A Reactance Theorem 



By RONALD M. FOSTER 



Synopsis: The theorem yivcs itu- most kcikt.iI form of the driving-point 

 impetlancc of any network coiii(x>sed of a finite niiniter of self-indurtances, 

 mutual iniluctanrcs, and cap.icitics. This impedance is a pure reactance 

 with a nun\ber of resonant and anti-resonant frequencies which alternate 

 with each other. Any such imjH-dancc may l)e physically realized (pro- 

 vide<l resistances can lie made ntKli.nihly small) by a network consisting of a 

 nunilvr of simple resonant circuits (inductance and capacity in series) in 

 parallel or a numlx-r of simple anti-resonant circuits (inductance and capac- 

 ity in parallel) in series. Formulas are given for the design of such net- 

 works. The variation cf the reactance with frequency for several simple 

 circuits is shown by curves. The proof of the theorem is based upon the 

 solution of the analogous dynamical problem of the small oscillations of a 

 system atxiut a [xisition of equilibrium with no frictional forces acting. 



AN imijortant theorem' gives the drixinvj-poini imiK'dance - of 

 an\- network composed of a finite numlier of self-inductanrcs, 

 mutual inductances, and capacities; showing tiiat it is a pure reactance 

 with a number of resonant and anti-resonant fre(|uencies which 

 alternate with each other: and also showing how any such impedance 

 may be physically reahzed by either a simple iniraliel-series or a 

 simple series-parallel network of inductances and capacities, pro- 

 vided resistances can be made negligibly small. The object of this 

 note is to give a full statement of the theorem, a brief di.scussion of 

 its ph>'sical significance and its applications, and a mathematical 

 proof. 



The TiiKdREM 



The most general driving-point impedance S obtainable by means of a 

 finite resistanceless network is a pure reactance which is an odd rational 

 function of the frequency p 2t and which is completely determined, 

 except for a constant factor II, by assigning the resonant and anti- 

 resonant frequencies, subject to the condition that they alternate and 

 include both zero and infinity. Any such impedance may be physically 



' The theorem was first stated, in an equivalent form and without his proof, by 

 George A. Campbell, Bell System Technical Journal, November, 1922, pages 13, 26, 

 and 30. By an oversight the theorem on page 26 was made to include unrestricted 

 dissipation. Certain limitations, which are now being investigated, are necessary 

 in the ^eneml case of dissipation. The theorem is correct as it stands when there is 

 no dissipiition, that is, when all the R's and G's vanish; this is the only case which is 

 considerefJ in the present paper. 



.\ corollary of the theorem is the mutual equivalence of simple resonant compo- 

 nents in (larallel and simple anti-resonant comp<ments in series. This corollary 

 had lx!en previously and independently disj-overed by Otto J. Zolx'l as early as 

 1919, and was sut)sequently publishefl by him, together with other reactance theorems. 

 Bell System Technical Journat, January, 102.?, pages 5-9. 



•The driving-point impedance of a network is the ratio of an impressed electro- 

 motive force at a point in a branch of the network to the resulting current at the 

 same point. 



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