Sec. 2.2] LOSSLESS TRANSFORMATIONS 13 



of Fig. 2.2 can be written in the form 



Va = Zaalo + Zafclft (2.2) 



V5 = Zjalo + Zbblfo (2.3) 



The column vectors Vo and \b comprise the terminal voltages applied to 

 the transformation network on its two sides, and the column vectors !„ 

 and Ift comprise the currents It flowing into it. The four Z matrices in 

 Eqs. 2.2 and 2.3 are each square and of nth order. They make up the 

 square 2wth-order matrix Zr of the lossless transformation network. The 

 condition of losslessness can be summarized in the following relations, 

 which express the fact that the total time-average power P into the 

 transformation network must be zero for all choices of the terminal 

 currents : 



therefore 



or 



Equation 2.5 does not require that the transformation network be recipro- 

 cal. 



The original w-terminal-pair network, with impedance matrix Z and 

 noise column matrix E, imposes the following relations between the 

 column matrices V and I of the voltages across, and the currents into, 

 its terminals: 



V = ZI + E (2.6) 



The currents I into the w-terminal-pair network are, according to Fig. 2.2, 

 equal and opposite to the currents !« into one side of the 2w-terminal-pair 

 network. The voltages V are equal to the voltages Vo. We thus have 



V = v.; I = -la (2.7) 



Introduction of Eqs. 2.7 into Eq. 2.6 and application of the latter to 

 Eq. 2.2 give 



la= -{Z + ZaaT^Zablb + (Z -f Z„o)-1E 



When this equation is substituted in Eq. 2.3, the final relation between 

 Vfo and Ift is determined: 



Vfe = Z'ift -h E' (2.8) 



where 



Z' = -Z6a(Z + ZaaT^Zai + Z^fc (2.9) 



