# A Wideband Power Amplifier in 65 nm CMOS Covering 25.8 GHz-36.9 GHz by Staggering Tuned MCRs
## 2 Analysis and Design
### Wideband Output Matching Network Design  
Wideband output matching network, which transfers the load impedance to \(Z_{opt}\) for the power stage in a wideband frequency range, is the key factor for large power  
Figure 1: (**a**) The whole three-stage wideband power amplifier. (**b**) The unit stage cell of the power amplifier. characteristics of a PA [23]. A symmetrical MCR could be used to achieve the broadband \(Z_{opt}\)[16]. In a symmetrical MCR, as shown in Figure 2,  
\[Q_{1}=Q_{2}=Q_{0} \tag{2}\]  
\[\omega_{1}=\omega_{2}=\omega_{0} \tag{3}\]  
in which  
\[Q_{0}=Q_{1}=Q_{2}=\omega_{1}R_{1}C_{1}=\omega_{2}R_{2}C_{2} \tag{4}\]  
\[\omega_{0}=\omega_{1,2}=\frac{1}{\sqrt{L_{1}C_{1}}}=\frac{1}{\sqrt{L_{2}C_{2}}} \tag{5}\]  
K is the coupling coefficient between the two inductors, as shown in Figure 2. In this work, the interest frequency band is from 26 GHz to 36 GHz. As the output power and power gain at low frequency is larger than that at high frequency, we choose 32 GHz rather than 31 GHz as the center operation frequency to calculate the parameters of the output-matching MCR. Based on the loadpull simulation, we choose \(Zopt=\) (26 + j*26) \(\Omega\) for the power stage at 32 GHz. The conjugate of Zopt is \(Zopt^{*}=\) (26 + j*26) \(\Omega\), which is equivalent to a resistor of \(R_{1}=\) 52 \(\Omega\) parallel with a capacitor of \(C_{11}=\) 95.65 fE, as shown in the lumped model of the output symmetrical MCR in Figure 3. The symmetrical MCR transfers the input impedance of the power combiner to \(Zopt\) for the power stage. The power combiner at the output, as shown in the dashed line on the top of Figure 1a, is designed to reactively transfer the load impedance to \(Z_{T}=\) (27.86 - j*25.96) \(\Omega\) for the output transformer by using microstrip transmission-line segments. In the output symmetrical MCR shown in Figure 3, \(C_{1}=C_{11}+C_{12}\), \(C_{2}=C_{21}+C_{22}\). \(C_{12}\) represents the extra added capacitor on the primary side if needed. \(C_{21}\) represents the equivalent paralleled output capacitance of the power combiner input impedance (\(Z_{T}\)), which is 89 fF in this work. \(C_{22}\) represents the extra added capacitor on the right side if needed. \(R_{2}\) represents the equivalent paralleled output resistor of the power combiner input impedance (\(Z_{T}\)), which is also 52 \(\Omega\) in this work. For a symmetrical MCR, the coupling factor k and Q factor satisfy the equations below under the no in-band ripple condition [16]:  
\[Q=\sqrt{\frac{-1+\sqrt{1+4*\left(\omega_{c}R_{1}C_{1}\right)^{4}}}{2}} \tag{6}\]  
\[k=\frac{1}{\sqrt{1+Q^{2}}} \tag{7}\]  
\[L_{1}=\frac{{R_{1}}^{2}C_{1}}{Q^{2}} \tag{8}\]  
\[Q_{1}=Q_{2}=Q_{0} \tag{2}\]0  
\[Q_{1}=Q_{2}=Q_{0} \tag{2}\]1 The lumped model of a MCR. \[Q_{1}=Q_{2}=Q_{0} \tag{2}\]2  
in which \(\omega_{c}\) is the center operation frequency with 32 GHz, as mentioned above. Assuming \(C_{12}\) is 19 fF (could be changed in a later design stage if needed), then the total left capacitance \(C_{1}=C_{11}+C_{12}\) is calculated to be 114.69 fF. An initial assumption of \(C_{12}\) of 19 fF is based on the fact that there is usually parasitic capacitance of the transformer [16], and extra added capacitor \(C_{12}\) can be used as an additional degree of freedom to facilitate the design of MCR. \(C_{12}\) should be as small as possible because a larger \(C_{12}\) (leading to a larger \(C_{1}\)) results in decreased GBW of the MCR [16]. According to Equations (6) and (7), \(Q=1.0116\), and k = 0.703 are obtained. According to Equations (8)-(10), we obtain that \(L_{2}=L_{1}=303.26\) pH, \(C_{2}=C_{1}=114.69\) fF, and \(C_{22}=25.69\) fF at 32 GHz, respectively. Some iterations of the calculation and design tuning of the values of Q, k, \(L_{1}\), \(L_{2}\), and \(C_{12}\) are necessary to make these parameters satisfy the no in-band ripple condition equations. Figure 4 presents (a) the 3D view of the output transformer (TF4 in Figure 1) in the symmetrical MCR and (b) its layer structure.