Design investigation of electromechanical generator for energy harvesting

. In this paper designs of electromechanical generator for low frequency energy harvesting have been investigated. Simulation with finite element method has been conducted in order to determine highest output voltage of simple and robust generator consisting of permanent magnet and windings. In first part round magnets have been used in spherical and cylindrical form, benefiting from their ability to roll through winding with almost no mechanical friction inducing voltage in into windings. In the second part spindles with smaller radius than circumference of magnet were added to axis to increase rotational velocity of magnet in ambition to further increase induced voltage. As a result of added spindles and use of different magnet shapes length of winding turn varied and resistance of winding varied with it. To ensure similar conditions, windings have been recalculated to lowest electrical resistance using same fill factor, resulting in less winding turns decreasing induced voltage. In case of same kinetic energy input, higher rotational velocity combined with lower inertia produced higher induced voltage output.


Introduction
Constant power supply offshore always presents challenge, especially for self-sustaining applications. Where during the night most of energy is consumed powering lights to ensure night visibility, for maritime safety. Also considering that sizable amount of vessels are anchored in natural bays used only occasionally or for few months a year, but they still need to retain minimum electricity supply for mast light and to ensure that starter battery does not fully discharge.
Hence, durable self-sustaining systems are fundamental for such applications. Since PV panels are one of more popular choices for self-sustaining systems, due to abundance of solar energy, such systems are not always optimal choice. Especially as solar powered systems have big disadvantage in cloudy/foggy conditions and at high latitudes, where daily solar energy over winters is very low.
Great deal of research has been done on providing power for self-sustainable systems. Wave energy harvesting sea buoys are one of them, although main focus is on providing energy for seabed anchored buoys, where linear generators are mostly used [1][2][3][4] offering high energy density. However harvesters as such are not feasible for moving objects, thus some research have been done for non-stationary applications [5][6][7][8][9][10][11][12].
In prospect to create simple, durable and cost efficient design of generator that can generate sufficient electrical energy on the move, design investigation with simulations, has been conducted. This paper proposes electromagnetic generator with varying geometries of magnets and windings, to determine combination with highest output voltage.

Principle of operation
Generator design consists of compact windings aligned with XY plane and two-pole NdFeB permanent magnet (PM), as presented Fig. 1 shows, which is in first case spherical and in second case cylindrical shape. Magnets magnetization orientation is perpendicular to the PM axis plane. Idea is that when the generator is subjected to the alternating inclination change, PM inside of device starts to roll through winding along Z axis. Thus inducing voltage into surrounding winding corresponding to Lenz's law.
Induced voltage waveform there by varies as result of magnetic flux time derivative, which depends on number of factors. Firstly on proximity of PM to observing winding, due to exponential weakening of magnetic field intensity with the distance. Secondly, depending on magnet's magnetization angle at given point compared to the winding's plane normal (Z axis), as it was already discussed for the spherical magnet in [9]. Thirdly, depending on the winding geometry and number of turns. And lastly, depending also on flux leakage in air gap and to the sides of winding.

Scope of investigation
Shape of magnets was limited to sphere and three dimensions of cylinder, as presented in Fig. 2, where spheres radius RS of 20 mm determined volume of PM for cylinder shapes. Cylinder radii wearied from 14 to 22 mm, where cylinder radius RC22 of 22 mm was presenting shortest wire loop around PM, cylinder with radius RC17.5 of 17.5 mm was presenting square axis intersection and RC14 was presenting cylinder PM with double height compared to diameter.
In first case, PM with dimensions stated in previous paragraph, were assumed to roll on their own base/circumference trough the windings. In second case spindles were added to magnets as shown on lower part of Fig. 2. Hence reducing rolling circumference to radius r of 2 mm, similarly as previously done in [9], forcing magnet to higher angular velocity and making less displacement along Z axis in one rotation. To establish comparable design conditions height hw and width ww (10 by 10 mm) of windings staid the same and contained 100 turns of wire (N). Presuming filling factor of 0.71 (100 turns with nominal diameter of 0,95 mm wire). Air gap agap between magnet and winding was equal to 1.5 mm.

Modelling
Due to the need for simplification we assumed that spherical PM magnetization vector was always aligned with XZ plane, which is not fully true for sphere without spindles, where orientation of magnetic field can deviate from alignment with XZ plane. With this presumption we ensure that magnetic field cut winding at right angle, thus generating maximum induced voltage.
Magnet's velocity v along Z axis was constant and equal to the velocity of rolling object after 0.5 m of height difference H, presented in equation (1), where m presents mass of magnet and g gravitational acceleration. Velocity of magnet varied correspondingly to magnets inertia I and rolling base radius rbase,, which was in case of no spindles determined with PM radius and in case of spindles with radius of spindle.
Sphere's velocity is slightly higher than cylinder's (2.65 m/s compared to 2.56 m/s), if rolling on their circumference. On the other hand with equal radius of spindles, velocity along Z axis is determined only by magnets inertia.
With previously stated conditions 3D magnetostatic model was created in ANSYS Maxwell using finite element method. Simulations output was winding's magnetic flux, which was then transformed to induced voltage using the ratio of flux difference and time difference between two sequential calculated points.

Models without spindles
As specified in section 2.2 we firstly simulated magnets rolling on their circumference, as suggested in upper half of Fig. 2, assuming their equal conditions of operation. Rotational motion and displacement along Z axis created periodical waveform, presented in Fig. 3. Where observed winding's induced voltage peak was generated when magnet was closest to it, taking in to consideration magnetic field alignment with Z axis.
Symmetry of induced voltages occurs only in case of magnet's field alignment with Z axis and at the same instance being in exact centre of winding. Hence producing equal positive and negative peaks and therefore maximum peak to peak induced voltage of one winding. Closely comparing symmetrical induced voltages in Fig. 4, for all four designs, the highest winding output can be determined. As shown in Fig. 4 highest peak to peak value of 59,42 V was achieved by cylinder with radius of 14 mm, mainly due to higher rotational velocity and consequently flux change rate.
Second highest peak to peak value of 46.88 V was produced by cylindrical magnet square cross section. Cylinder with smallest loop around axial cross section and spherical magnets generated lowest induced voltages of 39.6 and 37.09 V.

Models with spindle
To increase flux change rate, spindles with radius of 2 mm (Fig. 2) were added to magnet's axis to force magnet in turning with higher angular velocity, with smaller displacement in Z axis, therefore peak to peak induced voltages increased and were more in phase as seen in Fig. 5, compared to values without spindle (Fig. 3), where peaks of induced voltage were more evenly distributed. Biggest improvement was noticeable on sphere shape, where peak to peak voltage almost doubled and reached 73.26 V. Although highest peak to peak induced voltage of one winding was generated with 14 mm radius of cylinder with spindles (98.81 V) as seen in Fig. 6.   Fig. 6. Comparison of induced voltage for sphere and cylinders with radii from 14 to 22 mm rolling on spindle.
As consequence of using different shapes of magnets and added spindles, length of winding changed, as shown on Fig. 2, where average length of turn was presented with dashed line. Therefore with usage of same fill factor resistance changed as well. Total length of each winding was derived from average length of turn and written in equations (2)(3)(4)(5). Where Rmag and hmag stand for radius and height of magnet and lspin is the length of spindle.
have been already ideas how to solve this problem without need for spindles. On the other hand this design can make a good reference for comparison to other designs. To reach highest potential difference at any given moment, every second winding has been inverted, thus resulting in opposing polarities of neighbouring windings. From potential difference (solid line) of positive (dash-dot line) and negative voltage wave curve (dot line), root mean square (RMS) voltage (dash line) was calculated and presented in Fig. 7. Where designs without spindle are presented on top half of Fig. 7 and on the bottom half are designs with spindle.
Comparing induced voltages, it is obvious that higher angular speed increased induced voltage RMS value from 23% to 53%. Highest increase was observed with sphere and lowest with 22 mm radius cylinder. Highest voltage output designs in both cases were cylinders with radius of 14 mm, producing 48.31 V without spindles and 67.51 V with them. Second highest voltage was achieved in both cases by cylinder with radius of 17.5 mm, generating 39.97 V without spindles and 51.7 V with them. Lowest RMS voltages were produced with sphere and cylinder with square axis cross section. In case of cylinder without spindles 32.91 V was generated, 0.44 V more than by sphere. In other instance where spindles were used sphere produced 9.07 V more than 22 mm cylinder with 40.37 V. Taking into consideration differences in geometry, the use of spindles increased the length of windings, due to increase in average length of turn, as shown in Fig. 2. RMS induced voltages (E) were recalculated for the lowest resistance RRS (winding of sphere without spindles), taking into account the same fill factor (FF) of 0,71 and total length of winding lavg, thus changing number of turns and wire area using equation (6), where ρCu electrical resistivity and Aw presents cross section area of winding and equals to hw times ww. RS

RS w eqR
Cu avg Recalculated RMS voltages (EeqRRS) decreased, due to lower number of turns, although order of designs by EeqRRS did not change compared to previously stated one with voltages generated at 100 turns. Highest recalculated value decrease of 9.3 V was experienced by cylinder with radius of 14 mm, still having 10.5 V higher induced voltage compared to second highest cylinder with 17.5 mm radius.

Conclusion
In prospect to create feasible and simple designs of electromagnetic generator for low frequencies, eight proposed geometries were simulated and analysed. Focusing on highest voltage output at limited winding space induced voltage and internal resistance were observed. As a consequence of shape difference, velocity of rolling magnet was determined by the circumference and inertia, where both played an important role. Higher induced voltages were generated by faster rotating magnets. Therefore spindles were introduced to designs to further increase angular velocity increasing also axial length. Higher winding resistance caused additional voltage drop that had to be taken into account at power generation. Therefore recalculated RMS voltages have been introduced to ensure equal resistance. According to recalculated RMS voltages highest output was achieved by the smallest inertia design. Cylinder with four to one ratio between height and radius rolling on spindles and lowest voltages were obtained by the highest inertia cylinder without spindles.