in the crystal 

 vinder an applied constant voltage. The reactance characteristic of the 

 network is shown by Fig. 1.8 as a function of frequency. The reactance 

 starts out as a negative reactance at low frequencies, becomes zero at the 

 resonant frequency /k , becomes positive and very large at the anti-resonant 

 frequency Ja , then again becomes a negative reactance. Due to the high 

 ratio of Co to Ci existing in a crystal the separation between /a and Jr be- 

 comes very small. For example, for a,n AT crystal this ratio is around 200 

 and the separation of /a from/s is only a quarter of a per cent in frequency. 

 Since it can be shown that an oscillator will only oscillate on the positive 

 reactance part of the crystal characteristic, the narrow separation between 

 resonant and anti-resonant frequencies explains why a crystal can act as 

 such a good stabilizer for an oscillator. As long as the crystal resonance 

 itself does not change with temperature or other conditions, the very sharp 

 reactance frequency characteristic will not allow the oscillator frequency 

 to change much with a change in oscillator voltage, tube conditions, or any 

 other changes which are likely to cause a change in frequency for a coil and 

 condenser controlled oscillator. 



Strictly speaking, a resistance should be added in series with the induc- 

 tance Li to represent the internal losses in the crystal, the loss of energy 

 at the clamping points and the loss of energy due to setting up of air waves 

 by the crystal motion. However, the value of this resistance and the 

 amount of energy lost is very small in a crystal compared to what the losses 

 are in purely electrical elements. A demonstration which shows this 

 effect and shows that most of the losses of a well mounted longitudinally 

 vibrating crystal are acoustic losses caused by setting up air waves in the 

 vicinity of the crystal, can be made by using two oscillators, one a fixed 

 oscillator and the other one controlled by a resonant circuit or a crystal. 

 The fixed oscillator may be set at 99 kilocycles and the crystal oscillator 

 controlled by a 100-kc crystal. The two will beat together giving the 1000- 

 cycle note. When the battery is taken off the crystal oscillator, it continues 

 to oscillate till the energy built up in the crystal is dissipated in the internal 

 dissipation of the crystal. A good electrical circuit which has a ratio of 

 reactance to resistance, or Q of the coil of 300 dies down almost instantane- 

 ously. For a crystal mounted in air it takes about half a second to become 

 inaudible. This corresponds to a () of 30,000 where Q is defined as the ratio 

 of the reactance of the coil Zi of Fig. 1.8 to the resistance. For a crystal 

 mounted in a vacuum a much higher Q is obtained due to the elimination of 

 the loss of energy by acoustic radiation. For such a crystal it takes eight 



