Considerations on the $ {{ 4 {{NT}}_{\rm o} }/{ {T}_{\rm m}}}$ Ratio and the Noise Correlation Matrix of Active and Passive Two-Port Networks

Abstract

The constraint on the Wiatr–Pospieszalski (WP) parameter <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ \Theta _{\rm W\!P}= {{ 4 {NT}_{\rm o} }/{ {T}_{\rm m}}}\leqslant 2$</tex></formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ {\hbox{N}}$</tex></formula> is the Lange invariant and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ {T}_{\rm m}$</tex></formula> the minimum equivalent noise temperature and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ {T}_{\rm o}={\hbox{290}}~ \left ( {\hbox{K}} \right )$</tex> </formula> , has been suggested as a validation tool for the characterization of active microwave devices, despite its acknowledged limitations. This paper provides a solid theoretical background on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ \Theta _{\rm W\!P}$</tex></formula> that is used to investigate noisy linear two-port networks. In particular, parallel feedback is discussed thoroughly in order to understand its effect on the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ \Theta _{\rm W\!P}$</tex></formula> parameter and determine the conditions for which <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ \Theta _{\rm W\!P}$</tex></formula> is larger or smaller than 2. A suitable expression that explains the conditions for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ \Theta _{\rm W\!P}$</tex></formula> of on-chip devices to be less than 2 is obtained. The theoretical results are supported by a scalable GaN device model extracted from measurements providing further insight into the noise performance of microwave devices. Finally, the investigation is applied to passive two-port networks to highlight the singular properties of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ \Theta _{\rm W\!P}=2$</tex> </formula> . The broad result of this paper's investigation is that the constraint <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ \Theta _{\rm W\!P}\leqslant 2$</tex> </formula> is not a property of the noise parameter matrix as the constraint <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1\leqslant \Theta _{\rm W\!P} $</tex> </formula> is; yet, the threshold <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ \Theta _{\rm W\!P}=2$</tex></formula> is demonstrated to have unique features for both active and passive two-port networks. Alternative and original demonstrations of the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1\leqslant \Theta _{\rm W\!P} $</tex></formula> constraint are also obtained as a byproduct of the investigation.