654 BELL SYSTEM TECHNICAL JOURNAL 



consisting of resist ancfs, caparities, and induclanccs can hv replaced, 

 in so far as the impedance IxHween terminals is concerned, by any 



oiu- of ilie cli-\cii networks shown by Fig. 1, upon assigning the proper 

 \aiiies to tlie ek'nieiiis. l-'.acii of these networks consists of two 

 resistances, two capacities, and two self-indiictaiices witii nuitii.il 

 inductance between them. 



\-Aub of tliese ele\en ni'tworks realizes impedances with arbitr.iriK' 

 assigned roots and with poles an%where in the entire domain of pos- 

 sibilities, subject to the general conditions stated in Theorem I. 

 Special ca.ses of these networks realize, for arbitrarily assigned roots, 

 onl\ critical lines and points in the domain. All these special cases 

 are listed in Table III, with a specification of the lines or points in 

 the diiniaiii reaiiz.ibie b\- eacli, as illustrated by h'igs. 4 and ."). 



Certain Hniiied ri'iLjions (jf the domain cut be realizi-d b\- networks 

 which contain no miiiiial in(hictance ,ind whicii are not speci.d cases 

 of liie networks gi\en b\ Thecireni II. These ni'tworks are gi\en 

 iiy liie following theorem : 



Theorem III. Any driving-point iinpcilunce of a luv-niesh cireuit 

 consisting of resistiDiees. eiipacities, and self-indiietanees can be realized 

 by at least three and not more than five of the twelve networks shozvn by 

 /•/g. 2, upon assigning to the elements of each network the values given 

 by liable II. These twelve networks are the only networks without mutual 

 inductance and without superfluous elements by which any impedance 

 can. in '^owral, he realized. 



These twelve networks, taken together, cover that portion of the 

 domain realizable w ithoiit mutual inductance. Networks with mutual 

 inductance are needed in order to co\ er the entire domain. These 

 twelve are the onl>- networks, witiiout superfluous elements, realizing 

 limited regions in the domain. Kach of these networks consists ol two 

 resistances, two capacities, two self-inductances, and one additional 

 resistance, capacity, or self-indiui.inci.'. Thr iwiKe networks, wlili 

 their si)ecial cases, are all listed in T.ible 111, witli .i specilicaiion ot 

 the regions, lines, or i)oints ri'ali/.ilile ii\ each. 



In addition to the sjjccific formulas for the networks ol Figs. I ,md 2. 

 it is con\enienl to have general formul.is for the com|iutation of all 

 networks meeting the given conditions, including those networks 

 with superfluous elements as well as all special cases. The most 

 general two-mesh circuit is shown by Fig. (i; accordingh', tlu' most 

 general network under consideration is that shown by l"ig. 7. l-oiinnhiN 



