804 



BELL SYSTEM TECHNICAL JOURNAL 



proceed to prove, it is the common numerator of all of the resistance and 

 conductance expressions throughout the network. 



Let us suppose now that the first branch, iaix, of the network is 

 added in series. The admittance after its addition is 



1 



F{x) 



Fix) 



.Gi(x) . 



F2ix)C{x) ' 'G^ix) 



F^i 



^)^^^)[^ 



+ 



G^ix) 



+ aix 



— I- 



F2{x)C{x) ( -~-~-^ aix 



F2ix) C{x) 



\FAx) 

 I F^ix) 



+ 



) \G2ix) 



G,{x) . \ 2 



+ GiX 



Upon remembering the way in which C{x) was chosen we observe that 

 the expressions in the denominators of the conductance and susceptance 

 fraction and in the numerator of the susceptance fraction reduce to 

 polynomials. 



So far we have been able to show that the impedance of the load and 

 the admittance of the network after one branch is added can be so 

 expressed that (1) their imaginary components are rational functions, 

 (2) the numerators of their real components are equal to F{x), and (3) 

 the denominators of their real components are polynomials. It is also 

 possible, however, to show that if these statements hold for the im- 

 pedance and admittance at any two consecutive junctions they will 

 hold also at the next following junction. Referring to Fig. 3, let us 



ian+iX 



Yn + i 



Zn 



-n-i-2 



^Sn-t-2^ 



/ _F_ • NrvH\ / F , ■ Nn\ 



ian-ix 



I 



lanx 



Fig. 3 — Impedance and admittance relations at « + 1st branch of network. 



suppose that the impedance after n branches of the network have been 



added is 



__ F{x) . 7V„(x) 

 ^" D„{x)'^' Dn'{x) 



and that the admittance after w + 1 branches have been added is 



Fix) iV„+i(.v) 



"^"^^ Dr^+.ix) '^ ' Dn+lix) 



