Sec. 7.4] OPTIMIZATION, DEFINITE CASE 71 



Thus, according to the foregoing results and Eq. 7.38, 



M = M. = ^ = - ,/"^ ,, = -^, (7.44) 



_ l^ ^RikTo A/ kTo A/ 



G 



We have therefore proved that the circulator arrangement indeed 

 achieves the lowest possible noise measure. Since it also leads to a 

 unilateral amplifier with positive real input and output impedances, an 

 arbitrary gain can be achieved through cascading of such identical ampli- 

 fiers. We observe, however, that a lossy network (ideal lossless circulator 

 plus Rq) has been employed with the original amplifier to optimize its 

 noise performance. 



The optimization carried out in connection with Fig. 7.3 has a useful 

 corollary concerning circuit connections of maser amplifiers for optimum 

 noise performance. One of the forms of the maser has for an equivalent 

 circuit a one-terminal-pair negative resistance Ri in series with a noise 

 voltage generator Eni- To make a two-terminal-pair network, we may 

 consider as an artifice not only the noisy negative resistance Ri of the 

 maser but also another positive resistance R2 at a temperature T2. The 

 two resistances can be treated as the canonical form of a two-terminal-pair 

 network. Lossless imbedding of these two resistances therefore leads to a 

 two-terminal-pair amplifier with the eigenvalues Xi = — \Eni\/{4:Ri) > 

 and X2 = —kT2 A/ < 0. The best noise measure that can be expected 

 from the resulting amplifier is If opt = ^i/{kTo A/). The circulator 

 arrangement has been shown to achieve this noise measure. Thus, it 

 provides one of the optimum network connections of the maser with 

 regard to noise performance. It should be re-emphasized that the as- 

 sumed presence of a positive resistance R2 in the circuit is an artifice that 

 enables the use of the theory of two-terminal-pair networks for the noise 

 study of the one-terminal-pair maser. The assumed temperature of the 

 resistance is immaterial because it determines only the negative eigen- 

 value of the characteristic-noise matrix, which has no relation to the 

 optimum noise measure achieved with gain. 



The results of this chapter lead to the following theorem : 



1. Any unilateral amplifier with U > 1 may be optimized with input 

 mismatch alone. 



2. A nonunilateral amplifier with U > 1, which is also stable for all 

 passive source and load impedances, may be optimized with input mismatch 

 alone. 



3. Any amplifier with U > 1 may be optimized by first making it uni- 

 lateral, using lossless reciprocal networks, and subsequently employing input 

 mismatch. 



