Energy Conversion and Management 52 (2011) 2712–2723

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Analysis of brushless DC generator incorporating an axial field coil Hassan Moradi ⇑, E. Afjei Department of Electrical and Computer Engineering, Shahid Beheshti University, GC, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 6 April 2010 Received in revised form 27 January 2011 Accepted 29 January 2011 Available online 29 March 2011 Keywords: BLDC generator Without permanent magnet Finite element analysis Boundary element analysis

a b s t r a c t This paper describes the magnetic analysis and experiment of a three-phase field assisted brushless DC (BLDC) generator. Unlike conventional BLDC generators, the permanent magnet is replaced with an assisted field winding. The stator and rotor are constructed with two dependent magnetically sets, in which each stator set includes nine salient poles with coil windings, and the rotor comprises of six salient poles. Other pole combinations also are possible. This construction is similar to a homopolar inductor alternator. The DC current in the assisted field winding produces axial flux which makes the rotor magnetically polarized at its ends. The magnetic field flows axially through the rotor shaft and closes through the stator teeth and the machine housing. To evaluate the generator performance, two types of analysis, namely the numerical technique and the experimental study have been utilized. In the numerical analysis, 2-D finite element (FE) analysis has been carried out using a MagNet CAD package (Infolytica Corporation Ltd.), to confirm the accuracy of the predicted flux-linkage characteristics, whereas in the experimental study, a prototype BLDC generator was constructed for verifying the actual performance. Furthermore, the evaluation method based on a hybrid numerical method coupling the finite element (FE) analysis and boundary element (BE) method, has been carried out to confirm the accuracy of the 2-D FE analysis simulation results. It provides not only confirmations of the investigation in results but also exact illustration for magnetic field distribution for this complex generator geometry. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Variable speed permanent magnet brushless DC (BLDC) machines offer many advantages including compact form, high efficiency, robustness to harsh environment, low maintenance and easy manufacturability [1,2]. BLDC machines are used in industries for different applications such as, wind energy, automotive, aerospace, home appliances and many industrial equipment and instrumentation [3,4]. Fundamentally, a BLDC generator produces out-put power characteristics very similar to the classical separately excited DC generator. The stator of a BLDC generator consists of stacked steel laminations with windings placed in the slots that are axially cut along the inner periphery or around stator salient poles. In the conventional BLDC generator, excitation is provided by permanent magnets mounted on a solid iron rotor [5]. This structure has some inherent disadvantages such as: loss of flexibility of field flux control, changing the magnetic properties of permanent magnets when subjected to external magnetic fields and/or temperature changes, limited operating-speed range and high cost of the high flux density permanent magnets. These problems have been addressed by many researchers [6,7].

⇑ Corresponding author. Tel.: +98 9122937599. E-mail addresses: [email protected], [email protected] (H. Moradi). 0196-8904/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2011.01.027

In the case of variable speed application likes to wind energy and automotive applications [8], when conventional permanent magnet BLDC machine are used as generators, the output voltage will be of variable magnitude and frequency. Electronic converters are then necessary to obtain a regulated output voltage. The initial and maintenance costs of these converters are high [9]. Furthermore, wind energy systems based on synchronous generators [10,11], and also induction machines, especially the cage type [12,13] are widely used. For these kinds of electrical machines for a wide range of speed variation, an electronic converter as an interface between the generator and the grid, and also a voltage regulator to control the magnitude of the output voltage are needed. This is because of their structure with inherent inflexibility of field flux control. For such applications a machine with capability of field flux control to hold the output voltage at a desirable level for different speeds are suggested. This paper presents characteristic analysis of a field assisted Salient-Pole BLDC generator, which does not use a permanent magnet in the rotor. The generator configuration is modified and designed based on having double layers of the stator/rotor and a stationary field assisted coil placed between them for achieving maximum output voltage as well as power for a certain machine volume. The electric machine is operated as an electrical generator by rotating the shaft, varying the amplitude and direction of current in the assisted coil, and extracting electrical energy from the stator winding. Unlike a permanent magnet generator, the field

H. Moradi, E. Afjei / Energy Conversion and Management 52 (2011) 2712–2723

assisted coil brushless DC generator does not require complex circuitry for regulation output voltage, and it does not require a parasitic load for damping excess energy. For the two layers field assisted machine, the absence of windings and permanent magnets on the rotor support both high rotational speeds and high-temperature operation. Furthermore, the absence of permanent magnets on the machine structure reduces the machine cost. Varying the amplitude and direction of the current in the field coil can control the output voltage to a level between zero volts and the maximum voltage. Unlike a permanent magnet generator, the field assisted coil brushless DC generator does not require complex circuitry for regulation output voltage. The nature of this configuration makes it compatible with any application that requires variablespeed operation. Complex geometry and non-linear properties of the proposed Salient-Pole structure is the main reason for calculation and analysis of the flux distribution inside the machine for different excitation currents and rotor positions. So, an accurate knowledge of the magnetization characteristics is essential for the prediction and evaluation of machine performance. In the numerical analysis, 2-D FE analysis has been carried out to confirm the accuracy of the predicted flux-linkage characteristics. Furthermore, the evaluation method based on a hybrid numerical method coupling the FE analysis and boundary equation (BE) method, has been carried out to confirm the accuracy of the 2-D FE analysis simulation results. For the experimental determination of the magnetic characteristics of BLDC machine, the flux density are measured in the teeth, furthermore the Output parameters of generator are measured for No-Load and under-load by different speed. This paper is managed the following manner. Section 2 describes the principle of operation of the presented BLDC generator. Section 3 describes numerical analysis of magnetic field for BLDC generator. Next section describes methodology evaluation for flux density distribution. Section 6 puts forward an algorithm for design procedure of presented machine. Detailed experimental results on the prototype are presented in Section 7. Finally, conclusion is presented.

2. Two layer BLDC generator discription The proposed Field assisted generator can be considered the dual of the BLDC motor that is presented in [14,15], although there are some important differences in control objectives and control implementation also in the field and phases winding. The BLDC generator of Fig. 1 has steel laminations on the rotor and stator. Fig. 1 shows a structure and a field flux path of proposed configuration ((a) 3-D field flux path, (b) 3-D cut view and (c) 2-D front view of proposed configuration by magnetic flux path). The stator and rotor are constructed with two dependent magnetically sets, in which the two sets are exactly symmetrical with respect to a plane perpendicular to the middle of the machine shaft. Each layer consists of nine stator poles and six rotor poles, respectively. Every stator and rotor pole arcs are 30°. This is a three-phase machine, therefore, three coil windings from one layer is connected in series with the other three coil windings in the other layer. There are concentrated windings placed around each salient pole on the stator. The coils around the individual poles are connected to form the phase windings. Number of turns for each phase winding is calculated and optimized for generator mode and a wire with suitable diameter for up to 10 A load current is utilized. There is a stationary reel by a rotating cylindrical core, which has the field coils wrapped around it and is placed between the two-stator sets. The DC current in the assisted field coil produces an axial field which makes the ends of the rotor a north and south magnetically (Fig. 1c). The magnetic flux produced by the coils

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travels through the guide and shaft to the rotor and then to the stator poles, and finally closes itself through the machine housing (Fig. 1c). It is worth mentioning that, the number of stator poles and their configuration is completely different than that of the switched reluctance generator. A cut view and 3-D view of the machine are shown in Fig. 1a and b respectively. The basic dimensions are: a rotor pole arc of 30°, a stator pole arc of 30° (rotor and stator pole arc are like to BLDC motor presented in [14,15]), and an air-gap length of 0.6 mm. For ease of manufacture, the diameter of the shaft is chosen to be 9 mm, the outer diameter of the rotor is 59.4 mm, and the length of the stator pole is 15 mm. To sum up the above design, the geometric parameters, which are summarized as in Table 1, appear to be reasonable for the constraints previously described. The suggested configuration is to some extent similar to the switched reluctance (SR) machine, but it is worth mentioning that, the number of stator and rotor poles and their configuration, phases overlapping region, number of phases that can be excited simultaneously and commutator to excite the phases is completely different than that of the SR machine [16,17]. As mentioned before, for the proposed machine, the absence of windings and permanent magnets on the rotor support both high rotational speeds and high-temperature operation. Furthermore, the absence of permanent magnets on the machine structure reduces the machine cost. Varying the amplitude and direction of the current in the field coil can control the output voltage to a level between zero volts and the maximum voltage. Unlike a permanent magnet generator, the field assisted coil brushless DC generator does not require complex circuitry for regulation output voltage. The nature of this configuration makes it compatible with any application that requires variable-speed operation. 3. Numerical analysis of magnetic field for BLDC generator Because of inherent non-linear magnetic characteristics and the doubly Salient-Pole structure of presented two layers BLDC machine, numerical methods must be used for the calculation of the magnetic field distribution and the prediction of the machine magnetization characteristics. So it’s an essential part of design procedure [18]. The FE method, which has its root in variational calculus, and is a very useful engineering numerical method for the solution of boundary value problems defined over irregular boundaries. The method can be applied to solve both linear and non-linear problems, but is particularly powerful for non-linear problems that involve the field variables defined within a computational domain [19]. Due to FEM being a well known numerical method for field solution and design validation of Salient-Pole machine by complex geometry and non-linear properties [20]; it is used as efficient tools for accurate evaluation of the machine performance in the proposed configuration. 3.1. Field distribution background Complex geometry and non-linear properties of the Salient-Pole field assisted structure is main reason for calculation and analysis of the flux distribution inside the machine for different excitation currents and rotor positions. So, an accurate knowledge of the magnetization characteristics is essential for the prediction and evaluation of Machine performance. In this section, field distribution analysis background for the proposed BLDC machine is presented. 3.2. Magnetic field theory There are two common methods for solving magnetic field problems. One utilizes magnetic vector potential A, and the other

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Fig. 1. Structure and a field flux path of proposed configuration machine.

employs electric vector potential T. For the first method, 2-D magnetic field problems can be expressed by just one component of the vector potential A. Magnetic vector potential V can be derived using the following quasi-static Maxwell’s equation [19,20]:

~ ¼~ Curl H J; ~ J ¼ ez j and Div B ¼ 0

ð1Þ

where ~ J is the current density vector in the z direction. The magnetic flux density ~ B and potentials are related by:

Curl ~ A ¼~ B Bx ¼

@/ ; @y

ð2Þ By ¼

@/ @x

ð3Þ

By substituting ~ B in these equations the vector potential is governed by:

r  ðcr  AÞ ¼ j

ð4Þ

H. Moradi, E. Afjei / Energy Conversion and Management 52 (2011) 2712–2723 Table 1 Geometric parameters of proposed configuration. Symbol

Meaning

Value of unit

Machine parameters Rso Rsi Rro Rri Rsf Lg Lstk bs br ns nr q N

Stator outer diameter Stator inner diameter Rotor outer diameter Rotor inner diameter Shaft diameter Airgap Stack length Stator pole angle Rotor pole angle Stator pole number Rotor pole number Number of phases Number of turns per phase

80 mm 60 mm 59.4 mm 40 mm 9 mm 0.6 mm 146 mm 30° 30° 9 6 3 110 Turns

whereas using the T–X method, T can be simplified to produce a solution with only two components of T. Due to complex structure and inherent non-linear properties of the magnetic materials for the presented two layers field assisted BLDC machine, using of the numerical techniques such as FEM for the calculation of the magnetic field according to mentioned equations is suggested. The finite element method is used most often because of its ability to take magnetic saturation and complex geometries into account. 3.3. Two dimensional FEM analyses

where the magnetic reluctivity c is a non-linear function of the flux density in the iron part, and is taken to be constant, equal to that of air, through the nonferrous parts of the magnetic circuit. Neglecting end effect in a 2-D analysis, the magnetic vector potential satisfies the following equation, dependent on the region:  Non-magnetic, non-conducting region (airgap):

r2 A ¼ 0

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ð5Þ

FE technique gives precise information of the machine parameters such as magnetic flux density, inductances and electromagnetic torques based on magnetic field calculation using machine geometry dimensions and materials. FEM is capable to consider the magnetic field saturation effect based on machine performance. For 2-D analysis of proposed machine, only one side of the machine which is symmetric to the field coil is considered. The partial differential equation for the magnetic vector potential is [24]:

    @ @A @ @A þ ¼ jðx; yÞ c c @x @x @y @y

ð16Þ

 Stator windings:

r 2 A ¼ l0 j

ð6Þ

 Stator  and  rotoriron:

@ @A @ @A þ ¼ jðx; yÞ c c @x @x @y @y

ð7Þ

The magnetic vector potential can be computed by taking into consideration the saturation effects of the isotropic ferromagnetic stator and rotor material, or No. In T–X method, electric vector potential known as T–X formulation in which T defined as [21,22]:

j¼rT

ð8Þ

From Maxwell’s equation we have:

jþrT ¼rH

ð9Þ

Then:

r  ðH  TÞ ¼ 0

ð10Þ

Since the vector H–T can be expressed as the gradient of a scalar, i.e.,

H ¼ T  rX

ð11Þ

where X is a magnetic scalar potential. And, since

rE¼

@B @t

Then,

rE¼r

ð12Þ   1

r



  @B @ ðT  rXÞ ¼ l0 lr @t @t 

rT ¼

   @T @X r ¼ l0 lr @t @t

ð13Þ

which finally reduces to the following two scalar equations [23]:

    @T @X ¼ lrr @t @T

r2 T  lr

ð14Þ

And

r2 X ¼ 0

ð15Þ

When a two dimensional magnetic field problem is solved by A and V, the need to solve for all the two components of A arises,

where j is the current density value in the element under consideration (in A/m2), A is magnetic vector potential (in Wb/m). As mentioned before, magnetic flux density is defined as:

B¼rA

ð17Þ

In the variational method (Ritz) the solution tool is obtained by minimizing the following function [25]:

1 FðAÞ ¼ 2

Z Z Xe

 2  2 ! Z Z @A @A c þc AJdXe dXe  @x @y Xe

ð18Þ

where X is the problem region of integration. It is worth mentioning that, the natural Neumann condition @A ¼ 0 is applied to the contribution of all finite elements, except @n those sharing a side with the airgap ring. The contribution of a finite element ^e sharing a side with a circular boundary of the airgap is given by the following expression:

Z Z Xe

 2  2 ! Z @A @A @A dXe  c þc cA dC^e @x @y @n Xe

ð19Þ

The detailed derivation was presented in [24]. In the 2-D FE analysis, the cylindrical stator and rotor domains were meshed using eight-node quadrilateral and six-node triangular finite elements. It is mentioned that dense meshes are assigned to places where the field variation are significant. The field analysis has been performed using a MagNet CAD package (Infolytica Corporation Ltd., 2007). MagNet CAD package uses the finite element technique for an accurate and quick solution of Maxwell’s equations [26]. Each module is tailored to simulate different types of electromagnetic fields and is available separately for both 2-D and 3-D designs and also, analysis is based on the variational energy minimization technique to determine the magnetic vector potential. It should be mentioned that, this package utilizes T–X method which is described in last section. To discretize the functional (10), the interior region Xé is subdivided into N small triangular or quadrilateral elements, and consequently, the fictitious boundary C is broken into Ms short segments. On the assumption that the field distribution within each element is linear, or quadratic, or higher order function, the field of the eth element can be expressed as [25]:

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Ae ðx; yÞ ¼

3n X

Nei ðx; yÞAei

ð20Þ

i¼1

where Nei ðx; yÞ is the interpolation function, Aei represents the nodal field and n stands for the order of the interpolation function. The machine parameters such as magnetic flux density, inductances and electromagnetic torques are essential parameters for prediction of presented structure dynamic model, which all of these parameters can be computed based on magnetic field obtained from FE analysis. First the stator pole flux waveform is obtained. Then flux waveforms in all parts of the machine can be calculated. But, for calculation of the stator pole flux and current, voltage equation and static flux-linkage characteristic are used. Based on the energy conversion, co-energy is the integral of the flux-linkage versus current and torque is calculated from derivative of co-energy versus rotor angle. Based on flux calculation, flux linkage k, magnetic energy or co-energy Wco and torque T can be computed as follow [6]: The flux linkage in each phase is:

w¼

Z 1 ~~ jAdv i v

ð21Þ

which, after finite-element discretization, becomes:

w¼

n Nl X A k Sk s k¼1

ð22Þ

taking into account the machine axial length l, the number of turns N per phase, and the area S for phase winding. Magnetic co-energy W co ðh; iÞ can be calculated on the basis of flux linkage wðh; iÞ as:

W co ðh; iÞ ¼

Z

i

wðh; iÞdijðh ¼ const:Þ

ð23Þ

0

The most general expression for the torque produced by one phase is calculated from the co-energy derivative with respect to h angular position is given by:

Tðh; iÞ ¼

@W co ðh; iÞ jði ¼ const:Þ @h

ð24Þ

The detailed derivation was presented in [6]. A number of assumptions have been made to simplify the FEM modeling problem. The computed quantities were assumed to remain constant when considering different sections of the machine, thus allowing the problem to be solved with 2-D analysis. It has also been assumed that the materials of which the machine is made are isotropic. Regarding the ferromagnetic material characteristics of the

BLDC generator, it becomes necessary to specify its B–H curve, which includes magnetic saturation, to obtain a reliable FEM model. 4. Numerical results As mentioned before, MagNet CAD package has been used to solve non-linear Poisson’s equations and therefore to obtain the magnetic vector potential of each node element in the proposed configuration. The magnetic field distribution in the machine has been obtained from the computed values. For pre-processing stage of FEM analysis, the geometrical model, physical material properties, mesh generation are required. The boundary conditions have to be set up with geometrical model. The geometry section of the BLDC machine designed in MagNet CAD can be automatically discretized by adjusting the dimensions of the finite elements through a smart size option (according to the local machine geometry). The mesh grid size around the airgap must be smaller than the other adjacent regions, which allows greater mesh refinement, where one expects a higher degree of changing the magnetic quantities (Fig. 2a). In this study 2-D FEM model consists of 6520 elements and 16,120 meshes. The next step consists of specifying the phase current for which a solution is to be obtained. This is done by applying a current density on the elements that correspond to the windings of a certain phase. The value of the current density to apply is given by the quotient between the excitation current and the section area of one wire of the winding. The field excitation current values used to obtain the model are between 0.5 up to 2 A, in 0.5 A increments. A density current signal j must be defined in such a way that one establishes the direction and magnitude of the magnetic field in the respective stator pole. Besides, it is also necessary that the magnetic field direction generated by both phase windings be coherent. Once a solution is obtained, i.e., the magnetic potential vector distribution along the machine is known, it is possible to calculate the value of the magnetic flux in each phase. One side of the machine cross section is shown in Fig. 2b. As seen from this figure, this machine has nine stator poles as well as six rotor poles. To evaluate the proposed configuration performance, 2-D FE analysis with apply normal field boundary conditions over the inner and outer borders of the machine has been utilized. Magnetic field density contour and arrows of the machine that is utilizing 2-D FE analysis are shown in Figs. 3–8. In this study, several geometries have been set to be analyzed, one for each rotor angular position h. This is defined as the angle between a certain pole stator (which is taken as reference) and one of the rotor poles. The angular positions considered between 30° and +30° in 1.5° increment

Fig. 2. Cross section and mesh generation the studied BLDC machine.

H. Moradi, E. Afjei / Energy Conversion and Management 52 (2011) 2712–2723

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Fig. 3. Magnetic field density contour and arrows of the machine that is utilizing 2-D FE analysis full-align position, field current is 0.5 A.

Fig. 4. Magnetic field density contour and arrows of the machine that is utilizing 2-D FE analysis non-align position, field current is 0.5 A.

Fig. 5. Magnetic field density contour and arrows of the machine that is utilizing 2-D FE analysis half-align position, field current is 0.5 A.

steps. This covers the nonaligned, half-aligned and aligned sequence between machine poles. 2-D FE analysis for different value of assisted field current has been utilized. In this study, only assisted field considered to be turned on and having 0.5 up to 2 A, in 0.5 A increments. Figs. 3–8 ((a) arrows plot and (b) contour plot) show the magnetic field density for aligned, half-aligned and nonaligned cases for a field current of 0.5 A and 1 A. It is worth mentioning here that, the stator and rotor cores are made of a non-oriented silicon steel lamination. The magnetization curve is taken from the manufacturer’s data sheet for M-27 steel.

When an electrical current passed through the assisted field coil, an axial magnetic field is produced. The magnetic flux produced by the coil travels through the guide and shaft to the rotor and then to the stator poles, and finally closes itself through the machine housing. Produced magnetic flux moved through the three-phases winding, and a voltage is induced in the phases winding which value of induced voltage dependent to the angle between the stator pole (which is taken as reference) and the rotor pole. Result of computed flux linkage of 2-D magnetic field in the proposed machine for different phases winding are presented in Table 2. The obtained results in Table 2 are related to field current 0.5 A.

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Fig. 6. Magnetic field density contour and arrows of the machine that is utilizing 2-D FE analysis full-align position, field current is 1 A.

Fig. 7. Magnetic field density contour and arrows of the machine that is utilizing 2-D FE analysis non-align position, field current is 1 A.

Fig. 8. Magnetic field density contour and arrows of the machine that is utilizing 2-D FE analysis half-align position, field current is 1 A.

As mentioned before, to evaluate the proposed configuration performance, 2-D FE analysis with applies normal field boundary conditions over the inner and outer borders of the machine has been utilized. The angular positions considered between 30° and +30° in 1.5° increment steps. This covers the nonaligned, half-aligned and aligned sequence between machine poles. Results of computed flux linkage in the proposed BLDC generator for field current 1 A for different phases winding are presented in Table 3. 5. Methodology evaluation for flux density distribution The FEM is used most often because of its ability to take magnetic saturation into account. However, it sometimes presents

some implementation difficulties, in particular, with the mesh generation in the boundaries such as airgap, which is very small in comparison to the main dimensions of the stator and rotor [25]. Since the BE method, permits formulation of field variables along the boundaries only, then it can be very efficient for linear boundary value problems and for non-linear problems with nonlinearity occurring along the boundaries [27]. Therefore, a hybrid computational methodology based on coupling the FE and BE for the calculation of the magnetic field can be very efficient to confirm the accuracy of the 2-D FE analysis simulation results. It provides not only confirmations of the investigation results but also exact illustration for magnetic field distribution for this complex configuration [28]. This section describes a coupled FE and BE computational methodology for the solution the magnetic field problems. In

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H. Moradi, E. Afjei / Energy Conversion and Management 52 (2011) 2712–2723 Table 2 Results of computed flux linkage of 2-D magnetic field in the proposed BLDC generator. Phase A

Phase B

Flux linkage (Wb), field current = 0.5 A 0.00337 0.009212 0.00408 0.008348 0.00483 0.007877 0.00528 0.007557 0.00572 0.007310 0.00610 0.007165 0.00629 0.006818 0.00659 0.006505 0.00685 0.006197 0.00710 0.005889 0.00741 0.005512 0.00768 0.005105 0.00807 0.004488 0.00861 0.003375 0.00903 0.003907 0.00888 0.004601 0.00860 0.005057 0.00868 0.005607 0.00858 0.006080 0.00854 0.006274 0.00852 0.006538 0.00857 0.006831 0.00855 0.007116 0.00858 0.007347 0.00863 0.007630 0.00877 0.007960 0.00888 0.008400 0.00898 0.009336 0.00836 0.009033 0.00779 0.008710 0.00762 0.008641 0.00733 0.008576 0.00694 0.008446 0.00666 0.008431 0.00636 0.008420 0.00610 0.008463 0.00590 0.008631 0.00551 0.008653 0.00512 0.008713 0.00436 0.008995 0.00326 0.009315

Phase A 0.00279 0.00340 0.00366 0.00495 0.00464 0.00495 0.00540 0.00564 0.00651 0.00670 0.00733 0.00787 0.00866 0.00926 0.00956 0.00904 0.00917 0.00852 0.00856 0.00827 0.00840 0.00860 0.00819 0.00821 0.00866 0.00918 0.00947 0.00956 0.00873 0.00823 0.00740 0.00687 0.00693 0.00590 0.00551 0.00530 0.00443 0.00427 0.00434 0.00266 0.00288

Table 3 Results of computed flux linkage of 2-D magnetic field in the proposed BLDC generator.

Rotor angle (°)

Phase A

0.00 1.50 3.00 4.50 6.00 7.50 9.00 10.5 12.0 13.5 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 37.5 39.0 40.5 42.0 43.5 45.0 46.5 48.0 49.5 51.0 52.5 54.0 55.5 57.0 58.5 60.0

Flux linkage (Wb), field current = 1 A 0.00279 0.009973 0.00340 0.008927 0.00366 0.007986 0.00495 0.007722 0.00464 0.006915 0.00495 0.006646 0.00540 0.006107 0.00564 0.005426 0.00651 0.005033 0.00670 0.004463 0.00733 0.004318 0.00787 0.003742 0.00866 0.003079 0.00926 0.002798 0.00956 0.002986 0.00904 0.003101 0.00917 0.003465 0.00852 0.004400 0.00856 0.005015 0.00827 0.005125 0.00840 0.005403 0.00860 0.006534 0.00819 0.006421 0.00821 0.006796 0.00866 0.007352 0.00918 0.008019 0.00947 0.009071 0.00956 0.009930 0.00873 0.009511 0.00823 0.009273 0.00740 0.008710 0.00687 0.008613 0.00693 0.008727 0.00590 0.008649 0.00551 0.008376 0.00530 0.008320 0.00443 0.008153 0.00427 0.008631 0.00434 0.009080 0.00266 0.009161 0.00288 0.009864

Phase B

Phase A

Rotor angle (°)

0.009455 0.009445 0.009120 0.008897 0.008694 0.008502 0.008429 0.008515 0.008538 0.008511 0.008648 0.008747 0.009347 0.009506 0.008927 0.008143 0.007416 0.006929 0.006338 0.005802 0.005552 0.005645 0.004784 0.004515 0.003963 0.003481 0.003066 0.002462 0.002386 0.003807 0.00400 0.004287 0.005590 0.005183 0.005590 0.006092 0.006399 0.007119 0.007770 0.008671 0.009401

0.00 1.50 3.00 4.50 6.00 7.50 9.00 10.5 12.0 13.5 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 36.0 37.5 39.0 40.5 42.0 43.5 45.0 46.5 48.0 49.5 51.0 52.5 54.0 55.5 57.0 58.5 60.0

what follows, the FE–BE formulation of the magnetic field problems is described, along with the detailed procedure for the implementation of the FE–BE coupling. A stability analysis of these algorithms shows superior accuracy and convergence characteristics in comparison with conventional FEM formulations.

the results obtained are more stable than those of the finite element method. The contribution of all boundary elements to the magnetic field at a given point Kðxk ; yk Þ, in terms of the magnetic vector potential, is given by the following relationship [27,28]:

5.1. BE method formulation

C k Ak ¼

As mentioned before, BE method is based on the integral equation formulation of the fields and defines unknowns only on boundaries. This leads to a reduction of complexity in modeling and solving the problem. The boundary element method is applied to the all boundary regions, such as airgap which is an interface domain between the stator and rotor and on the external stator outline. These boundary regions are linear regions by constant magnetic permeability l0 which are geometrically approximated by three-node boundary elements extracted from the stator and rotor finite element meshes [25]. Enforcing prescribed boundary conditions, an equivalent source that would sustain the field is found with the use of a free space Green’s function. The effect of the equivalent source at any point on the boundary is found from the source location and Green’s function [27]. Computation of the potential and field at an arbitrary point involves direct integration of the equivalent source thus obtained. Since no interpolation is involved in these computations,

 X Z  @A @G A dCe G @n @n Ce e   XZ 1 @ 1  ln 1r 1 @A dCe  A 2p ¼  ln @n r @n Ce 2p e

ð25Þ

G is a Green function and r is the distance from the given point Kðxk ; yk Þ to a current point (x, y) on the airgap boundary. The nodal approximations of the magnetic vector potential A and its normal derivative @A on a quadratic boundary element are [29]: @n

A¼

s X i¼1

Ni ðnÞ  Ai ;

  s @A X @A ¼ Ni ðnÞ  @n @n i i¼1

ð26Þ

The application of this formulation to all boundary nodes, using adequate treatment, generates a linear boundary-element system of equations. The detailed derivation was presented in [27,28]. The resulting BE formulation matrix is dense and nonsymmetrical, which increases storage requirements and slows the convergence rate for

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Table 4 Results of coupled FE and BE method for computation of 2-D magnetic field in the proposed BLDC. Magnetic field

Bx(max)

By(max)

Bxy(max) (T)

Aligned Un-aligned

0.818 0.082

0.381 0.013

0.902 0.082

Table 5 Comparison of FEM simulation and hybrid field solution for the proposed BLDC.

Aligned Un-aligned

FEM Bt(max) (T)

Bxy(max) (hybrid method) (T)

Tolerance (%)

0.902 0.082

0.933 0.088

3.3 6.8

iterative solvers. It requires a preconditioning matrix more robust than the block Jacobi preconditioner to improve the convergence rate. The function of a preconditioning matrix is to make the origi-

nal matrix more diagonal-like, thus improving the convergence rate. 5.2. Coupling of finite and boundary elements While there are many different ways to couple the boundary and finite element methods, the simplest and yet natural way is to make use of the physical constraints for the flux and field variables along the common boundaries of the FE and BE regions. This approach will be taken in this paper. The coupling of the FE and BE formulations is done by expressing the continuities of the magnetic vector potential and of the tangential component of the magnetic field on the airgap interfaces. The consideration of the Dirichlet boundary condition on the external stator outline leads to a non-linear system of equations solved using the Newton–Raphson algorithm [28]. The hybrid coupling method is applied to the calculation of the magnetic field in proposed BLDG and its result presented in Table 4. By applying Eqs. (25) and (26) to our proposed structure, 2-D magnetic field distribution was calculated and developed under Matlab software ver. 2010 (MathWorks, Co., 2010) [30]. The obtained results are presented in Table 4. The results obtained by the coupled FE and BE Method are compared with the results based on FEM and indicates an excellent conformity which is shown in Table 5. This approach is a good alternative to survey a conventional FEM such as above mentioned FEM simulation, because the calculation of the magnetic field is done without any simplifying assumption. 6. Algorithm In this paper, new configuration of BLDC generator is presented. It provides not only good characteristics for the BLDC generator in design process but also appropriate cost for the generator. For this reason, FEM simulation for magnetic field analysis regarding to specification of BLDC generator is carried out. Even more important

Fig. 10. Proposed BLDC generator fabricated in the laboratory.

Table 6 Measuring results of GAUSSMETER BROCKHOUSE 460.

Fig. 9. Algorithm of design parameters for new configuration of BLDC generator based on hybrid solution verification.

Magnetic field

Bx(max)

By(max)

Bxy(max) (T)

Aligned Un-aligned

0.796882 0.078972

0.372389 0.012993

0.879599 0.07988

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Fig. 11. Output voltage from one phase of the generator: (a) field current equal 0.5 A and (b) field current equal 1 A.

than FEM simulation, the coupled FE–BE field analysis is offered as methodology evaluation. Then, vector potential and flux calculation according to Eqs. (25) and (26) are done for different angles. Comparison of calculated magnetic field and induced flux linkage by their requested values respectively illustrates the conformity of design parameters stage. In this level of machine design, if they obtained values are verified, construction of machine by optimized characteristics will be possible. These stages are presented in the algorithm as shown in Fig. 9.

7. Experimental results Fig. 10 illustrates the proposed 9/6 BLDC generator that is fabricated in the laboratory. Flux density measurement in the teeth of the machine is carried out by GAUSSMETER BROCKHOUSE 460. Results of this measurement are shown in Table 6. Measuring results are in good agreement with those obtained by the theoretical approach based on FEM simulation in Section 5. As mentioned before, the stator and rotor cores are made of a non-oriented silicon steel lamination. The basic dimensions are: a rotor pole arc of 30°, a stator pole arc of 30°, and an air-gap length of 0.6 mm. For ease of manufacture, the diameter of the shaft is chosen to be 9 mm, the outer diameter of the rotor is 59.4 mm, and the length of the stator pole is 15 mm. In experimental test, the shaft of the generator is connected to a motor to act as a prime mover. Output parameters of generator are measured for No-Load and under-load (RLoad = 27 X) by different speed from 200 rpm up to 3000 rpm. All these measurements are done for two different field currents equal by 0.5 A and 1 A. The results of these tests are shown in Fig. 12a–c. In these figures curve fitting (power) has been used for better presentation of the data points. The actual output voltage of one phase for two different field currents equal by 0.5 A and 1 A are also shown in Fig. 11a and b. The voltages have harmony which is due to the shape of the stator and rotor poles. Output voltage (Vp–p) of generator are measured for No-Load operation by different speed from 200 rpm up to 3000 rpm for two different field currents equal by 0.5 A and 1 A. Results of this measurement are presented in Table 7. Out-put power and voltage (Vp–p) of generator are measured for under-load mode by different speed from 200 rpm up to 3000 rpm for two different field currents equal by 0.5 A and 1 A. The load in this test is a resistive (RLoad = 27 X). Results of this measurement are shown in Table 8. Fig. 12a shows plot of generated peak to peak output voltage in No-Load mode for different speed. Maximum voltage obtained in speed 3000 rpm and field current 1 A.

Table 7 Measured output voltage of generator for No-Load mode. Speed (rpm)

No-Load

200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

I field = 0.5 A Output voltage (V)

I field = 1 A Output voltage (V)

1.82 2.81 5.62 8.01 10.4 13.4 16.1 18.3 21.2 23.4 26.1 28.5 31.3 34.1 36.5

3.61 6.82 9.81 14.2 18.3 21.5 24.2 28.1 31.5 36.2 40.3 44.1 46.2 50.2 54.1

Table 8 Measured output voltage (Vp–p) and power of generator for under-load condition (RLoad = 27 X). Speed (rpm)

RLoad=27 X I field = 0.5 A

200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

I field = 1 A

Output voltage (V)

Out-put power (W)

Output voltage (V)

Out-put power (W)

2.35 4.84 7.42 9.41 11.2 13.5 15.0 17.5 19.5 20.5 22.0 24.0 26.0 27.5 28.0

0.0256 0.1067 0.2535 0.4091 0.5807 0.8437 1.0417 1.4178 1.7604 1.9456 2.2407 2.6667 3.1296 3.5012 3.6296

3.21 6.23 9.62 13.0 15.7 19.0 21.1 23.5 26.2 28.4 30.2 32.1 34.0 36.1 37.5

0.0474 0.1784 0.4267 0.7824 1.1412 1.6713 2.0417 2.5567 3.1296 3.6296 4.1667 4.7407 5.3519 6.0013 6.5104

Measurement of generated voltage is done for under-load operation and Fig. 12b presented result of this test. As shown in Fig. 12b amplitude of generated voltage are dependent to shaft speed and value of field current.

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Fig. 12. (a) Output voltage for No-Load mode, (b) output voltage for under-load (RLoad = 27 X), and (c) out-put power for under-load.

8. Conclusion This paper has described the electromagnetic characterization and experiment of a field assisted 9/6 Salient-Pole BLDC generator, which does not use a permanent magnet in the rotor. To evaluate the generator performance, two types of analysis, namely 2-D FE analysis as the numerical technique and the experimental study have been utilized. Furthermore, a hybrid FE–BE field solution as methodology evaluation for 2-D FE analysis has been carried out. The result of FE analysis is in close agreement with the experimental results. The out-put power characteristics obtained from experimental tests approves the proper operation of novel proposed structure for BLDC generator. Test results confirmed the feasibility of the proposed generator design. References [1] Jang Seok-Myeong, Cho Han-Wook, Choi Sang-Kyu. Design and analysis of a high-speed brushless dc motor for centrifugal compressor. IEEE Trans Magn 2007;43(6). [2] Zhang XZ, Wang YN. A novel position-sensorless control method for brushless DC motors. Energy Convers Manage 2011;52:1669–76. [3] Gieras Jacek F, Wing Mitchell. Permanent magnet motor technology: design and applications. Marcel Dekker; 2002. ISBN: 0-8247-0739-7. [4] Asaei Behzad, Rostami Alireza. A novel starting method for BLDC motors without the position sensors. Energy Convers Manage 2009;50:337–43. [5] Hanselman Duane C. Brushless permanent-magnet motor design. McGrawHill; 1994. ISBN: 0-07-026025-7. [6] Eastham Miller TJ. Brushless permanent-magnet and reluctance motor drives. Clarendon Press; 1989. ISBN: 0198593694. [7] Shin Pan Seok, Woo Sung Hyun, Koh Chang Seop. An optimal design of large scale permanent magnet pole shape using adaptive response surface method with latin hypercube sampling strategy. IEEE Trans Magn 2009;45(3). [8] Fodorean D, Giurgea S, Djerdir A, Miraoui A. Numerical approach for optimum electromagnetic parameters of electrical machines used in vehicle traction applications. Energy Convers Manage 2009;50:1288–94. [9] Gieras Jacek F, Wang Rong-Jie, Kamper Maarten J. Axial flux permanent magnet brushless machines. Springer Science; 2008. ISBN: 978-1-4020-6993-2.

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ID 764480

Title Analysis of brushless DC generator incorporating an axial field coil

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