578 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



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of 4.01 is tlieu realizable by 3.1. Hence Mal> is a realizable reactance 

 2(n + ?/?)-pole, and so therefore is Mad , as we noted in discussing 

 Figure 7 (4.59). Therefore, if N^^' of Figure 5 is physically realizable, 

 so also is N and therefore N. We denote by Ni the N^^' obtained in this 

 way from N, and define IB as the operation which constructs Ni from N. 



We must still establish the claims made in 4.04 for IB. No properties 

 of N^^* = Ni have been proved beyond the existence of its impedance 

 matrix, Z*^*(p), but this is all that is claimed in the third column of 

 4.04. The fifth column is also established. We must now however com- 

 pare the degree of Ni , i.e., of Z^^'(p), with that of Z(p). 



By 2.13, 2.14 and 2.15 applied to (4), (5), and (0) of 4.G2, 



5(Z^'') = 5(Z) + rank (A) + rank (B), 



d{Z^'^) = 8{Y^'') = 5(F''') + 2 rank (G), 



5(F"^) - 6(Z''") = 5(Z*") + rank (H) + rank (F). 



We know m = rank (G) > 1. Let 



r = rank (A) + rank (B). 



Then from (23), and the relations above in order, 



5(Z) = 6(Z^") - r = (5(Z^") + 2m) - r 



= {8(Z^'^) + r) + 2m - r 



= 8(Z^*^) + 2m. 



Hence 8{Z) - 5(Z''') = 6(N) - 5(Ni) = 2m > 0. The claims of 4.04 

 are then established. 



4.71 We must yet verify 4.07 for IB. Let 8{M) be the degree of 



M{p) = i Mo + pM^ . 

 V 



Then by 3.21, Mad , whose matrix is M{p), can be realized with 8(M) 

 reactive elements. By Figure 7, then Mad can be so realized, and it 

 follows that exactly 8(M) reactive elements are comprised between N 

 and Ni under IB. 



Now by 2.14 and 2.15, 



8(M) = rank (Mo) + rank {MJ. 



We shall compute the second term. The first is obtained by an exactly 

 parallel calculation. 



