Sec. 6.2] MATRIX FORMULATION OF EXCHANGEABLE POWER 45 



We note that Eq. 6.7 is the one-terminal-pair form of Eq. 5.1, where all 

 the submatrices have become scalars. 



Since multiplication of Eq. 6.7 by a constant c does not alter it, we can, 

 purely as a matter of form, always make a new column vector 



B:] = - = UA 



y = 



and a new scalar 



7 = cEs 



so that the source equation (Eq. 6.7) becomes 



y\ = 7 (6.8) 



with 



^ = Zs* (6.9) 



yi 



where Zs is still the internal impedance of the source. This formal 

 multiplication feature of the source equation is helpful in interpreting the 

 following analyses. 



Now the exchangeable power Pg of the source may be written in 

 matrix form 



£|2 \T? \2 I-, 1 2 



„ = . S\ ^ 1-^51 ^ |7| (. .fxx 



^^ 2(Zs + Zs*) 2(xtPx) 2(y+Py) ^ ' ""^ 



where the square "permutation" matrix P is the two-terminal-pair form 

 of Eq. 5.266. 



P = 



c a ^«"' 



It has the properties P^ = P and P~^ = P as indicated in Eqs. 5.27. 

 The usefulness of the last expression in Eq. 6.10 lies in the fact that it 

 can be written by inspection for any two-terminal source with a source 

 equation in the form of Eq. 6.8. 



Exchangeable-Power Gain. Consider a linear source-free two- 

 terminal-pair network, described by its general-circuit constants A,B, 

 C, D, as shown in Fig. 6.2. If v is the "input" column vector defined by 

 Eq. 6.6, and we let u be the "output" column vector, 



u = p;] (6.12.) 



and T is the general-circuit matrix, 



