# Nonlinear Analysis and Closed-Form Solution for Overhead Line Magnetic Energy Harvester Behavior
## 7 Output Current and Power Considerations  
In DCM, CVL charging commences at \(t_{1}\) and terminates at \(t_{2}\). Hence, the time interval \([t_{1},t_{2}]\) is defined as the power transfer window. The charging current at the rectifier's output can be found as the per-half-cycle average of the difference between the secondary current, \(i_{2}(t)\), and the magnetizing currents, \(i_{m}(t)\), drawn by the OLMEH's magnetizing inductance:  
\[I_{b}=\frac{1}{T/2}\int_{t_{1}}^{t_{2}}(i_{2}(t)-i_{m}(t))dt \tag{22}\]  
Figure 7: OLMEH’s simulated waveforms in: DCM due to low current \(I_{1}=50\) A rms, \(V_{b}=23\) V (**a**); DCM due to high voltage \(I_{1}=100\) A rms, \(V_{b}=25\) V (**b**); DCM-CCM boundary \(I_{1}=100\) A rms, \(V_{b}=22\) V (**c**); CCM due to high current \(I_{1}=125\) A rms, \(V_{b}=23\) V (**d**); CCM due to low voltage \(I_{1}=100\) A rms, \(V_{b}=22\) V (**e**). Calculation of the average secondary current component available for charging, which is the first term in (22), is straightforward:  
\[I_{2 ch}=\frac{1}{T/2}\int_{t_{1}}^{t_{2}}i_{2}(t)dt=\frac{1}{T/2}\int_{t_{1}}^{t_{ 2}}(I_{2m}\sin\omega t)dt=\frac{I_{2m}}{\pi}(\cos\omega t_{1}-\cos\omega t_{2}) \tag{23}\]  
However, finding the per-half-cycle average of the magnetizing current, which is the second term in (22), requires additional consideration. Since the flux density changes sign at \(t=t_{Z}\), see Figure 5, two subintervals \([t_{1},t_{Z}]\) and \([t_{Z},t_{2}]\) each with appropriate models should be considered to produce the correct result. Therefore:  
\[I_{mag}=\frac{1}{T/2}\int_{t_{1}}^{t_{2}}i_{m}(t)dt=\frac{1}{T/2}\int_{t_{1}}^{ t_{2}}i_{m}(t)dt+\frac{1}{T/2}\int_{t_{Z}}^{t_{2}}i_{m}(t)dt=I_{mag1}+I_{mag2} \tag{24}\]  
The average of the negative (recycled) term of the magnetizing current, \(I_{mag1}\), during the time interval \([t_{1},t_{Z}]\), see Figure 5, should be calculated according to the negative flux density model, see (2), with \(B_{1}\) as the initial condition:  
\[I_{mag1}=\frac{1}{T/2}\int_{t_{1}}^{t_{2}}i_{m}(t)dt=\frac{1}{T/2}\int_{t_{1}} ^{t_{2}}\left(\frac{l_{c}}{N}\frac{b\left[\frac{V_{b}}{NA_{c}}(t-t_{1})+B_{1} \right]}{a+\left[\frac{V_{b}}{NA_{c}}(t-t_{1})+B_{1}\right]}\right)dt \tag{25}\]  
The (recharging) magnetizing current term, \(I_{mag2}\), should be calculated within the time interval \([t_{Z},\ t_{2}]\), see Figure 5, according to the positive flux density model, (2), with \(B=0\) as the initial condition:  
\[I_{mag2}=\frac{1}{T/2}\int_{t_{Z}}^{t_{2}}i_{m}(t)dt=\frac{1}{T/2}\int_{t_{Z} }^{t_{2}}\left(\frac{l_{c}}{N}\frac{b\left[\frac{V_{b}}{NA_{c}}(t-t_{Z}) \right]}{a-\left[\frac{V_{b}}{NA_{c}}(t-t_{Z})\right]}\right)dt \tag{26}\]  
Skipping the details of the derivation of (25) and (26), which is quite tedious, substitution into (24) yields the per half-cycle average magnetizing current:  
\[I_{mag}=\frac{\omega}{\pi}\frac{bl_{c}}{N}(2t_{Z}-t_{2}-t_{1})-\frac{\omega} {\pi}\frac{bl_{c}}{N}\ \frac{aNA_{c}}{V_{b}}ln\left(\left(1-\frac{V_{b}(t_{2}-t_{Z})}{ aNA_{c}})\left(1+\frac{V_{b}(t_{Z}-t_{1})}{aNA_{c}(1+B_{1}/a)}\right)\right) \tag{27}\]  
Applying (23) and (27), the OLMEH's per half-cycle average output power in the DCM mode can be found as:  
\[P_{o}=V_{b}\big{(}I_{2 ch}-I_{mag}\big{)} \tag{28}\]  
While in CCM, the magnetizing current is symmetric and has a zero average during half a cycle; see Figure 5. This implies that the energy stored in the core is recycled to the CVL within the half-cycle (note that in DCM part of the energy is recycled to the system during State 1, not charging the CVL). Therefore, the average output power in the CCM mode is simply:  
\[P_{o}=V_{b}I_{2 ch}\approx\frac{2\sqrt{2}}{\pi N}V_{b}I_{rms} \tag{29}\]  
The output power in CCM mode (29) is a linearly increasing function of the CVL voltage, \(V_{b}\), whereas in the DCM mode, the output power (28) appears as a bell-shaped function with a distinct extremum. This means that an increase of the CVL voltage does not necessarily bring an increase in the output power but depends on the operating point.