RESISTANCE COMPENSATED BAND-PASS CRYSTAL FILTERS 425 



the network of Fig. 2. In analyzing such networks it is usually more 

 convenient to reduce them to their equivalent lattice form and apply 

 network equivalences holding for lattice type networks. This can be 

 done by applying Bartlett's Theorem ^ which states that any network 

 which can be divided into two mirror image halves can be reduced to an 

 equivalent lattice network by placing in the series arms of the lattice a 

 two-terminal impedance formed by connecting the two input terminals 

 of one half of the network in this arm and short-circuiting all of the cut 

 wires of the network, and in the lattice arm placing the same network 

 with all its cut wires open-circuited. Applying this process to Fig. 1, 

 a lattice network equivalent to the network of Fig. 1 is that shown on 

 Fig. 3. In this network the capacitances can be considered as sub- 



Fig. 3 — Lattice equivalent of crystal filter of Fig. 1. 



stantially dissipationless and if the network representing the crystal can 

 also be considered dissipationless, the resistance introduced by the 

 coils can be effectively brought outside the lattice and incorporated 

 with the terminal resistances. This follows from the fact that an 

 inductance with an associated series resistance can just as well be 

 represented over the narrow-frequency range of the filter by an in- 

 ductance paralleled by a much higher resistance. The impedance of an 

 inductance and resistance in series and the impedance of an inductance 

 and resistance in parallel are given by the expressions 



Ri + jooLi = 



R^ijwL^ R<i,(x>^L^ + jwL^R^ 



i?2 + joiLi 



Ri^ + co^W 



(1) 



* "Extension of a Property of Artificial Lines," A. C. Bartlett. Phil. Mag., 4, pp. 

 902-907, Nov. 1927. 



