THEORY AND DESIGN OF WAVE-FILTERS 3 



Z\ and identical shunt impedances s 2 , each of which has a physically 

 realizable structure. For purposes of illustration, at the left end is 

 shown a mid-series section whose two series impedances hz± are each 

 half a full series impedance element. On the right is a mid-shunt 



section, its two shunt admittances — - each being half that of a full 



2s 2 



shunt admittance; its shunt impedances are therefore, 2z 2 . Cor- 

 responding to these two mid-point terminations are the mid-series 

 and mid-shunt characteristic impedances K x and K 2 , respectively. 



When any ladder type design has been obtained its mid-series 

 and mid-shunt sections, being respectively in the form of three star- 

 connected (T) and three delta-connected (II) impedances, may serve 

 as the basis of transformations by ordinary means to determine the 

 elements of other uniform types (such as the lattice type shown in 

 Fig. 6) having equivalent properties. Generally such equivalent 

 uniform types are not as economical as the ladder type either due to 

 difficulties of construction or a larger number of elements per section. 

 The theory of composite wave-filters is included in that of uniform 

 types as here presented and so does not require a separate treatment. 



Fundamental Formula 



The mathematical formulae upon which the design rests follows, 

 their derivation being given in Appendix I. 



cosh r = l+i-=l+i 7 2 , 



K _ Z1Z2 k _ = }?_ 



Vziz 2 + lzi VF+T? K~i 

 _r _ 



2K l -z 1 2z 2 -K 

 e- 1 - = 



2K\-\-Zi 2z 2 -\-K 2 ' 



(1) 



in which 



2i, 2 2 = series and shunt impedances per section, 

 r = A-\-iB = propagation constant per section, 

 K\, Ki = mid-series and mid-shunt characteristic im- 

 pedances. 



\z 2 



and k = -y/ 



Z\Z 2 , 



