## 2 The Applied Methodology
### Analysis: Power Transfer Window  
The following definitions are introduced: \(N\) is the number of OLMEH secondary turns, \(I_{1m}\) is the line (primary) current amplitude, and \(I_{2m}=I_{1m}/N\) is the amplitude of the secondary current. As usual, \(f=1/T=\omega/2\pi\) is the line frequency. Examining the waveforms in Figure 2 reveals that the magnetic flux density swings from its initial negative saturation value \(B(t_{1})=-B_{s}\) towards the positive saturation value \(B(t_{2})=+B_{s}\). The transition is linear since it results from the application of the constant voltage \(V_{b}\) to the secondary winding; see Figures 3b and 4c. Therefore  
\[B(t_{2})=+B_{s}=\frac{1}{NA_{c}}\int_{t_{1}}^{t_{2}}V_{b}dt-B_{s}, \tag{4}\]  
whence  
\[B_{s}=\frac{V_{b}}{2 NA_{c}}(t_{2}-t_{1}), \tag{5}\]  
The practical saturation level that the core can reach, \(B_{s}\), is presumed to be lower than the heavy saturation flux density level, \(B_{s}<B_{sat}\), defined by (2). The time interval that appears in (5)  
\[T_{w}=(t_{2}-t_{1}), \tag{6}\]  
is defined as the power transfer window. To allow the charging current pulse to terminate naturally, the pulse width, \(t_{pw}\), set by the controller, see Figure 2, has to be appropriately matched so that \(t_{pw}>T_{w}\). Switching the MOSFET on, see Figure 3d, decreases the resistance of the conductive path, minimizes the conduction losses during the SC states, and so increases the OLMEH's conversion efficiency. Combining (5) with the Froelich Equation (1) yields the magnetic field intensity attained by the core at the peak of saturation:  
\[H(t_{2})=H(B_{s})=\frac{bB_{s}}{a-B_{s}}=\frac{b\frac{V_{b}}{2N\lambda_{c}}(t _{2}-t_{1})}{a-\frac{V_{s}}{2 NA_{c}}(t_{2}-t_{1})}=\frac{N}{l_{c}}i_{m}(t_{2}), \tag{7}\]  
The right-hand side of (7) was attained by [PERSON]'s law. Further examination of the waveforms in Figure 2 suggests that charging State 2 is terminated when the magnetizing current equals the secondary current:  
\[i_{m}(t_{2})=i_{2}(t_{2})=I_{2m}\text{sin}(\omega t_{2}), \tag{8}\]Hence, substituting (8) into (7) yields  
\[\frac{b\frac{V_{b}}{2N\Lambda_{c}}(t_{2}-t_{1})}{a-\frac{V_{b}}{2N\Lambda_{c}}(t_ {2}-t_{1})}=\frac{N}{l_{c}}I_{2m}\text{sin}(\omega t_{2}), \tag{9}\]  
Since \(t_{1}\) is set by the controller and assumed to be known, (9) is an equation of a single variable \(t_{2}\). Yet, (9) is difficult to solve. A straightforward approach to obtain the solution of (9) is to proceed with numerical analysis. An alternative approach offered here is obtaining an analytical solution through approximation of the sine function on the right-hand side of (9) by a rational function, considering that the range of interest lies somewhere in \(\frac{2}{3}\pi<\omega t<\frac{5}{6}\pi\). As a compromise between complexity and accuracy, it is suggested to approximate the falling segment of the sine function using  
\[\text{sin}(\omega t)\approx\frac{K_{1}(\pi-\omega t)}{K_{2}-\omega t}, \tag{10}\]  
With \(K_{1}\) = 3.232 and \(K_{2}\) = 6.003, (10) agrees with the \(\text{sin}(\omega t)\) function at 120\({}^{\circ}\), 150\({}^{\circ}\), and 180\({}^{\circ}\), thus providing good accuracy within the range of interest. A comparison plot of the \(\text{sin}(\omega t)\) function to the chosen approximation function (10) is shown in Figure 5.  
Substitution of (10) into (9) yields  
\[\frac{l_{c}}{N}\frac{b\left[\frac{V_{b}}{N\Lambda_{c}}(t_{2}-t_{2})\right]}{a -\left[\frac{V_{b}}{N\Lambda_{c}}(t_{2}-t_{Z})\right]}=I_{2m}\frac{K_{1}(\pi- \omega t_{2})}{K_{2}-\omega t_{2}}, \tag{11}\]  
Introducing the definitions of the normalized time  
\[t_{N}=\frac{\omega}{\pi}t, \tag{12}\]  
the normalized secondary current  
\[B_{s}=\frac{V_{b}}{2 NA_{c}}(t_{2}-t_{1}), \tag{5}\]0  
the normalized load voltage  
\[B_{s}=\frac{V_{b}}{2 NA_{c}}(t_{2}-t_{1}), \tag{5}\]1  
\[B_{s}=\frac{V_{b}}{2 NA_{c}}(t_{2}-t_{1}), \tag{5}\]2 Comparison plot of the \(\text{sin}(\omega t)\) function to the chosen approximation function (10) (for \(K_{1}\) = 3.232 and \(K_{2}\) = 6.003).