# A Wideband Power Amplifier in 65 nm CMOS Covering 25.8 GHz-36.9 GHz by Staggering Tuned MCRs
## 2 Analysis and Design
### Interstage Matching Network Design  
For the interstage matching network between the power stage and driver stage 2, the target is to achieve broadband matching between driver stage 2 output impedance and power stage input impedance. In transformer matching, we uses a simplified lumped T-model, shown in Figure 12, to estimate the size/parameters of the transformers [30]. For a transformer-based MCR, both the bandwidth and in-band ripple increase with higher coupling factor k, which has been investigated previously in references [16; 31; 32; 33]. Meanwhile, the higher k results in lower insertion loss of the transformer. Considering that the TF1 and TF2 in MCR1 and MCR2 (shown in Figure 1) are used to adjust the small-signal bandwidth performance of the power amplifier, a higher k is preferred for TF3 (transformer 3 in MCR3 shown in Figure 1) regardless of the higher in-band ripple. Usually, the gate impedance has a higher Q factor than the output impedance of the transistor, and the higher Q results in increased gain ripple [16]. Adding paralleled resistors at the gate is an alternative way to reduce the Q factor and to extend the bandwidth [16; 34]. Two 150 \(\Omega\) resistors (Rg at the power stage) are added at the gate of the power stage. Then, the input impedance of the power stage after adding resistors at the gate is ZM1 = \(16-\)j*43.6. The output impedance of driver stage 2 is ZM2 = \(33.56-\)j*62.78. The equivalent T matching of TF3 and the specific structure of MCR3 are shown in Figure 13a,b, respectively. In the MCR3, the output impedance of diver stage 2 (ZM2) is equivalent to a paralleled tank including a resistor \(R_{1}\) and a capacitor \(C_{1}\). The input impedance of the power stage (ZM1) is equivalent to a paralleled tank including a resistor \(R_{2}\) and a capacitor \(C_{2}\). The ZM2 is affected by the input-matching condition of driver stage 2 due to the non-zero S12 of the transistors. Considering that the neutralization technique is used for all three stages to cancel out the effect of Cgd of the transistors, the S12 for all three stages is small enough so that the ZM2 of driver stage 2 is not severely affected by the previous matching condition. Also, simulation iterations can be set to reset the value of ZM2. The TF3 is used to match ZM1 and ZM2. According to reference [35], when the two resonators of an MCR satisfy the condition of \(\omega_{1}=\omega_{2}\), the peak Z21 of an asymmetrical MCR is almost the same as that of a symmetrical MCR, and the mismatch of \(\omega_{1}\) and \(\omega_{2}\) will deteriorate the insertion loss, especially at a higher frequency. In MCR3, \(Q_{1}\) = 1.87 and \(Q_{2}\) = 2.725 are obtained at 35 GHz. With a lower \(Q_{1}\) and \(\omega_{1}\) = \(\omega_{2}\), the MCR3 achieve a better flat response and the mismatch of \(Q_{1}\) and \(Q_{2}\) could be accepted [35]. With the help of TF3, the matching between ZM1 and ZM2 has lots of matching trajectories, only including the T-model or T-model with extra capacitors, as shown in Figure 14a. Considering that the gain and output power of the transistors decrease as the operating frequency increases, the matching trajectory of TF3 is designed to achieve a good match at 35 GHz, and also \(\omega_{1}\) = \(\omega_{2}\) is set at 35 GHz, as shown in Figure 14b. Thus, a good match between ZM1 and ZM2 at 35 GHz and \(\  
Figure 14: (**a**) Different TF matching trajectories from ZM1 to ZM2 on Smith Chart and (**b**) the chosen trajectory that both satisfies matching and \(\omega_{1}=\omega_{2}\). Figure 12: Lumped model of a transformer for matching. Figure 13: (**a**) The equivalent T matching of TF3 in MCR3 and (**b**) the specific structure of MCR3. For the interstage matching network between driver stage 1 and driver stage 2, which is part of MCR2, it aims to achieve a high-pass form Z21 to compensate for the gain drop in MCR3 and MCR4 at high frequency.