332 



BELL SYSTEM TECHNICAL JOURNAL 



Reactive Impedance Characteristics 



All non-dissipative impedances have reactances which can be 

 separated into four forms of impedance functions, each of which can be 

 expressed as the ratio of two frequency-polynomials in if, where 

 i = V— 1, and / is frequency. It is known that such a reactance 

 necessarily has a positive slope with frequency and hence the resonant 

 and anti-resonant frequencies alternate on the frequency scale. The 

 four mathematical forms may be separated on the basis of the general 

 location of their resonant frequencies and have finite resonant fre- 

 quencies with or without zero and infinite resonant frequencies. These 

 reactive impedance forms are as follows: 

 Form 1. Resonant at zero and w finite frequencies. 



_ a,if + a.iify + ■■■ -f fl,„+,(-//)^» 



1 + b.xifr + • • • + bu^ir-" 



+1 



= tx. 



Form 2. Resonant at n finite and infinite frequencies 



1 + aodfY- + h a^ndfy" 



= tx. 



bvlf + b,{ify + • • • + 62n+l(//)^"+^ 



Form 3. Resonant at zero, n finite and infinite frequencies. 



_ a,if+aSff + h n2n+x{iff"+' 



^ 1 + b.iiff + • • • + 62n+2(^/)2"+2 



Form 4. Resonant at n finite frequencies. 



1 + aSfY + • • • + a2n((f)-" 



= IX. 



z = 



b^f+b^{ifY+ ••• + b,n-,iiff--' 



(51) 



(52) 



(53) 



(54) 



Each of these forms has a simple frequency relation which is expressible 

 as a theorem. 



Reactance Frequency Theorems 



The product F of the frequencies at which the reactance x is ± c in each 

 of the four reactive impedance forms is the folloiving: 



Form 1. 

 Form 2. 

 Form 3. 



F'^n+i = , proportional to c. 



a^n+i 



F2n+i = -T . inversely proportional to c. 



COoji+i 



F-2n+2 = -J , independent of c. 



0-271+2 



