xr.riroRKS coxt.iimxg rii'o reactances .wi 



Hut in .1 symmrtric.il (Icti-rniiii.iiit 



• J 1.-1.= -. 1 1.- =.1.1,.....; (i:j) 



tliiTi'furt' 



w llCIII 



ab^Kaxh (14) 



° < ,' 



<i 05) 



R., < R,. (1(>) 



To tiiul till' \>ilur of .V wlifii Zt lias some \aliK-, say Zi = iXi, it s 

 only necessary to mark the circular locus with a scale in terms of Z^. 

 This may be done directly by using (9) to determine the angle, <<>, 

 which the radius of the circle makes when Z2 = /A'2. It is simpler to 

 use the fact that a line passing through Rh and the point S has an 

 intercept on the reactance axis of 



X\ = kX2 (17) 



where x =6 «.. 



The factor k is determined by the resistances; therefore the scale, 

 as well as the locus, is completely fixed by the resistances. .Since k is 

 always positive, as X; is increased the circle is traversed in a clock- 

 wise sense; for positive values of Xi the upper semi-circle is covered: 

 for negative values, the lower. That is, when Zn is an inductance 

 the impedance of the network varies on the upper semi-circle from 

 R„ to Rb as the frequenc\- is increased from zero to infinity. When 

 the magnitutle of Zj is changed the same semi-circle is described but 

 each point (except the initial and final ones) is reached at a different 

 frequency. When Zj is a capacity the lower semi-circle, from Ri, 

 to Ra, is traced out. 



We know that, in general, the value of a pure reactance ■* increases 

 algebraically with frequency, and that its resonant and anti-resonant 

 frequencies alternate, beginning with one or the other at zero fre- 

 quency. When Z: is a general reactance, therefore, as the frequency 

 increases the entire circle is described in a clockwise sense between 

 each consecutive pair of resonant (or anti-resonant) frequencies. 

 For example, if Zj is made up of w branches in parallel, one being an 

 inductance, one a capacity and the others inductance in series with 

 capacity, as the frequency increases from zero to infinity the circle is 

 traced out completely «— 1 times commencing with Ra. 



'See: A Reactance Theorem, R. M. Foster, Bell System Technical Journal, .April, 

 1924, pages 250-267; also: Theory and Design of Uniform and Composite Electric 

 Wave-Filters, (). J. Zobel, Bell System Technical Journal, January, 1923, pages 

 1-47, especially pages 35-37. 



