Question 12.3: Design an amplifier for maximum gain at 4 GHz using single-s...

In practice, scattering parameters are usually provided by the manufacturer over a wide frequency range, and it is prudent to check stability over the entire range. Here we have limited the data to three frequencies to illustrate the point with-out undue computational burden. Using (12.28) and (12.29) to calculate K and Δ from the scattering parameters at each frequency in the above table gives  the following results:

K = \frac{1-\left|S_{11}\right| ^{2} – \left|S_{22} \right|^{2} + \left|\Delta \right| ^{2} }{2 \left|S_{12} S_{21} \right| } \gt 1 ,

We see that K > 1 and |Δ| < 1 at 4 and 5 GHz, so the transistor is unconditionally stable at these frequencies, but it is only conditionally stable at 3 GHz. We can proceed with the design at 4 GHz, but should check stability at 3 GHz after we find the matching networks (which determine Γ_{S} and Γ_{L}).

For maximum gain, we should design the matching sections for a conjugate match to the transistor. Thus, Γ_{S} = Γ_{in}^{∗} and Γ_{L} = Γ_{out}^{∗}, and Γ_{S}, Γ_{L} can be deter-mined from (12.40):

\Gamma _{S}=\frac{B_{1}\pm \sqrt{ B_{1}^{2}- 4\left|C_{1}\right|^{2} }}{2C_{1}}=0.872\angle 123^\circ ,

\Gamma _{L}=\frac{B_{2}\pm \sqrt{ B_{2}^{2}- 4\left|C_{2}\right|^{2} }}{2C_{2}}=0.876\angle 61^\circ ,

The effective gain factors of (12.16) can be calculated as

G_{S}=\frac{1}{1-\left|\Gamma _{S}\right|^{2} }=4.17 = 6.20  dB ,

G_{0}=\left|S_{21}\right|^{2}=6.76 = 8.30  dB,

G_{L}=\frac{1-\left|\Gamma _{L}\right|^{2} }{\left|1-S_{22}\Gamma _{L}\right|^{2} } = 1.67=2.22  dB,

Then the overall transducer gain is
G_{Tmax} = 6.20 + 8.30 + 2.22 = 16.7  dB.

The matching networks can easily be determined using the Smith chart. For the input matching section, first plot Γ_{S}, as shown in Figure 12.7a. The impedance, Z_{S}, represented by this reflection coefficient is the impedance seen looking into the matching section toward the source impedance, Z_{0}. Thus, the matching section must transform Z_{0} to the impedance Z_{S}. There are several ways of doing this, but we will use an open-circuited shunt stub followed by a length of line. We convert to the normalized admittance y_{s}, and work backward (toward the load on the Smith chart) to find that a line of length 0.120λ will bring us to the 1 + jb circle. Then we see that the required stub admittance is + j3.5, for an open-circuited stub length of 0.206λ. A similar procedure gives a line length of 0.206λ and a stub length of 0.206λ for the output matching circuit.

The final amplifier circuit is shown in Figure 12.7b. This circuit only shows the RF components; the amplifier will also require bias circuitry. The return loss and gain were calculated using a CAD package, interpolating the necessary scattering parameters from the data given above. The results are plotted in Figure 12.7c, and show the expected gain of 16.7 dB at 4 GHz, with a very good return loss. The bandwidth where the gain drops by 1 dB is about 2.5%. With regard to the potential instability at 3 GHz, we leave it to the reader to show that the designed matching sections present source and load impedances that lie within the stable regions of the appropriate stability circles. Note that the matching sections are frequency dependent, so the impedances and reflection coefficients are different at 3 GHz than their design values at 4 GHz. The fact that CAD simulation did not show any indication of instability over the frequency range of 3–5 GHz is evidence that the circuit is stable over this frequency range.