## 2 Circuit Analysis and Design
### Frequency Staggering Technique  
To achieve a flat gain over a wide frequency range, the center frequencies of three gain stages (M\({}_{1}\), M\({}_{2}\), and M\({}_{3}\)) are intentionally staggered. Through the interstage and output impedance matching described below, the peak-gain frequencies of the first, second, and third stages are tuned to 42, 25.5, and 30.5 GHz, respectively, as shown in Figure 6.  
Figure 4: Input matching network of the first gain stage. Figure 5: Loci of the input impedances from 26 to 40 GHz. The ac equivalent circuit of the amplifier is drawn in Figure 7a. For the first interstage matching between M\({}_{1}\) and M\({}_{2}\), the impedances seen looking into the output of M\({}_{1}\) (Z\({}_{\text{out1}}\)) and the input of M\({}_{2}\) (Z\({}_{\text{in2}}\)) are simulated as Z\({}_{\text{out1}}\) = 48.3 - \(j\)89.2 \(\Omega\) and Z\({}_{\text{in2}}\) = 14.2 - \(j\)71.9 \(\Omega\) at 42 GHz. For conjugate matching, Z\({}_{\text{in2}}\) is transformed to Z\({}_{\text{out1}}\) * by an L-section network consisting of L\({}_{3}\), L\({}_{4}\), and TL\({}_{1}\). As shown in Figure 7b, the series inductive components (L\({}_{4}\) and TL\({}_{1}\)) bring Z\({}_{\text{in2}}\) to Z\({}_{\text{in2},1}\), which is located on a constant conductance circle of Re{1/Z\({}_{\text{out1}}\) *}. Then, a shunt inductor (L\({}_{3}\)) transforms Z\({}_{\text{in2},1}\) to Z\({}_{\text{in2},2}\), which completes the conjugate matching to Z\({}_{\text{out1}}\) *. It should be noted that the value of L\({}_{3}\) determines the frequency at which the conjugate matching is satisfied. In Figure 7b, the locus of Z\({}_{\text{in2},2}\) at 42 GHz is depicted as L\({}_{3}\) decreases from infinity to 0.1 nH. According to the plot, L\({}_{3}\) is chosen as 0.8 nH to achieve conjugate matching at 42 GHz. The final matching point of Z\({}_{\text{in2},2}\) slightly deviates from Z\({}_{\text{out1}}\) * after fine optimization of the whole amplifier. The conjugate interstage matching between M\({}_{1}\) and M\({}_{2}\) at 42 GHz results in the gain peak of the first gain stage at the same frequency as that shown in Figure 6.  
The second interstage matching between M\({}_{2}\) and M\({}_{3}\) is performed in a similar way to the first interstage matching. However, the matching frequency is intentionally changed to 26 GHz. As shown in Figure 7c, the input impedance of M\({}_{3}\) (Z\({}_{\text{in3}}\)) is transformed to the conjugate of the output impedance of M\({}_{2}\) (Z\({}_{\text{out2}}\)). The center frequency of the matching is tuned to 26 GHz by choosing the value of a shunt inductor (L\({}_{5}\)) as 0.3 nH. It is noted that the second interstage matching is supposed to have the matching performance sensitive to the adjacent stages more than the first interstage matching. This is because the adjacent impedances are seen through C\({}_{\text{gd}}\) of M\({}_{2}\) and M\({}_{3}\), while C\({}_{\text{gd}}\) of M\({}_{1}\) is diminished by the neutralization inductor L\({}_{\text{neut}}\). Therefore, an extra shunt inductor (L\({}_{6}\)) is added as an additional degree of design freedom in the second interstage matching while minimizing the influence on the adjacent matching. The second interstage matching results in the gain peak of the second stage at 25.5 GHz, as observed in Figure 6.  
The output matching is fulfilled by L\({}_{8}\) and L\({}_{9}\) at 34 GHz, which leads to the peak gain of the final stage at 30.5 GHz shown in Figure 6. It is noted that there is a slight difference between the matching frequency and the peak gain frequency of each individual stage. This is because the gain and bandwidth of the whole amplifier were re-optimized after combining the individual stages. Figure 6: Gain of the three individual stages and the whole amplifier. Figure 7: Interstage matching: (**a**) AC equivalent circuit of the amplifier; (**b**) first interstage matching at 42 GHz; (**c**) second interstage matching at 26 GHz.