MAXIMALLY-FLA T FILTERS IN WA VEGUIDE 693 



frequency is increased it becomes difficult to isolate the branches and unde- 

 sirable mutual impedances arise which complicate the problem. In par- 

 ticular, in the microwave region, where waveguides are used, the physical 

 size of each branch may be large compared with the wavelength and it is 

 then impossible to lump all the branches at one place in the waveguide with- 

 out encountering the complicated effect of mutual impedances. 



A practical way of circumventing this difficulty is to distribute the branch 

 circuits along the transmission hne or waveguide at such distances that the 

 mutual impedances become negligible. Then, however, the lengths of 

 transmission Hne act as transducers, but since their properties are well under- 

 stood and readily calculable this appears to be a practical solution. As a 

 matter of fact, the impedance transforming properties of a length of trans- 

 mission line can be used to advantage.^- ^'^' ^"^ For instance, it is well 

 known that a quarter wavelength of lossless line transforms a load imped- 

 ance according to the relation 



Z=^ (18) 



where Zq is the surge impedance of the line and Zz, is the load impedanc \ 



Hence if the load impedance consists of a series resonant circuit contain- 

 ing an inductance, a capacity and a resistance equal to Zq in series, the im- 

 pedance at the input end of the quarter wavelength of line is given 



The input admittance is 



F= ^»[l+i2e(^^--j)]- (20) 



As can be seen from equation 7, this is identical with the input admittance 

 of a parallel tuned circuit whose terminating conductance is 



G = Fo. (21) 



The quarter-wave line likewise transforms a parallel circuit to a series 

 circuit, as is illustrated in Fig. 6. This property of the quarter-wave line 

 thus makes it possible to simulate a ladder network of alternate series and 

 shunt branches by spacing shunt branches (or series branches) at quarter 

 wavelength intervals along a transmission Une, as illustrated in Fig. 7. 

 The resonant frequencies and the selectivities of the branches are chosen as 

 before. 



