ELECTRICAL WAVE FILTERS EMPLOYING CRYSTALS 111 



The reactance characteristic of the crystal, as shown in Fig. 1, is a 

 negative reactance at low frequencies up to a resonant frequency /r. 

 For frequencies greater than //j, the reactance becomer positive up to 

 the anti-resonant frequency /a, above which the reactance is again 

 negative. The ratio of the anti-resonant frequency to the resonant 

 frequency is determined directly by the ratio r of Co to C\ existing in 

 the crystal. As shown by Fig. 1, 



f:=x/'+7- ■ (^) 



This ratio is usually greater than 125 for a quartz crystal and hence 

 the anti-resonant frequency is less than .4 per cent higher than the 

 resonant frequency. 



The previous papers considered mainly band-pass filters and dis- 

 cussed briefly low and high-pass crystal filters. It is also possible to 

 obtain band elimination and all -pass crystal filters by combining 

 electrical elements with the crystals in the proper manner. We 

 consider, first, all the types of filters which can be obtained by using 

 a single crystal in one arm of a lattice filter and electrical elements in 

 the other arms. Figure 2 shows all the possible single-band character- 

 istics which can be obtained by using a crystal in one arm and an 

 electrical impedance, or crystal impedance, in the other lattice arm. 

 For example, the first filter of the table shows a filter with a crystal in 

 one arm and a capacitance in the other arm. Column B shows the 

 reactance characteristic of each arm. A lattice filter will have a pass 

 band when the reactances are of opposite sign and will attenuate when 

 the reactances are the same sign. When the two reactances are equal 

 the filter will have an infinite attenuation. This result follows from 

 the expressions for the propagation constant and characteristic 

 impedance of a balanced lattice network which are 



tanh ^ = ^|i ; Z, = -{Z^-,, (3) 



where Z\ is the impedance of the series arm of the lattice and Zi that 

 of the shunt arm. The third column shows the attenuation character- 

 istic of this filter. It is a narrow band filter having a pass band 

 between the resonant and anti-resonant frequencies of the filter. 

 There is a peak of attenuation either above or below the band de- 

 pending on the value of the capacitance C\ in the lattice arm. The 

 last column shows the value of the characteristic impedance of the 

 filter as a function of the frequency. The dotted line indicates a 



