# Nonlinear Analysis and Closed-Form Solution for Overhead Line Magnetic Energy Harvester Behavior
## 3 Review of the Froelich Equation  
The Froelich equation is a first-order rational polynomial function that uses only two parameters, \(a\) and \(b\), and provides a wide-range approximation to the non-hysteretic \(B(H)\) characteristic of a magnetic core material:  
\[B(H)=\frac{aH}{b+|H|}=\begin{cases}\frac{aH}{b+|H|}&H>0\\ \frac{aH}{b-H}&H<0\end{cases} \tag{1}\]  
here, \(a\) is given in [T] units, whereas \(b\) stands in [At/m]. The inverse function, \(H(B)\), can be derived as:  
\[H(B)=\frac{bB}{a-|B|}=\begin{cases}\frac{bB}{a-B}&B>0\\ \frac{bB}{a+B}&B<0\end{cases} \tag{2}\]  
In practice, the parameters \(a\) and \(b\) can be established by curve-fitting to the experimental data. Measured BH characteristics of a silicon steel core sample (EILOR MAGNETICCORES) vs. the fitted BH curve, drawn by using the extracted Froelich parameters, are shown in Figure 2. Here, \(a\) = 2.05 [T] and \(b\) = 109.4 [A/m].  
It can be seen that Froelich equation parameter a represents the saturation flux density of the core, \(B_{s}\). Yet, this value is unattainable in practice since it suggests imposing an infinite magnetic field intensity on the magnetic material:  
\[B_{s}=\lim_{H\to\infty}\frac{aH}{b+H}=a. \tag{3}\]  
For the sake of a brief comparison, one possible approach to attaining a linearized BH curve is suggested here. Other approaches are also possible. Consider a piecewise linear approximation of the BH characteristic that includes the (\(B^{\prime}\), \(H^{\prime}\)) operating point where the flux density at \(B^{\prime}=B_{s}/2=a/2\). Substituting in (2) yields the magnetic field intensity \(H^{\prime}=b\). Thus, the equivalent magnetic permeability within the linear segment of the core can be approximated as the ratio:  
\[\mu_{eq}=\frac{B^{\prime}}{H^{\prime}}=\frac{a}{2b}, \tag{4}\]  
The considerations above also suggest that the saturation magnetic field intensity of the piecewise-linear model, \(H_{s}\), is:  
\[H_{s}=\frac{B_{s}}{\mu_{eq}}=2b \tag{5}\]  
Simulation is a helpful tool that can facilitate a detailed study of OLMEHs. Although PSIM requires a tedious trial-and-error procedure to adjust the magnetic core parameters to fit the experimental BH curve, once completed, this proved to be a well-spent effort. As shown in Figure 3a, the comparison of the simulated BH curve (obtained by PSIM v. 9.1) stands in good agreement with the fitted Froelich approximation (1).  
A triple comparison of the approximated BH curve is shown in Figure 3b. The Froelich approximation, here with the mentioned parameters, vs. the \(B=\frac{2}{\pi}B_{s}atan(\frac{H}{\beta})\) approximation used by [23], here \(B_{s}\) = 2.05 and \(\beta\) = 126, and vs. the piecewise-linear approximation used by [26], is calculated by (3) and (5). As expected, the comparison reveals a significant error of the piecewise-linear approach in the vicinity of the saturation knee, above \(B_{s}/2\), whereas the Froelich and \(atan(x)\) approximations stand in excellent agreement with each other. As is shown in this paper, the advantage of the Froelich approximation is that it  
Figure 2: Comparison of the measured BH curve of the silicon steel core sample (EILOR MAGNETIC CORES) to its Froelich approximation.  
lends itself to analytical treatise and results in an analytical solution, whereas using the \(atan(x)\) approximation necessitates the application of numerical analysis [23].