TRANSMISSIOX CIl.lR.ICir.RlSTICS OT W.ll-F.-PILTERS 581 



That cithor factor is (k-pciuU-nl only upon its own type of tcrmina- 

 ^tion and not ii[>on its position at the sending or receiving end, caa 

 reailil\- i)e shown. H\ the recii)rocal theorem tlic product F,FaFi,F, 

 is intle[XMulent of the direction of current propagation, and from 

 the forms of Fi and /•", the latter are also, whence the product Fa Fi, is 

 independent of direction. Since in addition Fa and Fb are inde- 

 pendent of each other they cannot depend upon position. This is 

 equivalent to the statement that the ratios £„/£ and I h which 

 any particular termination would give at the sending and receiving 

 ends, respectivel>-, arc equal. It will then be sufficient to consider 

 the factor for a gi\en termination at either end, say the receiving end. 



The four terminations found practical give terminal losses which 

 are reducible to two, namely, Lm and Lx now to be derived. 



Terminal Losses. L„, 'with Mid-M-type Terminations. These 

 terminations, already mentioned, are 



1, mid-shunt of a mitl-series "constant k" efiui\alent .l/-tvpe, 

 (A:„(m)):and 



2, mid-series of a mid-shunt "c<instaiit k" ecjuivalent .l/-tvpe, 

 (A',.(m) ). 



The relations between the ^l/-type characteristic impedances Knim) 

 and Kii^m), the parameter w, and the variables Uk and F* of the 

 "constant k" prototype are, from formulae ' in a pre\ious paper 



R _ K,,(m) _ ±Vl + Uk + iV, 



Since Knim) • KzAtn) =R', Kuim) and Kiiim) are inverse networks 

 of imjiedance product R'. As either of these terminations is at a 

 mid-fxjint, it forms an end for the wave-filter mid-part and in the 

 terminal factor Fi,, arbitrarily chosen, Zi, = R and 7/4=1, leaving 



„ 2VWR ,...,, 



In this factor the image impedance Wi, is either Knitn) or A'i;(m), 

 tiepending upon the type of termination. By (32) the factor is the 

 same for both tyjies provided they have the same parameter m, so 



'The radicals which occur in this and succeeding formulae are proportional to 

 physical impedances with positive resistance components. Hence, in each case 

 the double sign is to be interpreted such as to make the real part of the radical 

 positive. 



