## 2 Design of Dual-Band Matching Networks
### Implementation of Dual-Band Input Matching Network  
Different from the output matching, input matching is more sensitive to inductor and capacitor values. Small variations can have a significant impact on the reflection coefficient, making it challenging to achieve good matching within dual bands at high output power conditions. To address this issue, this article proposes a novel reconfigurable input matching circuit. The schematic is shown in Figure 8, and the input matching circuit consists of the switching capacitors and reconfigurable transformers. Applying a reconfigurable transformer significantly enhances dual-band input matching under high output power conditions. At the high-frequency band (5 GHz), the inductance values of the reconfigurable transformer decrease, ensuring effective input matching. Conversely, at the low-frequency band (2.4 GHz), the inductance values increase, facilitating optimal input matching at the lower band. Figure 6: On-resistance and Coff of switch transistor versus size. Figure 7: Achieved insertion loss of the output matching network. (a) 2.4 GHz. (b) 5 GHz. The simplified schematic of the reconfigurable transformers is shown in Figure 9:  
The reconfigurable transformer is composed of the main transformer windings (\(L_{p}\), \(L_{s}\)), tuning inductor \(L_{t}\), and tuning capacitor \(C_{t}\). The tuning inductor \(L_{t}\) is magnetically coupled to the main transformer with coupling coefficients \(k_{p}\) and \(k_{s}\), which is also connected directly with the tuning capacitor \(C_{t}\). The reconfigurable transformer is based on a tunable inductor, which is shown in Figure 10. The \(R_{t}\) is the parasitic resistance of the entire tuning branch, which includes the parasitic resistances of both the resonant inductor \(L_{t}\) and the switching capacitor \(C_{t}\). When the main inductor \(L_{1}\) inputs a signal, the induced current \(I_{t}\) through \(L_{t}\) and \(C_{t}\) will generate an induced magnetic field. Thus, the total magnetic flux of the main inductor is influenced by the induced magnetic field, and so the equivalent inductance of the main inductor will be changed and can be derived as Equation (8). \[L_{eq}=L_{1}+\frac{L_{1}L_{t}C_{t}k_{t}^{2}\omega^{2}\big{(}1-L_{t}C_{t}\omega ^{2}\big{)}}{\big{(}L_{t}C_{t}\omega^{2}-1\big{)}^{2}+R_{t}^{2}C_{t}^{2}\omega ^{2}} \tag{10}\]  
\[R_{eq}=R_{1}+\frac{C_{t}^{2}L_{1}L_{t}R_{t}k_{t}^{2}\omega^{4}}{\big{(}L_{t}C_ {t}\omega^{2}-1\big{)}^{2}+R_{t}^{2}C_{t}^{2}\omega^{2}} \tag{11}\]  
\[Q_{eq}=\frac{\big{(}1-C_{t}^{2}L_{t}^{2}\big{(}k_{t}^{2}-1\big{)}\omega^{4}+C _{t}\big{(}R_{t}^{2}C_{t}+L_{t}\big{(}k_{t}^{2}-2\big{)}\big{)}\omega^{2}\big{)} \omega L_{1}}{C_{t}^{2}L_{t}\big{(}L_{1}R_{t}k_{t}^{2}+R_{1}L_{t}\big{)}\omega^ {4}+C_{t}\big{(}R_{t}^{2}C_{t}-2L_{t}\big{)}\omega^{2}+R_{1}} \tag{12}\]  
Figure 8: Reconfigurable input matching circuit. Figure 9: Simplified schematic of the reconfigurable transformer. Meanwhile, due to magnetic coupling, the quality factor of the main inductor is also manipulated and is derived as Equation (10). Figure 11a illustrates the variations in main inductance value and \(Q\) factor when the tuning capacitor \(C_{t}\) changes. Figure 7b shows the impact of \(R_{t}\) on the main quality factor. It can be seen that the values of the main inductor (\(L_{1}\)) and quality factor (\(Q\)) are significantly influenced by the tuning capacitor, while quality factor has low relativity to \(R_{t}\), indicating that the tuning capacitor should be within an appropriate range to prevent a severe reduction in \(Q\). On the other hand, changing the tuning capacitor within the appropriate range allows for changes in inductance with minimal impact on the \(Q\) factor. The reconfigurable transformer in this work is formed by one tunable inductor \(L_{t}\). Thus, the values of the main reconfigurable transformer windings are influenced by \(L_{t}\), \(C_{t}\) with coupling factor \(k_{p}\), \(k_{s}\).