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

The constraint on the Wiatr–Pospieszalski (WP) parameter
$ \Theta _{\rm W\!P}= {{ 4 {NT}_{\rm o} }/{ {T}_{\rm m}}}\leqslant 2$
, where
$ {\hbox{N}}$
is the Lange invariant and
$ {T}_{\rm m}$
the minimum equivalent noise temperature and
$ {T}_{\rm o}={\hbox{290}}~ \left ( {\hbox{K}} \right )$
, 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
$ \Theta _{\rm W\!P}$
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
$ \Theta _{\rm W\!P}$
parameter and determine the conditions for which
$ \Theta _{\rm W\!P}$
is larger or smaller than 2. A suitable expression that explains the conditions for
$ \Theta _{\rm W\!P}$
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
$ \Theta _{\rm W\!P}=2$
. The broad result of this paper's investigation is that the constraint
$ \Theta _{\rm W\!P}\leqslant 2$
is not a property of the noise parameter matrix as the constraint
$1\leqslant \Theta _{\rm W\!P} $
is; yet, the threshold
$ \Theta _{\rm W\!P}=2$
is demonstrated to have unique features for both active and passive two-port networks. Alternative and original demonstrations of the
$1\leqslant \Theta _{\rm W\!P} $
constraint are also obtained as a byproduct of the investigation.