## 3 Mathematical Modelling and Analysis
### System Sensitivity  
Figure 4 illustrates the frequency response of the output power with different duty cycles. The instantaneous real power transferred to the load is provided in Equation (7) and the average power in Equation (8). By reducing \(D\), the output will have a wider response that is reflected in lower sensitivity. The quality factor of the proposed system (\(Q\)) as expressed in Equation (9) decreases proportionally to \(D^{2}\), as shown in Figure 3. By lowering \(Q\), the sensitivity decreases and tuning at resonant frequency will be more obtainable practically. For a less sensitive system, we recommend using \(Q\leq 6\), which can be easily achieved by choosing \(D\leq 50\%\) for the chosen values of this design example. The recommended \(D\) will change depending on the situation and the system design parameters.  
\[P_{\text{load}}=\frac{\Big{(}\frac{C_{c}\ R\ V_{in}\ D\ s}{C_{c}\ L\ s^{2}+C_{c}\ R\ s+1}\Big{)}^{2}}{2\ R_{\text{load}}}, \tag{7}\]  
\[|P_{\text{load}}|=\frac{{C_{c}}^{2}\ D^{2}\ R^{2}\ V_{in}^{2}\ \omega^{2}}{2 \Big{(}R_{\text{load}}\ +\ C_{c}^{2}\ L^{2}\ R_{\text{load}}\ \omega^{6}\Big{)}}, \tag{8}\]  
\[Q=\frac{1}{R}\sqrt{\frac{L}{C_{c}}}=Q_{\text{conv}}\ D^{2}, \tag{9}\]  
where \(s\) is \(j\omega\), \(P_{\text{load}}\) is the real power delivered to the load for the CPT system with the DC-DC buck converter added, and \(Q_{\text{conv}}\) denotes the quality factor of the conventional system.  
Figure 3: Input voltage needed to deliver 10 W to the load and the change in the quality factor of the circuit \(Q\) with different duty cycles \(D\).  
By reducing the system sensitivity, the system's ability to transfer power becomes impervious to large parametric variations. Figure 5 shows the impacts on output power with changes in the capacitive coupling interface. For the conventional system, by changing the interface value by only 10% of its original value, no power will transfer to the load and resonance frequency will vary considerably due to the high \(Q\). Conversely, by using the DC-DC buck converter on the power receiving side, the system becomes immune to parameter changes and the ability to transfer the power increases even under extreme conditions. For instance, if \(D=10\%\) is used, the power can be transferred with almost the same amount for any increase in the interface value from 250% more to 55% less than the original interface value by using DC-DC buck converter as a part of the secondary side. As such, the misalignment issue is of less concern.  
By adding the buck converter, the \(Q\) value remained low for any value of the interface, as shown in Figure 6, which guarantees the system robustness. On the contrary, the previous system \(Q\) is  
Figure 4: The frequency response of the output power at different duty cycles \(D\).  
Figure 5: The tolerance of the output power with the change in the capacitive interface value using different duty cycles \(D\).  
susceptible to variations in the coupling interface, especially when the coupling capacitance is reduced to less than 0.5 nF.  
The load can be a parameter that affects the system by changes that may occur due to the change in the battery under charge at the DC level, for example. Figure 7 shows that the \(Q\) value is stable and low for a wide range of changes in the load, unlike the case without DC-DC buck converter added where the \(Q\) reaches to more than 100 for some load values.