## 2 Design of Dual-Band Matching Networks  
### Design of Dual Band Output Matching Networks  
The design of the dual-band output matching network is crucial to achieving high output power across dual frequency bands. To achieve this goal, the dual-band output matching network should transform the 50 ohm impedance to the optimum load impedance of the power amplifier at both frequency bands. Additionally, the dual-band output matching network is required to maintain minimal loss at both frequency bands. This paper employs the reflection coefficient and insertion loss of the dual-band output matching network as guidance.  
The optimal load impedance of the PA is given by a load resistance in parallel with an equivalent negative capacitance [17] and determined through load-pull simulations, as  
\[Z_{opt}(\omega)=R_{opt}/\left(\frac{-1}{j\omega C_{out}}\right) \tag{1}\]  
In this design, the simplified output matching schematic is shown in Figure 1. The reconfigurable output matching network comprises a fixed transformer and switching capacitors \(C_{suss}\), which can adjust the load impedance seen in the matching network to achieve the optimum impedance in the two bands of interest. Ideally, the reflection coefficient \(\Gamma_{ins}=0\) and \(IL_{output}=0\). However, due to the Bode-Fano limit [25] and transformer parasitic resistance, the ideal matching network is unachievable. Thus, in practice, we set \(|\Gamma_{ins}|\leq-20\) dB and \(|IL_{output}|\leq 1.5\) dB as the target.  
The \(Z\)-parameter matrix for the transformer is as follows [26]:  
\[Z=\begin{bmatrix}R_{p}+j\omega L_{p}&-j\omega M\\ j\omega M&-(R_{s}+j\omega L_{s})\end{bmatrix} \tag{2}\]  
where  
\[M=k\sqrt{L_{p}L_{s}} \tag{3}\]  
and \(k\) represents the coupling factor between primary and secondary coils.  
Therefore, the equivalent load impedance for the power amplifier (PA) can be derived as Equation (4).  
\[Z_{load}=\Bigg{\{}R_{p}+\frac{\left(\omega M\right)^{2}\left(R_{s}+R_{eq} \right)}{\left(R_{s}+R_{eq}\right)^{2}+\left(\omega L_{s}+X_{eq}\right)^{2}} \Bigg{\}}+j\Bigg{\{}\omega L_{p}-\frac{\left(\omega M\right)^{2}\left(\omega L _{s}+X_{eq}\right)}{\left(R_{s}+R_{eq}\right)^{2}+\left(\omega L_{s}+X_{eq} \right)^{2}}\Bigg{\}} \tag{4}\]  
where  
\[\text{R}_{eq}=\frac{R_{L}}{1+\left(\omega R_{L}C_{s\text{sus}}\right)^{2}} \tag{5}\]  
\[\text{X}_{eq}=\frac{-{\omega R_{L}}^{2}C_{s\text{sus}}}{1+\left(\omega R_{L} C_{s\text{sus}}\right)^{2}} \tag{6}\]  
So \(\left|\Gamma_{ins}\right|\) and \(lL\) can be derived as  
\[\left|\Gamma_{ins}\right|=20 log\left|\frac{Z_{ins}-R_{opt}}{Z_{ins}+R_{opt}} \right| \tag{7}\]  
\[{lL_{output}=20 log\left|\frac{j\omega M}{\left(R_{s}+R_{eq}\right)+j\left( \omega L_{s}+X_{eq}\right)}\cdot\frac{C_{out}-Z_{ins}}{\left(Z_{ins}+R_{opt} \right)C_{out}}\cdot\sqrt{\frac{R_{opt}}{R_{L}}}\right|} \tag{8}\]  
where  
\[Z_{ins}=C_{out}//Z_{load} \tag{9}\]  
It can be seen that there are three main parameters (\(L_{p},\ L_{s}\) and \(C_{s\text{sus}}\)) to be determined. Considering the matching target, we calculate the minimum value of the \(\left|\Gamma_{ins}\right|\) and \(lL\) by sweeping values of the main parameters to find the optimum value range of \(L_{p}\) and \(L_{s}\)  
Figure 1: Simplified schematic of output matching circuit.  
where the value of \(C_{\text{sites}}\) is chosen to minimize the reflection coefficient and insertion loss. During this process, the quality factors of transformer windings are both assumed to be 12 at 2.45 G and 14 at 5.5 G, while the coupling factor of transformer windings is assumed to be 0.75, which are reasonable values for practical transformers in CMOS process. In addition, the value range of \(C_{\text{sites}}\) is limited to 0-3 pF, and the value of \(C_{\text{sites}}\) above 3 pF is regarded as unacceptable and is abandoned because of the low quality factor and large off-state capacitance (\(C_{off}\)). So far, the values of the components in the proposed dual-band output matching network have been preliminarily determined.