722 BELL SYSTEM TECHNICAL JOURNAL 



Fig. 3 — Electromechanical representation of a piezoelectric crystal 

 clamped on one end. 



are the same except Le, which is twice as large since twice the mass is 

 moved from the clamping position. 



When the direction of motion is parallel to the direction of the applied 

 field, the same networks hold but the elements have different lengths 

 entering into their determinations. The direction of the applied field 

 and of the motion is designated by h. The other two axes are still 

 designated by Im and U. The resulting constants are 



^ _ Klmlo . ^ _ Sle ^ ^ _^ -ti5 



^irle Imlo 47rrf 



_ Iplehlm J _ ^pleldm 



where Le, is the mechanical inductance for the symmetrical case (Fig. 

 2) and Le^ for the dissymmetrical case (Fig. 3). 



A simple example of the use of Fig. 2 in determining the effect of a 

 mechanical load on the impedance of a crystal is the problem of finding 

 out how much the energy radiated by a crystal to the surrounding air 

 affects the decrement of a quartz crystal vibrating longitudinally. 

 When a crystal vibrates, energy is radiated to the surrounding medium 

 by the motion of the ends of the crystal. If the dimensions of the 

 ends of the crystal are comparable to a wave-length or greater — which 

 they will ordinarily be for a crystal vibrating at a high frequency — it 

 is well known ^ that the radiating surface experiences a resistance to 

 motion equal to 



Rr = P.ih (mechanical ohms) (14) 



per square centimeter, where pa is the density of the medium and h 

 the velocity of propagation. For air Rr is about 41 ohms per square 

 centimeter. Hence the equivalent circuit for a crystal vibrating in 

 air is Fig. 2 terminated at the terminals 3-4 and 3-5 by the mechanical 



" See "Theory of Vibrating Systems and Sound," I. B. Crandall, Chap. 4, D. Van 

 Nostrand Co. 



