66 NETWORK REALIZATION OF OPTIMUM PERFORMANCE \Ch 7 



figure of the cascade is then equal to the minimum noise measure of each 

 stage, which, with the aid of Eq. 7.8 for N, is 



M, 



Xi 



1 



e.opt 



kTo A/ 8kTo A/ 



^ + ^ ^2h^^^^____ 



- 1 



' +^4^ ^ 2 Re (£„i/.i*) 



u' - 1 



"^ 1^ I 12 _ 1 (l-Snin/nir ~ l-Enl-^nl*r) 



(7.29) 



The proof of the inequality in Eq. 7.28 can easily be extended to cover 

 the case of nonunilateralized amplifiers of the class of Fig. 7.1a, provided 

 they have passive conjugate-image impedances. We start with the net- 

 work in the form that has the scattering matrix 



r 5i2"| 

 [-§•21 J 



with 

 and 



\S2l\ > 1 

 1^121 < 1 



The only differences occur as minor modifications in Eqs, 7.25 ff., where 

 |52il^ replaces \u\^ and (1 — |5'i2|^) appears in other terms. Thus, 

 unilateralization is not a necessary step to achieve optimum noise measure 

 with input mismatch. Amplifiers that have passive conjugate-image 

 impedances can be optimized for noise measure by an input mismatch 

 alone. However, the output impedance under optimized conditions is 

 guaranteed to have a positive real part only if the amplifier is also stable 

 under arbitrary passive input and output loading. Most vacuum-tube 

 and transistor amplifiers meet these conditions over a significant fre- 

 quency range. 



7.3. The Optimum Noise-Measure Expression for the Conventional 

 Low-Frequency Vacuum Tube 



We shall now derive from Eq. 7.29 the expression for the minimum 

 noise measure of a conventional low-frequency vacuum tube. The noise 

 in the tube is characterized by a grid noise resistance Rn, the input imped- 

 ance is Ri, and the plate resistance is rp. The noise-voltage column 



