S parameters of properly biased HFET- $1101$ measured using $50 \Omega$ netwook analyser at $6 \mathrm{GH}_2$

$ \begin{aligned} \\ &s_{11}=0.614 L-167.4^{\circ} \\\\ &s_{12}=0.046 L 65^{\circ} \\\\ &s_{21}=2.18 L 32.4^{\circ} \\\\ &s_{22}=0.716 L-83^{\circ}\\ \end{aligned}\\ $

Design an amplifier using this for maximum possible gain.

Solution:

Stability checks:-

$ \begin{aligned}\\ \Delta &=s_{11} s_{22}-s_{12521} \\\\ &=(0.614 \angle-167.4^{\circ} \times 0.716 -83^{\circ})-(0.046 L 65^{\circ} \times 2.18\angle32.4^0).\\\\ \Delta &=0.34 \angle 113^{\circ} \quad \therefore|\Delta|=0.34\lt1\\ \end{aligned}\\ $

$ \begin{aligned}\\ &K=\frac{\left.|-| s_{11}\right|^2-\left|s_{22}\right|^2+|\Delta|^2}{2\left|s_{12} . s_{21}\right|} \\\\ &.K=\frac{1-0.614^2-0.716^2+0.34^2}{2 \mid 0.046 \angle 65^{\circ} \times 2.18\langle 32.4^{\circ}|}=1.126\gt 1 \\ \end{aligned}\\ $

As $|\Delta k| \& k\gt1$ transistor is unconditionally stable of can be used for amplifier design.

Bilateral design

$ \left(\left|s_{12}\right|=0.046\lt1 \&\left|s_{12}\right| \neq 0\right) \\ $

We follow bilateral amplifier design for maximum possible gaín.

$ \begin{gathered} \\ B_1=1+\left|s_{11}\right|^2-\left|s_{22}\right|^2-|\Delta|^2=1+0.614-0.716^2-0.34^2=0.748 \\\\ B_2=1+\left|s_{22}\right|^2-\left|s_{11}\right|^2-|\Delta|^2=1+0.716^2-0.614^2-0.34^2=1.02 \\\\ c_1=s_{11}-\Delta s_{22}^*=0.614 L-167.4-\left(0.34 L 113 \times 0.716 L 83^{\circ}\right) \\\\ =0.37 L-169^{\circ} \\ \end{gathered} \\ $

$ \begin{aligned}\\ C_2=s_{22}-\Delta s_{11}^* &=0.716 \angle-83^{\circ}-\left(0.34 \angle 113^{\circ} \times 0.614 \angle 167.4^{\circ}\right) \\\\ &=0.5 \angle-84.39^{\circ} \\ \end{aligned} \\ $

$ \begin{aligned} \\ &\therefore \Gamma_{\mathrm{ms}}=\frac{B_1 \pm \sqrt{B_1^2-4\left|C_1\right|^2}}{2 C_1} \\\\ &\therefore \Gamma_{\mathrm{ms}}=\frac{0.748 \pm \sqrt{0.748^2-4(0.37)^2}}{2 \times 0.37 \angle-169^{\circ}} \\\\ &\therefore \Gamma_{\mathrm{ms}}=\frac{0.748 \pm 0.109}{0.74 \angle-169^{\circ}}=\frac{0.857}{0.74} .169^{\circ} \text { or } \frac{0.639}{0.74} \angle 169^{\circ} \\\\ &\therefore \Gamma_{\mathrm{ms}}=1.15 \angle 169^{\circ} \text { or } \Gamma_{\mathrm{ms}}=0.86 \angle 169^{\circ} \\\\ \end{aligned}\\ $

We choose $\Gamma_{\text {ms }}=0.86 \mathrm{~ \angle} 169^{\circ}$

$ \begin{aligned} \\ &\Gamma_{\mathrm{mL}}=\frac{B_2 \pm \sqrt{B_2^2-4\left|C_2\right|^2}}{2 C_2} \\\\ &\therefore \Gamma_{\mathrm{mL}}=\frac{1.02 \pm \sqrt{1.02^2-4 \times 0.5^2}}{2 \times 0.5 \angle-84.39}=\frac{1.02 \pm 0.2}{\angle-84.39} \\\\ &\Gamma_{\mathrm{mL}}=(1.02+0.2) \angle 84.39^{\circ} \text { or } \Gamma_{\mathrm{mL}}=(1.02-0.2) \angle 84.39^{\circ} \\\\ &\therefore \Gamma_{\mathrm{mL}}=1.22 \angle 84.39^{\circ} \text { or } \Gamma_{\mathrm{mL}}=0.82 \mathrm{~\angle} 84.39^{\circ} \\ \end{aligned} \\ $

We choose $\Gamma_{\mathrm{mL}}=0.82 \mathrm{~ \angle} 84.39^{\circ}$

$ \begin{aligned} \\ &\therefore \text { Bilateral maximum gain }=G=\left|\frac{S_{21}}{S_{12}}\right|\left[K-\sqrt{K^2-1}\right] \\\\ &\therefore G=\left|\frac{2.18 \angle 32^{\circ} .4}{0.046 L 65^{-0}}\right|\left[1.126-\sqrt{1.126^2-1}\right]=47.39 \times 0.6=28.83 \\\\ &\therefore G(d B)=10 \log (28.83)=14.59 \mathrm{~dB} \\ \end{aligned} \\ $

Design of matching circuit:

we plot $\Gamma_{m s}=0.86 L 169^{\circ}$ and $\Gamma_{\mathrm{mL}}=0.82 \mathrm{~\angle} 84.39^{\circ} \mathrm{g}$ match them with $z_0=50 \Omega$ using single short circuit parallel stub matching circuits.

Гms matching circuit:-

location of stub $=0.044 \lambda-0.016 \lambda=0.028 \lambda=d_1$

at point $1 \quad \bar{y}_1=1+j 3.25$

$\therefore$ locating $-j 3.25$ with $0.452 \lambda$ (point $l'$ )

$\therefore$ length of shout circuit parallel stub (from s.c. admittance point) $l_1=0.452 \lambda$

Гml matching circuit:-

location of stub $=0.452 \lambda-0.133 \lambda=0.319 \lambda=d_2$ at point $2 \quad \bar{y}_2=1-j 03.0$

$\therefore$ locating $+j 3.0$ with WTG $0.051 \lambda$ (point $\left.2^{\circ}\right)$

$\therefore$ length of shoat circuit parallels stub (from s.c. admittance point) $l_2=0.051 \lambda$