26 CHARACTERISTIC-NOISE MATRIX [Ch. 3 J 



Pe,i must remain constant at the foregoing value. Thus, from Eq. 3.9, 



x^[EEt - 2kT A/(Z + Zt)]x = 0, for all x 



or 



EE^ = 2kT A/(Z + Z+) 



a result proved previously by Twiss.^ In terms of the characteristic- 

 noise matrix N, we have 



N = - J EE^iZ + Z+)-i = -kT A/1 



an equation indicating that a passive dissipative network at equilibrium 

 temperature T always has a diagonal N matrix, with all the eigenvalues 

 equal to — ^r A/. 



3.4. Lossless Reduction in the Number of Terminal Pairs 



An w-terminal-pair network has a characteristic noise matrix of nth. 

 order, with n eigenvalues. If k of the n terminal pairs of the network are 

 left open-circuited and only the remaining {n — k) terminal pairs are 

 used, the original network is reduced to an {n — ^) -terminal-pair net- 

 work. This operation may be thought of as a special case of a more 

 general reduction, achieved by imbedding the original w-terminal-pair 

 network in a lossless {2n — ^) -terminal-pair network to produce {n — k) 

 available terminal pairs (see Fig. 3.2a). The case for n — k = 1 was 

 considered in Fig. 2.4. 



The characteristic-noise matrix of the {n — ^) -terminal-pair network 

 has {n — k) eigenvalues, which determine the extrema of the exchange- 

 able power Pe,n-k+\ obtained in a subsequent (variable) reduction to one 

 terminal pair (Fig. 3.2&). The successive reduction of the w-terminal- 

 pair network first to (n — k) terminal pairs, and then to one terminal 

 pair, is a special case of a direct reduction of the original network to one 

 terminal pair. Comparison of the dotted box in Fig. 3.2b with Fig. 2.4 

 shows that the exchangeable power Pe,n-k+i obtained by the two succes- 

 sive reductions, with variation of only the second network (Fig. 3.2b), 

 must lie within the range obtained by direct reduction with variation of 

 the entire transformation network (Fig. 2.4). Hence the stationary 

 values of Pe,n-k+i found in Fig. 3.2& must lie within the range prescribed 

 for Pe,n+i by Fig. 2.4. It follows that the eigenvalues of N for the 

 (n — ^) -terminal-pair network must lie within the allowed range of Pe,i 

 for the original w-terminal-pair network, illustrated in Fig. 3.1. 



^ R. Q. Twiss, "Nyquist's and Thevenin's Theorem Generalized for Nonreciprocal 

 Linear Networks," /. Appl. Phys. 26, 599 (1955). 



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