# Retractable Wireless Charging Windings for Inspection Robots
## 2 Principle and Modeling of WPT System  
### Magnetic Resonant Inductive Coupling Power Transfer  
The study is based on the foundation of inductive coupling power transmission and the symmetrical configuration of the SS topology magnetic resonance. The energy efficiency is primarily influenced by the load impedance, along with other operational factors, including frequency, tolerances of inductive and capacitive components, and coil alignment. The coupling coefficient (\(k\)) is equal to the ratio of the mutual inductance (\(M\)) to the square root of the product of primary coil self-inductance (\(L_{p}\)) and secondary coil (\(L_{s}\)). Figure 3 illustrates the circuit topology used for the modeling of the WPT system in the simulation. The parameters used in the simulation and experimental analysis are derived from this configuration.  
The power delivered to the load (\(P_{R_{L}}\)) is calculated using root mean square (rms) values of voltage and current, where \(V_{L}\) is the voltage across the load, \(R_{L}\) is the load resistance, and \(I_{s}\) represents the AC current in the secondary coil, as shown in Equation (1).  
\[P_{R_{L}}=\frac{|V_{L}|^{2}}{R_{L}}=|I_{s}|^{2}R_{L} \tag{1}\]  
The currents \(I_{p}\) and \(I_{s}\) can be determined through matrix computation as shown in Equation (3), using the impedance components defined in the \(Z\)-matrix (Equation (2)), where \(R_{p}\) and \(R_{s}\) represent the resistances on the primary and secondary sides, respectively, and \(V_{p}\) is the external source voltage.  
Figure 3: SS-compensated inductive coupling power transfer circuit model.  
Figure 2: (a) Octagon coils, (b) irregular shape coils.  
\[\left[Z\right]=\begin{bmatrix}R_{p}+j\omega L_{p}+\frac{1}{j\omega C_{p}}&j\omega M \\ j\omega M&R_{s}+R_{L}+j\omega L_{s}+\frac{1}{j\omega C_{s}}\end{bmatrix} \tag{2}\]  
The mutual impedance components satisfy the condition \(Z_{ps}=Z_{sp}\) due to the reciprocity of the inductive coupling network. Under resonance conditions, where the reactive components cancel out, this relationship simplifies as:  
\[\begin{bmatrix}R_{p}&j\omega M\\ j\omega M&R_{s}+R_{L}\end{bmatrix}\begin{bmatrix}I_{p}\\ I_{s}\end{bmatrix}=\begin{bmatrix}V_{p}\\ 0\end{bmatrix} \tag{3}\]  
The current at the primary side of the coil (\(I_{p}\)) is given in Equation (4), while \(V_{p}\) is given in Equation (5) as the following:  
\[I_{p}=\frac{j(R_{s}+R_{L})\cdot I_{s}}{\omega M} \tag{4}\]  
\[R_{p}\cdot I_{p}+j\omega M\cdot I_{s}=V_{p} \tag{5}\]  
Substituting Equation (4) into Equation (5):  
\[R_{p}\cdot\left(\frac{j(R_{s}+R_{L})\cdot I_{s}}{\omega M}\right)+j\omega M \cdot I_{s}=V_{p} \tag{6}\]  
Hence, \(I_{s}\) is obtained as follows:  
\[I_{s}=-V_{p}\left(\frac{j\omega M}{R_{p}(R_{s}+R_{L})+(\omega M)^{2}}\right) \tag{7}\]  
Energy efficiency, defined as the ratio of the output power across the load \(P_{R_{L}}\) to the input power delivered to the primary coil (\(P_{in,AC}\)), can be expressed as follows:  
\[\eta=\frac{P_{R_{L}}}{P_{in,AC}}=\frac{(\omega M)^{2}R_{L}}{(R_{s}+R_{L})(R_{p }(R_{s}+R_{L})+(\omega M)^{2})} \tag{8}\]  
The self and mutual inductances of the coils, along with their equivalent series resistances, have a direct effect on coupling efficiency and energy efficiency. This analysis relies on the mathematical modelling presented in [26].