## 3 Simulation  
The mutual inductance represented in Figure 3 is the operating principle of all electrical components interacting with the magnetic field. In this episode, all the components from the primary (transmitter) coil interact with the components of the secondary (receiving) coil, as presented in the block diagram of the WPT charging circuit (Figure 1). The mutual inductance of coils was taken into consideration to analyze the WPT circuit by applying Equation (1). \[M=k\sqrt{LpLs},\ \ 0\leq\ k\leq 1 \tag{1}\]  
where \(Lp\) and \(Ls\) represent the equivalent inductance in the primary (transmitter) and secondary (receiver) coils, respectively; \(k\) is the coupling coefficient that measures the magnetic coupling between the two coils; and \(M\) is the mutual inductance. The magnetic flux (\(\emptyset\)) common to both the transmitting and receiving coils are represented by \(k\emptyset_{\text{T}}\), where \(\emptyset_{\text{T}}\) is the total magnetic flux at a \(0^{0}\) angle. The magnetic flux is only presented in one  
Figure 1: Block diagram for the wireless power transfer prototype circuit. Figure 2: Magnetic resonance coupling circuit. coil, it is not included in the energy transfer and is represented by \((1-k)\diameter_{\text{T}}\). The overall inductance \(L\) can be calculated using the following equation. \[L=\frac{n^{2}\times\mu_{0}\times A}{L_{e}} \tag{2}\]  
where \(n\) is the number of turns in the coil, \(\mu_{0}\) is the permeability of free space as measured in the unit of Henry (H). The parameter \"\(A\)\" is the cross-sectional area of the coil measured in square meters (m\({}^{2}\)) and \(L_{e}\) is the length of coil measured in the unit of a meter (m). The coupling coefficient, \(k\), can be calculated from Equation (1) as follows. \[k=\frac{M}{\sqrt{LpLs}} \tag{3}\]  
The possible conclusion for the value of \(k\) satisfying \(0\leq k\leq\)1 is that when the value is less than 0.5, the coils are loosely coupled; when the value is greater than 0.5, the coils are tightly coupled; and when the value equals 1.0, the coils are magnetically coupled. It was necessary to separate the coupled circuit and to include the dependent voltage sources to each coupling circuit in further analyses. The dependent voltage sources are replacing the coupling between the two circuits. The current running through the secondary coil (receiving), \(I_{2}\), can be calculated as follows. \[I_{2}=\frac{-j\omega MI_{1}}{Z_{s}} \tag{4}\]  
where \(I_{1}\) is the current running through the primary coil (transmitting), \(j\omega MI_{2}\) is the dependent voltage of the primary coil, \(j\omega MI_{1}\) is the dependent voltage of the secondary coil, and \(Z_{s}\) is the reflective impedance that considers \(L_{s}\), as given by Equations (5)-(8). The parameter \(C_{s}\) represents the capacitance at secondary coils; \(R_{L}\) represents the output voltage load passed to the rectifier, storage capacitor, and/or voltage regulator to charge the battery of a mobile device; \(Rs\) represents the resistance at the secondary coil; and \(Z_{Ri}\) is the overall reflective impedance of both coils. \[Z_{s}=R_{L}+\frac{1}{j\omega C_{s}}+j\omega L_{s}+R_{s} \tag{5}\]  
\[j\omega MI_{2}=\frac{\omega^{2}M^{2}}{Z_{s}}+I_{1} \tag{6}\]  
\[Z_{Ri}=\frac{\left(\omega M\right)^{2}}{Z_{s}} \tag{7}\]  
\[Z_{Ri}=\frac{\left[\omega k\sqrt{LpLs}\right]^{2}}{Z_{s}} \tag{8}\]  
Figure 1:0 Mutual inductance of the primary and secondary coils. The value at \(I_{1}\) is at a maximum when \(Z_{s}\) is at a minimum. The total resistance (\(R_{L}\) ) of the circuit can be obtained based on the total load (\(R_{L}\)) and resistance at the secondary coil (\(R_{s}\)) according to the following equations with \(V_{in}=\omega_{o}\), where \(Cp\) represents the capacitance at the primary coil. Figure 1:1  
Figure 1:2  
Figure 1:3  
where the parameter \(P\) is the power given by the Equations (12) and (13).