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BELL SYSTEM TECHNICAL JOURNAL 



The constants a, /S, 7 and b may be written immedately from (15.56); 

 however, for purposes of simplification let 



= _L 



cogCo 



z = y — - (co — C02) 



2/" 

 M = ^- (co2 — coi) (See equation 15.50) 



(15.57) 



then, 



Z. = 



[r + jMr] + [t]Z 

 J + JZ 



(15.58) 



Now if Zc may be represented by W in (15.52) the remaining constants 

 must be 



a = r + JtM 7 = i 

 /3 = T 5 = i 



(15.59) 



Now Z represents the frequency variable and graphically represents a line 

 coincident with the F-axis in the Z-plane, and IF, the corresponding im- 

 pedance variable, represents a circle in the TF-plane. The coordinates of 

 the center of the circle, Wq , in the TF-plane is given by 



(15.60)= 



when 7 and 5 represent the conjugate functions ot 7 and 5 respectively. 

 The radius of the circle is given by 



(15.61)= 



*E. C. Titchmarch, "Theory of Functions," Oxford 1932, pp. 191-192. Note: These 

 equations are not derived in Titchmarch; however, by taking the limit as the diameter of 

 the circle in the Z-plane approaches infinity, (15.60) and (15.61) result. 



