394 HELL SYSTEM TECHNICAL JOURNAL 



stituted, the impedance locus may be marked with the frequency 

 scale for each reactance network in some distinctive manner. 



However, to show in the usual way some of the types of R and A' 

 curves represented by the locus of Fig. 3, as well as to avoid needless 

 complication of what is intended as an illustrative rather than a 

 working drawing, Figs. 3a and 3b have been prepared by direct pro- 

 jection from Fig. 3. In Fig. 3a are shown the R and A' curves plotted 

 against frequency when Zj is an inductance. In Fig. 3b are shown 

 similar cur\es when Z« is a doubly-resonant reactance. The R com- 

 ponent has a minimum at each resonant frequency and a maximum 

 at each anti-resonant frequency, while the A' component becomes 

 zero at resonant and anti-resonant frequencies alike. The number 

 of examples from this one resistance network might be multiplied 

 endlessly; it is believed, however, that these are sufficient to show the 

 great amount of information to be obtained in very compact form 

 from one simple figure in the complex plane, and the especial superior- 

 ity of the complex plane in displaying the characteristic common to 

 all the curves of F-igs. 3a and 3b: namely, that R and A at any fre- 

 quency, with any reactance network, are such that the impedance 

 lies on one circle. 



Two V.'\RiABLE Reactances Giving Eccentric 

 Annular Domain 



Returning to the more general impedance of (4) it is seen that in 

 each case short-circuiting and open-circuiting the terminals (2) and 

 (3) one at a time, and varying the reactance across the other termi- 

 nals, yields a locus for 5 which is a circle of the type just discussed. 

 These circles are determined as follows : 



where R, =c/c\ and Rj = d/d\. .\n examination similar to that in 



(13)-(15) shows that 



R^<.Rk<Ri, (18) 



R.<Rc<Rd. (19) 



