EVOLUTION OF QUAJiTZ CRYSTAL CLOCK 



521 



cases the difficulties introduced by such factors may be more easily and 

 more positively controlled. 



In every primary clock mechanism the resonant governing device must 

 be sustained in oscillation, and the manner in which this is done has a strong 

 bearing on its rate regardless of the quality of the governing element. The 

 basic requirements are the same for any kind of oscillator, whether a pen- 

 dulum, an electrically resonant circuit comprising inductance and capaci- 

 tance, a steel tuning fork, or a quartz crystal resonator. The requirements 

 were first stated for the case of the pendulum by Sir George Airy in 1827 and . 

 it has always been the aim in the design of every good pendulum driving 

 means to satisfy Airy's condition. 



A^ 



REAL VELOCITY 



Fig. 5 — Amplitude-phase diagram for resonant element. 



This condition is conveniently illustrated by the diagram of Fig. 5 which 

 shows the two most familiar representations of damped sinusoidal motion. 

 In order to provide a convenient scale in the drawing an impractically 

 large damping is represented, corresponding to a Q of 20. The ^ of a 

 resonant circuit is related to the logarithmic decrement, 8, by the relation 

 Qd = IT. The factor 8 is the logarithm, to base e = 2.718 • • •, of the ratio 

 of the amplitudes at any two successive periods. It should be noted that 

 the (3 of a good electrically resonant circuit is in the order of 200, that of a 

 good pendulum from 10,000 to 100,000 and that of a good quartz resonator 

 from 100,000 to 5,000,000. The significance of these higher values of Q 

 will be evident from the following discussion. 



In Fig. 5 the damped sine wave shown corresponds, point by point, to 

 the phase diagram, which is simply a logarithmic spiral. By suitable choice 

 of scale the spiral can be interpreted to represent either the amplitude or 

 the velocity^in which case the real amplitude is vertical and the real 

 velocity horizontal. In this representation the velocity is shown maximum 

 when the amplitude is zero, which is a very close approximation to fact 

 for all practicable values of Q. The discussion will center on the velocity 

 spiral. 



