98 BELL SYSTEM TECHNICAL JOURNAL 



The application of circuit condition I, (coi = co2 = coo) to (4) gives, 



EJE2 = 



RK^oVCiCiff 1 



V 



\COoCli?l OJQC2R2/ 



(5) 



_ h\ (1 - k^) (1 + 1 ) 



-i[(-Y(i-*^)--(2 + wf¥) + -l 



LV^O/ COoV COoClC2i?lA2/ CO J 



Circuit condition II (zero transmission loss at co = coo) is satisfied if 

 I Ex/Ei 1= 2 \/Ri/R2 . Substituting this condition in (5) and solving for 

 k gives: 



2 _ coo Cl C2 i?l i?2 + 1 ±i(cOoCli?l — COoC2i?2) /^x 



" (cOoCli^l)' + (cOoC2i?2)2 + MCiC2i?li?2)2 + 1 



Since k must be real, 



COqCiRi = COoC2i?2 



and from (6) and (7) 



COoCi?. = COoC2i?2 = Vl - Wk 



From (5) and (8) the transmission formula for this transformer is, 



(7) 



(8) 



El ^ Vl - k^ 

 Er k' 



ViLvrbO -©■<■-«) 



i ((-)' (1 - *^) - ^ (2 + k^) + ^»)1 



\\COo/ COo CO /J 



(9) 



El 



El 



E, 



El 

 Ei 



e' , where, 



L\"0/ 0)0 w J J 



(10) 



^ _ arctan ^^ " *' • W'^o)^ - *') - (V<^)'(2 + ^') + I ,^^. 



