658 liLiU. SYSTEM TECHNICAL JOURX.IL 



that an\- such impedance may he realized by a network consisting 

 of a number of simple resonant elements {inductance and capacity 

 in series) in parallel or a number of simple anti-resonant elemeiUs 

 (inductance and ca|)acit\ in parallel) in series. 



With risi>t,uices added In the circuit, the impedance is, in .general, 

 (-(implex; that is. it has both resistance and reactance components. 

 I'lir ,1 two-mesh circuit the impedance is expressed as ,i luiirtioii of 

 tin- time coefl'icient b\- Theorem 1. 



l-'ormula ( 1 ) jji\es the clri\ing-iKjint impedance of a two-mesh circuit 

 for any electrical cwcillation of the form e", where the time coefficient 

 X may ha\e an\- \.iltii\ real or complex. The time coefficients for 

 the free oscillations of the circuit with the driving branch closed are 

 the roots of the numerator (ai, ao, as, at), as given by (la); the free 

 periods of the circuit with the driving branch opened are the roots 

 of the denominator (182, 03). that is, the poles of the impedance func- 

 tion. For a complex value of the time coefficient, X = Xi-f/\;, Xi is 

 the (l,un|)in.n factor and X^ is the frequency- multiplied In- '2w. 



The two forms of fornnil.i (1) are equivalent, but each has its 

 sjjecial achantages. Sometimis one, sometimes the other, form is 

 more coinenient; the>- will bi- used interchangeabh- throughout the 

 paper. 



Formula ( la) gixes the impedance directK- in terms ol the roots ,nul 

 lioles. I-"ormula (lb) gives the im])edance in terms of the s\-mmetric 

 functions of the roots and poles, with the addition of an arbitrar>- 

 factor. Thus, without changing the impedance, all the coefficients 

 of the numerator and denominator of (lb) ma\- be multiplied by the 

 same constant factor ha\-ing any value other th.m zero. Formulas 

 slated in terms of the coefficients of (ibi .u-c in homogeneous and 

 s\-mmetrical form, .uid h,i\e the added ,i(l\,uii,im- nf inxoKing real 

 (|uantities only. 



Tin- spe(-ial (-ase of one i-ixit e(|U,d to /-vvn i> cibt. lined b\' setting 

 a, =0 in (la) and iit=0 in (lb). l-"or one root inl'mile, howe\t-r, 

 ill ( la 1 it is necessary to set a\ = -c and 11 = 0, with the pro\-isioii tli.ii 

 Jhu be liiiite; whereas in ( lb) it is simpl\- necessary to set flo = <l. 



It is sometimes (-oinenii-iii tn .idd the not.itinn )i\=0 .md /ii =^ "x . 

 cori-esponding to tlu' poles at /i-rn .md iiilinitN . In lormul.i (Ibi the 

 corresponding addition to the iKit.itinn i-onsists of the (-oi'tVuients 

 60=0 and 64=0. 



By the general restrictions (2) the constant 11 is positive or zero, 

 and the roots ;ind jioles are arranged in three (lairs, (ai, a-j), («», wi). 

 and f^;, fin), each i)air being the roots of a ijuadratic equation wiili 

 positi\-e real coelficienis. Thus each pair of the roots and jjoU-s is 



