36 BELL SYSTEM TECHNICAL JOURNAL 



the result of a positive total reactance slope in every case, due to the 

 above relations. That this property is not limited to such combina- 

 tions is seen from the general impedance expression for a non-dissipa- 

 tive reactance network, 4 



..iy/W-ff-fA-^, (12 ) 



Here M is a positive real, and the resonant and anti-resonant fre- 

 quencies, /i . . . f-2 nt alternate and are in the order of increasing 

 magnitude. The exponent, u, is unity or zero according as a resonant 

 or an anti-resonant frequency is the last of the series. Assuming 

 without loss of generality that /i is not zero, the reactance increases 

 with frequency from zero frequency up to f=fi, since all the factors 

 are positive. As / passes thru this anti-resonant frequency the re- 

 actance changes abruptly from positive to negative infinity and when 

 / increases to the resonant frequency ji the negative reactance in- 

 creases to zero. As / increases beyond the value ji the reactance is 

 again positive and the cycle of reactance changes with frequency 

 begins over again. 



The possibility of representing such a general reactance identically 

 at all frequencies by a network constructed of either a number of 

 simple resonant components in parallel, or simple anti-resonant com- 

 ponents in series, follows from the fact that in any particular case the 

 number of inductances and capacities involved is always equal to 

 the total number of conditions which this network must satisfy to 

 obtain such equality. Thus, its reactance must be zero and infinite 

 at the given resonant and anti-resonant frequencies, respectively, and 

 must have a definite magnitude at some one other frequency, which 

 conditions are sufficient to determine all the impedance elements. 

 In general, other equivalent combinations of inductances and capaci- 

 ties are also possible. 



Theorem 2, relating to inverse networks, will be proved by an 

 inductive method in which the given reactance network is assumed to 

 have the form of series and parallel combinations of inductances and 

 capacities, a form which by the first theorem can be taken to represent 

 the reactance of any non-dissipative reactance network. Let z { and 

 32 be one pair of impedances which are inverse networks of impedance 

 product D 2 to each other, and let z^ and z>, be another pair so that 



z\z'i = z'\Zt = D 2 = a constant positive real. 



Then z[ and z" in series, and z 2 and z 2 in parallel are a pair of inverse 

 4 See paper by G. A. Campbell, Vol. I, No. 2, p. 30, this Journal. 



