Nonlinear Analysis and Closed-Form Solution for Overhead Line Magnetic Energy Harvester Behavior

Featured Application

Magnetic energy harvesting of an overhead line.

Abstract

Recently, much attention has been given to the development of various energy harvesting technologies to power remote electronic sensors, data loggers, and communicators that can be installed on smart grid systems. Magnetic energy harvesting is, perhaps, the most straightforward way to capture a significant amount of power from a current-carrying overhead line. Since the harvester is expected to have a small size, the high currents of the distribution system easily saturate its magnetic core. As a result, the operation of the magnetic harvester is highly nonlinear and makes precise analytical modeling difficult. The operation of an overhead line magnetic energy harvester (OLMEH) generating significant DC power output into a constant voltage load was investigated in this paper. The analysis method was based on the Froelich equation to analytically model the nonlinearity of the core’s BH characteristic. The main findings of this piecewise nonlinear analysis include a closed-form solution that accounts for both the core and rectifiers’ nonlinearities and provides an accurate prediction of OLMEH transfer window length, output current, and harvested power. Continuous and discontinuous operational modes are identified and the mode transition boundary is obtained quantitatively. The theoretical investigation was concluded by comparison with a computer simulation and also verified by the experimental results of a laboratory prototype harvester. A good agreement was found.

1. Introduction

Advanced ultra-low power VLSI and wireless communications techniques spurred the development of various sensor networks and wearable electronics [1,2,3]. Such devices can be used to great advantage in various domestic, commercial, vehicular, military, and health applications.

A common concern of these technologies is their dependence on power sources. Batteries, biofuel cells, and supercapacitors have to be periodically replaced or recharged, posing certain application difficulties, whereas energy sources like solar cells, thermoelectric generators, piezoelectric and triboelectric generators, and various energy scavengers/harvesters can allow for a long service life while avoiding the need for maintenance and battery replacement. Energy harvesting is a particularly attractive approach to developing miniature self-powered systems designed to operate in remote locations or locations with restricted access [4].

The power distribution network crisscrossing the urban environment is a readily available, predictable, and reliable energy source. Therefore, recent studies have investigated the possibility of energy scavenging from power lines [5].

Thermal energy harvesting exploits the temperature difference developed by the current-carrying conductor and the ambient. DC electric current can be attained by applying thermo-electric generators clamped to the cable [6,7].

Energy can also be harvested from the electric field of the power line [8,9,10,11]. This suggests creating a capacitor-like structure by wrapping a conductive surface around the (isolated) line conductor and taking advantage of the displacement currents. Alternatively, the harvester can be designed to operate at a safe standoff distance from a high-voltage line. Since the line voltage is well stabilized, the electric field harvesters have the advantage of a nearly constant output current.

Magnetic energy harvesters (MEHs) rely on the magnetic field generated by a current-carrying conductor. Several types of MEHs were developed [4] that exploit different physical principles of interaction with the magnetic field such as variable reluctance [12,13], magnetostrictive [14,15], and ferrofluid [16,17,18] generators. The magnetic field generated by the line current can also be converted into the mechanical vibration of a miniature resonant cantilever. The output power can then be obtained by using a piezoelectric [19,20] or inductive [21] transducer mounted on the free end of the cantilever beam. Yet, the mentioned harvester types can develop only a tiny amount of power in the range of up to a few mW. This may be sufficient to operate miniature wireless sensors; however, in case a substantially higher power is needed, a different approach is needed. A current transformer (CT) type MEH is, perhaps, the most promising concept that can allow the harvesting of a significant amount of power from a power line. Two configurations are mainly considered in the current literature: the “stick on” or “freestanding” harvester, which is mounted in the close vicinity of a bus bar, and the “clamp on”-type harvester, which is configured to “embrace” the current-carrying conductor. Although the output power of MEHs varies with the line current amplitude, the average daily current profile can be measured and the system can be designed to its minimum expected value. Thus, MEHs can be considered as a quite dependable energy source.

Traditionally, CTs are used as sensors to accurately measure high AC currents. The most desirable feature of a CT is its high linearity in a wide range of line currents. Thus, the preferable operating conditions of a CT’s magnetic core are in the linear region of its BH characteristic. To help attain such a condition, the CT is usually operated under a nearly short-circuited secondary. The construction of an overhead line magnetic energy harvester (OLMEH) is quite similar to that of a CT; yet, since the task of an OLMEH is often charging a battery, the operating conditions of an OLMEH are quite different. Figure 1 shows an OLMEH as a part of a simple power processing scheme operating with a constant voltage load (CVL). The circuit employs a full-wave bridge rectifier to perform the AC to DC conversion. The rectifier’s action imposes a square wave voltage (whose amplitude equals the CVL voltage) across the OLMEH’s secondary winding. This may drive the magnetic core into the saturation region of the BH curve. While saturated, the OLMEH can provide no output current for a considerable portion of a line cycle. Thus, predicting the output power of OLMEHs is a challenging problem.

Furthermore, the increased power comes at the expense of increased volume and weight. To minimize the mentioned physical parameters while attaining the desired output power, OLMEHs’ core material, geometry, size, and number of turns have to be properly selected. Therefore, the aim of this article is to reexamine the workings of OLMEHs and obtain a complete analytical description of device operation, which can further serve as a theoretical foundation to develop a reliable set of engineering design rules. The applied methodology is as follows.

2. The Applied Methodology

This paper is concerned with the analysis of an overhead line energy harvester (OLMEH) illustrated in Figure 1. The OLMEH in Figure 1 is of a clamped-on CT type, the concept for which can be found in [22,23,24,25,26]. An analysis of the OLMEH’s behavior, while modeling the nonlinear BH characteristic using the atan(x) function, was described in [23]. The choice of such an approximation function results in analytical complications and necessitates the application of numerical solutions to attain the final results. Alternatively, ref. [26] opted to model the BH curve by a simple piecewise linear function for the sake of deriving crude analytical results. In the low flux density region, the linear approximation allows easy analytical description of the device; however, in the high flux density region, where the core approaches the saturation knee, the linear approach is inadequate.

One can argue that, in our age, a researcher assisted with modern computational tools can crack any nonlinear problem. To attain the solution, it is sufficient to just apply simulation or to properly introduce basic laws and system descriptions to the computer and then obtain the desired numerical results and plots. Yet, there is a clear and undisputable advantage to the old school approach of deriving an analytical solution to the problem. The latter can show the explicit effect each variable has on the end result and so provide a deeper understanding of the physical process. The analytical solutions can also facilitate the derivation of straightforward designing rules. Hence, an extra effort was put in to revisit the earlier findings, particularly of [23]. This paper aims to develop a complete analytical model that faithfully describes OLMEH operation. The proposed modeling approach is based on the Froelich equation, which represents the nonlinearity of the BH curve by a first-order rational polynomial function. The choice of the Froelich equation as the approximation function is imperative. The advantage of the undertaken approach over the numerical [23] is that the Froelich approximation allows deriving closed-form analytical solutions that consider the complex behavior of the magnetic core, yet provides meaningful analytical results suited for engineering applications. In summary, the contribution of this paper over the earlier counterpart is the analytical solutions for the: (a) power transfer window; (b) charging output power, and (c) prediction of the operational modes boundary.

The rest of this paper is organized as follows: Section 3 reviews the modeling of the BH curve by the Froelich equation. Section 4 presents basic assumptions, preliminary simulation waveforms, and OLMEH models. Section 5 presents the analysis of the discontinuous current mode. The continuous current mode of OLMEHs is described in Section 6. Section 7 discusses the boundary conditions for continuous to discontinuous current mode change. Analytical solutions for output current and power are derived in Section 8. Verification and experimental results are given in Section 9. Finally, conclusions are given.

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 $B\left(H\right)$ characteristic of a magnetic core material:

$$B\left(H\right)={\displaystyle \frac{aH}{b+\left|H\right|}}=\left\{\begin{array}{c}{\displaystyle \frac{aH}{b+H}} H>0\\ {\displaystyle \frac{aH}{b-H}} H<0\end{array}\right.,$$

(1)

here, a is given in [T] units, whereas b stands in [At/m]. The inverse function, H(B), can be derived as:

$$H\left(B\right)={\displaystyle \frac{bB}{a-\left|B\right|}}=\left\{\begin{array}{c}{\displaystyle \frac{bB}{a-B}} B>0\\ {\displaystyle \frac{bB}{a+B}} B<0\end{array}\right.$$

(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 MAGNETIC CORES) 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:

$$B_s=\underset{H\to \infty }{\lim }{\displaystyle \frac{aH}{b+H}}=a.$$

(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′, H′) 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:

$${\mu }_{eq}={\displaystyle \frac{B^{\prime }}{H^{\prime }}}={\displaystyle \frac{a}{2b}},$$

(4)

The considerations above also suggest that the saturation magnetic field intensity of the piecewise-linear model, $H_s$, is:

$$H_s={\displaystyle \frac{B_s}{{\mu }_{eq}}}=2b$$

(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 $B={\displaystyle \frac{2}{\pi }}{B}_{s}atan({\displaystyle \frac{H}{\beta }})$ approximation used by [23], here B_s = 2.05 and β = 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 lends itself to analytical treatise and results in an analytical solution, whereas using the atan(x) approximation necessitates the application of numerical analysis [23].

4. The Basic Assumptions and the Equivalent Circuit

As simple as the OLMEH’s power processing circuit in Figure 1 may look, the nonlinear properties of the magnetic core and the rectifiers make an exact analysis prohibitive. Therefore, the following simplifying assumptions were adopted:

(1)

Since clamping the OLMEH on the line does not change the line current, and since the output power drawn by the OLMEH is negligible in comparison to the power carried by the high voltage transmission line, the OLMEH can be considered as fed by a sinusoidal current source.

(2)

The hysteresis phenomenon is neglected and the nonlinearity of the core is modeled by the Froelich Equations (1) and (2).

(3)

A simplified cantilever model is adopted to represent the OLMEH magnetic circuit with a nonlinear magnetizing inductance, L_m, placed on the secondary winding, whereas the leakage inductance and the winding resistance are considered negligible.

(4)

Also, ideal rectifiers with no voltage drop and no reverse recovery are assumed.

The assumptions above lead to the OLMEH model illustrated in Figure 4a. The model was simulated in PSIM v 9.1. The key simulated waveforms shown in Figure 5 reveal the intricate details of OLMEH operation.

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.

State 1: Commences at the instance the rectifiers are cut off at zero current; thus, the CVL receives no current from OLMEH, i_b = |i_in| = 0. This occurs because the magnetizing current steers all the secondary current away from the rectifier, $i_m\left(t\right)=i_2\left(t\right)$; 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:

$$i_m\left(t\right)=i_2\left(t\right)=I_{2m}sin\omega t\approx I_{2m}\omega t,$$

(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:

$$H\left(t\right)\approx {\displaystyle \frac{I_{1m}\omega t}{l_c}}={\displaystyle \frac{N{I}_{2m}\omega t}{l_c}}<0$$

(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:

$$v_{in}\left(t\right)=N{A}_c{\displaystyle \frac{dB}{dt}}=N{A}_c{\displaystyle \frac{d}{dt}}{\displaystyle \frac{aH}{b-H}}={\displaystyle \frac{ab{l}_c{A}_c{N}^2\omega I_{2m}}{{\left(b{l}_c-N{I}_{2m}\omega t\right)}^2}}$$

(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, t₁, 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, $v_{in}\left(t_1\right)=V_b$ to (8), whence

$$t_1={\displaystyle \frac{b{l}_c}{\omega N{I}_{2m}}}-\sqrt{{\displaystyle \frac{ab{A}_c{l}_c}{\omega I_{2m}{V}_b}}}$$

(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 5.

Combining (1), (7), and (9) yields the value of the flux density, $B_1=B\left(t_1\right)$, at which State 1 is terminated:

$$B_1=B\left(t_1\right)\approx {\displaystyle \frac{a\frac{N{I}_{2m}\omega t_1}{l_c}}{b-\frac{N{I}_{2m}\omega t_1}{l_c}}}$$

(10)

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 5:

$$B\left(t\right)={\displaystyle \frac{1}{N{A}_c}}{{\displaystyle \int }}_{t_1}^t{V}_{b}dt+B_1={\displaystyle \frac{V_b}{N{A}_c}}\left(t-t_1\right)+B_1$$

(11)

The flux density crosses zero at the instant:

$$t_Z=t_1-{\displaystyle \frac{N{A}_c}{V_b}}{B}_1$$

(12)

It is worthwhile noting that during State 2 the winding voltage constrains the flux density; hence, the magnetic field intensity follows according to (2). By Ampere’s law, the rising magnetic field intensity generates the magnetizing current:

$${\left.i_m\left(t\right)\right|}_{t>t_Z}={\displaystyle \frac{l_c}{N}}H\left(t\right)={\displaystyle \frac{l_c}{N}}{\displaystyle \frac{b\left[\frac{V_b}{N{A}_c}\left(t-t_Z\right)\right]}{a-\left[\frac{V_b}{N{A}_c}\left(t-t_Z\right)\right]}}$$

(13)

State 2 is terminated at $t_2$, see Figure 5, when the rectifier’s input current is reduced to zero:

$$i_{in}\left(t_2\right)=i_2\left(t_2\right)-i_m\left(t_2\right)=0$$

(14)

The Condition (14) can be rewritten using (13):

$$i_m\left(t_2\right)={\displaystyle \frac{l_c}{N}}{\displaystyle \frac{b\left[\frac{V_b}{N{A}_c}\left(t_2-t_Z\right)\right]}{a-\left[\frac{V_b}{N{A}_c}\left(t_2-t_Z\right)\right]}}=I_{2m}sin\omega t_2$$

(15)

The solution for the State 2 termination instant, $t_2$, can be found by solving (15) numerically, whereas the analytical solution of (15) can be obtained by approximating the sine function on the right-hand side. Approximation is expected to represent a sinusoidal segment in the range of interest which, according to Figure 5, is located on the falling slope of the sine function. The first that comes to mind is to apply a truncated Taylor series. However, using the high-order polynomial makes the analytic solution of (15) an impossible task. Therefore, several options may be considered to approximate the sine function in the range ${\displaystyle \frac{2}{3}}\pi <\omega t<{\displaystyle \frac{5}{6}}\pi$. The simple linear approximation of the type $\left(\pi -\omega t\right)$ cannot provide sufficient accuracy within the desired range. A downgoing parabola of the type ${\displaystyle \frac{4}{{\pi }^2}}\omega t\left(\pi -\omega t\right)$ is another candidate that can provide acceptable accuracy in the full range of $0<\omega t<\pi$. When substituted in (15), this will result in a cubic equation, which can be solved, however, with some difficulties. As a compromise between complexity, accuracy, and range, an approximation based on a “mirrored Froelich equation” is suggested here to approximate the falling segment of the sine function as:

$$sin\omega t\approx {\displaystyle \frac{K_1\left(\pi -\omega t\right)}{K_2-\omega t}}$$

(16)

with K₁ = 3.232 and K₂ = 6.003, (16) agrees with the $sin\left(\omega t\right)$ function at 120°, 150°, and 180°, thus providing good accuracy within the range of interest. A comparison plot of the mentioned approximation functions is shown in Figure 6.

Substitution of (16) into (15) yields:

$${\displaystyle \frac{l_c}{N}}{\displaystyle \frac{b\left[\frac{V_b}{N{A}_c}\left(t_2-t_Z\right)\right]}{a-\left[\frac{V_b}{N{A}_c}\left(t_2-t_Z\right)\right]}}=I_{2m}{\displaystyle \frac{K_1\left(\pi -\omega t_2\right)}{K_2-\omega t_2}}$$

(17)

As shown in Appendix A, the solution for (17), i.e., State 2 termination instant, $t_2$, is quite tedious. The result can be written using the normalized solution, $t_{2N}$, (A21), as:

$$t_2={\displaystyle \frac{T}{2}}{t}_{2N}$$

(18)

6. Continuous–Discontinuous Mode Boundary

The discontinuous current mode (DCM) regime described above is the result of a relatively high volt per turn applied across the harvester’s winding that brings the core to saturation before the end of a half-cycle. In DCM, the instant the rapidly rising magnetizing current intercepts the secondary current the rectifiers are cut off at zero current at $t_2<T/2$; see Figure 5. The rectifiers resume conduction at the instant the voltage induced in the winding (due to the varying magnetic field generated by the line current) equals the CVL voltage; see (8). However, for lower values of the applied voltage, the build-up of the magnetizing current is slower and the interception point occurs at the instant, $t_2$, at which the induced voltage is already of a sufficient magnitude to initiate the immediate rectifier’s conduction (yet, in the reversed polarity). Thus, the dead time in the rectifiers’ input current, i_in, see Figure 5, vanishes. As the result, for lower CVL voltages the rectifier can conduct for the entire half-cycle. Therefore, this operational regime is dubbed the continuous current mode (CCM). Since in CCM the rectifiers’ conduction lasts for the entire half-cycle and the operation is symmetrical, the magnetic flux density in the core becomes a true triangular waveform. Simulated waveforms of an OLMEH at the CCM-DCM boundary are shown in Figure 7c. Here, the applied CVL voltage, $V_b^{*}$, brings the positive peak value of the magnetic flux density infinitesimally close to $-B_1$ (recall that B1 is negative). Applying the CCM condition $t_2-t_1=T/2$ to (11) yields the DCM-CCM boundary condition:

$${\displaystyle \frac{V_b^{*}}{N{A}_c}}\left({\displaystyle \frac{T}{2}}\right)+2B_1=0$$

(19)

Substitution of (9) and (10) into (19) and further manipulation gives:

$$V_b^{*}={\displaystyle \frac{2\omega aN{A}_c}{{\pi }^2}}\left[\pi +{\displaystyle \frac{b{l}_c}{I_{1m}}}-\sqrt{2\pi {\displaystyle \frac{b{l}_c}{I_{1m}}}-{\left({\displaystyle \frac{b{l}_c}{I_{1m}}}\right)}^2}\right]$$

(20)

However, under a constant voltage load condition the CCM-DCM mode transition is determined by the line current, which can be derived from (20):

$$I_{1m}^{*}={\displaystyle \frac{4\omega abN{A}_c{l}_c{V}_b}{{\left(\pi V_b-2\omega aN{A}_c\right)}^2}}$$

(21)

The mode changes under varying CVL voltage and line current conditions are illustrated in Figure 7. In the given simulation example, the calculated DCM-CCM boundary occurs at $V_b^{*}=23.16 V$ for $I_1^{*}=100 A$ rms; see Figure 7c. DCM prevails in the low-line current range, $I_1<I_1^{*}$, as in Figure 7a or for a high voltage range $V_b>V_b^{*}$ as in Figure 7b, whereas CCM prevails in the high line current range, $I_1>I_1^{*}$, as in Figure 7d or for a low voltage range $V_b<V_b^{*}$ as in Figure 7e.

7. Output Current and Power Considerations

In DCM, CVL charging commences at $t_1$ and terminates at $t_2$. Hence, the time interval $\left[t_1,t_2\right]$ 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, $i_2\left(t\right)$, and the magnetizing currents, $i_m\left(t\right)$, drawn by the OLMEH’s magnetizing inductance:

$$I_b={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_1}^{t_2}\left(i_2\left(t\right)-i_m\left(t\right)\right)dt$$

(22)

Calculation of the average secondary current component available for charging, which is the first term in (22), is straightforward:

$$I_{2ch}={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_1}^{t_2}{i}_2\left(t\right)dt={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_1}^{t_2}\left(I_{2m}\sin \omega t\right)dt={\displaystyle \frac{I_{2m}}{\pi }}\left(\cos \omega t_1-\cos \omega t_2\right)$$

(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 $\left[t_1,t_Z\right]$ and $\left[t_Z,t_2\right]$ each with appropriate models should be considered to produce the correct result. Therefore:

$$I_{mag}={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_1}^{t_2}{i}_m\left(t\right)dt={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_1}^{t_Z}{i}_m\left(t\right)dt+{\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_Z}^{t_2}{i}_m\left(t\right)dt=I_{mag1}+I_{mag2}$$

(24)

The average of the negative (recycled) term of the magnetizing current, $I_{mag1},$ during the time interval $\left[t_1,t_Z\right]$, see Figure 5, should be calculated according to the negative flux density model, see (2), with $B_1$ as the initial condition:

$$I_{mag1}={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_1}^{t_Z}{i}_m\left(t\right)dt={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_1}^{t_Z}\left({\displaystyle \frac{l_c}{N}}{\displaystyle \frac{b\left[\frac{V_b}{N{A}_c}\left(t-t_1\right)+B_1\right]}{a+\left[\frac{V_b}{N{A}_c}\left(t-t_1\right)+B_1\right]}}\right)dt$$

(25)

The (recharging) magnetizing current term, $I_{mag2}$, should be calculated within the time interval $\left[t_Z, t_2\right]$, see Figure 5, according to the positive flux density model, (2), with $B=0$ as the initial condition:

$$I_{mag2}={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_Z}^{t_2}{i}_m\left(t\right)dt={\displaystyle \frac{1}{T/2}}{{\displaystyle \int }}_{t_Z}^{t_2}\left({\displaystyle \frac{l_c}{N}}{\displaystyle \frac{b\left[\frac{V_b}{N{A}_c}\left(t-t_Z\right)\right]}{a-\left[\frac{V_b}{N{A}_c}\left(t-t_Z\right)\right]}}\right)dt$$

(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:

$$I_{mag}={\displaystyle \frac{\omega }{\pi }}{\displaystyle \frac{b{l}_c}{N}}\left(2t_Z-t_2-t_1\right)-{\displaystyle \frac{\omega }{\pi }}{\displaystyle \frac{b{l}_c}{N}} {\displaystyle \frac{aN{A}_c}{V_b}}ln\left(\left(1-{\displaystyle \frac{V_b\left(t_2-t_Z\right)}{aN{A}_c}}\right)\left(1+{\displaystyle \frac{V_b\left(t_Z-t_1\right)}{aN{A}_c\left(1+B_1/a\right)}}\right)\right)$$

(27)

Applying (23) and (27), the OLMEH’s per half-cycle average output power in the DCM mode can be found as:

$$P_o=V_b\left(I_{2ch}-I_{mag}\right)$$

(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:

$$P_o=V_b{I}_{2ch}\approx {\displaystyle \frac{2\sqrt{2}}{\pi N}}{V}_b{I}_{1rms}$$

(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. Hence, the OLMEH designer should properly select the OLMEH parameters while aiming at the maximum power point. OLMEH design issues are out of scope of this paper and will be examined elsewhere.

8. Verification and Experimental Results

Verification of the theoretical predictions was conducted first by simulation. The theoretical and simulated results for key parameters were found to be in excellent agreement; see

Table 1. Comparison of the simulated vs. the calculated key parameters at different peak line current levels and at a load voltage of V_b = 24 V.

I1pk [A]	t1Sim [mS]	t1Calc [mS]	t2Sim [mS]	t2Calc [mS]	B1Sim [T]	B1Calc [T]	POSim [W]	POCalc [W]
50	−1.35	−1.43	7.05	7.07	−1.25	−1.29	12.87	13.08
100	−1.15	−1.19	8.23	8.27	−1.51	−1.52	33.05	33.21
200	−0.92	−0.93	9.05	9.11	−1.66	−1.67	73.25	73.34

and

Table 2. Comparison of the simulated vs. the calculated borderline voltage V_b* for different turns number N, at line current I₁ = 100 A rms.

N	40		50		60		70
	Calc	Sim	Calc	Sim	Calc	Sim	Calc	Sim
	23.16	23.25	28.95	29.5	34.75	35.2	40.54	41.14

.

An experimental laboratory prototype OLMEH shown in Figure 1 was constructed by stacking together three pairs of 10H10 C-cores made of silicon steel (EILOR MAGNETIC CORES). The magnetic path length was l_c = 120 mm, and the total core cross-section was Ac = 1800 mm². The total number of secondary turns was N = 40. The SB256 bridge rectifier was used.

To test the performance of the prototype OLMEH, an experimental test bed was constructed, see Figure 8, using two back-to-back connected isolation transformers capable of sustaining a low-side current of up to 200 A rms. The OLMEH line current was regulated using a VARIAC and a high-power resistive load. In Figure 8, the OLMEH is illustrated successfully charging a battery.

Magnetic properties of the prototype OLMEH were tested and Froelich coefficients were extracted from the experimental data as mentioned.

The prototype OLMEH was tested at several line currents and CVL voltage levels. During the experiments, electronic load was used to emulate the voltage sink load. Key waveforms of the OLMEH are shown in Figure 9. Here, the line current was I₁ = 100 A rms. The cusp portion of the rectifiers’ input voltage waveform, Vin, manifests State 1, whereas the flat portion reveals State 2. Accordingly, the CVL current is zero during State 1, whereas CVL is provided with a pulse of current during State 2. This stands in agreement with the theoretical prediction above.

For V_b = 25 V, the measured output power was 49.4 W, see Figure 9a, whereas for V_b = 35 V the output power was 44.6 W, see Figure 9b. Also note that for V_b = 35 V the duration of State 1 is relatively long and is seen in Figure 9b. However, at the reduced CVL voltage, V_b = 25 V, the duration of State 1 is shrinking; see Figure 9a. This shows that OLMEH is approaching the CCM-DCM boundary.

The plot of the OLMEH’s output power vs. the CVL voltage at a line current of 100 A rms and 180 A rms is shown in Figure 10.

A comparison of the calculated vs. the measured output power as a function of the line current at fixed CVL voltage V_b = 30 V is presented in Figure 11. An excellent match was obtained.

A comparison of the calculated and the experimental results for line current I₁ = 100 A rms is shown in Figure 12. CCM mode is observed at V_b < 20 V, whereas DCM settles in at a higher CVL voltage level.

9. Conclusions

This paper offers a model, analysis, and analytical solutions for a clamped-type overhead line energy harvester with a full-wave rectifier with natural commutation loaded by a constant voltage load.

This paper also reviews the Froelich equation and adopts it as a main tool to investigate harvester operation. This paper further suggests a model and qualitative description of the energy conversion process, and presents the simulation results followed by a full nonlinear analysis supported by experimental results. The contribution of this paper is the complete analytical solutions for charging and idle time intervals, the solution for the average output current, and the solution for the average output power.

The proposed theory was verified by comparison to simulation and experimental data collected from a laboratory prototype. An excellent match was observed. The prototype harvester was shown to provide a substantial charging power of up to 50 W in 25 V CVL at 100 A rms line current.

Compared to the earlier models [23,26], the choice of the Froelich equation as the approximation function to describe the core BH characteristic is critical, and proved to be well-suited to facilitate the derivation of complete analytical solutions to the key parameters of OLMEHs. The proposed approach offers both meaningful engineering insight and an accurate analytical description of the OLMEH’s performance. The undertaken approach also allowed finding the CCM-DCM regime boundary that has not yet been discussed in the literature.

Yet, the solutions are still an approximation. Firstly, because the Froelich equation does not consider the hysteresis feature and, secondly, since (16) is valid in a limited range of the power line angle the analysis result can be applied in cases where the width of the power transfer window exceeds 100°.

This experimental work revealed that handling the OLMEH prototype affects its magnetic properties. This is mainly attributed to the variations of the technological air gap remaining between the two C-core halves of a split magnetic core that depends on the mounting force, surface imperfections, etc. Therefore, in the authors’ opinion, the derivation of a more precise and, thus, more complex theoretical model (that includes the hysteresis effect) is of questionable practical value due to the mentioned, and unavoidable in practice, core parameter variations.