## 2 Results
### Small-Signal Investigation for LNA Design
#### 2.1.1 Reverse Isolation and Stability  
Along with power gain and noise figure, reverse isolation and stability are very important parameters when designing the LNA. They are evaluated in term of scattering parameter \(S_{12}\) and \(\mu\) criteria of a two-port network, respectively. Expression for \(S_{12}\) can be derived by using the formulas in the previous section:  
\[S_{12}\approx\frac{2g_{\text{d1}}g_{\text{d2}}g_{\text{d3}}}{g_{\text{m1}}g_{ \text{m2}}g_{\text{m3}}}Z_{0}j\omega C_{\text{gs1}} \tag{18}\]  
From the above expression for \(S_{12}\), it can be seen that the isolation of the proposed configuration is very high due to the triple contribution of the \(g_{\text{d}}/g_{\text{m}}\) with \(g_{\text{m}}>>g_{\text{d}}\). Moreover, the expression also implies that \(S_{12}\) is dominated by the non-linear characteristic of \(C_{\text{gs1}}\). This means that \(S_{12}\) of the LNA can be improved by adjusting the first gate bias voltage \(V_{\text{g1}}\).  
The \(\mu\) criteria is a measure of stability and it indicates that if one two-port has larger \(\mu\), it is more stable. Moreover, if one two-port is stable, \(\mu\) is greater than unity. Therefore, this criteria is usually used to compare stability among two-port networks. Here, \(\mu\) is described through scattering parameters of a two-port network by the following formula [35]:  
\[\mu=\frac{1-|S_{11}|^{2}}{\left|S_{22}-\Delta S_{11}^{*}\right|+|S_{12}S_{21}|} \tag{19}\]  
A two-port which has small \(S_{12}\) and high \(\mu\) will be a great candidate for the LNA design. Thanks to the independently biased technique of the proposed 3-stack GaN HEMT topology, it is possible to adjust the bias condition of each transistor to improve both the reverse isolation and stability. Due to the fact that isolation of the proposed configuration is very high as mentioned earlier, \(S_{12}=0\) can be assumed. Substituting the above derived S-parameters into this formula and using the approximations as mentioned in the last section and assumption \(S_{12}=0\), the expression of \(\mu\) can be derived as below:  
\[\mu\approx\frac{1}{\left|S_{22}\right|}\approx\left|\frac{1-jZ_{0}\omega C_{\rm g 1}}{1+jZ_{0}\omega C_{\rm g1}}\right| \tag{20}\]  
The above formula indicates that stability of the LNA is also dominated by the first gate bias voltage \(V_{\rm g1}\) which contributes to the non-linear behavior of \(C_{\rm g1}\). However, it is obvious that the effect of \(C_{\rm g1}\) on the stability is not significant due to the similar terms in both numerator and denominator.  
From the above discussions, it can be concluded that \(V_{\rm g1}\) is the dominant factor which mainly contributes to the improvement of both the isolation and stability of the LNA.  
As mentioned, among the bias terminals including gate and drain bias terminals, the first gate bias (\(V_{\rm g1}\)) plays a key role for the improvement of the isolation and stability. This is demonstrated in Figure 4, which shows the isolation and stability investigation with the variation of \(V_{\rm g1}\) from \(-3.0\) V to \(-1.0\) V. As can be seen in the figure, isolation can be improved remarkably just by adjusting this bias terminal while the stability can be also slightly improved as expected. When \(V_{\rm g1}\) decreases, not only can the \(S_{12}\) reach very low values below \(-40\) dB in the entire frequency range, but the stability can be also improved.