A Robust Generator-Harvester for Independent Sensor Systems

Citation: Zukal, J.; Szabó, Z.; Kˇríž, T.; Kadlec, R.; Dˇedková, J.; Fiala, P. A Robust Generator–Harvester for Independent Sensor Systems. Appl. Sci. 2024,14, 1246. https://doi.org/ 10.3390/app14031246 Academic Editor: Alessandro Lo Schiavo Received: 13 November 2023 Revised: 8 January 2024 Accepted: 15 January 2024 Published: 2 February 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article A Robust Generator–Harvester for Independent Sensor Systems † JiˇríZukal, Zoltán Szabó, Tomáš Kˇríž, Radim Kadlec, Jamila Dˇedkováand Pavel Fiala * Department of Theoretical and Experimental Electrical Engineering, Brno University of Technology, Technická12, 616 00 Brno, Czech Republic; [email protected] (J.Z.); [email protected] (Z.S.); [email protected] (T.K.) *Correspondence: [email protected]; Tel.: +420-604-076-280 † This paper is an extended version of our paper published in PIERS 2023 conference, Zukal, J.; Szabo, Z.; Pernica, R.; Kadlec, R.; Dedkova, J.; Klima, M.; Fiala, P. Designing a Robust Model of a Linear Motion-driven Harvester. In Proceedings of the 2023 Photonics & Electromagnetics Research Symposium (PIERS), Prague, Czech Republic, 3–6 July 2023, pp. 732–738, https://doi.org/10.1109/PIERS59004.2023.10221336. Abstract: The research is centered on energy production and harvesting to facilitate the transformation of electrical energy with energy-independent sensor systems, using powering devices in the expected power range of P= 10–10,000 W. A model application case for a harvester is the conversion of energy stored in the compressed gas during expansion; such gas embodies the energy stored in scenarios such as braking a car using an auxiliary pump. Similar systems find use in sensing various quantities in the transport sector (bridge structures, infrastructural components, cars, and other objects). The proposed theoretical harvester models describing the transformation of linear motion energy into electricity provide relevant support for the experiments. In the given context, the results obtained in the designing and construction of a robust motion generator with a primarily linear geometry-based system technology are presented, too. The expected output of electrical power of an N-segment harvester within the tested type is variable, and the design exploits the rectilinear motion generated by an engine using compressed air, a small fuel system, and similar options to obtain an expected/adjustable N-segment power in the range of P sm = 10–500 W. The fundamental structure of the generator core has been continuously numerically modeled, and an experimental setup has been developed to analyze the specific parts and variations in order to validate the concept and to achieve the most suitable parameters with the selected construction materials (a power yield increase of up to 2000 times). A scaled-down version of the model principle was tested in the experiments, and the parameters and results were compared with the predicted theoretical analyses. Generally, the conceptual layout of an enhanced magnetic circuit layout transforming motion energy into electricity was presented and verified. Keywords: harvesting; electromagnetic field; numerical model; renewable energy; linear motion; sensor systems 1. Introduction Between the years 2000 and 2020, the trends of energy saving, energy system independence, energy processing efficiency, and energy harvesting based on electromagnetic field principles [ 1 ] reached almost all industries. The field of energy harvesting, or the concept of energy transduction [ 2 – 4 ], does not cover the entire area of energy production and processing to convert diverse forms of energy into electricity; this domain includes only a limited part of the principles [ 5 – 8 ] and corresponding device designs [ 9 – 12 ] that allow exploiting unused types of energy for a given purpose [ 13 , 14 ]. In power generation, specific energy conversion principles still remain to be included, such as the conversion of energy from flowing media (water, air) and the conversion of incident RF electromagnetic waves, namely, photovoltaic systems, into electricity [15–17]. Considering the scarcity of alternative energy sources, these may find application mainly in mobile or wireless devices [ 18 – 20 ], sensor-independent systems [ 3 , 4 ], autonomous Appl. Sci. 2024,14, 1246. https://doi.org/10.3390/app14031246 https://www.mdpi.com/journal/applsci Appl. Sci. 2024,14, 1246 2 of 16 sensor instruments, and systems where a stable power supply is disadvantageous or impossible to secure [ 3 , 4 ]. The requirement for the development and use of alternative energy source principles involves, among other factors, estimating the minimum operating time without a stable/permanent (non-mobile) distribution system. A narrower, more tightly defined segment of power harvesting and generation utilizes Faraday’s law of induction [ 1 ] in devices that employ vibration [ 3 ], rotational motion, and/or linear motion [ 2 , 4 – 6 ]. These principles are further modified through various concepts of micro-, mini- [ 3 ], and large-sized generators [ 2 ]. In the last-named option, the power delivered is expected to be in the order of tens to thousands of watts, either discontinuously or in a continuous time. Such generators and design approaches must meet the desired lifetime requirements, and if they satisfy this prerequisite, they are referred to as robust. Designing conceptual solutions to enable a highly efficient conversion of motion into electrical energy is an interesting—and already partially solved—problem. The principle of converting motion energy into electricity is based on the understanding and consistent use of the possibilities of Faraday’s induction law [ 1 ], as presented in sources [ 3 , 4 ] and [ 18 – 20 ] and as outlined hereabove. Researching and exploring suitable harvester concepts are steps that generally allow us to identify a highly efficient device to transform kinetic energy, potentially enabling the operator to power components such as sensors with permanently supplied electrical energy that may be of rare origin. In sensors and measuring systems, special transportation equipment, and cars, meaning items that feature a higher power consumption ( P = 10–1000 W ), the power supply of electricity from conventional sources such as batteries, internal combustion engines, and related sources is either disadvantageous in terms of the solution and cost or difficult to implement. For this reason, residual or alternative forms of energy are then widely sought for, including, but not limited to, the expansion of compressed air and various types of linear movement. The options of converting kinetic energy into electricity comprise, for example, the electromagnetic principle based on the full Faraday effect, the result being high conversion efficiency [1,18–20]. The project presented herein expands on the concepts in paper [ 21 ], namely, efficient methods for generating and exploiting linear motion. In this article, however, the parameters of selected vibration harvesters are discussed and compared in greater detail, and, importantly, an estimation the expected characteristics inherent with the proposed implementation is outlined. Further, the relevant numerical model based on Maxwell’s equations is rendered more extensively, both in the derivation and the results, and can be applied using powerful numerical modeling tools in the Finite element system The novel, expanded diagrams that display three experimentally tested concepts of one cell of the periodic arrangement of the harvester, then allow us to represent the symbolic connection of the magnetic circuit in one element of the harvester and express the periodic structure of the experimental generator. The practical measurements centered on three novel configurations of the magnetic circuit then show that a suitably arranged harvester cell circuit facilitates in achieving a major increase in the movement energy extraction (up to 2000 times the power obtained under comparable electrical load setting conditions). These experiments and evaluation tasks have been performed to refine and verify the previous outcomes. Regarding the conference paper “Designing a Robust Model of a Linear Motion-driven Harvester”, the authors showed how changing the arrangement of the magnetic circuit can fundamentally affect the output parameters, namely, the produced electricity yield; this step was carried out by measuring along the outlines of the documented experiment. To the best of our knowledge, no extensive research on designing the magnetic circuit in linear generators to achieve the maximum possible rate in converting motion energy into electricity has been conducted to date. To facilitate such work, it is necessary to know the principle and mathematical model that lead to formulating the coupling and energy conversion as well as the technological feasibility of the design (detail of the magnetic flux Appl. Sci. 2024,14, 1246 3 of 16 changes). A large portion of the tasks is nevertheless proposed in this article, which thus may support further development in the field. 2. State of the Research Field and Topics Analyzed We discuss the designing and selection of a concept to convert various forms of energy into electromagnetic energy, the central aim being to transform mechanical linear motion into electricity. In this context, a broad range of approaches are examined to yield instantaneously delivered/transmitted electrical power in the range of P = 10–10,000 W. The published articles and papers can be classified into several groups. The first one embraces the area of designing and solving the electrical, electronic, and electromagnetic parts of the projected devices; prominent sources include, above all, [ 5 – 8 ]. Another group addresses the harvester/generator drive issues, often in relation to combustion engine functions and parts [ 9 – 12 ]. Yet, another set comprises articles on motion generation, the drifting of the electromagnetic part of the generator [ 13 , 14 ], models [ 18 – 20 ], motion principles, and appropriate experiments [ 22 – 37 ]. Further categories cover the following methods and processes, respectively: the linear motion of the power unit based on the hydraulic transfer of dynamic energy to the motion element within the linear generator configuration, the generator system (kinetic-to-electrical energy conversion) and the control, modeling, simulation, measurement, and evaluation of relevant model parameters. Detailed insights into the analysis and solution of the generator concept and more serious attempts to create a mathematical model and simulation of the hybrid motor are provided in, for example, articles [ 38 , 39 ], which focus on designing the purely electromagnetic part, i.e., on electromagnetic conversion using a generator. Another research direction leads to the designing and solution of the combustion power unit in linear engines, namely, an engine with free pistons. This mechanism, relative to a rotary engine with a crank mechanism, has only one or two pistons connected on a common shaft [ 26 ] and features precisely the controlled detonation and dynamics of the moving mass. A linear motor with free pistons uses the motion of a common shaft, on which a linear electric generator is mounted [ 40 ], or, alternatively, a hydraulic piston mounted on a common shaft is employed to generate hydraulic pressure by moving the piston. Only a limited number of authors [ 41 ] have hitherto simulated a setup to control a motor with a hydraulic pump. The publications, to date, on the construction and testing of a linear combustion engine in connection with a generator include articles and theses from West Virginia University (the USA) [ 2 , 4 ], in which using the engine is also considered for an electric power generator applicable in relatively remote locations, thanks to fewer moving parts and greater reliability, efficiency, and compactness [ 5 ] compared to the rotary engine concept. Similarly, the authors of [ 6 , 42 ] propose employing a linear combustion engine as an electric power generator to replace the current gasoline or diesel generators. By extension, the basic areas that host linear generators in conjunction with a free-piston internal combustion engine subsume automotive technologies and independent power systems. The basic elements of a linear generator are a linear internal combustion or a free-piston engine and a linear electric motor generating electrical energy (linear generator). The requirements of a linear electric generator resulting from a basic analysis of a linearly arranged internal combustion engine with a generator, all in terms of the output power achieved for charging the traction battery Q= 21 kWh and the electric drive P = 80 kW , are published in [ 43 ]. Other published articles or papers discuss, for instance, an approach that relies on deploying a linear generator to convert the mechanical vibrations of a car into electrical energy [ 44 ]. Some of the researchers also analyze the performance and quality of a linear microgenerator, including testing to exploit the energy from the mechanical propagation of waves with free pistons [7]. 2.1. Problems Relating to the Electromagnetic Component of the Designed Harvester A specific area of research within the above power generation procedures encompasses the designing of the electromagnetic part of the device or apparatus, completed with Appl. Sci. 2024,14, 1246 4 of 16 associated steps and stages such as coupling to a motion source and interaction with the system that is being powered, namely, a sensor or a sensor system. A detailed discussion of linear machine concepts suitable for use in linear motor/generator configurations, including an interpretation in terms of the geometry (planar or tubular), is available in [ 22 ]. A comparison involving linear generators is detailed in [23]. Furthermore, an interesting area lies in modeling specifically designed devices, as is presented in articles [24,25]. The decisive parameter for comparing the efficiency of the design and implementation of generators and their transformation characteristics, the parameters for the choice of the downstream concept, is the quantity effective volumetric power density p efd [W/m 3 ], shown in Table 1[ 20 ]. Using this parameter, the efficiency rates of an energy source with respect to its volume are easily comparable, and such a procedure thus facilitates the comparison between selected concepts and designs, which may be dimensionally different and difficult to measure. Table 1. The parameters of selected mini-generators Reprinted/adapted with permission from Ref. [20]. Copyright 2024, copyright Pavel Fiala [20]. Reference Permanent Magnet Type Generator Body Size x,y,z [m] Output Power Pout [W] Output Voltage Uout [V] Effective Power Density pefd [W/m3] Beeby et al. [28], 2007 −375 mm32×10−60.428 RMS ≈6 Zhu et al. [29], 2010 FeNdB 2000 mm361.6–156.6 ×10−6− ≈30−80 Kulkarni et al. [ 27 ], 2008 FeNdB 3375 mm30.6 ×10−60.025 ≈0.2 Wang et al. [31], 2007 FeNdB 256 mm30.06 - Lee et al. [33], 2012 FeNdB 1.4 ×10−4m31.52 ×10−34.8 ≈10 Yang et al. [32], 2014. −50,000 mm313.4 ×10−30.7−2.0 ≈270 Elvin et al. [30], 2011 −15,000 mm34×10−60.007 ≈0.26 MG I [18], 2006 FeNdB 90, 40, 30 mm 70 ×10−34−60 (300) p-p ≈650 MG II [18], 2006 FeNdB 50, 27, 25 mm 19.5 ×10−36−15 ≈60 MG III FeNdB 50, 25, 25 mm 5.0 ×10−31.0−2.5 ≈15 MG IV FeNdB 50, 35, 25 mm 8.0 ×10−31.0−2.5 ≈18 * Lith. battery [34], 2018 * Lith. battery ≈40 ×106 * Supercap [35], 2010 * Supercap ≈3−5 * Fuel * Fuel ≈4×109 * U235 * U235 ≈9×1016 The asterisk indicates a note for conventional energy sources (batteries, fuel, nuclear reaction) for comparison with harvesters. 2.2. Designing an Electromagnetic Transformation Method: Renewable Energy The first article written by prominent researchers in linear motion generation was published at West Virginia University [4], focusing on the engine and alternator sections. The direct design of a linear synchronous generator and the magnetic flux density distribution via a finite element analysis are outlined in [ 36 ]. The generator consists of a stator that comprises coils and a rotor made of permanent magnets. 3. Mathematical–Physical Model The basic formulation for deriving the mathematical expression to characterize the electromagnetic part of the model is based on Faraday’s induction law (1); see, for example, the interpretation by J.A. Stratton [ 1 ]. The use of 3D modeling and simulation, analysis, and comparisons with experimental models of a linear generator is discussed in multiple sources, including [ 8 , 18 – 20 ]. The simulation and experimental verification of the parameters of the equivalent circuits and the rotor/stator core losses via the FEM are addressed in article [ 37 ]. An evaluation of the related impact of the electrical parameters, load, and drive dynamics is proposed in [7,8]. Appl. Sci. 2024,14, 1246 5 of 16 According to [ 8 , 20 ], the change in the magnetic flux Φ according to Faraday’s induction law is I ℓ E(t)·dℓ | {z } Φ =−Z S ∂B(t) ∂tdS +I ℓ (v(t)×B(x,y,z,t)) ·dℓ, (1) where E(t) is the electric field intensity vector, B(t) denotes the magnetic flux density vector (induction), v(t) indicates the speed of the shift of the generator core position in time (instantaneous speed), Sdenotes the cross-section of the magnetic flux regions, and λ denotes the curve along the boundary of area S. The simple concept, which was experimented with previously [ 2 – 4 , 8 , 20 ], can be applied to the electromagnetic part of the harvester. The equation of the motion relating to the arrangement of the electromagnetic part of the generator/harvester, the force field couplings, and the linear motion of the moving part are expressible as m.. x+lc . x+k x =fmagB,. x,t+fmech(t), (2) where mis the mass, m m denotes the mass of the moving part of the generator system, l c is the damping coefficient, krepresents the stiffness coefficient, xstands for the position of the body, . x expresses the velocity of the moving part (dx/dt), .. x is the acceleration of the moving part (d 2 x/dt 2 ), f mag stands for the force acting on the moving part via interaction with the magnetic field, and f mech denotes the force of the mechanical motion of the moving part of the linear actuator. In order to obtain a concept to deliver an efficient harvester yield with linear motion, we have to accept the dynamic parameters of the system to prevent the the conceptual design from markedly reducing the energy conversion efficiency of the generator system already at the beginning of the actual designing task. As already determined via a comparison with other motion/vibration harvester designs [ 20 ], the moving segment of the electromagnetic part of the generator for the zenith linear actuator concept must, in the moving part, exhibit a mass parameter m m close to the minimum value (solution of Equation (2), Figure 1a). In the case of the moving part of the generator model, the mass must be close to the minimum. The instantaneous value of the electric current through the winding i(t) at an electric load on the terminals of the winding R z can be derived and formulated according to the procedure in [2]. Appl. Sci. 2024,14, 1246 6 of 16 Appl. Sci. 2024, 14, x FOR PEER REVIEW 7 of 17 with the magnetic field, and fmech denotes the force of the mechanical motion of the moving part of the linear actuator. In order to obtain a concept to deliver an efficient harvester yield with linear motion, we have to accept the dynamic parameters of the system to prevent the the conceptual design from markedly reducing the energy conversion efficiency of the generator system already at the beginning of the actual designing task. As already determined via a comparison with other motion/vibration harvester designs [20], the moving segment of the electromagnetic part of the generator for the zenith linear actuator concept must, in the moving part, exhibit a mass parameter mm close to the minimum value (solution of Equation (2), Figure 4a). In the case of the moving part of the generator model, the mass must be close to the minimum. The instantaneous value of the electric current through the winding i(t) at an electric load on the terminals of the winding Rz can be derived and formulated according to the procedure in [2]. (a) B, © Pavel Fiala u(t)  pohyb Moving part Fixed part Magnetic resistance (electric generator winding) Magnetic flux changer Magnetic circuit Permanent magnet Magnetic flux Mass of the moving part mm changer u1(t) v(t) Permanent magnet Pole extension Moving part Fixed part Electric coil Magnetic circuit u2(t) B, © Pavel Fiala  Magnetic flux changer (b) Figure 4. (a) A modified magnetic circuit: a model with concentrated parameters (left) and its design scheme (right) in a linear motion system with minimized dynamic losses (double-acting arrangement). (b) A modified magnetic circuit: a model with concentrated parameters (left) and its design scheme (right), representing an arrangement for the maximum yield efficiency of the linear motion system with minimized dynamic losses (a double-acting arrangement). If the electromagnetic arrangement of a linear generator is considered, as shown in, for example, Figure 3, it will be driven by the corresponding unit (an internal combustion engine) in the dynamic non-volatile state (2); to characterize such an application, we employ a general mathematical model. Faraday’s law of induction (1) is employed to formulate the principle and the design of the active part of the generator, also enabling the motion energy of the system to transform into electrical energy, through induction, to the electrical voltage in the inserted conductor of the electric winding; the relevant part is described as B, © Pavel Fiala u(t) Permanent magnet  Magnetic resistance (electric coil) Magnetic circuit Magnetic flux changer Fixed part Moving part Mass of the moving part mm u(t) Permanent magnet Magnetic circuit g(t) Pole extension Moving part Fixed part Magnetic flux changer Mass of the moving part mm Permanent magnet B, © Pavel Fiala  Electric coil Figure 1. (a) A modified magnetic circuit: a model with concentrated parameters (left) and its design scheme (right) in a linear motion system with minimized dynamic losses (double-acting arrangement). (b) A modified magnetic circuit: a model with concentrated parameters (left) and its design scheme (right), representing an arrangement for the maximum yield efficiency of the linear motion system with minimized dynamic losses (a double-acting arrangement). If the electromagnetic arrangement of a linear generator is considered, as shown in, for example in [ 36 ], it will be driven by the corresponding unit (an internal combustion engine) in the dynamic non-volatile state (2); to characterize such an application, we employ a general mathematical model. Faraday’s law of induction (1) is employed to formulate the principle and the design of the active part of the generator, also enabling the motion energy of the system to transform into electrical energy, through induction, to the electrical voltage in the inserted conductor of the electric winding; the relevant part is described as I ℓ E(t)·dl =−dΦ(t) dt (3) If a thin conductor in the form of a closed loop is inserted into such a variable electric field, an electric current i(t) starts to flow therein. Now, let us denote the current that flows through the closed loop without the presence of an external source of electric voltage as the induced current. The magnetic flux Φ is generated by an external magnetic field. We then have u(t)=−dΦ(t) dt (4) u(t)=u1(t)+u2(t)(5) u(t)=Umag|g(t)|(6) Appl. Sci. 2024,14, 1246 7 of 16 where U mag is the maximum value of the induced coil winding voltage for a single-line arrangement (Figure 1a) under the given conditions, and g(t) denotes a function of the time dependence of the magnetic field corresponding to the system parameters and the dynamics of the moving part of the generator. The resulting induced voltage for the double-acting arrangement (Figure 1b) changes with respect to the single magnetic circuit arrangement into the form (6). The energy stored in the magnetic field source (a permanent magnet) is written as Wm=Z Vm 1 2BMHMdV, (7) where B M ,H M are the magnetic flux density and magnetic intensity at the working point of the permanent magnet, and V m denotes the volume of the magnet. The energy, converted into heat (Joule heat), in the winding of the loaded coil (for our problem, an irreversible form of energy) reads WJ=Z VJc 1 2 J2 γdV, (8) where γ is the specific conductance of the coil conductor, Jrepresents the current density vector, and V Jc represents the volume of the coil conductors. The energy that dampens the moving oscillatory motion of the generator core due to the electrical load on the coil terminals is written as WV=Z ℓ Z VJ fmdV ·ndℓ=Z ℓ Z VJ (J×B)dV ·ndℓ(9) where f m is the specific force acting on the motion part of the generator, nstands for the normal vector in the direction of the electric current flow i(t), d λ denotes the displacement length due to the specific force, Bis the magnetic flux density vector, and V J expresses the volume of the electrically conductive components. From Equations (7)–(9), the equation of state is obtained as follows: mma dx −Z ℓ Z VJ (J×B)dV ·ndℓ−Z VJc 1 2 J2 γdV =1 2md x dt 2 , (10) ηZ Vm 1 2BMHMdV =1 2mmd x dt 2 , (11) where dx is the deflection of the generator core, η represents the energy utilization efficiency of the permanent magnet module, and adenotes the acceleration (d 2 x/dt 2 ). After being modified, and when the expressions for voltage and current have been inserted, Equation (11) reads mma dx −Z ℓ Z VJ I Sv n×BdV ·ndℓ−Z VJc 1 2I Sv2 γdV =1 2mmd x dt 2 , (12) ηZ Vm 1 2BMHMdV =1 2mmd x dt 2 , (13) Appl. Sci. 2024,14, 1246 8 of 16 where Iis the maximum value of the amplitude of the electric current flowing through the coil conductor, S v is the cross-section of the winding conductor, and dtis the time change. Then, we have mma dx −Z ℓ Z VJ P U Sv n×BdV ·ndℓ−Z VJc 1 2P U Sv2 γdV =1 2mmd x dt 2 (14) By comparing Equations (12)–(14), an expression is obtained from which the order of magnitude of the moving part (core) of the generator can be determined depending on the pre-specified parameters. mma dx −Z ℓ Z VJ P U Sv n×BdV ·ndℓ−Z VJc 1 2P U Sv2 γdV =ηZ VM 1 2BMHMdV (15) According to Equations (2)–(14), the addition of the braking forces F br gives the basic formula that characterizes the generator system. We have md2x dt2+lcdx dt +k x =mmd2x dt2−Z VJ (Jv×B)·ndV | {z } Fbr −R VJc (Jcirc ×B)·ndV, md2x dt2+lcdx dt +k x =mmd2x dt2−R VJdx dt ux×Bbr(t)×B·ndV −R VJc i(t) Svn×B·ndV, md2x dt2+lcdx dt +k x =mmd2x dt2−R VJdx dt ux×Bbr(t)×B·ndV −R ℓJc (i(t)n×B)·ndℓ, (16) where B br is the braking component of the magnetic induction vector, J v denotes the current density vector of the electrically conductive components due to eddy currents, J circ represents the current density vector in the coil winding, i(t) stands for the instantaneous value of the coil electric current, and u x is the unit vector of the coordinate system. Interestingly, in this context, the approaches described in research articles [ 45 , 46 ] can be considered inspiring for further characterization of the modeling procedure. The model described above was compiled using the finite element method (FEM) in ANSYS [ 47 ]. Both the dynamics of the model in the Ansys Multiphysics module [ 47 ] and the magnetic field distribution subproblems for the static and the quasi-stationary arrangements were analyzed, as shown below To increase the yield (efficiency) of the dynamic energy transformation (Figure 1a) generated by the motion, the magnetic circuit layout, according to Figure 1b, is applicable. In order to achieve a high efficiency in the Faraday induction law-based conversion (1) of the kinetic energy of the generator into electricity, the arrangement of the magnetic circuit, shown schematically in Figure 1a, is utilized. Due to the arrangement of the magnetic circuit in Figure 1b, the change in the flux d Φ /dx or d Φ /dt in Figure 1b is multiplied. A significant increase in the magnitude of the induced electric voltage u(t) can therefore be expected in a comparable winding of the electric generator. This corresponds to a change in the proposed arrangement (Figures 2and 3) for the resulting electric voltage u(t). The segment of the magnetic arrangement of the linear generator will be periodically repeated during the designing part, and therefore the sections and subchapters below will deal with the fundamental parts of the periodic arrangement of the generator elements. Appl. Sci. 2024,14, 1246 9 of 16 Appl. Sci. 2024, 14, x FOR PEER REVIEW 10 of 17 be expected in a comparable winding of the electric generator. This corresponds to a change in the proposed arrangement (Figures 5 and 6) for the resulting electric voltage u(t). The segment of the magnetic arrangement of the linear generator will be periodically repeated during the designing part, and therefore the sections and subchapters below will deal with the fundamental parts of the periodic arrangement of the generator elements. u1(t) v(t) Moving part Fixed part u2(t) M M Segment I Segment II Position II. B, © Pavel Fiala  Magnetický obvod Magnetic flux changer Permanent magnet Magnetic circuit v(t) u2(t) B, © Pavel Fiala u(t)  pohyb Moving part Fixed part Magnetic resistance (electric generator winding) Magnetic flux changer Magnetic circuit Position I. Segment I Segment II Position II.  Segment I Segment II (a) (b) Figure 5. The segments I and II of a single-acting magnetic circuit arrangement of the periodic structure of the linear motion shown below. Such configurations allow us to achieve the maximum change in the magnetic flux of the moving and fixed parts: (a) a principal geometrical arrangement, and (b) a symbolic representation of the magnetic circuit (a model with concentrated parameters). Figure 2. The segments I and II of a single-acting magnetic circuit arrangement of the periodic structure of the linear motion shown below. Such configurations allow us to achieve the maximum change in the magnetic flux of the moving and fixed parts: (a) a principal geometrical arrangement, and (b) a symbolic representation of the magnetic circuit (a model with concentrated parameters). Appl. Sci. 2024, 14, x FOR PEER REVIEW 11 of 17 u1(t) v(t) Moving part Fixed part u2(t) M Segment I Segment II Position II. u21(t) u11(t) M B, © Pavel Fiala  Magnetický obvod Magnetic flux changer Permanent magnet Magnetic circuit v(t) u2(t) B, © Pavel Fiala u(t)   pohyb Moving part Fixed part Magnetic resistance (electric generator winding) Magnetic flux changer Magnetic circuit Position I. Segment I Segment II Position II.   Segment I Segment II (a) (b) Figure 6. The segments I and II of the double-acting magnetic circuit of the periodic structure of the linear motion to achieve the maximum change in the magnetic flux of the moving and fixed parts: (a) principal geometrical arrangement, and (b) a symbolic representation of the magnetic circuit (a model with concentrated parameters). The magnetic circuit arrangement and efficiency are related to the arrangement of the transition of the moving part of the magnetic circuit to the circuit of the fixed part of the generator. The detail is provided in Figure 4b—magnetic flux converter. Several pole mounting configurations for transmitting the magnetic flux through the air gap of the moving and the static parts of the generator are illustrated in Figure 7. Figure 3. The segments I and II of the double-acting magnetic circuit of the periodic structure of the linear motion to achieve the maximum change in the magnetic flux of the moving and fixed parts: (a) principal geometrical arrangement, and (b) a symbolic representation of the magnetic circuit (a model with concentrated parameters). Appl. Sci. 2024,14, 1246 16 of 16 42. Zou, H.; Wang, M.; Tang, M.; Li, C.; Tian, C. Experimental investigation and performance analysis of a direct-driven linear generator. Energy Procedia 2017,142, 284–290. [CrossRef] 43. Seo, U.-J.; Riemer, B.; Appunn, R.; Hameyer, K. Design considerations of a linear generator for a range extender application. Arch. Electr. Eng. 2015,64, 581–592. [CrossRef] 44. Takahara, K.; Ohsaki, S.; Itoh, Y.; Ohyama, K.; Kawaguchi, H. Characteristic analysis and trial manufacture of permanent-magnetic type linear generator. Electr. Eng. Jpn. 2008,166, 94–100. [CrossRef] 45. Nguyen, V.T.; Ta, Q.T.H.; Nguyen, P.K.T. 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