THÔNG TIN TÀI LIỆU
Proceedings of the 2020 International Conference on Advanced Mechatronic Systems, Hanoi, Vietnam, December 10 - 13, 2020 OPTIMIZING LINEAR GENERATOR DESIGN’S PARAMETERS FOR OUTPUT POWER USING MIX NUMERICAL AND ANALYTICAL TECHNIQUE Do Huy Diep Faculty of Engineering Mechanics and Automation University of Engineering and Technology,VNU Hanoi, Vietnam e-mail: diepdh@vnu.edu.vn Nguyen Van Duc Faculty of Engineering Mechanics and Automation University of Engineering and Technology,VNU Hanoi, Vietnam e-mail: ducvn2@gmail.com Nguyen Xuan Quynh Institute Of Mechanical Engineering Hanoi University of Science & Technology (HUST) Hanoi, Vietnam email: quynhctm@gmail.com Dang The Ba Faculty of Engineering Mechanics and Automation University of Engineering and Technology,VNU Hanoi, Vietnam e-mail: badt@vnu.edu.vn tubular generator As the force is perpendicular to the direction of the translator motion, the translator must be perfectly aligned for such type of movement Rare earth magnets have a high value of residual induction, thus possess high magnetic field strength and perfect for linear permanent generators Neodymium magnets (NdFeB) are widely used for this purpose In linear PM tubular generator, with different types of shapes of the coils, the output power can be improved A few type of such coil has been proposed and compared in [7] Linear generators are designed by utilizing an iron core or an air-cored [8, 9] Tubular PM linear generators having iron cored stator are proposed in [10] This type of linear generator shows a significant harmonic content in the induced voltage in the generator To reduce the energy losses, air-cored machines have been investigated in [11] The results proved that the air cored machines have an advantage in mechanical design In [12] variable air gap is considered which gives better results in terms of preventing demagnetization Variable air gap linear generator gives more electrical power as compared to the fixed air gap length generator under the same operating conditions Abstract—Permanent magnet linear devices have wide applications in various fields In the field of wave energy conversion, the use of linear generator has earlier been regarded as difficult and uneconomical due to technical problems Researches on wave energy converters have been carried out, but the conversion efficiency is still limited Studies of the magnetic field of a linear generator have shown the ability to significantly improve performance when using the Halbach array magnets structure In this study, a mixed numerical and analytical technique is presented to optimize linear generator design’s parameters for wave energy converter power performance At first, numerical method used for maximize of magnetic field’s strength inside linear generator Then, a matlab-simulink program use the simulated magnetic field result to optimize the power of the linear generator Keywords—Wave energy converter, linear generator, Halbach arrays, numerical simulation, matlab-simulink I INTRODUCTION Permanent magnet linear machine generates linear motions directly without rotation to translation conversion mechanisms, which significantly simplifies system structure, and improves system working efficiency, dynamic response and control performance [1] In the field of wave energy conversion, the use of linear generator has earlier been regarded as difficult and uneconomical due to technical problems A brief description of various types of generators with their characteristics is listed here The first common and simplest linear generator is linear Faraday induction generator The disadvantage of this machine is that a high excitation current is required due to the low reactance of the winding This, in turn, reduces the efficiency of the machine [2,3] In a permanent magnets (PM) synchronous generator, the excitation is provided by the PMs The translator and magnetic fields move at the same speed, which is referred to as synchronous.[4-6] Another generator type is Linear PM 978-1-7281-6530-1/20/$31.00/ ©2020 IEEE Several machines have been designed with different magnetization patterns, which certainly improve the magnetic flux density and output power The magnetization patterns of PMs in linear generators have been designed to be radial, axial, Halbach, and quasi-Halbach [13-15] With all type of generators listed above, the output power of generators can be improved by optimize the geometry structures, increase magnetic flux density or winding turns number etc…To calculate and maximize output power, an analytical method or numerical method is usually applied In this study, a mixed numerical and analytical technique is presented to optimize linear generator design’s parameters for wave energy converter out power performance The magnetic field inside linear generator is examined by using the finite element method Then, the numerical results are used as initial Authorized licensed use limited to: Queens University Belfast Downloaded on May 17,2021 at 05:52:37 UTC from IEEE Xplore Restrictions apply parameters for a Matlab-simulink program to calculate the output power of the linear generator The optimal result will be drawn with proper parameters of linear generator II MAGNETIC FIELD INSIDE LINEAR GENERATOR Based on PM arrangement, magnetic field distribution in the machine is formulated with Laplace’s and Poisson’s equations analytically In this part, an examination of magnetic field depending on the magnet’s geometric dimensions is carried on, from that reasonable magnet’s size can be chosen to maximize the magnetic flux density inside generator We will study a six-sides linear generator model improves based on model was developed in project QG14.01 of VNU In formulation of the magnetic field, the generator space under study is divided into two regions bases on magnetic characteristics The air gap or coil space that has permeability of 1.0 is denoted as Region The permanent magnet volume filled with rare-earth magnetic material is denoted as Region The magnetic field property of Region and is characterized by the relationship between magnetic field intensity, H (in A/m) and flux density, B (in Tesla) as: ଵ ൌ ߤ ࡴଵ (1) ൌ ߤ ߤଵ ࡴ ߤ ࡹ (2) Fig Cross section and longitudinal section of the linear generator In order to solve Laplace and Poisson equations, generally we use numerical method or analytical technique However, the finite element method gives more accurate results than that of the analytical calculation especially in the solving field problems of complicated shape objects In case of a symmetrical problem with a simple shape, a 2D representation gives a sufficient result The computational simulations by finite element method (FEM) uses Ansys Maxwell tool to assist calculation Magnetic material in the simulation is NdFe35 with the following features (Table 1) Where μ0 is the permeability of free space with a value of 4S.10-7 H/m, μr is the relative permeability of permanent magnets, M = Brem/μ0 is the residual magnetization vector in A/m, and Brem is the remanence TABLE SPECIFICATIONS OF THE MAGNETIC MATERIAL The governing equations of magnetic field, i.e Laplace’s and Poisson’s equations, are significant for the solution of magnetic field The Gauss’s law for magnetisms is state that .Bi=0 where i = 1,2 Relative permeability 1.0997785406 Magnetic coercivity (A/m ) -890000 Bulk conductivity (Siemens/m) 625000 Remanence Br (Tesla) 1.23 (3) To validate simulation method, configurations of magnets in the generator are applied The first configuration is arranged as in [16], the second configuration is arranged as in Fig Therefore the equation can be written as ߘ ൈ ൌ െߘ ଶ Parameter Thus, we can have a magnetic vector potential, Ai, so that ൌ ߘ ൈ Magnet’s Specifications (4) In region 1, the combination of Maxwell’s equation and (1) gives ߘ ൈ ൌ ߘ ൈ ߤ ࡴ ൌ ߤ ࡶ (5) Substituting (4) into (5) yields ଶ ൌ െߤ ࡶ where J (A/m2) is current density in the field In permanent magnet J = 0, therefore the Laplace’s equation for Region is obtained as ߘ ଶ ൌ Ͳ Fig Scheme of dual Halbach arrays magnets In configuration 1, there is only a polarized array of magnets along the y direction spaced 7mm apart, and the magnets dimensions are: 25mm long, 10mm wide The magnet arranged in configuration has the polarizations of Fig 2, in which the magnets along the y direction have the same size as configuration The magnet configuration differs from the magnets by the presence of magnets polarizing along the X direction that fill the gap between the linearly polarized magnets The magnets are 7mm long, 10mm wide The distance between the two magnets is 16mm (6) For Region 2, the combination of Maxwell’s equation and (2) gives ߘ ൈ ൌ ߤ ߤଵ ࡶ ߤ ߘ ൈ ࡹ (7) Similarly, (4) and (7) yield the Poisson equation for Region ଶ ൌ െߤ ൈ ࡹ (8) Authorized licensed use limited to: Queens University Belfast Downloaded on May 17,2021 at 05:52:37 UTC from IEEE Xplore Restrictions apply The Fig shows that the magnetic flux density is increase when Halbach arrays structure is used The blue-dot line shows the magnetic flux density at the center line of generator which is introduced in VNU project-QG.14.01, and red line shows the magnetic flux density of generator when double Halbach arrays structure is used The maximum value of magnetic flux density at center can improved around 10.8% therefore the output performance can be significantly increase 0.67 B max Fitting line 0.665 B max (Tesla) 0.66 0.655 0.65 0.645 0.64 0.635 10 15 20 25 30 Length of X-axis polarized PMs b (mm) 35 40 Fig Magnetic flux density vs length b of X axis polarization From simulation results, we can choose value pairs based in length of magnet polarized vertically and horizontally so that the maximum magnetic flux value is obtained when the magnitude of the magnet along the Y axis is 32mm and the magnitude of the magnets along the X axis is 25 mm Fig Magnetic flux density at center of generator in two types For the purpose of increasing magnetic flux field, magnetic flux field is investigated with various length of Yaxis polarization and X-axis polarization PMs when the width is set at 10mm In the next part, we investigate the maximum value of magnetic flux density by changing the length of Y axis polarization magnets a (mm) from 10mm to 40mm, and its dependence on the fixed length b (mm) of X axis polarization magnets (Fig 4) The numerical simulation results show that magnetic flux’s density is symmetric about the central axis along the Ox direction and the periodic period Hence the distribution of the magnetic flux field along the Ox axis can be described by the equation: ʹߨ ߮௫ ሺݕሻ ൌ ߮ොሺ ݔሻ (9) ߣ With ߮ො is the value of the average maximum magnetic flux intensity from the face of the core close to the outside magnet surface, λ is the magnetic wavelength of the magnets structure Then the magnetic flux distribution field can be expressed as the sum of the sine and cosine functions whose period is equal to a multiple of the magnet angle frequency (Fig 6) 0.66 0.64 0.62 B max (Tesla) 0.6 B max Fitting line 0.58 0.56 ߮௫ ሺݕሻ ൌ ߮ො ൬ 0.54 0.52 0.5 (10) ܤ
ሺ݊߱ݔሻሽ 0.48 0.46 10 ʹߨ ݔ൰ ߣ ൌ ሼܣ ሺ݊߱ݔሻ 15 20 25 30 Length of Y axis polarized PMs a (mm) 35 40 Fig Magnetic Flux density vs length a of Y axis polarization The value of magnetic flux density increases as the magnitude of the magnets increases, and asymptotically approaches a value that cannot be increased Using this table, we can optimize the magnitude of the polarization length along the Y direction Next, with the magnitude of the magnets with the longitudinal polarization determined a = 32mm, we continue to investigate the change of B max with the size change of the horizontal polarization magnet (Fig 5) Fig Numerical solution and approximation function of magnetic field The maximum value of the magnetic field obtained through the above simulation results is used as the source signal for the calculation of the electromotive force in the generator Authorized licensed use limited to: Queens University Belfast Downloaded on May 17,2021 at 05:52:37 UTC from IEEE Xplore Restrictions apply Then L (Henry) is the inductance of the coil, calculated by the expression L = Po.P.N2.S/lống where S is the coil crosssection, Po is the permeability of the air, P is the coil core permeability, N is the number of turns, and lống is the length of the coil (Fig.8) III ELECTROMOTIVE FORCE AND POWER OF LINEAR GENERATOR To calculate the electromotive force and power of the linear generator, a Matlab-simulink program has been programmed to calculate more smoothly and quickly The resistance R of the winding is calculated using the formula ܴ ൌ ߩ where ρ is the resistivity of the conductor, ld Relative motion of the coils (connected to the first buoy) relative to the magnets (attached to the housing of generator and attached to the second buoy), calculated by the movement of the two buoys x(t)=s1(t)-s2(t) (s1(t) is the motion of the first buoy, s2(t) is the motion of the second buoy) However, to simplify this problem, the relative motion of the coil with the magnet is assumed to be periodic oscillation with given wave amplitude and frequency x(t)= s1(t)-s2(t)=dsin(wmt) Where d is the wave oscillation amplitude, ωm is the angular frequency of the collector buoy based on the wave interaction ௦ is the length of the conductor, sd is the cross-section of wire (Fig 9) The flux distribution of the magnets along the generator against the winding oscillations has the shape: ߮௫ ൌ ߮ො ቆ ʹߨ ݔሺݐሻቇ ߣ (11) ൌ ሺܣ ൫݊ݔݓሺݐሻ൯ ܤ
ሺ݊ݔݓሺݐሻሻሻ The sum of magnetic flux due to wire swept over area Ȱ ൌ ߮ ௫ ݔሶ ݀ݐ Fig Block diagram of the coil inductance When the winding is connected to an external circuit with pure resistance, according to Kirchhoff's law, the equation for the current of circuit generated by a winding coil has the form (Fig 10): Thus, a winding coil contains N wires when moving in magnetic field will produce an electromotive force, according to Faraday's law: ݀Ȱ (12) ݀ݐ With each selected magnet size and period, the crosssection area of the inductor will change, and a change in diameter wire will changes the turns number in winding coils ߝ ൌ െܰ L dI L (t ) dt RL e(t ) § R Ri à ă L RL I L (t ) L â L (13) The number of turns is calculated using the formula N = round(l/ddd) x round(p/ddd) where l is the magnet length, p is the magnet's period, and ddd is the diameter of the wire (Fig 7) Fig 10 Diagram generator with external circuit Consider a linear generator with the following specifications (Table 2) TABLE SPECIFICATION OF THE LINEAR GENERATOR Fig Cross section of winding coil Specifications Of The Linear Generator Magnet size (mm) Cross section of winding coil (mm) Wire’s diameter (mm) The resistivity of conductor (10-7 Ω.m) Rotor’s radius (mm) Coil numbers Magnet sides Relative permittivity (10-7 T.m/A) Parameter 12x30x50 12x60 0,7 1,72 53 4S The incident wave causes relative motion between inductor and magnets has the parameters (Table 3) Fig Block diagram to calculate the number of turns of wire and the resistance of an inductor Authorized licensed use limited to: Queens University Belfast Downloaded on May 17,2021 at 05:52:37 UTC from IEEE Xplore Restrictions apply TABLE INCIDENT WAVE PARAMETERS Waves Characteristics Amplitude wave (m) Wave period (s) decrease, and the resistance of the coil will increase In addition, the inductance of the coil will increase cause an increase in the coil resistance Parameter 0.5 Conversely, as the diameter of the wire increases, the number turns of winding in a phase decreases, resulting in a decrease in the total amount of the resulting flux as the coils move However, an increase in wire diameter results in a decrease in coil resistance and a decrease in magnetic resistance a) Currents sperately in winding coils Change the wire size parameters lead to the datasheet (Table 4): TABLE THE INFLUENCE OF WIRE’S DIAMETER ON WINDING PARAMETERS AND OUTPUT POWER The Influence Of Wire’s Diameter On Winding Parameters And Output Power b) The total electromotive forces in winding coils Wire’s diameter (mm) Turns number in a winding coil (rounds) Resistance of a winding coil (Ω) Inductance of a coil (10-5H) Power (W) 0,5 0,6 0,7 0,8 0.9 2000 1445 1125 858 720 540 58,343 29,273 16,744 9,777 6,483 3,938 6.660 4,812 3,746 2,857 2,398 1,798 46,939 69,053 86,036 87,465 86,748 67,488 From the results obtained, the wave energy conversion efficiency of the device can be enhanced when choosing the right value for the wire diameter In this case, the wire diameter = 0.8mm will give the max power (Fig 12) c) power line of generator 100 90 80 Power(W) 70 60 50 40 Fig 11 Result a) Currents sperately in winding coils b) The total electromotive forces in winding coils c) power line of generator 30 The calculation results are shown in the Fig.11 as the currents in coils (Fig 11a), the sum of electromotive forces in coils (Fig 11b) and the output power on external load (Fig 11c) 20 0.4 0.5 0.6 0.7 0.8 wire diameter(mm) 0.9 1.1 Fig 12 The output power with wire’s diameter IV OPTIMIZING WINDING COIL’S DIAMETER TO INCREASE V CONCLUSION OUTPUT POWER In order to overcome the disadvantages of low power linear generators in wave energy converters, we have proposed to use dual Hallbach arrays to enhance the flux for ironless linear generators Ansys program has been used for simulated the magnetic flux field in the generator The numerical results show that the magnetic flux strength of generator can be improved significantly when using dual Halbach array with proper magnet size parameter Then the electromotive and output power of generator are calculated analytical by a Matlab-simulink program From the result of simulink program, the output power can be maximized by Through the calculation example above, the change in winding parameters will gradually change the internal resistance of the generator, also change the number of turns of winding in a phase, and at the same time change the inductance of the winding As the diameter of winding coils decreases, the number of turns of the conductor in a phase increases, resulting in more magnetic flux being obtained as the coil moves, thereby increasing the electromotive force However, the diameter of a small wire leads to an increase in the length of the conductor in one phase, and at the same time the wire section will Authorized licensed use limited to: Queens University Belfast Downloaded on May 17,2021 at 05:52:37 UTC from IEEE Xplore Restrictions apply suitable wire’s diameter The optimal result will be tested with a protype generator in next research ACKNOWLEDGMENT This work has been supported/partly supported by VNU University of Engineering and Technology under project number CN20.15 REFERENCES [1] Introduction [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] of linear machine, https://en.wikipedia.org/wiki/Linear_motor R.E Phillips, “Linear faraday induction generator for the generation of electrical power from ocean wave kinetic energy and arrangements thereof,” Google Patents, 2014 N P Gargov, A F Zobaa and G A Taylor, “Direct drive linear machine technologies for marine wave power generation,” 47th IEEE Int Universities Power Engineering Conf (UPEC), London, UK, 2012, pp 1–6 H Polinder, M.E.C Damen and F Gardner, “Linear PM generator system for wave energy conversion in the AWS,” IEEE Trans Energy Convers., 2004, 19, (3), pp 583–589 X Sun, S Luo, L Chen, H Jiang and X Xu, “A linearization control scheme for permanent magnet 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Bolognani, D D Corte and F Tonel, "Tubular linear permanent magnet motors: an overall comparison," in IEEE Transactions on Industry Applications, vol 39, no 2, pp 466-475, March-April 2003, doi: 10.1109/TIA.2003.809444 D.T Ba, N D Anh and P V Ngoc, “Numerical Simulation and Experimental Analysis for a Linear Trigonal Double-Face Permanent Magnet Generator Used in Direct Driven Wave Energy Conversion,” ISSN: 1876 – 6169 Procedia Chemistry Volume 14, 2015, pp 130137 Authorized licensed use limited to: Queens University Belfast Downloaded on May 17,2021 at 05:52:37 UTC from IEEE Xplore Restrictions apply ... number of turns, and lống is the length of the coil (Fig.8) III ELECTROMOTIVE FORCE AND POWER OF LINEAR GENERATOR To calculate the electromotive force and power of the linear generator, a Matlab-simulink.. .parameters for a Matlab-simulink program to calculate the output power of the linear generator The optimal result will be drawn with proper parameters of linear generator II MAGNETIC... INFLUENCE OF WIRE’S DIAMETER ON WINDING PARAMETERS AND OUTPUT POWER The Influence Of Wire’s Diameter On Winding Parameters And Output Power b) The total electromotive forces in winding coils Wire’s diameter
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