104 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1951 



where 



P + Q = Ro+ {1 - n{)S + X2(l + M2)(l + k2) Eq. (10) 

 Substituting for P + ^ a-nd rearranging 



Zi2 = Rq -\- Xz 



R, + X,{1 + M2)(l + fe - 2kl) + ^^^ ^^ff" 



Z,N -f- ZO 



Ro + X,(l + M2)(l + fe) + ^\ ^'^^^^ 



Eq. (11) 



Equation (11) can be rewritten in a form from which the equivalent 

 circuit can readily be constructed. This form is given below in Equation (12) 



1 



Zi2 = i?6 + 



Xs 



+ 



1 



L2£JU(i+iU2)J 



Xsd + k. 



2kl) 



.m + M2) 

 + 



2^1 



[ 1 - Ml ] r X3 1 r 252^ "I 

 Li + M2jL4^i^2jU^ + 25j 



Eq. (12) 



From Equation (12) the equivalent circuit of Fig. 9(a) can be developed. 

 Starting at the lower right-hand side of Equation (12) it will be observed 

 that 2SZn/{Zn + 25) represents Zn and 2S in parallel. This parallel 

 combination is multiplied by (1 — jUi)^3/(l + M2)(4^3 ^2)- In series with 

 the parallel combination is the leakage inductance of transformer T equal 

 to X3(l + ^2 — 2^3)72^3; and the term containing Rq, which stands for 

 Rp -\- i?2(l + M2). In parallel with all of this is X3. The resistance Re is 

 simply a series resistance as evident from Fig. 8. 



It can be seen from Fig. 9(a) that X3 14X2^3 is the impedance ratio of 

 transformer T, and that — (jui — 1)/()U2 + 1) is the transformation ratio of 

 an ideal converter. The resulting circuit can be represented by Fig. 9(b). 



Next consider the impedance Z34, which is seen looking into terminals 

 3 and 4 of Fig. 8. 



£4 



Z34 — 



Zn 



Eq. (13) 



I A = 



£4 



Eq. (14) 



(Zs + 25) [(Zz. + i?6 + X3)(P + - 2M3'(1 -h M2)] 



E, __ - 2(1 - ixi)S\Zj^ + R, + X3) Eq. (15) 



h~ (P + Q){Z!. + i?6 + X3) - 2^3^(1 + M2) 



