236 BELL SYSTEM TECHNICAL JOURNAL 



inductance can be evaluated by equation (7) and subtracted from the 

 impedance Z. This leaves an impedance 



Z' = Z - jcoL, (8) 



from which can be evaluated the constants of the two crystals in 

 parallel by employing equation (6). In this way the constants of 

 any filter can be evaluated when the desired attenuation and im- 

 pedance characteristic are specified. Several of the band-pass filters 

 are discussed in detail in a former paper.^ 



IV. Application of Divided Plate Crystals to Balanced and 

 Unbalanced Filters 



The use of a divided plate crystal to cut the number of crystals in 

 half in a balanced lattice filter has been mentioned previously.^- s. *. lo 

 The theory of this use has not been previously discussed and since it 

 results in further applications it seems worth while to present it here. 



In order to use the divided plate crystal in filters it is necessary to 

 find an equivalent circuit for such a crystal which will hold for measure- 

 ments between any pair of the four terminals. It was shown in a 

 previous paper ^ that an electro-mechanical equivalent of a fully 

 plated crystal free to vibrate on both ends could be represented as 

 shown in Fig. 14A. In this figure the capacitance Co is the static 

 capacitance of the crystal, the condenser Cm represents the effective 

 compliance of the crystal at the resonant frequency, and the inductance 

 Lm represents the effective mass. A perfect transformer of im- 

 pedance ratio 1 to <p^, where 



if- = 



(522') 



(9) 



represents the coupling from electrical to mechanical energy, (p in 

 effect is the ratio of the force exerted by the crystal when it is clamped, 

 to the applied voltage or it is the force factor of the system. If now 

 we use only half the plating on the crystal, for example the plates 1, 

 2 of Fig. 14B, the same representation will hold. The static capaci- 

 tance Co will be divided by 2, and the force applied by a given voltage 

 will also be divided by 2 or the transformer ratio will be <pl2. The 

 same compliance and mass will be operative. Hence the equivalent 

 circuit of a crystal with plates covering half the crystal will be as 

 shown on Fig. 14C. For a crystal with two sets of plates, the repre- 



2, 3, 4 LoC. Cit. 



"See patent 2,094,044, W. P. Mason, issued Sept. 28, 1937. 

 " Loc. cit. 



