Sec, 6.4] ALLOWED RANGES OF NOISE MEASURE 53 



ventional sense, greater than unity (G = Gg, if Gg > 1 ; G = <» , if 

 Ge < 0). 



We observe from Eq. 6.30 that the numerator is never negative. 

 Therefore, changes in sign of Mj occur with those of the denominator. 

 In Case c, which includes most conventional amplifiers, M/ changes 

 sign only at a zero of the denominator. Thus, the values of M/ cannot 

 lie between \i/{kTo A/) and \2/{kTo A/). 



The two cases of interest. Cases b and c of Fig. 6.6, have a least positive 

 eigenvalue of N which we call Xi > 0. We have therefore proved the 

 following theorem : 



Consider the set of lossless transformations that carry a two-terminal- 

 pair amplifier into a new two-terminal-pair amplifier with a conventional 

 available gain G greater than 1 . When driven from a source that has an 

 internal impedance with positive real part, the noise measure of the trans- 

 formed amplifier cannot be less than \i/{kTo A/), where Xi is the smallest 

 positive eigenvalue of the characteristic-noise matrix of the original amplifier. 



The fact that Me' > Xi/ikTo A/) also puts a lower limit upon the 

 excess-noise figure. Suppose that the amplifier is imbedded in a lossless 

 network and then connected to a source with an internal impedance 

 having a positive real part. Let the resulting exchangeable power gain 

 be Ge. Then, the excess-noise figure of the resulting ampHfier has to 

 fulfill the inequality 



kTo^fV "Ge) ^^'^^^ 



F..-l> 



Consequently, an amplifier has a definite lower Hmit imposed on its 

 excess-noise figure, and this Hmit depends upon the exchangeable-power 

 gain achieved in the particular connection. 



If Ge > 0, that is, the output impedance of the amplifier has a positive 

 real part, the excess-noise figure can be less than \i/(kTo A/) only to the 

 extent of the gain-dependent factor (1 — l/Gg). 



If Ge < 0, that is, the output impedance has a negative real part, the 

 lower limit to the excess-noise figure is higher than \i/{kTo A/) by 



(1 + WGe\). 



Thus, if two amplifiers with the same eigenvalue Xi of their characteristic- 

 noise matrix are driven with a positive source impedance, one of which has a 

 positive output impedance, the other a negative one, then the minimum 

 excess-noise figure of the latter cannot be less than that of the former. 



Equation 6.31 has established a gain-dependent lower bound for the 

 excess-noise figure achievable with lossless imbeddings of a given amplifier. 

 At large values of |Ge|, the excess-noise figure is evidently equal to the 



