Sec. 7.4] 



OPTIMIZATION, DEFINITE CASE 



69 



In the special case of a negative-resistance amplifier, the eigenvalues Xi 

 and X2 are both positive. Accordingly, resistances R\ and R2 of the 

 canonical form are both negative. We suppose now that the eigenvalue 

 Xi has the smaller magnitude. According to the theory of Chap. 6, this 





^ 



i?] 



R2 



EnlEn2 = 



J 



Fig. 7.2. Canonical form of two-terminal-pair amplifier. 



eigenvalue determines the lowest achievable value of the noise measure 

 Me. We shall now prove that this lowest value, Xi/{kTo A/), can indeed 

 be achieved using only that terminal pair of Fig. 7.2 which contains the 

 negative resistance Ri and noise generator Eni- 



As shown in Fig. 7.3, the terminal pair {Ri, En\) of the canonical form 

 is connected to terminal pair 2 of an ideal lossless circulator with the 

 scattering matrix 



S = 



(Transmission lines with 1-ohm characteristic impedance are connected 

 to all four terminal pairs of the circulator.) Terminal pair (4) of the 

 circulator is matched to a 1-ohm load at a temperature Tq, terminal pair 

 (1) is used as the input, and terminal pair (3) is used as the output 

 (Fig. 7.3). 



The equations for the resulting two-terminal-pair network can easily 

 be derived using the scattering-matrix representation. We find that the 



