396 BELL SYSTEM TECHNICAL JOi'RX.IL 



Similarly, if Z; is held constant while Z3 \'aries, the locus of 5 is one 

 of another family of circles. 



By the use of (9), keeping (5) in mind, it ma>' be shown that the 

 circles of each of these families are tangent to two circles determined 

 by the resistance network alone. Both families are tangent inter- 

 nallj- to a circle centered on the resistance axis, extending from Ra to 

 Rd. Both are tangent to a circle centered on the resistance axis, ex- 

 tending from Rb to Re, in such a way that the Zj-constant circles are 

 tangent externally and the Z2-constant circles are tangent enclosing the 

 circle from Ri to Re- These relationships are illustrated in Fig. 4. 



The circles Ra to R,i and Rb to Rt are, therefore, outer and inner 

 boundaries, respectively, of the region mapped out by the two families 

 of circles generated when tirst one and then the other reactance is 

 treated as a parameter while the remaining reactance is treated as 

 the variable. No matter what reactances ma>- be attached to termi- 

 nals (2) and (3), the resistance component R, measured at terminals 

 (1), is not greater than the resistance when terminals (2) and (3) are 

 open and not less than the resistance when terminals (2) and (3) are 

 short-circuited, and the reactance component A', measured at termi- 

 nals (1), is not greater in absolute value than half the difference of 

 the resistances measured when terminals (2) and (3) are open and 

 short-circuitcfi. That is, 



Ra<R<Rd. (22) 



\X\<hiRj-Ra). (23) 



The two families of circles (Zj-conslant and Zs-constant) intersect 

 and may be used as a coordinate system from which the components 

 of 5 may be read for any pair of values Zj, Z.i. To avoid inter- 

 sections giving extraneous values of 5 resort is made to a doubly- 

 sheeted surface, analogous to a Riemann surface, for which the two 

 boundary circles are junction lines. That is, the impedance plane is 

 conceived of as two superposed sheets, transition from one to the 

 other being made at the boundary circles. Thus, in Fig. 5, where 

 the two sheets are separated, each Z2-constant circle is shown run- 

 ning from the outer to the inner boundary in Sheet I (using the clock- 

 wise sense), and from the inner to the outer boundary in Sheet II, 

 while the Zs-constanl circles run from the inner to the outer boundary 

 in Sheet I and are completed in Sheet II.'' 



' It may be mentioned that the inner and outer boundaries are iinpedaiu'e curves 



traced out when ZiZj = - — '-^ and -^ = ." ,"" , respectively. 



A ij.u A ii.:j ^) A 12 A ij.jj 



