PARALLEL-TUNED TRANSFORMER DESIGN 



97 



Derivation of Design Formulas 



(a) Matched, or Symmetrically Loaded Transformer. 

 In the circuit of Fig. 1: 



£i = I.{R. + 1/jcoCO + h{- lAcoCO 

 = /:(- 1/VcoCO + /sO'coL: + l/iojC) + h{-joiM) 

 = /2(- icoM) + /30'a^i:2 + l/ya;C2) + /4(- l/iccCz) 



= /3(- l/yC0C2) + /4(i?2 + l/icoCs) 



T 



= K'VL, Lg 

 Fig. 1. — Parallel tuned, matched, or symmetrically loaded transformer. 



Eliminating Ii, h and I3 among these equations gives: 



^i^^^ + .^ [(m - ^^ ) iC.R. + C2i..)] 



£l//4 = 



(1) 



) (2) 



-(m- '^)] - i . ^l^^") 

 V ilf /J CO M / 



If the coefficient of mutual coupling is k, and the first and second induct- 

 ance and capacitance are resonant at co 1 and C02 radians per second, respec- 

 tively, we can substitute in (2) the expressions, 



M = WlJ:^ ,U= \loy,C. , U = I/C02C2 , h = E2/R2 (3) 



This gives the general expression, 



E,/E2 = ^'^^^^ /[^ . _i_ + ^2 . 1 "I 

 k \L'«'2 R2C2 wi RiCij 



L coiaj2 \R2C2 RiCi/ J 



I'^ + a^ + LjLi!_.] 



[_COi 002 COiCli?! aJ2C2-K2j 



+ i 1-^(1 



\CO1CO2 



^ ) + W I — + — + 



(4) 



COi Ci i?i aj2 C2 -^2 



_ Wl£02\ 

 CO / 



