Optimizing linear generator design’s parameters for output power using mix numerical and analytical technique45069

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

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

Dang The Ba

Faculty of Engineering Mechanics and

Automation

University of Engineering and

Technology,VNU

Hanoi, Vietnam

e-mail: badt@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

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

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

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



Proceedings of the 2020 International Conference on Advanced Mechatronic Systems, Hanoi, Vietnam, December 10 - 13, 2020

978-1-7281-6530-1/20/$31.00/ ©2020 IEEE

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

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 1 The permanent magnet volume

filled with rare-earth magnetic material is denoted as Region

2 The magnetic field property of Region 1 and 2 is

characterized by the relationship between magnetic field

intensity, H (in A/m) and flux density, B (in Tesla) as:

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

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

Thus, we can have a magnetic vector potential, Ai, so that

Therefore the equation can be written as

In region 1, the combination of Maxwell’s equation and

(1) gives

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 1 is obtained

as

For Region 2, the combination of Maxwell’s equation and

(2) gives

Similarly, (4) and (7) yield the Poisson equation for

Region 2

Fig 1 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)

TABLE 1 SPECIFICATIONS OF THE MAGNETIC MATERIAL Magnet’s Specifications Parameter Relative permeability 1.0997785406

Bulk conductivity (Siemens/m) 625000 Remanence Br (Tesla) 1.23

To validate simulation method, 2 configurations of magnets in the generator are applied The first configuration

is arranged as in [16], the second configuration is arranged as

in Fig 2

Fig 2 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 2 has the polarizations of Fig 2, in which the magnets along the y direction have the same size as configuration 1 The magnet configuration 2 differs from the magnets 1 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



The Fig 3 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

Fig 3 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

Y-axis polarization and X-Y-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)

Fig 4 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 5 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

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:

߮௫ሺݕሻ ൌ ߮ො•‹ሺʹߨ

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)

߮௫ሺݕሻ ൌ ߮ො •‹ ൬ʹߨ

ߣ ݔ൰

൅ ܤ௡…‘•ሺ݊߱ݔሻሽ

(10)

Fig 6 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

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

Length of Y axis polarized PMs a (mm)

B max Fitting line

0.635 0.64 0.645 0.65 0.655 0.66 0.665 0.67

Length of X-axis polarized PMs b (mm)

B max Fitting line



III ELECTROMOTIVEFORCEANDPOWEROFLINEAR

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

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

The flux distribution of the magnets along the generator

against the winding oscillations has the shape:

߮௫ ൌ ߮ො •‹ ቆʹߨ

ߣ ݔሺݐሻቇ

ൌ ෍ሺܣ௡•‹൫݊ݓݔሺݐሻ൯

൅ ܤ௡…‘•ሺ݊ݓݔሺݐሻሻሻ

(11)

The sum of magnetic flux due to 1 wire swept over area

Ȱ୺ൌ ׬ ߮௫ݔሶ݀ݐ

Thus, a winding coil contains N wires when moving in

magnetic field will produce an electromotive force, according

to Faraday's law:

ߝ ൌ െܰ݀Ȱ஻

With each selected magnet size and period, the

cross-section area of the inductor will change, and a change in

diameter wire will changes the turns number in winding coils

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 7 Cross section of winding coil

Fig 8 Block diagram to calculate the number of turns of wire and the

resistance of an inductor

Then L (Henry) is the inductance of the coil, calculated by the expression L = Po.P.N2.S/lống where S is the coil cross-section, 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)

The resistance R of the winding is calculated using the formula ܴ ൌ ߩ௟೏

௦ ೏ where ρ is the resistivity of the conductor, ld

is the length of the conductor, sd is the cross-section of wire (Fig 9)

Fig 9 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):

( )

L L



Fig 10 Diagram generator with external circuit

Consider a linear generator with the following specifications (Table 2)

TABLE 2 SPECIFICATION OF THE LINEAR GENERATOR Specifications Of The Linear

Generator

Parameter

Cross section of winding coil (mm) 12x60

The resistivity of conductor (10 -7 Ω.m) 1,72

Relative permittivity (10 -7 T.m/A) 4 S

The incident wave causes relative motion between inductor and magnets has the parameters (Table 3)



TABLE 3 INCIDENT WAVE PARAMETERS

Waves Characteristics Parameter

a) Currents sperately in 3 winding coils

b) The total electromotive forces in winding coils

c) power line of generator

Fig 11 Result a) Currents sperately in winding coils b) The total

electromotive forces in winding coils c) power line of generator

The calculation results are shown in the Fig.11 as the

currents in 3 coils (Fig 11a), the sum of electromotive forces

in 3 coils (Fig 11b) and the output power on external load

(Fig 11c)

IV OPTIMIZING WINDING COIL’S DIAMETER TO INCREASE

OUTPUT POWER

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

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

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

Change the wire size parameters lead to the datasheet (Table 4):

TABLE 4 THE INFLUENCE OF WIRE’S DIAMETER ON WINDING

PARAMETERS AND OUTPUT POWER The Influence Of Wire’s Diameter On Winding Parameters And

Output Power Wire’s

diameter (mm)

Turns number in a winding coil (rounds)

Resistance

of a winding coil (Ω)

Inductance of

a coil (10- 5 H)

Power (W)

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)

Fig 12 The output power with wire’s diameter

V CONCLUSION

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

20 30 40 50 60 70 80 90 100

wire diameter(mm)



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

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