A REACTANCE THEOREM 261 



Physical Discussion 



The variation of the reactance X = S/i with frequency is illustrated 

 by the curves of Fig. 1 in all the typical cases of formula (1) for n = 1 

 and for « = 2. For ever>' curve the reactance increases with the 

 frequency,' except for the discontinuities which carry it back from 

 a positive infinite value to a negative infinite value at the anti-reso- 

 nant points. Thus between every- two resonant frequencies there is an 

 anti-resonant frequency, no matter how close together the two resonant 

 frequencies may be. The effect of increasing n by one unit is to add 

 one resonant point, and thus to introduce one additional branch to 

 the reactance curve, this branch increasing from a negative infinite 

 value through zero to a positive infinite value. 



That formula (1) includes several familiar circuits is seen by con- 

 sidering the most general network with one mesh, that is, an induct- 

 ance and a capacity in series, with the impedance iLp+itCp)"^. 

 This expression is given immediately by (1) upon setting n = l, II = L, 

 and Pi = l/vLC. Since L and Care both positive these constants 

 satisfy the conditions stipulated under (1), thus verifying the theorem 

 for circuits of one mesh. This general one-mesh circuit includes as 

 special cases a single inductance L by setting II = L and pi=0, and a 

 single capacity C by setting 11 = and pi = oo such that Hp\=\/C. 



In Fig. 1 the reactances shown by the curves on the right are the 

 negative reciprocals of those on the left. Fig. 1 also shows networks 

 which give the several reactance curves, the networks being computed 

 by means of formulas (2) and (3). The networks are arranged in 

 pairs with reciprocal driving-point impedances and with the networks 

 themselves reciprocally related, that is, the geometrical forms of the 

 networks are conjugate,' and inductances correspond to capacities 

 of the same numerical value and vice versa. This relation is a natural 

 consequence of the reciprocal relation between an inductance and a 

 capacity of the same numerical value, these being the elements from 

 which the networks are constructed. 



For M = l, formulas (2) and (3) give identical networks, as illus- 

 trated by the reactances A, B, A', and B' of Fig. 1, each of which is 

 realized by a single network. For the reactances C and C the two 

 formulas give distinct networjcs, ci and Cj, c[ and Cj, respectively, these 



• This has been proved by Otto J. Zobel (loc. cit., pp. S, 36), using the formula 

 for the most general driving-point impedance given by George A. Campbell (loc. 

 cit.. p. 30). 



• For a further treatment of conjugate or inverse networks, see P. A. MacMahon, 

 EUctricinn, April 8, 1892, pages 601, 602, and Otto J. Zobel, loc. cit., pages S, 36, 

 and 37. 



