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MAGNETIC FIELD CALCULATION OF CLAW POLE PERMANENT MAGNET MACHINESUSING MAGNETIC NETWORK METHOD

Y.G. Guo and J.G. ZhuFaculty of Engineering, University of Technology, Sydney

P.O. Box 123 Broadway NSW 2007

Abstract

This paper reports the three-dimensional magnetic field analysis using the magnetic network method for evaluation of parameters and performance of a claw pole permanent magnet machine.The magnetic network model is built based on the principle of equivalent magnetic flux pipes andthe network equations are derived to solve the nodal magnetic scalar potentials, which are used tocalculate the magnetic flux flowing through different cross-sectional areas. The results arevalidated by 3D finite element method. The magnetic network technique has the advantage of combining the speed of the conventional magnetic circuit method and the computational accuracyof finite element analysis.

1. INTRODUCTION

Claw pole machines are commonly used asautomobile, wind and hydro generators due to theirsimplicity and low manufacturing cost. The uniquearmature construction with one circumferentiallywound global excitation coil for all poles in one clawpole stack implies a peculiar armature pole structure,which results in a truly three-dimensional (3D)magnetic field in the machine. In contrast toconventional types of electrical machines, whereanalysis can be conducted in a two-dimensional (2D)plane perpendicular to the machine shaft, the claw

pole machine model must take into account all threefield components: radial, tangential, and axial. Someparts of the core structure cannot be modelled to anequivalent 2D frame because they are not invariantalong one of the axes and hence only 3D analysis canprovide accurate results. For this reason, it is usuallyimpossible or very difficult to accurately predict theparameters and performance of such complexstructured devices by the conventional magneticcircuit (CMC) method. The flux through one part isconsidered to flow in the direction of one axis only,which is not true for the 3D flux electromagneticdevices. In addition, the consideration for the effect of

saturation and leakage fields relies heavily onexperience or experimental data. Therefore, theconventional empirical method cannot provide reliableresults in situation where complex structures orcompletely new designs are being developed althoughit may be very useful for analysing small changes orrapidly determining the machine dimensions for givenspecifications.

Theoretically, the 3D finite element method (FEM) iscapable of accurately analysing magnetic fielddistributions in any devices since it can include all thestructure, material and excitation details. In order toobtain acceptable accuracy of finite element analysis,

the machine needs to be split into a huge number of geometrical elements. The field values are assumed tobe a simple function of position within these elements,enabling interpolation of results. The time required forcalculating the field distribution, however, could bevery long, especially in design optimisation stage,which needs a great number of iterations to reach anoptimum design. This is only possible with high-speedcomputers with large RAMs and it suggests that thefinite element analysis (FEA) may not be the mosteffective method for solving the everyday problems inelectrical machine design.

In order to overcome the shortcomings of the FEMand the CMC, the magnetic network method (MNM),based on the principle of equivalent magnetic fluxpipes [1,2] is used in this paper for calculating themagnetic field distribution in a claw pole permanentmagnet machine. Similar to the FEM, the solutionregion can be chosen according to the symmetry of thestructure and the whole solution region is divided intoa number of elements with quasi-regular shape andrelatively uniform field distribution. The equivalentmagnetic permeance (or reluctance) of an element canbe evaluated by the magnetic Ohms law. Theelements are connected with nodes. The nodalmagnetic scalar potential, the magnetic flux flowingthrough the elements and various parameters arecomputed based on the similarity between themagnetic and electrical networks.

Compared with the FEM, the MNM is capable of analysing electrical machines of any structure with anacceptable accuracy with a much smaller number of elements (allowing rapid iteration of computation).The compromise between the computational accuracyand the CPU time is good enough to be appealing.Accuracy can be improved by concentrating elementsin critical and saturated parts of the machine, e.g. inthe main air gap, the claw pole parts, and the sub-air

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gap between claw pole pieces [3].

Opposite to the triangle (2D) and tetrahedra (3D)commonly used in the FEM, each permeance in themagnetic network has distinct physical meaning and

has direct influence on the output results. The flux inthe magnetic network can flow through a magneticpermeance in the defined direction only, whereas inthe FEM there is no restriction on the direction of fluxthrough any element. In a magnetic network, resistivecomponents represent magnetic reluctances orpermeances. The values depend on the material andgeometry, and for ferromagnetic parts, the flux densityin the region as well.

To compute all the three components of the magneticfield at one point, the magnetic property of theelement with a block shape, in which that point islocated, can be modelled by 6 reluctances and 7nodes, as shown in Fig.1. Any irregular hexahedronused for the solution region meshing can betransformed to an regular equivalent block.

Fig.1 3D reluctance model of a solid element

In Fig.1, R x, R y, and R z are the equivalent reluctanceson the X , Y , and Z axes, respectively, and can becalculated as following

Z Y X

R

x

x =

02

(1)

Z X Y

R y

y =

02 (2)

Y X Z

R z

z =

02 (3)

where x, y, and z are the relative permeabilities onthe X , Y , and Z axes, 0 is the permeability of air, X ,Y , and Z are the side lengths of the solid element.The soft magnetic composite material and the mildsteel, used as the stator core and the rotor corerespectively, are magnetically isotropic and hence

z y x == (4)

This paper introduces how a 3D magnetic network model of a claw pole permanent magnet machine is

built, considering all the peculiarities of machineconstruction and material properties. The MNM isvalidated by the 3D finite element analysis withANSYS 5.6, a powerful commercial finite elementsoftware package. The computation shows that theMNM can be a very effective design tool forelectromagnetic devices with complex structures.

2. THE MNM MODEL

2.1 The Claw Pole Machine Structure

Fig.2 shows the structure of a claw pole permanent

magnet (PM) machine prototype. This machine has anouter rotor structure, which comprises 20 surface-mounted NdFeB PMs and a mild steel cylinder. Thestator consists of two claw-pole pieces mounted on ashaft. The stator winding is located between the twoclaw pole pieces and not shown in the figure forclarity.

(a)

(b)Fig.2 Magnetically relevant parts of (a) rotor and

(b) stator of a claw pole PM machine

The rotor is similar to that of a common PM acmachine, but the stator structure is more complex and

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its magnetic field is truly 3D. Therefore, the magneticnetwork model must be 3D as well. Only 3D magneticcircuit network can correctly estimate the influence of the stator iron parts (particularly the tapered claws)and the air gaps with irregular shapes on the machine

performance. Each magnetic flux path in the machineis represented by a magnetic permeance or reluctancein the magnetic network, which takes into account thegeometry of the path and magnetic properties of thematerial.

For the convenience of analysis, the claw pole statorand rotor iron cores are divided into several segmentsalong both axial and circumferential directions, asshown in Fig.3. Each segment has a reasonablyregular shape. The magnetic field path in any segmentis assumed to be independent and connected to theothers by non-linear magnetic reluctance.

Fig.3 Claw-pole surface segments along bothaxial and circumferential directions

2.2 The Magnetic Network

The magnetic network model is built with flux tubesdescribed by magnetic reluctance or permeancevalues, and magnetomotive forces or flux sources. Aflux tube is a 3D solid in which the flux is constrainedto flow in one direction and may be identified by afinite element analysis [4,5]. The network topology ischosen according to the geometrical considerationbased on the general knowledge of the direction of theflux tubes. The three dimensional magnetostatic finiteelement analyses can be used, assisting to achieve thisgoal, especially for the main air gap and the leakagereluctances.

In the case of 3D magnetic field analysis using themagnetic scalar potential:

mBmAlmF F F == dlH (5)

= Am dAB (6)

m

mm

F R

= (7)

where F m is the magnetomotive force applied alongthe length L of the tube, or the magnetic scalarpotential difference between the two equipotential

surfaces at the extremities, m the flux though the tube,and Rm the equivalent reluctance, which can be also

estimated by the magnetic Ohms law as in thefollowing equation when the tube has a regular shape:

A L

Rr

m0

= (8)

where A is the equivalent or computational area, 0 thepermeability of air, and r the relative permeability of the tube depending on the material, and forferromagnetic parts, the flux flowing through it.

When the flux tube is made of the permanent magnet,it is modelled as a combination of a magnetomotiveforce and a magnetic reluctance in series. Themagnetomotive forces ( mmf ) of the magnet iscalculated by

mcm L H F = (9)

where H c is the coercive force of the magnet and Lm the length along the field direction.

The demagnetisation curve of the NdFeB magnets canbe modelled as a straight line. The magnet reluctanceis calculated by (8) where the magnetisation is equalto zero and the reluctivity is a constant. The stator androtor reluctances are also calculated with constantreluctivities. The saturation phenomena of theferromagnetic materials are taken into account as avariation of the relative reluctivity, which is a functionof the flux flowing through them based on the given B-H curve.

When the rotor rotates, the magnetic field in themachine changes periodically. This is taken intoaccount by the main air gap reluctance, which is afunction of the rotor position.

Fig.4 illustrates one magnetic network model of theclaw pole PM machine prototype at the original rotorposition. The so-called original rotor position (0 o) iswhere the major path of the flux produced by the PMshas the highest permeance, namely the rotor magnetsand the stator claw poles share the same radial axes,respectively. In Fig.4, Rry, Rrm , Rml, Rg, Rcp, Rcl, Rsy and Rsh are the magnetic reluctances of one rotor yoke

segment, the magnet, the leakage flux path betweenthe rotor magnets, the main air gap flux path from themagnet to one claw-pole segment, the claw-polesegment, the leakage flux paths between claw-polesegments, the stator yoke, and the shaft, respectively.F rm is the mmf of the PM and F s the mmf per pole of the stator winding.

Fig.4 shows the basic structure of the magneticnetwork, but the practical model used in the magneticfield calculation may have much more nodes andelements. For example, the claw pole segment is asub-network consisting of a number of radial, axial

and circumferential reluctances, based on the accuracyrequirement.

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Fig.4 Equivalent reluctance network

3. FIELD ANALYSIS WITH MNM MODEL

3.1 Network Equations

When the magnet permeances (or reluctances) andmmf s are known, the nodal magnetic scalar potentialscan be calculated by the following network equations:

}{}]{[ smG = (10)

where [G] is the magnetic permeance matrix, { m} thenodal magnetic scalar potential vector, and { s} theflux source vector. Then, the magnetic flux, the fluxdensity and field strength in each element, which is

required for the calculation of the electrical equivalentcircuit parameters and performance of the electricalmachine, can be determined from the nodal magneticpotentials.

Because the permeance matrix [G] includes some non-linear elements G i

(i=1,2, ,n), these equations aresolved by the Newton-Raphson iteration method.These elements form a non-linear network and thevalues can be determined by the magnetic potentialdifferences across them. The network equations aresolved by the following steps:

(1) A series of nodal magnetic potentials are assumedas the initial values under a certain mmf Fm (e.g. themagnet mmf for no-load analysis), and the potentialdifferences F i across the two terminals of theequivalent permeance network G i

are considered asthe equivalent magnetic potential sources. When thetotal flux i

flowing through G i are obtained by

solving the modified network equations, where G i are

replaced by the equivalent magnetic potential sourcesF i , the G i elements can be determined by

i

ii

F G

= (11)

(2) The Newton-Raphson method is used to solve the

nodal magnetic potentials in the network equations.Assume m

(k) to be the k th solutions after k timesiterations, the (k+1) th solutions are achieved by thefollowing linear equations:

)()()(

}{}{][k

s

k

m

k

J = (12)where [J] is the Jacobian matrix and

)()1()( }{}{}{ k mk

mk

m = + (13))()()( }{][}{}{ k m

k s

k s G = (14)

After a number of iterations, the right-hand side vectorof (12) should approach zero if the iteration isconvergent. The solution reaches the requiredaccuracy when the following condition is satisfied:

2)()( }]}{][[{ k mk s G (15)where is a small enough positive quantity.

3.2 Flux Calculation

When the nodal magnetic potentials are solved, theflux flowing through the branches can be obtainedwith the magnetic Ohms law:

m

smm

m

mm R

F F F RF == 21 (16)

where F m1 and F m2 are the magnetic scalar potentialsat the two terminal nodes, F s is the mmf produced bythe permanent magnets and/or the stator current, along

the direction from node 2 to node 1, and Rm is theequivalent reluctance of the branch. The magneticreluctance is flux-dependent when the branch consistsof ferromagnetic materials. In this case, the numericaliteration is needed as discussed in Section 3.1.

The flux flowing through several typical cross-sections at no-load operation by the magnetic network model in Fig.4 are listed in Table 1. The resultsobtained by the FEA, which is to be presented in thenext section, are also listed in the table forcomparison.

Table 1. Calculated magnetic fluxes through variouscross-sections (in mWb)

Cross-sections MNM FEMStator yoke and shaft 0.503 0.489Rotor yoke 0.0480 0.0468One rotor magnet 0.123 0.115Main airgap per pole-pitch 0.118 0.114

3.3 Calculation of Stator EMF Waveform

When the rotor rotates, the flux linking the statorwinding varies and an electromotive force ( emf ) isinduced. The emf frequency depends on the rotor

speed while the emf waveform is determined by thewaveform of the flux. The total flux flowing though

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the stator yoke and shaft can be considered as the totalflux linking the stator coil. At the different rotorpositions, the air gap reluctances have different valuesand the magnetic network model may need to bemodified. For example, when the rotor moves 180 o

(electrical), the magnetic network model is the sameas that shown in Fig.4, but that the direction of themmf s produced by the rotor magnets changes. Theflux 1 linking the stator winding at different rotorangles from 0 o to 180 o (electrical) are calculated andthe corresponding emf magnitude induced can bedetermined by

111 2 fN E = (17)

where N 1 is the number of turns of the stator coil and f the emf frequency determined by the rotor speed, viz.

60

pn f = (18)

where p is the number of pole pairs of the machineand n the rotor speed in rev/min. When the machineruns at 300 rev/min, the emf frequency is 50 Hz.

The emf waveform of the stator winding is plotted inFig.5. It is almost sinusoidal versus the rotor position(in electrical degrees). It should be noted that the fluxlinking the stator winding and the induced emf havethe same shape of waveforms, but shifted by 90 o (electrical).

Fig.5 EMF waveform of the stator winding

4. FIELD ANALYSIS WITH 3D FEM

To validate the MNM method, 3D finite elementanalysis has been performed to calculate the magneticfield distribution in the claw pole permanent magnetmachine prototype as shown in Fig.2. ANSYS 5.6, apowerful commercial finite element software package,was employed for the calculation. Considering thesymmetrical nature of the structure, it is only requiredto analyse the magnetic field in one pole pitch [6].

To determine the 3D magnetic field distribution, thescalar magnetic potential was employed with half

periodic boundary conditions such that the fluxdensity on the two half periodic boundary planes obeythe following equations:

),,(),,( zr B zr B r r = (19)

),,(),,( zr B zr B = (20)),,(),,( zr B zr B z z = (21)

and the degrees of freedom or the scalar magneticpotentials have to satisfy

),,(),,( zr zr mm = (22)

Fig.6 illustrates the magnetic flux density vector plotsat the original rotor position when the machineoperates at no-load. It can be found that the major pathfor the magnetic flux produced by the permanentmagnets is along one of the permanent magnets, themain air gap, one of the claw pole stator pole pieces,the stator yoke and shaft, the other claw pole statorpiece, the main air gap, another permanent magnet,and then the stator yoke to form a closed loop. It canalso be seen that a large amount of leakage flux existsbetween the claw poles and the magnets. This plotproves the general knowledge about the fluxdirections used in the magnetic network method.

The flux flowing through different cross-sectionalareas are calculated and compared with those obtainedby the MNM, as shown in Table 1. The results by twomethods have a reasonably good agreement.

Fig.6 Magnetic flux density vectors at no-load

The theory applied to the calculation of the fluxpassing through one cross-sectional area is

= Am d AB (23)

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For numerical calculation, the cross-section is dividedinto a number of small sub-areas and the flux densityin the middle of each sub-area can be considered asthe value through all the sub-area if the division is fineenough. Then, equation (23) can be rewritten as

=i

iim A B (24)

The flux flowing through the stator yoke and shaft iscalculated as the effective flux of the stator winding.When the rotor rotates, the flux varies and an emf isinduced in the winding. The flux waveform wascalculated by rotating the rotor magnets for one polepitch in 20 steps. The corresponding emf waveform at300 rev/min, computed from (17) and (18), is plottedin Fig.5. The results by two methods have reasonablygood agreement.

5. CONCLUSIONS

The magnetic network method is in fact a distributedmagnetic circuit method. The MNM can achievecomparable computational accuracy to the FEM withmuch fewer elements and shorter CPU time. The factthat the nodes are concentrated in the crucial regionssuch as the air gap enables accurate calculation of magnetic fields. On the other hand, owing to itsdistributed nature, the MNM is far more accurate thanthe conventional magnetic circuit method when theelectrical machine structure is very complex.Therefore, the MNM can be a practical and useful

method for the optimum design of electromagneticdevices with complex structures.

REFERENCES

[1] Ostovic, V., Dynamics of Saturated ElectricMachines, New Yoke, Springer-Verlag, 1989

[2] Ostovic, V., Miller, J.M., Garg, V.K., Schultz,

R.D., and Swales, S.H., A Magnetic-Equivalent-Circuit-Based Performance Computation of aLundell Alternator, IEEE Transactions onIndustry Applications, Vol. 35, No. 4, July/August1999, pp825-830

[3] Rasmussen, C.B., and Ritchie, E., A MagneticEquivalent Circuit Approach for Predicting PMMotor Performance, Proceeding of IEEE IASAnnual Meeting, New Orleans, Louisiana, 5-8October 1997, pp10-17

[4] Roisse, H., Hecquet, M., and Brochet,Simulations of Synchronous Machines Using anElectric-Magnetic Coupled Network Model,IEEE Transactions on Magnetics, Vol. 34, No. 5,September 1998, pp3636-3659

[5] Hecquet, M. and Brochet, P. Time Variation of Forces in a Synchronous Machines Using ElectricCoupled Network Model, IEEE Transactions onMagnetics, Vol. 34, No. 5, September 1998,pp3214-33217

[6] Guo, Y.G., Zhu, J.G., and Ramsden, V.S.,Calculation of Cogging Torque in Claw PolePermanent Magnet Motor, Proceeding of Australasian Universities Power EngineeringConference, Darwin, Northern Territory,Australia, 26-29 September, 1999, pp343-347