## 4 Modified Topology with Harmonic Control  
The following section explores three strategies for incorporating harmonic tuning as shown in Figure 8.  
Introducing a series resonator tuned to the 2 nd and 3 rd harmonics before the load resistance \(R\). Transforming \(L_{0}\) and \(C_{0}\). Alternatively, a combination of both strategies. These approaches aim to improve harmonic suppression while preserving the advantages of Topology 2.2.1 in balancing efficiency and bandwidth response. For the transformation of \(L_{0}\) and \(C_{0}\), there are four mathematically possible cases. Of these, only two yield valid solutions, as outlined below:  
The inductor \(L_{0}\) and capacitor \(C_{0}\) can be converted to an equivalent form as shown in Figure 9. This equivalent form consists of a parallel \(LC\) tank circuit, which can be tuned to resonate at \(n\omega_{0}\) by applying Equation (42). \[1-n^{2}\omega_{0}^{2}L_{pn}C_{pn}=0 \tag{42}\]  
Since there are multiple unknowns, additional equations can be constructed by applying the harmonic shorting condition at \(m\), as given in (43). \[\text{Im}(Z_{\text{net}}(m\omega_{0}))=0 \tag{43}\]  
For most practical scenarios, considering harmonics up to the third order is sufficient [30]. With this, the following modifications are possible:  
1. An inductor can be converted into an equivalent circuit consisting of another inductor in series with a parallel \(LC\) resonant circuit. The net impedance shown by Figure 9a is given by Equation (44) \[Z_{\text{net}}(j\omega)=j\omega L_{an}+\frac{j\omega L_{pn}^{L}}{1-\omega^{2} L_{pn}^{L}C_{pn}^{L}}\] (44) Two cases are possible: 1. The parallel resonator is tuned to the 2 nd harmonic and the net impedance is shorted at the 3 rd harmonic. The different components are given by the following equations (refer to Appendix A.1 for the calculations): \[L_{a2}=\frac{3}{8}L_{0}\] (45) \[L_{p2}^{L}=\frac{15}{32}L_{0}\] (46)  
Figure 8: Possibilities of incorporating the harmonic tuning network. \[C_{p2}^{L}=\frac{8}{15\omega_{0}^{2}L_{0}}\] (47) (b) The parallel resonator is tuned to the 3 rd harmonic, and the net impedance is shorted to the 2 nd harmonic. This condition is not possible, as shown in Appendix A.1. 2. Similar to an inductor, a capacitor can also be converted to a parallel \(LC\) resonant circuit connected by capacitance in series, as shown in Figure 9b. In this condition, the net impedance is given by Equation (48). \[Z_{\text{net}}(j\omega)=\frac{1}{j\omega C_{an}}+\frac{j\omega L_{pn}^{C}}{1- \omega^{2}L_{pn}^{C}C_{pn}^{C}}\] (48) (a) The parallel resonator is tuned to the 2 nd harmonic and the net impedance is shorted at the 3 rd harmonic. This condition is also not possible as shown in Appendix A.2. 2. The parallel resonator is tuned to the 3 rd harmonic and the net impedance is shorted at the 2 nd harmonic. The equivalent parameters are given by Equations (49)-(51) (Appendix A.2 contains the calculations): Figure 9.0 (49) Figure 9.1 (50) Figure 9.2 (51)  
The comparative trade-off table evaluates various terminations, with performance metrics analyzed and presented in Figure 9.3 The terminologies used in this paper are as follows:  
1. 2.2.2: \(L_{0}\) and \(C_{0}\) with 2 nd-harmonic tuning. 2. 2.2.3: \(L_{0}\) and \(C_{0}\) with 3 rd-harmonic tuning. 3. 2.2.3: \(L_{0}\) and \(C_{0}\) with both 2 nd and 3 rd-harmonic tuning. 4. 2.2.1: \(L_{0}\) transformed as per Equations (45)-(47) with \(C_{0}\) unchanged. 5. 2.2.C: \(C_{0}\) transformed as per Equations (49)-(51) with \(L_{0}\) unchanged. 6. 2.2.13: Final topology with \(L_{0}\) transformed with \(C_{0}\) unchanged while also adding 3 rd-harmonic tuning before the load resistor, as shown in Figure 9.4  
Figure 9.5 Final topology. Figure 9.6 Transformation of components (**a**) \(L_{0}\) transform, and (**b**) \(C_{0}\) transform.