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## Q. 13.3

TRANSISTOR OSCILLATOR DESIGN

Design a transistor oscillator at 4 GHz using a GaAs MESFET in a common gate configuration, with a 5 nH inductor in series with the gate to increase the instability. Choose a load network to match to a 50 Ω load, and an appropriate terminating network at the input to the transistor. The scattering parameters of the transistor in a common source configuration are ($Z_{0} = 50 Ω$) $S_{11} = 0.72∠ −116^{◦},S_{12} = 0.03∠57^{◦}, S_{21} = 2.60∠76^{◦},$ and $S_{22} = 0.73∠ −54^{◦}.$

## Verified Solution

The first step is to convert the common source scattering parameters to the scattering parameters that apply to the transistor in a common gate configuration with a series inductor. (See Figure 13.9a.) This is most easily done using a microwave This is most easily done using a CAD package. The new scattering parameters are

$S^{′}_{11} = 2.18 ∠−35^{◦}$,

$S^{′}_{12} = 1.26∠18^{◦}$,

$S^{′}_{21} = 2.75∠96^{◦}$,

$S^{′}_{22 }= 0.52∠155^{◦}$.

Note that |$S^{′}_{11}$| is significantly greater than $|S_{11}|$, which suggests that the configuration of Figure 13.9a is more unstable than the common source configuration.

Calculating the output stability circle ($Γ_{L}$ plane) parameters from (11.25) gives

$C_{L}=\frac{(S^{\prime }_{22}-\Delta^{\prime }S^{\prime *}_{22})^{*}}{\mid S^{\prime }_{22} \mid ^{2}-\mid \Delta^{\prime }\mid ^{2} } =1.08\angle 33^{\circ }$

$R_{L}=\left|\frac{S^{\prime }_{12} S^{\prime }_{21}}{\mid S^{\prime }_{22} \mid ^{2} -\mid \Delta^{\prime }\mid ^{2}} \right| =0.665,$

Since$\mid S^{′}_{11}\mid$= 2.18 > 1, the stable region is inside this circle, as shown in the Smith chart in Figure 13.9b. There is a great amount of freedom in our choice for $Γ_{L}$, but one objective is to make |$Γ_{in}$| large. We therefore try several values of $Γ_{L}$ located on the opposite side of the chart from the stability circle, and select $Γ_{L}= 0.59∠−104^{◦}.$ Then we can design a single-stub matching network to convert a 50 Ω load to $Z_{L} =20 − j35 Ω$, as shown in Figure 13.9a. For the given value of  $Γ_{L}$, we calculate  $Γ_{in}$ as

$\Gamma _{in}=S^{\prime }_{11}+\frac{S^{\prime }_{12} S^{\prime }_{21}\Gamma _{L} }{1-S^{\prime }_{22}\Gamma _{L}}=3.96\angle-2.4^{\circ }$

or $Z_{in }= −84 − j1.9 Ω.$Then, from (13.31), we find $Z_{S}$ as

$R_S=\frac{-R_{in}}{3 }$              (13.31a)

$X_S=-X_{in}$                  (13.31b)

$Z_{s}=\frac{-R_{in}}{3} -jX_{in}=28 + 1.9 \Omega$

Using $R_{in}/3$ should ensure enough instability for the startup of oscillation. The easiest way to implement the impedance $Z_{S}$ is to use a 90 Ω load with a short length of line, as shown in the figure. It is likely that the steady-state oscillation frequency will differ from 4 GHz because of the nonlinearity of the transistor parameters.  