# Nonlinear Analysis and Closed-Form Solution for Overhead Line Magnetic Energy Harvester Behavior
## 5 Analysis of the Discontinuous Current Mode  
An inspection of the OLMEH's key simulated waveforms, presented in Figure 5, reveals that the rectifier's input current, \(i_{in}\), vanishes for a substantial time interval. Henceforth, this operational mode is dubbed the discontinuous current mode (DCM). While in DCM, each line half-cycle is comprised of two distinct time subintervals that emerge according to the conduction state of the rectifier; see Figure 5. These are designated as State 1 and State 2. The equivalent circuits of State 1 and State 2 are shown in Figure 4b and Figure 4c, respectively. Figure 4: Simplified OLMEH model (**a**) and its DCM equivalent circuits during the positive half-cycle with: rectifier OFF (State 1) (**b**); rectifier ON (State 2) (**c**). Figure 5: OLMEH’s key simulated waveforms in the discontinuous mode. State 1: Commences at the instance the rectifiers are cut off at zero current; thus, the CVL receives no current from OLMEH, \(i_{b}=\mid i_{in}\mid=0\). This occurs because the magnetizing current steers all the secondary current away from the rectifier, \(i_{m}(t)=i_{2}(t)\); see Figure 5. Hence, for the low power line angle, \(\omega t\), values, see Figure 5, the expression for the magnetizing current can be approximated by:  
\[i_{m}(t)=i_{2}(t)=I_{2m}sin\omega t\approx I_{2m}\omega t, \tag{6}\]  
where \(I_{1m}\) is the peak line current, \(I_{2m}=I_{1m}/N\) is the peak secondary current, and \(N\) is the secondary number of turns. During State 1, the magnetic field intensity in the core is dictated by the line current constrain:  
\[H(t)\approx\frac{I_{1m}\omega t}{l_{c}}=\frac{NI_{2m}\omega t}{l_{c}}<0 \tag{7}\]  
where \(l_{c}\) is the magnetic path length of the OLMEH's core. Therefore, according to (7) the line current generates the magnetic field, \(H\), that, according to (1), forces the magnetic flux density, \(B\), to decrease. Hence, the core is taken out of deep saturation towards a shallower saturation; see Figure 5. Flux density variation induces a voltage in the OLMEH's secondary winding, which appears across the rectifier's AC terminals as:  
\[v_{in}(t)=NA_{c}\frac{dB}{dt}=NA_{c}\frac{d}{dt}\frac{aH}{b-H}=\frac{abl_{c}A_ {c}N^{2}\omega I_{2m}}{\left(b_{c}-NI_{2m}\omega t\right)^{2}} \tag{8}\]  
Here, \(A_{c}\) is the cross-section of the harvester's core. Note, that for negative, \(H<0\), values, the Froelich Equation (1) with the \((b-H)\) term in the denominator is used in (8). Equation (8) describes the concave segment seen in the \(V_{in}\) voltage waveform in Figure 5.  
State 1 terminates at the instant, \(t_{1}\), when the voltage developed by the OLMEH's winding equals the CVL voltage, thus turning the rectifiers on. The time instant, \(t_{1}\), can be found by applying the condition, \(v_{in}(t_{1})=V_{b}\) to (8), whence  
\[t_{1}=\frac{bl_{c}}{\omega NI_{2m}}-\sqrt{\frac{abA_{c}l_{c}}{\omega I_{2m}V_ {b}}} \tag{9}\]  
Note that (9) returns a negative result, \(t_{1}<0\), because State 1 terminates before the secondary current zero crossing, designated as time reference \(t=0\); see Figure 4:0  
Combining (1), (7), and (9) yields the value of the flux density, \(B_{1}=B(t_{1})\), at which State 1 is terminated:  
Figure 4:1  
State 2 commences at \(t=t_{1+}\). Here, the rectifiers' conduction imposes a constant CVL voltage, \(V_{b}\), across the harvester's secondary. As a result, the magnetic flux density in the core starts increasing linearly with time; see Figure 4:2  
Figure 4:3  
The flux density crosses zero at the instant:  
Figure 4:4It is worthwhile noting that during State 2 the winding voltage constrains the flux density; hence, the magnetic field intensity follows according to (2).