Analytical electromagnetic field and forces calculation for linear, cylindrical and spherical...

8/10/2019 Analytical electromagnetic field and forces calculation for linear, cylindrical and spherical electromechanical conver

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BULLETIN OF THE POLISH ACADEMY OF SCIENCESTECHNICAL SCIENCESVol. 52, No. 3, 2004

Analytical electromagnetic field and forces calculation forlinear, cylindrical and spherical electromechanical converters

D. SPAEK*

Institute of Theoretical and Industrial Electrotechnics, Silesian University of Technology, 10 Akademicka St, 44100 Gliwice, Poland

Abstract.The paper deals with the problem of force and torque calculation for linear, cylindrical and spherical electromechanical converter.The electromagnetic field is determined analytically with the help of separation method for each problem. The results obtained can be used astest tasks for electromagnetic field, force and torque numerical calculations. The analytical relations for torque and forces are also convenientfor analysis of material parameters influence on electromechanical converter work.

Keywords: electromagnetic forces, anisotropy, analytical solutions.

1. Introduction

The determination of forces induced by electromagneticfield in electromechanical converter is an important prob-lem. For electromechanical converters the acting force ortorque values play deciding role for users. Modern tech-nologies enable us to construct electromechanical convert-ers with parts having wide range of properties betweenthem anisotropic. Especially, the induction motors lin-ear, cylindrical and spherically shaped have got oftenmagnetically anisotropic parts. The magnetic anisotropycould increase the value of forces and torques arising inelectromechanical converter. The intention of this paperis to present the analytical solutions for linear, cylindrical

and spherical induction motor problems. Basing on themain equation for electromagnetic forces density there arepresented and discussed forces and torques values actingin electromechanical converters. The analytical relationsfor force and torque are the base for material parame-ters influence analysis. The presented below solutions foranisotropic and isotropic cases for magnetic circuit of in-duction motor can be used as a test-task for numericalpackages, too.

2. Force and torques in electromagneticfield

The first pair of Maxwell equations [13] take the well-known form

curlE= B div B= 0. (1)The second pair of Maxwell equations can be presentedin vector notation as follows

div D= curl H= j+ D. (2)The constitutive relation for electromagnetic field vectorsfor non-hysteresis medium are

Hu=uvBv, (3)

Du= uvEv, (4)

* email: [email protected]

whereuv denote dielectric permittivity, uv are magnetic

reluctivity, u, v, w are curvilinear system co-ordinates(summation due to twice appearing indices is accepted).The volume density of the total force [4, 5] can be

written in the form of (Appendix)

f= fL+ N+ M, (5)

that point out the physical reasons for arising of theelectromagnetic force component and is widely used forcalculation of forces that appear in electromagnetic fieldregion [69]. On the right hand side of Eq. 5 there arethree components of electromagnetic force as follows

acting on external currents/charges the Lorentzforce equals to

fL= E+ j B, (6) originated by change of magnetic reluctivity and

electric permittivity [10] the so-called nonhomogenous

component N

N = 1

2BuBwgrad(uw) 1

2EuEwgrad(uw), (7)

and asymmetry component M. (Eq. 66).The forces/torques described mathematically by Lo-

rentz formula are called Lorentz forces/torques. Theforces/torques given by non-homogeneous and asymmetrycomponents are called material forces/torques.

The tensor notation presented in Appendix isconvenient for forces components evaluation in curvilinearco-ordinate systems e.g. cylindrical and spherical. Thetotal force density can be also presented with the help ofMaxwell stress tensor in vector notation as follows

div(eu) =efLu+ eNu+ eMu, (8)where

u= BHu+ iueV, (9)fLu= jvBwjwBv, (10)Nu=

1

2

ueu

B2u+1

2

uveu

BuBv+1

2

wveu

BvBw

+12

veu

B2v , (11)

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D. Spaek

Mu=1

2(uv vu)Bu Bv

eu +

1

2(vu uv)Bv Bu

eu. (12)

For Cartesian co-ordinate systeme= 1,u = x (Section 3);for cylindrical e = r, u = (Section 4) and sphericalsystem e = r2 sin , u = (Section 5), subsequently.

So, the virtue of tensor notation is evident: it leads tothe vector notation easily. Moreover, the tensor notationcan be easily implemented in any programming language.The components of tensors - presented in matrix form- are very convenient in use for any loop algorithms,especially.

3. Linear electromechanical converter(1 dimensional field)

The simplest linear electromechanical converter [11] foranalytical analysis of electromagnetic field and force cal-culation is presented. For describing Lorentz FL and

anisotropy forces FFe the main Eq. (5) will be applied.The model of the electromechanical converter consid-ered is an induction linear motor with conducting andanisotropic carriage (Fig. 1). The model can representlinear motor used in traction. The motor carriage issmooth and magnetically homogeneous. The conductingrotor carriage has the width a, and the air-gap width isg= const.

Fig. 1. Model of linear induction motor

For the linear induction motor the analytical solutionfor the electromagnetic field distribution and forces can

be found. The correctness of the analytical analysis givesthe possibility to demand for the highest accuracy offorce values. Additionally, the solutions obtained for forcesdecomposition confirm the form of Eq. (5) mathematicallyproved (in Appendix) . The total force decomposition forcomponents called is satisfied as shown in this paragraph.

The electromagnetic field calculation will be providedunder the following assumptionsa) electric displacement current vanishes due to the small

field frequency [11],b) Maxwells stress tensor part connected with the elec-

tric field can be omitted due to considered range of

the frequency DuEv HuBv ,c) rail windings induce the sinusoidal mmf (magnetomo-

tive force [10], [12])

s(x) =scos(ky 2f t), (13)where s stands for magnitude of mmf, y is theposition on the infinitely long carriage towards move

direction, k propagation constant, f means statorsupply frequency,

d) reluctivities for motor carriage, are as followsx xyyx y

,

e) conductivity of motor carriage is (isotropic parame-ter).

The assumed symmetry of linear motor leads to theone-dimensional analysis. Field change in perpendiculardirection to the move direction is neglected. The magneticflux density of vertical component of magnetic potentialAz can be presented as follows

B= ixAzy iy Az

x . (14)

The magnetic field strength components for the anisotro-pic region can be shown in the form of

Hx = xBx+ xyBy, Hy =yxBx+ yBy. (15)

The Maxwell equation

curl( H) = j = E= A (16)and Eqs. (14)(16) lead to the main equation in the formof

y 2

Azx2

(xy+ yx) 2

Azxy

+ x 2

Azy2

=Az. (17)

The time-partial derivative of Az as a multiplication ofthe operandi and the complex magnetic potential A atthe steady state could be represented as follows

A i A, (18)where means carriage currents pulsation.

Equation (17) leads to either generalized Helmholtzdifferential equation for conducting carriage

y2A

x2 (xy+ yx)

2A

xy+ x

2A

y2 =iA, (19)

or the Laplace differential equation for the air-gap region

2A

x2 +

2A

y2 = 0. (20)

Equations (19) and (20) will be solved by the separationof the variables in the form

A= X(x)Y(y) =X Y. (21)

At the steady-state function Y(y) for monoharmonicstator mmf has the given below form

Y(y) = exp(iky). (22)

Rearranging the Eq. (19) with due to separation schemefor X(x) and Y(y) leads to differential equation for the

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Analytical electromagnetic field and forces calculation

layer

d2X

dx2 + ik

xy+ yxy

dX

dx

xk2

y+ 2

X= 0, (23)

and for the air-gap Eq. (20) takes the form given below:

d2

Xdx2 k2X= 0. (24)The solutions of the Eqs. (23) and (24) are given inTable 1. The four unknown constants aa, ba, a , b canbe evaluated by formulating the boundary conditions andthere are grouped in Table 2.

The evaluation of magnetic field potential distributionis followed by the electromagnetic torque componentsanalytical calculations. The Maxwell stress tensor leadsto the total electromagnetic torque by means of thewell-known formula

F =oV

BBrdS. (25)

The electromagnetic torque component forced by machine

rotor currents can be evaluated by the Lorentzs forcedensity as follows

FL = V jzBrdV . (26)The equation shows the total torque decomposition, whichcan be written in the shortened form of

F =FL+ FFe, (27)

where FFe denotes the so-called material torque. Thiscomponent is calculated with the help of the Eq. (12) asfollows (u= x)

FFe =V

MxdV. (28)

All torques are calculated for chosen linear motor. Theaccuracy of the calculation provided with MathcadTM

equals to 1015 (data shown in Fig. 2 are for SI UnitSystem).

Table 1

Solution of the differential equations

Region anisotropic carriage Eq. (23)

(index a)

air-gap Eq. (24)

(index )

Solutions X(x) X(x) = aaexp(1x) +baexp(2x) X(x) = aexp(kx) +bexp(kx)

constants aa, ba a , b

Table 2

The boundary conditions for magnetic field

Boundary condition Field excited by stator currents Constants for solutions

Rail mmfy = a+g oBy =sy

aa = s1o {Ue

1(a+g) W e2(a+g)}1

Carriage surface y = a Bx = BaxoBy =yBay+ yxBax

a1 = ikxy + yx

2y

a0 =

k2 xy

+ i y

1,2 = a1

a21+ a20

ba = aaS, a =aaU, b =aaW

Inner layer surface y = 0 yBy+ yxBx = 0 where

S= y1+ikyx

y2+ikyx

P = yko

(1e1a S2e

2a) +iyx

o(e1a Se2a)

Q= e1a Se2a

U= 12 (P+ Q)eka

W = 12 (Q P)eka

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D. Spaek

Linear induction machine anisotropic carriage (SI unit system)

o := 41 1 107x :=

o

3 y :=

o

1.5 xy := 0.5 y + 0.0 y yx:= 0.0 y 0.0 y i

:= 7 106

a1 := yx + xy

2 y k i s:= 2 i 10 a0 :=

k2 xy

+ s + s2

y k:= 0.70

g := 0.005 l:= 10 a:= 0.1 s:= 2000 1 := a1 a12 + a02 2 := a1 + a12 + a02FORCE COMPONENTS

Maxwell method x:= a + 0.3 g Lorentz and material forceF := 1 Re(Hy(x) Bx(x)) F = 1.234 F L:= Im(s) k l C F L= 1.244

F F e:= l k Im a

0(y x xy ) (Bxa(x) Bya(x))dx

Forces ratio

F LF

= 1.008 F F eF

= 0.008 F L + F F eF

= 1.000

Fig. 2. Torque calculations for anisotropic rotor MathcadTM program

4. Cylindrical electromechanical converter(2 dimensional field)

For describing Lorentz TeL and anisotropy TeFe torquesthe proved method for induction motor with solid rotorwill be applied [1215]. Let us consider the model of theinduction motor with conducting and anisotropic rotor

presented in Fig. 3. The model can represent the 2-phaseferrite induction machine widely used in the industry.The motor rotor is magnetically homogeneous, so thetangential component of non-homogeneous componentvanishes N = 0. The magnetic rotor does not exhibithysteresis phenomenon. The machine rotor is cylindrical.Its outer radius is R. The conducting rotor layer has thewidtha, and the air-gap width is g = const.

Fig. 3. Model of induction motor with solid rotor

For the simplified model of the induction motor theanalytical solution for the electromagnetic field distri-

bution can be found. The correctness of the providedanalysis gives the possibility to demand for the accuracy

almost of 100% for torque components values. The nu-merical analysis does not ensure such the level of torquecalculations. The electromagnetic torque decompositionbalance is satisfied, too.

Fig. 4. Induction motor model the cross-section and dimensions

The electromagnetic field calculation will be provided

under the following assumptionsa) electric displacement current vanishes,

b) Maxwells stress tensor part connected with the elec-tric field can be omitted due to considered range ofthe frequency,

c) stator windings induce the sinusoidal 2p-pole mmf

s() =scos(p 2f t), (29)where s stands for the magnitude of mmf, is theposition angle,f means the stator supply frequency,

d) reluctivities for motor rotor, are given by the reluctiv-ity matrix generally

r rr

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e) conductivity of machine rotor is (isotropic parame-ter).The assumed symmetry of motor geometry and mmf

source lead to the two-dimensional analysis. The magneticflux density in terms of an axial component of magnetic

potential Az (z is symmetry axis of solid rotor) can bepresented as follows

B= ir1

r

Az

i Azr

. (30)

The magnetic field strength components for the anisotro-pic region can be shown in the general form of

Hr =rBr+ rB, H= rBr+ B. (31)

Combining the Maxwell equation with the Eqs. (30)and (31) one obtains for steady-state the differentialHelmholtz equation for the anisotropic solid rotor

r

r

(rA

r

)

r+ r

r

2A

r

+r

r2

2A

2

=iA, (32)

and Laplace differential equation for the air-gap

1

r

r(r

A

r) +

1

r22A

2 = 0. (33)

The Eqs. (32) and (33) will be solved by the separationof the variables in the following form

A= R(r)S() =RS, (34)

at the steady-state when the function S() for monohar-monic stator mmf takes the form

d2R

dr2 +

[1 2c]r

dR

dr

rp2

r2+ 2

R= 0, (35)

wherep is a pair-pole number for stator windings.

Equation (32) with respect to the given above sepa-ration scheme for the functions R(r) and S() gives thedifferential equation for the layer

d

2

Rdr2 +[1 2c]r dRdr

rp

2

r2 + 2

R= 0, (36)

and for the air-gap

r

R

d

dr

r

dR

dr

=p2. (37)

The solutions of the Eqs. (36) and (37) [16] are groupedin Table 3. The four unknown constantsaa,ba, a,b canbe evaluated by formulating the boundary conditions andthere are grouped in Table 4.

Table 3

Solution of the differential equations

Region anisotropic layer Eq. (36)(index a)

air-gap Eq. (37)(index )

Solutionsfor R(z),z= r .

R(z) = aazcIpB(z)

+bazcKpB(z)

c= ip(r+ r)/2

pB =

c2 +p2r/

R(r) = arp +br

p

constants aa, ba a , b

Table 4

The boundary conditions for magnetic field

Boundary condition Field excited by stator currents Constants for solutions

Stator current mmfr = R+g = Rg oB = 1Rs

s

aa = s1o {U R

ps W R

ps }

1

ba = aaS, a = aaU, b =aaW

where

Rotor outer surfacer = R Br =Bar, oB = Ba+rBar S= IpB(Ra)

KpB

(Ra), =

i/

Inner layer surface r = Ra= Ra B+rBr = 0

P = po

IpB(R)SK

pB(R)

Q= IpB(R)SKpB(R)

U= 0.5

P R

p+1

+QR

pW = 0.5

P Rp+1 +QRp

After evaluating the magnetic field potential distri-bution both the magnetic flux density components andthe electromagnetic torque components can be evaluated,analytically. The Maxwell stress tensor leads to the to-tal electromagnetic torque by means of the well-knownformula

Te= orVBBrdS. (38)

The electromagnetic torque component forced by machinerotor currents can be evaluated by Lorentzs force density

as follows

TeL=V

rjzBrdV . (39)

There is a question: whether the both torques are equalfor electromechanical converter which rotor exhibits themagnetic anisotropy features? According to Eq. (5) after applying the Gaussian theorem [17] for a cylin-drical surface V which is situated in the air-gap it is

satisfiedTe= TeL+ TeFe. (40)

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D. Spaek

The equation shows the total torque decomposition (non-homogeneous component vanishes), where TeFe denotesthe so-called material torque. The material torque canbe physically interpreted as a part of total torque thatcan be exerted only if either directional nonhomogenity

or forced magnetic anisotropy appears. The electromag-netic torques at the steady state are calculated exemplaryelectromechanical converter. The considered cases of theanisotropy are grouped in Table 5.

Table 5

Rotor magnetic anisotropy cases and results of calculations for the induction motor

l= 0.3, R = 0.1, a = 0.07, s = 500, p = 2, = 7105, g = 0.001, s = 23.0i

The case The reluctivities

o = (4107 H/m)1

The current torque ratio

Te,L/Te [%]

The material torque ratio

Te,Fe/Te [%]

The total torque value

Te [Nm]

a) r = = 0.04or = r = 0

100,0 0,0 4,29

b) = 0.04o r = 0.05or = r = 0

100,0 0,0 3,46

c) r = = 0.04or = r = 0.1r

100,0 0,0 4,34

d) = 0.04o r = 0.05or = r = 0.1r

100,0 0,0 3,50

e) = 0.04o r = 0.05or = 0.10rr = 0.15r

91,4 8,6 3,83

f) = 0.04o r = 0.05or = 0.10irr = +0.10ir

100,0 0,0 4,01

g) = 0.04o r = 0.05or = {0.10i0.10}rr = {0.15 +i0.10}r

91,3 8,7 4,44

Induction motor with solid anisotropic rotor

r :=o

30 :=

o

35 r:= o := o rx:= 0.1 r := 0.2

c:= r + r2 p i s:= 2 i 5 pB:=

p2 r

+ c2 p:= p

r

p:= 2

g := 0.0015 l:= 0.35 R:= 0.1 a:= 0.07 s:= 800 := 9 106c= 0.3i pB= 2.139 =

(s + s2 ) 1 :=

s 1

Torque components r:= R + 0.4g

Maxwell method Lorentz and material torques

T e:= r2

l

Re(H(r)

Br(r)) T e= 12.612

T eL:= p l Im(s) R

Ra

(|Aa(r)|)2 rdr T eL= 12.451

TeFe:= p l Im RRa

(r r)

Bra(r) Ba(r) rdr

TeFe= 0.161

Torques ratios

T eL

T e = 0.98723

TeFe

T e = 0.01277

T eL + TeFe

T e = 1.00000

Fig. 5. Torque calculations for anisotropic rotor MathcadTM program

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From the analyses carried out and grouped in Table 5it is apparent, that the material torque component canappear for induction motor with round, homogeneous andnon-hysteresis rotor if the matrix of magnetic reluctivitiesis not Hermitan conjugate.

For the electromechanical converter the electromag-netic torque components have been evaluated Fig. 5. Itis significant that for induced anisotropy (the cases e) andg)) appears the material torque TeFe. For Hermitian con-jugate reluctivity matrix case f) (or for symmetricalmatrix particularly) the material torque vanishes.

Hence, there is possible to express the total torque Te(calculated by Maxwell stress surface integral) caused byLorentz force in the form of surface integral for Hermitianconjugate reluctivity matrix (e.g. symmetrical matrix).

For the analysis of influence of normal magneticanisotropy (r = ar = 0) on the torque-slip curve

shape there are considered three cases of rotor magneticanisotropy

with radial-dominate magnetic reluctivity T r andr > a,

with angular-dominate magnetic reluctivity T anda > r,

compared with the case of isotropic rotor T0.

For the three specified above cases (at the conditionr+ = const = 30 o, = 7 106 S/m, R= 0.05 m,a= 0.03 m, g = 0.001 m, = 500 A, p = 1, l = 0.3 m,f1 = 50 Hz) the electromagnetic steady-state torque-slipcurve (s= (f1np)/f1) have been obtained with the helpof the presented magnetic field distribution Fig. 6.

Fig. 6. Torque-slip curves for different anisotropy: T r for r > solid line, T for r = dot line (isotropic rotor), T r > dash-dot line (for all cases r+ a = const, r = r = 0)

The analyses carried for cylindrically shaped elec-tromechanical converter have brought out to the issues,such as

For the electromechanical converter with roundrotor, which does not exhibit both the nonhomogenity ofmagnetic reluctivity and the hysteresis phenomenon onlythe anisotropy of the magnetic reluctivity may cause the

material torque. Indeed, the material torque appears forthe cylindrical-shaped electromechanical converter in the

case

r=r

For the electrical machine rotor with radial-dominate magnetic reluctivity the electromagnetic torqueis the greatest one for stable (arising) part of torque-slipcurve.

The obtained results enable one to state thatthe electromagnetic torque of electromechanical convertercan be represented by the surface integral for both theisotropic and the normal magnetic anisotropy of rotor.

5. Spherical electromechanical converter(3 dimensional field)

For the electromechanical converter with spherical rotor Fig. 7 (e.g. spherical motor [1822]) the electromag-netic field could be determined with the help of separationof variable method. The book [20] presents the main solu-tions for spherical converter by means of circuit analysis.The further works present the field analysis of the prob-lem in theoretical form [18, 19] or numerical [21, 22].

Fig. 7. Spherically shaped electromechanical converter-view

The non-standard separation proposed leads to theanalytical solution for the spherical symmetry problemfor anisotropic region. The solution obtained can beapplied for analysis of electromechanical converters with

spherically shaped moving part. The magnetic flux densityfor spherical co-ordinate can be presented in the form

B=1r

r sin

A

Asin

1r

rA

r Ar

+1

r sin

rrAsin Ar

. (41)

For simple case A = A = A1 = A1 (is motivatedphysically by the construction of the spherical motor Figs. 7 and 8) it is satisfied

B=1r

r sin

A

1

r

rA

r

. (42)

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D. Spaek

Fig. 8. Spherical co-ordinate system

The Maxwells Eq. (16) and constitutive relation forfield (42) in the form of

HrH

H

=

rrBr+ rBrBr+ B

0

(43)

lead for-component to the relation given below

1

r

rr

rr sin

A

r

rA

r

1r sin

rrr sin

A

r

r

rA

r

=j. (44)

The standard separation method in the well-known formof

A= A = R(r)()() =R , (45)does not lead to the easily solvable equation, but theproposed below non-standard separation

A= A =R(r)F(, ) =R F, (46)results in an easy separable equation

rR

2rR

r2 +

rr

r2Fsin2

2F

2

rrR

R

r +

rr2R

rR

r

1

Fsin

F

=i. (47)

For the separated function F(, ) it is assumed theseparation with the constantp2 in the form of

1

Fsin2

2F

2 = p2 (48)

with the analytical solution

F =A exp(ip sin ) + B exp(ip sin ). (49)

The solution (exemplary for constant B = 0 due totechnical aspects) leads to equation

2rR

r2 ip

r

r

R

r +

rr2

rR

r

=p2rr

r2 + i

R, (50)

that confirms the proposed non-standard separation (46)is successful (the solution forA = 0 andB= 0 is obtainedby substituting p p).

The equation obtained has the analytical solution foranisotropic region as follows [16]

R(r) = (r)h1

2 (C1I(r) + C2K(r)) , (51)

where it was denoted

2 = i 2

, (52)

h= ip r+ r

2, (53)

=

(h 12)2 +p2r+ ipr

. (54)

For the air-gap (= 0) it is satisfied

2rR

r2 (+ 1) R

r = 0 (55)

( + 1)= p2 (56)

with the solution given below

R= R(r) =ar

1

+ br

2

. (57)The analytical solution for the spherical motor can

be presented in terms of separated function R(r) andF(, ) obtained with the help of separation proposed.The exemplary spherical induction motor with anisotropicand spherical rotor is considered and shown in Figs. 7and 8. For the monoharmonic magnetomotive force ofstator the magnetic field distribution and electromagnetictorque are calculated. The data for analysis are presentedin Fig. 9 (radii notation is adequate to Fig. 4).

The accuracy for both partial differential solutionsare checked and presented in Fig. 10 for conducting and

nonconducting region for separated ordinary differentialequations. The solutions for ordinary differential equation

Spherical induction motor with anisotropic rotor (SI units)

r := o

15 :=

o

20 r :=

o

1 :=

o

1 r:= 0.3 r:= 0.3 p:= 1

h:= zr + r

2 p i 1 :=

(0.5 h)2 + (r p2 + z p i r) 1 2 := 1:= 1

s:= 2 i 10g:= 0.0002 R:= 0.045 a:= 0.03 s:= 5000 := 10 106 = 1.22202

Fig. 9. The data set for the exemplary spherical motor

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Z( ) and D( ) denote solution for vector magnetic poten-tial separated functionR( ) for conducting region (= 0)and air-gap ( = 0), respectively. In Fig. 10 LZ (RZ)means value of left-hand (right-hand) side of ordinarydifferential Eq. (50) for conducting region. In Figure 10

LD (RD) means value of left-hand (right-hand) sides of

ordinary differential Eq. (55) for the air-gap region, re-spectively. The accuracy for both ordinary differentialequations are checked and presented in Fig. 10, for ex-emplary data. For exact solutions it should be satisfiedLZ/RZ= 1, and LD/RD= 1 as it is shown in Fig. 10.

r:= Ra + 0.5 a Accuracy for partial differential equationsLZ:=

2

r2Z( r) +2(l h)

r r

r(Z( r))

RZ :=

2 +

r p2 + (z p i r) r2

Z( r) LZ

RZ = 1.00000

LD:=

r

rD(r)

+

2

r

rD(r)

RD:=

p2r2 r

D(r) LD

RD = 1.00000

Fig. 10. Accuracy for partial differential equations solutions

r:= R + 0.5 g Electromagnetic torque total valueMaxwells method I:= 0.5 ( o)(cos() cos(o))

Lorentz and material torques

T e:= r3 I Re(H(r) Br(r)) Cr :=R

Ra

(|Aa(r)|)2 r2dr

T e= 0.020 TeCu:= z p I Im(s) Cr TeCu= 0.020

TeFe:= p I Im RRa

(r r) (Bra(r) Ba(r)) r2dr

Torques ratios

TeCuT e

= 1.000 TeFeT e

= 0.000 TeCu + TeFeT e

= 1.00000

Fig. 11. Electromagnetic torque calculation for spherical motor

Power balance Poynting vector, power losses, magnetic energy

Er := r

2 k

R

Ra(|Z( r) p|)2dr

E :=

2 k R

Ra(|dxZ( r)|)

2

dr

Ca :=

0 if (r= 0) (r = 0)

0.5 k z p i

R

Ra

Z( r) dxZ( r)dr otherwise

Er := r C a Er := r CaE := Er+ E + Er +Er E:= 0.5 k (|s|)2 Cr0 P q:= (|s|)2 k Cr0E := 0.01 E := 0 P q:= 1.238

Sc := s k R D(R) dxD(R) Q:= 2 Im(s E + s E P e:= 2 Re(s E+ s E)Sc = 1.238 + 1.309i Q= 1.309 P e= 0

Er+ Er

Er + E = 0.078

E

E= 0.000

P q+ P e + Q iSc

= 1.00000

Fig. 12. Electromagnetic field energy balance calculation

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D. Spaek

For the evaluated magnetic filed distribution and elec-tromagnetic torque has been calculated as shown inFig. 11. There are presented the electromagnetic torquestwo components: Lorentz TeL and material TeFe that to-gether constitute the total electromagnetic torqueTe. The

torque decomposition is presented for checking analysiscorrectness, too. It should be pointed out that for eitherisotropic or normally anisotropic (r = r) rotor theanisotropy torque component disappears (in the case ofHermite conjugate reluctivity matrix, generally). The con-dition for this statement is the same as for cylindricallyshaped rotor (Section 2).

Moreover, it was controlled accuracy of electromag-netic field solutions by means of the energy-power balance.The Poynting vector [15] is used for evaluating the mag-netic energy and power losses in rotor. In order to checkthe accuracy of electromagnetic field solutions the energy-power balance has been checked. The Poynting vector isused for evaluating the magnetic energy and power lossesin rotor. For checking the correctness of the obtained elec-tromagnetic field distribution the power balance has beenchecked with the help of MathCadTM program. The com-plex power is calculated with the help of Poynting vectoras follows

Sc = ioS

BAzdS. (58)

The fulfilment of electromagnetic field power balanceensures us that the analytical solutions are correct. More-

over, the power analysis leads to the power losses valuein solid rotor

Pq =V

E2zdV . (59)

The exemplary calculations are shown in Fig. 12.

For technical purposes the analysis of influence ofnormal magnetic anisotropy (r = r = 0) on thetorque-slip curve is carried out for two cases of rotormagnetic anisotropy

with radial-dominate magnetic reluctivity T r andr > ,

with angular-dominate magnetic reluctivity T and > r,

in comparison with the case of isotropic rotor T.

For the three specified above cases (at conditionr+ = const = 30 o, = 7 106 S/m, R= 0.05 m,a = 0.03 m, g = 0.002 m, = 1000 A, p = 1 forradii notation see Fig. 4) the electromagnetic steady-statetorque-slip curve have been obtained with the help ofthe presented magnetic field distribution Fig. 13. Thetorque-slip curves show that electromechanical converterwith radial-dominate magnetic reluctivity develops elec-

tromagnetic torque at stable part of torque-slip curvegreater than for other anisotropy and isotropic rotor.

Fig. 13. Torque-slip curves for different anisotropy: T r for r > solid line, T for r = dot line (isotropic rotor), T for

r > dash-dot line (for all cases r + = const,r = r = 0) for spherical motor

6. ConclusionsThe proposed method for electromagnetic torque/forcecalculation described in analytical way enables us tocalculate its total and component values. The torque/forcecomponents indicate the physical reason for theirs

induction i.e Lorentz force and

material force component (nonhomogeneous and ani-sotropy components).

The mathematical proof for the main equation forforce density has been provided with the help of tensornotation in Appendix. The tensor notation is convenientdue to its mathematical form not only for curvilinear

co-ordinate system but also for numerical algorithms.For the chosen electromechanical converters linear,

cylindrical and spherically shaped electromagnetic fieldand force/torque have been calculated. The accuracy ofthe calculation carried out has been checked by totalforce/torque decomposition correctness, too.

The solutions obtained lead to the conclusions that theradial-dominate anisotropy increases the electromagnetictorque value at steady-state of work.

For spherically shaped electromechanical converterthe non-standard separation is proposed that leads tothe analytical solutions. The mathematical form of non-

standard separation is given by Eq. (46).It is pointed out, that the Lorentz force can be

presented by means of surface integral for homogeneousregions if its reluctivity matrix is either isotropic oranisotropic Hermite-conjugate.

Appendix

The energy tensor, which includes the stress tensor, canbe defined as follows

ki = GklFil+ 14kiGlmFlm. (57)The excitation tensorGik has the following form

Gik =oFik + Wik,

248 Bull. Pol. Ac.: Tech. 52(3) 2004


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where

Wik =

0 cPx cPy cPzcPx 0 +Iz IycPy Iz 0 +IxcPz +Iy Ix 0

,

Gik =

0 cDx cDy cDzcDx 0 Hz +HycDy +Hz 0 HxcDz Hy +Hx 0

.

The tensor density of the energy tensor can be defined asfollows

ki = GklFil+ 14kiGlmFlm, (58)The first pair of the Maxwells equations for the

curvilinear co-ordinate system can be written in the form

Fik,l+ Fkl,i+ Fli,k = 0. (59)

The second pair of the Maxwells equations where the

tensor density of the excitation tensor appears has theformGik

xk =Gik,k = ji. (60)

Due to the skew-symmetry of tensors Fik and Gik onegets

Fik,lGik + 2Kl+ 2(FklG

ik),i= 0.

where

Kl = Fliji, Kl =

|g|Kl.

The excitation tensor and the electromagnetic field tensorare connected by material parameters as follows

Gik =ik

pq

Fpq + Gik, (61)

where the material coefficients constitute the tensor

ikpq =

gikpq.

The material relation (63) took the form known forGik = 0 and for symmetrical material coefficients.

Consecutive, it can be written

(FikGik),l =FikG

ik,l + Fik,lG

ik, (62)

hence, according to the Eqs. (61) and (62), there aresatisfied the following relations:

(FikGik),l =

pqikFpqFik,l+ ikpqFpqFik,l+ Fik

ikpq,lF

pq

+ [FikGik

,l + Fik,lG

ik].

The material tensor pqik in terms of the symmetricaland the asymmetrical parts can be represented as follows:

pqik = 12

(pqik + ikpq) + 12

(pqik ikpq) = pqiks

+pqik

a

.

Hence

(FikGik),l = 2G

ikFik,l 4Ml 4Ql 4Nl,where there were defined

Ql = 12 [FikGik,l Fik,lGik], (63)

Ml = 12ikpq

a

FpqFik,l = 12

ikpq

a

FpqFik,l, (64)

Nl = 12Fikikpq,lFpq. (65)

The tensor force density satisfies the following relations:12

(FikGik),l+ 2Nl+ 2Ql+ 2Ml+ 2Kl+ 2(FklG

ik),i = 0,

thus finally with the help of Kroneckers delta symbol:

Kl = {GkpFlp+ 14klGpqFpq},k Nl Ql Ml.The divergence of the energy tensor is equal to

( ki ),k =Ki+ Ni+ Qi+ Mi, (66)The non-vector notation for the tensor density of gener-alised force volume density enables one to exchange thevolume integral into the surface one with the help ofGausss theorem.

References

[1] R. S. Ingarden and A. Jamiokowski, Classical Electrody-namics, Elsevier-PWN, Warsaw, 1985.

[2] G. Joos, Lehrbuch der Theoretischen Physik, Leipzig, 1954,

(in German).[3] I. E. Tamm, The Fundamentals of Electric Theory, WNT,Warszawa, 1965, (in Polish).

[4] L. D. Landau and E. M. Lifszyc, The Classical Theory ofFields, Pergamon, New York, 1951.

[5] L. D. Landau and E. M. Lifszyc, Electrodynamics ContinuousRegions, PWN 1960, (in Polish).

[6] Z. Ren, Comparison of different force calculation methodin 3D finite element modelling, IEEE Transaction on Magnetics30(5), 34713474 (1994).

[7] R. Rosensweig, Ferrohydrodynamics, Cambridge UniversityPress, Cambridge 1985.

[8] D. Spaek, Anisotropy component of electromagnetic torquein electrical machines, Archives of Electrical Engineering1, 109126(1999).

[9] D. Spaek, Fast analytical model of induction motor forapproaching rotor eccentricity, COMPEL International Journal forComputation Mathematics in Electrical & Electronics Engineering18(4), MCB University Press, 570586 (1999).

[10] B. Adkins and P. G. Harley, The General Theory ofAlternating Current Machines, Chapman and Hall, London, 1978.

[11] J. Turowski, K. Zakrzewski and R. Sikora, Analysis andSynthesis of Electromagnetic Field, Ossolineum, Warszawa, 1990 (inPolish).

[12] Kent R. Davey, Analytic analysis of single and three phaseinduction motors, Problem 30a, Workshop TEAM, http://ics.ec-lyon.fr/team.html.

[13] A. Nethe, T. Scholz and H-D. Stahlmann, An analyticalsolution method for magnetic fields using the Fourier analysis andits application of ferrofluid driven electric machines, Conference

Proceedings ISTET 2003, Warsaw, Vol. II, 421424 (2003).[14] A. Demenko, Numerical Approach to Electrical Machines

Dynamics, Pozna University of Technology, Pozna, 1997 (inPolish).

[15] A. Demenko, Finite element analysis of electromagnetictorque saturation harmonics in a squirrel cage machine, COMPEL18(4), 619628, (1999).

[16] I. S. Gradsztajn and I. M. Ryzik, Tablicy Intjegralow, Sum,Rjadow i Proizwjedjenij, Moskwa, 1962, (in Russian).

[17] S. Kastner, Vectors, Tensors, Spinors, Akademie-Verlag,Berlin, 1960 (in German).

[18] K. Davey, G. Vachtsevanos and R. Powers R., The anal-ysis of fields and torques in spherical induction motors, IEEETransaction on MagneticsMag-23 (1), (1987).

[19] J. Purczyski and L. Kaszycki, Power losses and elec-

tromagnetic torque of spherical induction motor, Rozprawy Elek-trotechniczne34(3), 819838 (1988) (in Polish).

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[20] A. N. Miliach, Osnowy Tjeorii Elektrodinamiczjeskich Sist-jem s Tremija Stepjenjami Swobody Dwizjenija, Akadjemija NaukYkrainskoj SSR, Kijew, 1956.

[21] K.-M. Lee, R. Roth and Z. Zhou, Dynamic modellingand control of a ball-joint-like variable reluctance spherical motor,ASME Journal Of Dynamics Systems, Measurements, And Control

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degrees-of-freedom variable-reluctance spherical motor, J. of SystemsEngineering4, 6069 (1994) (also presented in Proceedings of Japan-U.S.A. Symposium On Flexible Automation, Kobe, Japan, July 1118,1994).

250 Bull. Pol. Ac.: Tech. 52(3) 2004

Analytical electromagnetic field and forces calculation for linear, cylindrical and spherical...

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