research article mutual inductance and magnetic...
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Research ArticleMutual Inductance and Magnetic Force Calculations forBitter Disk Coil (Pancake) with Nonlinear Radial Current andFilamentary Circular Coil with Azimuthal Current
Slobodan Babic1 and Cevdet Akyel2
1Departement de Genie Physique Ecole Polytechnique CP 6079 Succ Centre Ville Montreal QC Canada H3C 3A72Departement de Genie Electrique Ecole Polytechnique CP 6079 Succ Centre Ville Montreal QC Canada H3C 3A7
Correspondence should be addressed to Slobodan Babic slobodanbabicpolymtlca
Received 12 June 2016 Revised 8 August 2016 Accepted 22 August 2016
Academic Editor Gorazd Stumberger
Copyright copy 2016 S Babic and C Akyel This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Bitter coils are electromagnets used for the generation of extremely strongmagnetic fields superior to 30 T In this paper we calculatethe mutual inductance and the magnetic force between Bitter disk (pancake) coil with the nonlinear radial current and the circularfilamentary coil with the azimuthal currentThe close form expressed over complete elliptic integrals of the first and second kind aswell as Heumanrsquos Lambda function is obtained for this configuration either for themutual inductance or for themagnetic forceTheresults of thismethod are comparedwith those obtained by the improvedmodified filamentmethod for the presented configurationAll results are in an excellent agreement
1 Introduction
In the literature and scientific papers the calculation of themutual inductance and the magnetic force between ordinarycircular coils with the uniform current densities have beengiven by different methods either analytical or numeri-cal [1ndash15] However there are not a lot references aboutthe mutual inductance and the magnetic force calculationbetween circular coils in which the current densities are notuniform The coils with this characteristic are Bitter coils orBitter solenoids Bitter coils are used in high magnetic fieldapplications and they differ from the ordinary coils in thatthey have an inverse radial distribution of current [16 17]Theinteresting point is the calculation of the mutual inductanceand the magnetic force between Bitter coils or between oneBitter coil and an ordinary coil Generally the calculation ofthe magnetic force between circular coils is closely related tothe calculation of theirmutual inductance Since theirmutualenergy is equal to the product of their mutual inductanceand the currents in the coils the component of the magneticforce of attraction or repulsion in any direction is equal tothe product of the currents multiplied by the differential
coefficient of themutual inductance takenwith respect to thatcoordinate The mutual inductance and the magnetic forceare obtained over multiple integrals with the different kernelfunctions Thus in the calculation of the mutual inductanceand the corresponding magnetic force between two coils weneed to integrate their kernel functions depending on coilconfigurations Hopefully these kernel functions are Greenfunctions 119903minus05 and 119903minus15 where the ldquo119903rdquo is the distance betweentwo coils This integral approach much easier than thedifferential approach leads to relatively simple expressionswhich are incorporated in these two physical quantities Inthis paper we calculate the mutual inductance between theBitter disk coil with nonlinear radial current and a circularfilamentary coil with the azimuthal current (ordinary coil)Coils are coaxial and in air Either the mutual inductanceor the magnetic force is obtained in closed form expressedover complete elliptic integrals of the first and second kind aswell as over Heumanrsquos Lambda function [18 19] The resultsof these calculations will be compared to those obtained bythe improved filament method for concerned configurationThe results obtained by these twomethods are in an excellentagreement
Hindawi Publishing CorporationAdvances in Electrical EngineeringVolume 2016 Article ID 3654021 6 pageshttpdxdoiorg10115520163654021
2 Advances in Electrical Engineering
z
y
x
z
y
N1
z1
z2
z1
z2
R1
R2
R
Figure 1 The thin Bitter disk coil (pancake) and the filamentarycircular coil
2 Basic Expressions
The mutual inductance and magnetic force between a diskcoil (pancake) with the uniform azimuthal current density1198691
= 1198731
1198681
(1198772
minus1198771
) and a filamentary coil with the azimuthalcurrent 119868
2
(see Figure 1) can be calculated respectively by[1 2]
119872 =
1205830
1198731
119877
(1198772
minus 1198771
)
int
120587
0
int
119877
2
119877
1
119903119868
cos 120579 119889119903119868
119889120579
1199030
(1)
119865 = minus
1205830
1198731
1198681
1198682
119877 (1199112
minus 1199111
)
(1198772
minus 1198771
)
int
120587
0
int
119877
2
119877
1
119903119868
cos 120579 119889119903119868
119889120579
1199033
0
(2)
where
1199030
= radic(1199112
minus 1199111
)2
+ 1199032
119868
+ 1198772
minus 2119903119868
119877 cos 120579 (3)
and1198731
is the number of turns of the disk coil (pancake)1205830
= 4120587 times 10minus7Hm is the permeability of free space
(vacuum) and 119903 120579 119911 are the cylindrical coordinates Bothcoils are in air or in a nonmagnetic and nonconductingenvironment
In [2] the magnetic force is obtained in the close formexpressed over the complete elliptic integral of the first kindandHeumanrsquos Lambda function and themagnetic force in thesemianalytical form expressed over complete elliptic integralof the first and second kind Heumanrsquos Lambda function plusone simple integral which has to be evaluated by some of thenumerical integrations [1]
Let us suppose the disk coil be the Bitter coil in which thecurrent density is not uniform [16 17] and given by
1198691
=
1198731
1198681
119903119868
ln (1198772
1198771
)
(4)
From (1) (2) and (4) we obtain the expressions of the mutualinductance and themagnetic force between the Bitter coil andthe filamentary coil as follows
119872119861
=
1205830
1198731
119877
ln (1198772
1198771
)
int
120587
0
int
119877
2
119877
1
cos 120579 119889119903119868
119889120579
1199030
(5)
119865119861
= minus
1205830
1198731
1198681
1198682
119877 (1199112
minus 1199111
)
ln (1198772
1198771
)
int
120587
0
int
119877
2
119877
1
cos 120579 119889119903119868
119889120579
1199033
0
(6)
where
1199030
= radic(1199112
minus 1199111
)2
+ 1199032
119868
+ 1198772
minus 2119903119868
119877 cos 120579 (7)
3 Calculation Method
Integrating in (5) and (6) over 119903119868
and 120579 (substituting 120579 = 120587minus120573)the mutual inductance and the magnetic force between theBitter disk and the filamentary circular coil can be expressedrespectively in an analytical form as follow (see AppendicesA and B)
119872119861
=
1205830
1198731
2 ln (1198772
1198771
)
119899=2
sum
119899=1
(minus1)119899minus1
119879119899
(8)
119865119861
=
1205830
1198731
1198681
1198682
ln (1198772
1198771
)
119899=2
sum
119899=1
(minus1)119899minus1
119878119899
(9)
where
119879119899
=
119896119899
radic119877119877119899
[2119877radic1198772
+ 1199112
119876
minus 1198772
119899
minus 1198772
minus 1199112
119876
]119870 (119896119899
)
+
119896119899
radic119877119877119899
[(119877 + 119877119899
)2
+ 1199112
119876
] 119864 (119896119899
) minus 1205871003816100381610038161003816119911119876
1003816100381610038161003816119881119899
119878119899
=
119896119899
119911119876
radic119877119877119899
119877
radic1198772
+ 1199112
119876
+ 119877
119870 (119896119899
) minus
120587
2
sgn (119911119876
) 119881119899
119881119899
= 1 minus Λ0
(1205791119899
119896119899
)
+ sgn (radic1198772 + 1199112119876
minus 119877119899
) [1 minus Λ0
(1205792119899
119896119899
)]
119911119876
= 1199112
minus 1199111
1198962
119899
=
4119877119877119899
(119877 + 119877119899
)2
+ 1199112
119876
le 1
119898 =
2119877
radic119877 + 1199112
119876
+ 119877
le 1
1205791119899
= arcsin1003816100381610038161003816119911119876
1003816100381610038161003816
radic1198772
+ 1199112
119876
+ 119877
1205792119899
= arcsinradic 1 minus 1198981 minus 1198962
119899
1198962
119899
le 119898
(10)
Thus all expressions are obtained in the closed formexpressed over complete elliptic integrals of the first andsecond kind as well as Heumanrsquos Lambda function [18 19]
31 Special Cases and Singularities
Mutual Inductance In the mutual inductance calculationspecial cases and singular cases appear for 119911
119876
= 0 and 1198962119899
= 1
or 119911119876
= 0 and 1198962119899
= 1
Advances in Electrical Engineering 3
311 119911119876
= 0 and 1198962119899
= 1 This case is the regular case coveredby (8) In this case the term 119879
119899
can be given in the followingsimplified form
119879119899
=
1198961198990
(119877 + 119877119899
)
radic119877119877119899
[(119877 minus 119877119899
)119870 (1198961198990
)
+ (119877 + 119877119899
) 119864 (1198961198990
)]
(11)
1198962
1198990
=
4119877119877119899
(119877 + 119877119899
)2
le 1 (12)
312 119911119876
= 0 and 1198962119899
= 1 This case is the singular case (119877 =1198771
or 119877 = 1198772
) for which the term 119879119899
is equal either 41198771
or41198772
Magnetic Force In the magnetic force calculation for 119911119876
= 0
its value is zero because the coils are in the same plane
Notation Above Λ0
corresponds to Heumanrsquos Lambda func-tion as defined in [18 19] and ldquosignrdquo function returnsinteger indicating the sign of a number Functions 119870 and119864 correspond to complete elliptic integrals of the first andsecond kind [18 19]
4 Modified Filament Method
In [17] the mutual inductance is calculated between the Bittercoil and the superconducting coil with uniform current den-sity by using the filament method Here we give the modifiedformulas for the mutual inductance and the magnetic forcefor the treated configuration (see Figure 2) using the filamentmethod
Applying some modification in the mutual inductancecalculation given in [17] we deduced the mutual inductanceand the magnetic force between the Bitter coil and thefilamentary coil as follows
119872BF =1198731
(1198772
minus 1198771
)
(2119899 + 1) ln (1198772
1198771
)
119897=119899
sum
119897=minus119899
119872(119897)
119903119868
(119897)
119865BF =1198731
1198681
1198682
(1198772
minus 1198771
)
(2119899 + 1) ln (1198772
1198771
)
119897=119899
sum
119897=minus119899
119865 (119897)
119903119868
(119897)
(13)
where
119872(119897)
=
21205830
radic119903119868
(119897) 119877
119896 (119897)
[(1 minus
1198962
(119897)
2
)119870 (119896 (119897)) minus 119864 (119896 (119897))]
119865 (119897)
= minus
1205830
1198681
1198682
119888119896 (119897)
4radic119903119868
(119897) 119877
[
2 minus 1198962
(119897)
1 minus 1198962
(119897)
119864 (119896 (119897)) minus 2119870 (119896 (119897))]
z
c
R
cells
R4
R3
2n + 1
RII
hII
Figure 2 Configuration of mesh matrix Filamentary coil-thin diskcoil (pancake)
119903119868
(119897) =
1198772
+ 1198771
2
+
1198772
minus 1198771
2119899 + 1
119897 (119897 = minus119899 0 119899)
119888 = 1199112
minus 1199111
= 119911119876
1198962
(119897) =
4119903119868
(119897) 119877
(119903119868
(119897) + 119877)2
+ 1198882
(14)
Expressions (13) will be used to confirm the validity ofanalytical formulas (8) and (9)
5 Examples
To verify the validity of the new formulas we apply them tothe following set of examples
Example 1 Calculate the mutual inductance and the mag-netic force between the Bitter disk coil and a filamentarycircular coil with the following dimensions and the numberof turns Currents in coils are unit
Filamentary Circular Coil 119877 = 20mm 1199111
= 0mm
Bitter Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 50mm1198731
= 100
Applying (8) and (9) the mutual inductance and themagnetic force are respectively
119872119861
= 52147220251309271 nH
119865119861
= minus16016662833788816 120583N(15)
The computational time is about 0037677 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 5214722025131178 nH
119865BF = minus1601666283378895 120583N(16)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0362069 seconds
Example 2 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
4 Advances in Electrical Engineering
Filamentary Circular Coil 119877 = 20mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100Applying (8) and (9) the mutual inductance and the
magnetic force are respectively
119872119861
= 17442876932070914 120583H
119865119861
= 0N(17)
The computational time is about 0020657 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 1744287693207058 120583H
119865BF = 0N(18)
The number of subdivision was 119899 = 20000 and thecomputational time about 0030564 seconds
Example 3 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
Filamentary Circular Coil 119877 = 40mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100
This case is the singular case because 119877 = 1198771
Applying (8) (9) and (11) the mutual inductance and the
magnetic force are respectively
119872119861
= 10931677144024915 120583H
119865119861
= 0N(19)
The computational time is about 0020223 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 1093167597572536 120583H
119865BF = 0N(20)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0513017 seconds
Example 4 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
Filamentary Circular Coil 119877 = 60mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100
This case is also the singular case for which 119877 = 1198772
Applying (8) (9) and (11) the mutual inductance and the
magnetic force are respectively
119872119861
= 14062056002507577 120583H
119865119861
= 0N(21)
The computational time is about 003929 seconds
By using the modified filament (13) and (14) we obtain
119872BF = 140620549882661 120583H
119865BF = 0N(22)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0487595 seconds
In comparative calculations the significant figures whichagree are bolded
Thus with these examples we confirmed the validity ofthe present analytical method for calculating the mutualinductance and the magnetic force between the Bitter diskwith the nonlinear radial current distribution and an ordi-nary filamentary circular coil All calculations were executedin Matlab programming
6 Conclusion
The new accurate mutual inductance and magnetic forceformulas for the system of the Bitter disk with the nonlinearradial current and the filamentary circular with the azimuthalin air are derived and presented in this paper All expressionsfor either the mutual inductance or the magnetic force areobtained in the close form expressed over complete ellipticintegrals of the first and second kind and Heumanrsquos Lambdafunction Also we gave in this paper the improved formulasfor the mutual inductance and the magnetic force betweentreated coils by using the filament method All singular casesare obtained in a closed form The presented method can beused to calculate these important electrical quantities suchas the mutual inductance and the magnetic force for coilscombinations comprising Bitter coils of rectangular crosssection and the circular coils with uniform current densities
Appendix
A Mutual Inductance Integrals
The first integral in (5) is 11986810
which can be solved analytically[19]
11986810
= int
119877
2
119877
1
119889119903119868
radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
= aresh119903119868
+ 119877 cos 2120573
radic1198772sin22120573 + 1199112
119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(A1)
The last integral in (5) is
11986820
= int
1205872
0
cos 2120573 sdot aresh119877119899
+ 119877 cos 2120573
radic1198772sin22120573 + 1199112
119876
119889120573 (A2)
Advances in Electrical Engineering 5
which can be obtained in the following form
11986820
=
119896119899
2
radic119877119899
119877
int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
119896119899
2
radic119877119899
119877
1198680
(A3)
where
119911119876
= 1199112
minus 1199111
1198962
119899
=
4119877119877119899
(119877 + 119877119899
)2
+ 1199112
119876
le 1
Δ = radic1 minus 1198962
119899
sin22120573
(A4)
The integral 1198680
can be obtained in its final form expressed overcomplete elliptic integrals of the first and second kind 119870(119896)119864(119896) as well as Heumanrsquos Lambda function Λ
0
(120576 119896) [18 19]
1198680
= int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 = int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
Δ
119889120573
+
1199112
119876
1198772
int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
2119877radic1198772
+ 1199112
119876
minus 1198772
119899
minus 1198772
minus 1199112
119876
2119877119877119899
119870(119896119899
) +
(119877 + 119877119899
)2
+ 1199112
119876
2119877119877119899
119864 (119896119899
)
minus
1205871003816100381610038161003816119911119876
1003816100381610038161003816
2119896119899
radic119877119877119899
119881119899
(A5)
where
119881119899
= 1 minus Λ0
(1205791119899
119896119899
)
+ sgn (radic1198772 + 1199112119876
minus 119877119899
) [1 minus Λ0
(1205792119899
119896119899
)]
(A6)
Replacing 11986820
in expression (5) we obtain (8)
B Magnetic Force Integrals
The first integral in (6) is 11987110
which has the analytic solution[19]
11987110
= int
119877
2
119877
1
119889119903119868
(1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
)
32
=
119903119868
+ 119877 cos 2120573
(1198772sin22120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(B1)
The last integral in (6) is
11987120
= 2119877119911119876
int
1205872
0
cos 2120573 (119877119899
+ 119877 cos 2120573) 119889120573
(1198772 sin2 2120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
(B2)
which can be given in the following form
11987120
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+ int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573]
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+
119877119899
119877
1198680
]
= minus
119896119899
119911119876
radic119877119899
119877
[119870 (119896119899
) +
119877119899
119877
1198680
]
(B3)
Integral 1198680
appears in Appendix A so that replacing 11987120
in (6)we obtain (9)
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Akyel S I Babic and S Kincic ldquoNew and fast proceduresfor calculating the mutual inductance of coaxial circular coils(circular coil-disk coil)rdquo IEEE Transactions on Magnetics vol38 no 5 pp 2367ndash2369 2002
6 Advances in Electrical Engineering
[2] C Akyel S I Babic S Kincic and P J Lagace ldquoMagnetic forcecalculation between thin circular coils and thin filamentarycircular coil in airrdquo Journal of Electromagnetic Waves andApplications vol 21 no 9 pp 1273ndash1283 2007
[3] S Babic C Akyel Y Ren and W Chen ldquoMagnetic forcecalculation between circular coils of rectangular cross sectionwith parallel axes for superconducting magnetrdquo Progress InElectromagnetics Research B no 37 pp 275ndash288 2012
[4] S I Babic and C Akyel ldquoNew analytic-numerical solutions forthe mutual inductance of two coaxial circular coils with rectan-gular cross section in airrdquo IEEE Transactions on Magnetics vol42 no 6 pp 1661ndash1669 2006
[5] S I Babic and C Akyel ldquoMagnetic force calculation betweenthin coaxial circular coils in airrdquo IEEE Transactions on Magnet-ics vol 44 no 4 pp 445ndash452 2008
[6] J T Conway ldquoInductance calculations for noncoaxial coilsusing bessel functionsrdquo IEEE Transactions onMagnetics vol 43no 3 pp 1023ndash1034 2007
[7] R Ravaud G Lemarquand S Babic V Lemarquand andC Akyel ldquoCylindrical magnets and coils fields forces andinductancesrdquo IEEE Transactions onMagnetics vol 46 no 9 pp3585ndash3590 2010
[8] R Ravaud G Lemarquand V Lemarquand S Babic and CAkyel ldquoMutual inductance and force exerted between thickcoilsrdquo Progress in Electromagnetics Research vol 102 pp 367ndash380 2010
[9] A Shiri and A Shoulaie ldquoA new methodology for magneticforce calculations between planar spiral coilsrdquo Progress inElectromagnetics Research vol 95 pp 39ndash57 2009
[10] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets Part 1 axial magnetizationrdquo IEEE Transactions onMagnet-ics vol 45 no 7 pp 2996ndash3002 2009
[11] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets part 2 radial magnetizationrdquo IEEE Transactions on Mag-netics vol 45 no 9 pp 3334ndash3342 2009
[12] J Coulomb and G Meunier ldquoFinite element implementation ofvirtual work principle for magnetic or electric force and torquecomputationrdquo IEEETransactions onMagnetics vol 20 no 5 pp1894ndash1896 1984
[13] A Benhama A C Williamson and A B J Reece ldquoForceand torque computation from 2 D and 3-D finite elementfield solutionsrdquo IEE ProceedingsmdashElectric Power ApplicationsJournal vol 146 no 1 pp 25ndash31 1999
[14] E P Furlani ldquoA formula for the levitation force betweenmagnetic disksrdquo IEEE Transactions on Magnetics vol 29 no 6pp 4165ndash4169 1993
[15] E P Furlani ldquoFormulas for the force and torque of axialcouplingsrdquo IEEE Transactions on Magnetics vol 29 no 5 pp2295ndash2301 1993
[16] J T Conway ldquoNon coaxial force and inductance calculationsfor bitter coils and coils with uniform radial current distribu-tionsrdquo in Proceedings of the International Conference on AppliedSuperconductivity and Electromagnetic Devices (ASEMD rsquo11) pp61ndash64 Sydney Australia December 2011
[17] Y Ren F Wang G Kuang et al ldquoMutual inductance and forcecalculations between coaxial bitter coils and superconductingcoils with rectangular cross sectionrdquo Journal of Superconductiv-ity and Novel Magnetism vol 24 no 5 pp 1687ndash1691 2011
[18] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Series 55 National Bureau of Standards AppliedMathematics Washington DC USA 1972
[19] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press New York NY USA 1965
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 Advances in Electrical Engineering
z
y
x
z
y
N1
z1
z2
z1
z2
R1
R2
R
Figure 1 The thin Bitter disk coil (pancake) and the filamentarycircular coil
2 Basic Expressions
The mutual inductance and magnetic force between a diskcoil (pancake) with the uniform azimuthal current density1198691
= 1198731
1198681
(1198772
minus1198771
) and a filamentary coil with the azimuthalcurrent 119868
2
(see Figure 1) can be calculated respectively by[1 2]
119872 =
1205830
1198731
119877
(1198772
minus 1198771
)
int
120587
0
int
119877
2
119877
1
119903119868
cos 120579 119889119903119868
119889120579
1199030
(1)
119865 = minus
1205830
1198731
1198681
1198682
119877 (1199112
minus 1199111
)
(1198772
minus 1198771
)
int
120587
0
int
119877
2
119877
1
119903119868
cos 120579 119889119903119868
119889120579
1199033
0
(2)
where
1199030
= radic(1199112
minus 1199111
)2
+ 1199032
119868
+ 1198772
minus 2119903119868
119877 cos 120579 (3)
and1198731
is the number of turns of the disk coil (pancake)1205830
= 4120587 times 10minus7Hm is the permeability of free space
(vacuum) and 119903 120579 119911 are the cylindrical coordinates Bothcoils are in air or in a nonmagnetic and nonconductingenvironment
In [2] the magnetic force is obtained in the close formexpressed over the complete elliptic integral of the first kindandHeumanrsquos Lambda function and themagnetic force in thesemianalytical form expressed over complete elliptic integralof the first and second kind Heumanrsquos Lambda function plusone simple integral which has to be evaluated by some of thenumerical integrations [1]
Let us suppose the disk coil be the Bitter coil in which thecurrent density is not uniform [16 17] and given by
1198691
=
1198731
1198681
119903119868
ln (1198772
1198771
)
(4)
From (1) (2) and (4) we obtain the expressions of the mutualinductance and themagnetic force between the Bitter coil andthe filamentary coil as follows
119872119861
=
1205830
1198731
119877
ln (1198772
1198771
)
int
120587
0
int
119877
2
119877
1
cos 120579 119889119903119868
119889120579
1199030
(5)
119865119861
= minus
1205830
1198731
1198681
1198682
119877 (1199112
minus 1199111
)
ln (1198772
1198771
)
int
120587
0
int
119877
2
119877
1
cos 120579 119889119903119868
119889120579
1199033
0
(6)
where
1199030
= radic(1199112
minus 1199111
)2
+ 1199032
119868
+ 1198772
minus 2119903119868
119877 cos 120579 (7)
3 Calculation Method
Integrating in (5) and (6) over 119903119868
and 120579 (substituting 120579 = 120587minus120573)the mutual inductance and the magnetic force between theBitter disk and the filamentary circular coil can be expressedrespectively in an analytical form as follow (see AppendicesA and B)
119872119861
=
1205830
1198731
2 ln (1198772
1198771
)
119899=2
sum
119899=1
(minus1)119899minus1
119879119899
(8)
119865119861
=
1205830
1198731
1198681
1198682
ln (1198772
1198771
)
119899=2
sum
119899=1
(minus1)119899minus1
119878119899
(9)
where
119879119899
=
119896119899
radic119877119877119899
[2119877radic1198772
+ 1199112
119876
minus 1198772
119899
minus 1198772
minus 1199112
119876
]119870 (119896119899
)
+
119896119899
radic119877119877119899
[(119877 + 119877119899
)2
+ 1199112
119876
] 119864 (119896119899
) minus 1205871003816100381610038161003816119911119876
1003816100381610038161003816119881119899
119878119899
=
119896119899
119911119876
radic119877119877119899
119877
radic1198772
+ 1199112
119876
+ 119877
119870 (119896119899
) minus
120587
2
sgn (119911119876
) 119881119899
119881119899
= 1 minus Λ0
(1205791119899
119896119899
)
+ sgn (radic1198772 + 1199112119876
minus 119877119899
) [1 minus Λ0
(1205792119899
119896119899
)]
119911119876
= 1199112
minus 1199111
1198962
119899
=
4119877119877119899
(119877 + 119877119899
)2
+ 1199112
119876
le 1
119898 =
2119877
radic119877 + 1199112
119876
+ 119877
le 1
1205791119899
= arcsin1003816100381610038161003816119911119876
1003816100381610038161003816
radic1198772
+ 1199112
119876
+ 119877
1205792119899
= arcsinradic 1 minus 1198981 minus 1198962
119899
1198962
119899
le 119898
(10)
Thus all expressions are obtained in the closed formexpressed over complete elliptic integrals of the first andsecond kind as well as Heumanrsquos Lambda function [18 19]
31 Special Cases and Singularities
Mutual Inductance In the mutual inductance calculationspecial cases and singular cases appear for 119911
119876
= 0 and 1198962119899
= 1
or 119911119876
= 0 and 1198962119899
= 1
Advances in Electrical Engineering 3
311 119911119876
= 0 and 1198962119899
= 1 This case is the regular case coveredby (8) In this case the term 119879
119899
can be given in the followingsimplified form
119879119899
=
1198961198990
(119877 + 119877119899
)
radic119877119877119899
[(119877 minus 119877119899
)119870 (1198961198990
)
+ (119877 + 119877119899
) 119864 (1198961198990
)]
(11)
1198962
1198990
=
4119877119877119899
(119877 + 119877119899
)2
le 1 (12)
312 119911119876
= 0 and 1198962119899
= 1 This case is the singular case (119877 =1198771
or 119877 = 1198772
) for which the term 119879119899
is equal either 41198771
or41198772
Magnetic Force In the magnetic force calculation for 119911119876
= 0
its value is zero because the coils are in the same plane
Notation Above Λ0
corresponds to Heumanrsquos Lambda func-tion as defined in [18 19] and ldquosignrdquo function returnsinteger indicating the sign of a number Functions 119870 and119864 correspond to complete elliptic integrals of the first andsecond kind [18 19]
4 Modified Filament Method
In [17] the mutual inductance is calculated between the Bittercoil and the superconducting coil with uniform current den-sity by using the filament method Here we give the modifiedformulas for the mutual inductance and the magnetic forcefor the treated configuration (see Figure 2) using the filamentmethod
Applying some modification in the mutual inductancecalculation given in [17] we deduced the mutual inductanceand the magnetic force between the Bitter coil and thefilamentary coil as follows
119872BF =1198731
(1198772
minus 1198771
)
(2119899 + 1) ln (1198772
1198771
)
119897=119899
sum
119897=minus119899
119872(119897)
119903119868
(119897)
119865BF =1198731
1198681
1198682
(1198772
minus 1198771
)
(2119899 + 1) ln (1198772
1198771
)
119897=119899
sum
119897=minus119899
119865 (119897)
119903119868
(119897)
(13)
where
119872(119897)
=
21205830
radic119903119868
(119897) 119877
119896 (119897)
[(1 minus
1198962
(119897)
2
)119870 (119896 (119897)) minus 119864 (119896 (119897))]
119865 (119897)
= minus
1205830
1198681
1198682
119888119896 (119897)
4radic119903119868
(119897) 119877
[
2 minus 1198962
(119897)
1 minus 1198962
(119897)
119864 (119896 (119897)) minus 2119870 (119896 (119897))]
z
c
R
cells
R4
R3
2n + 1
RII
hII
Figure 2 Configuration of mesh matrix Filamentary coil-thin diskcoil (pancake)
119903119868
(119897) =
1198772
+ 1198771
2
+
1198772
minus 1198771
2119899 + 1
119897 (119897 = minus119899 0 119899)
119888 = 1199112
minus 1199111
= 119911119876
1198962
(119897) =
4119903119868
(119897) 119877
(119903119868
(119897) + 119877)2
+ 1198882
(14)
Expressions (13) will be used to confirm the validity ofanalytical formulas (8) and (9)
5 Examples
To verify the validity of the new formulas we apply them tothe following set of examples
Example 1 Calculate the mutual inductance and the mag-netic force between the Bitter disk coil and a filamentarycircular coil with the following dimensions and the numberof turns Currents in coils are unit
Filamentary Circular Coil 119877 = 20mm 1199111
= 0mm
Bitter Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 50mm1198731
= 100
Applying (8) and (9) the mutual inductance and themagnetic force are respectively
119872119861
= 52147220251309271 nH
119865119861
= minus16016662833788816 120583N(15)
The computational time is about 0037677 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 5214722025131178 nH
119865BF = minus1601666283378895 120583N(16)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0362069 seconds
Example 2 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
4 Advances in Electrical Engineering
Filamentary Circular Coil 119877 = 20mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100Applying (8) and (9) the mutual inductance and the
magnetic force are respectively
119872119861
= 17442876932070914 120583H
119865119861
= 0N(17)
The computational time is about 0020657 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 1744287693207058 120583H
119865BF = 0N(18)
The number of subdivision was 119899 = 20000 and thecomputational time about 0030564 seconds
Example 3 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
Filamentary Circular Coil 119877 = 40mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100
This case is the singular case because 119877 = 1198771
Applying (8) (9) and (11) the mutual inductance and the
magnetic force are respectively
119872119861
= 10931677144024915 120583H
119865119861
= 0N(19)
The computational time is about 0020223 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 1093167597572536 120583H
119865BF = 0N(20)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0513017 seconds
Example 4 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
Filamentary Circular Coil 119877 = 60mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100
This case is also the singular case for which 119877 = 1198772
Applying (8) (9) and (11) the mutual inductance and the
magnetic force are respectively
119872119861
= 14062056002507577 120583H
119865119861
= 0N(21)
The computational time is about 003929 seconds
By using the modified filament (13) and (14) we obtain
119872BF = 140620549882661 120583H
119865BF = 0N(22)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0487595 seconds
In comparative calculations the significant figures whichagree are bolded
Thus with these examples we confirmed the validity ofthe present analytical method for calculating the mutualinductance and the magnetic force between the Bitter diskwith the nonlinear radial current distribution and an ordi-nary filamentary circular coil All calculations were executedin Matlab programming
6 Conclusion
The new accurate mutual inductance and magnetic forceformulas for the system of the Bitter disk with the nonlinearradial current and the filamentary circular with the azimuthalin air are derived and presented in this paper All expressionsfor either the mutual inductance or the magnetic force areobtained in the close form expressed over complete ellipticintegrals of the first and second kind and Heumanrsquos Lambdafunction Also we gave in this paper the improved formulasfor the mutual inductance and the magnetic force betweentreated coils by using the filament method All singular casesare obtained in a closed form The presented method can beused to calculate these important electrical quantities suchas the mutual inductance and the magnetic force for coilscombinations comprising Bitter coils of rectangular crosssection and the circular coils with uniform current densities
Appendix
A Mutual Inductance Integrals
The first integral in (5) is 11986810
which can be solved analytically[19]
11986810
= int
119877
2
119877
1
119889119903119868
radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
= aresh119903119868
+ 119877 cos 2120573
radic1198772sin22120573 + 1199112
119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(A1)
The last integral in (5) is
11986820
= int
1205872
0
cos 2120573 sdot aresh119877119899
+ 119877 cos 2120573
radic1198772sin22120573 + 1199112
119876
119889120573 (A2)
Advances in Electrical Engineering 5
which can be obtained in the following form
11986820
=
119896119899
2
radic119877119899
119877
int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
119896119899
2
radic119877119899
119877
1198680
(A3)
where
119911119876
= 1199112
minus 1199111
1198962
119899
=
4119877119877119899
(119877 + 119877119899
)2
+ 1199112
119876
le 1
Δ = radic1 minus 1198962
119899
sin22120573
(A4)
The integral 1198680
can be obtained in its final form expressed overcomplete elliptic integrals of the first and second kind 119870(119896)119864(119896) as well as Heumanrsquos Lambda function Λ
0
(120576 119896) [18 19]
1198680
= int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 = int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
Δ
119889120573
+
1199112
119876
1198772
int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
2119877radic1198772
+ 1199112
119876
minus 1198772
119899
minus 1198772
minus 1199112
119876
2119877119877119899
119870(119896119899
) +
(119877 + 119877119899
)2
+ 1199112
119876
2119877119877119899
119864 (119896119899
)
minus
1205871003816100381610038161003816119911119876
1003816100381610038161003816
2119896119899
radic119877119877119899
119881119899
(A5)
where
119881119899
= 1 minus Λ0
(1205791119899
119896119899
)
+ sgn (radic1198772 + 1199112119876
minus 119877119899
) [1 minus Λ0
(1205792119899
119896119899
)]
(A6)
Replacing 11986820
in expression (5) we obtain (8)
B Magnetic Force Integrals
The first integral in (6) is 11987110
which has the analytic solution[19]
11987110
= int
119877
2
119877
1
119889119903119868
(1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
)
32
=
119903119868
+ 119877 cos 2120573
(1198772sin22120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(B1)
The last integral in (6) is
11987120
= 2119877119911119876
int
1205872
0
cos 2120573 (119877119899
+ 119877 cos 2120573) 119889120573
(1198772 sin2 2120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
(B2)
which can be given in the following form
11987120
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+ int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573]
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+
119877119899
119877
1198680
]
= minus
119896119899
119911119876
radic119877119899
119877
[119870 (119896119899
) +
119877119899
119877
1198680
]
(B3)
Integral 1198680
appears in Appendix A so that replacing 11987120
in (6)we obtain (9)
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Akyel S I Babic and S Kincic ldquoNew and fast proceduresfor calculating the mutual inductance of coaxial circular coils(circular coil-disk coil)rdquo IEEE Transactions on Magnetics vol38 no 5 pp 2367ndash2369 2002
6 Advances in Electrical Engineering
[2] C Akyel S I Babic S Kincic and P J Lagace ldquoMagnetic forcecalculation between thin circular coils and thin filamentarycircular coil in airrdquo Journal of Electromagnetic Waves andApplications vol 21 no 9 pp 1273ndash1283 2007
[3] S Babic C Akyel Y Ren and W Chen ldquoMagnetic forcecalculation between circular coils of rectangular cross sectionwith parallel axes for superconducting magnetrdquo Progress InElectromagnetics Research B no 37 pp 275ndash288 2012
[4] S I Babic and C Akyel ldquoNew analytic-numerical solutions forthe mutual inductance of two coaxial circular coils with rectan-gular cross section in airrdquo IEEE Transactions on Magnetics vol42 no 6 pp 1661ndash1669 2006
[5] S I Babic and C Akyel ldquoMagnetic force calculation betweenthin coaxial circular coils in airrdquo IEEE Transactions on Magnet-ics vol 44 no 4 pp 445ndash452 2008
[6] J T Conway ldquoInductance calculations for noncoaxial coilsusing bessel functionsrdquo IEEE Transactions onMagnetics vol 43no 3 pp 1023ndash1034 2007
[7] R Ravaud G Lemarquand S Babic V Lemarquand andC Akyel ldquoCylindrical magnets and coils fields forces andinductancesrdquo IEEE Transactions onMagnetics vol 46 no 9 pp3585ndash3590 2010
[8] R Ravaud G Lemarquand V Lemarquand S Babic and CAkyel ldquoMutual inductance and force exerted between thickcoilsrdquo Progress in Electromagnetics Research vol 102 pp 367ndash380 2010
[9] A Shiri and A Shoulaie ldquoA new methodology for magneticforce calculations between planar spiral coilsrdquo Progress inElectromagnetics Research vol 95 pp 39ndash57 2009
[10] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets Part 1 axial magnetizationrdquo IEEE Transactions onMagnet-ics vol 45 no 7 pp 2996ndash3002 2009
[11] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets part 2 radial magnetizationrdquo IEEE Transactions on Mag-netics vol 45 no 9 pp 3334ndash3342 2009
[12] J Coulomb and G Meunier ldquoFinite element implementation ofvirtual work principle for magnetic or electric force and torquecomputationrdquo IEEETransactions onMagnetics vol 20 no 5 pp1894ndash1896 1984
[13] A Benhama A C Williamson and A B J Reece ldquoForceand torque computation from 2 D and 3-D finite elementfield solutionsrdquo IEE ProceedingsmdashElectric Power ApplicationsJournal vol 146 no 1 pp 25ndash31 1999
[14] E P Furlani ldquoA formula for the levitation force betweenmagnetic disksrdquo IEEE Transactions on Magnetics vol 29 no 6pp 4165ndash4169 1993
[15] E P Furlani ldquoFormulas for the force and torque of axialcouplingsrdquo IEEE Transactions on Magnetics vol 29 no 5 pp2295ndash2301 1993
[16] J T Conway ldquoNon coaxial force and inductance calculationsfor bitter coils and coils with uniform radial current distribu-tionsrdquo in Proceedings of the International Conference on AppliedSuperconductivity and Electromagnetic Devices (ASEMD rsquo11) pp61ndash64 Sydney Australia December 2011
[17] Y Ren F Wang G Kuang et al ldquoMutual inductance and forcecalculations between coaxial bitter coils and superconductingcoils with rectangular cross sectionrdquo Journal of Superconductiv-ity and Novel Magnetism vol 24 no 5 pp 1687ndash1691 2011
[18] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Series 55 National Bureau of Standards AppliedMathematics Washington DC USA 1972
[19] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press New York NY USA 1965
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Electrical Engineering 3
311 119911119876
= 0 and 1198962119899
= 1 This case is the regular case coveredby (8) In this case the term 119879
119899
can be given in the followingsimplified form
119879119899
=
1198961198990
(119877 + 119877119899
)
radic119877119877119899
[(119877 minus 119877119899
)119870 (1198961198990
)
+ (119877 + 119877119899
) 119864 (1198961198990
)]
(11)
1198962
1198990
=
4119877119877119899
(119877 + 119877119899
)2
le 1 (12)
312 119911119876
= 0 and 1198962119899
= 1 This case is the singular case (119877 =1198771
or 119877 = 1198772
) for which the term 119879119899
is equal either 41198771
or41198772
Magnetic Force In the magnetic force calculation for 119911119876
= 0
its value is zero because the coils are in the same plane
Notation Above Λ0
corresponds to Heumanrsquos Lambda func-tion as defined in [18 19] and ldquosignrdquo function returnsinteger indicating the sign of a number Functions 119870 and119864 correspond to complete elliptic integrals of the first andsecond kind [18 19]
4 Modified Filament Method
In [17] the mutual inductance is calculated between the Bittercoil and the superconducting coil with uniform current den-sity by using the filament method Here we give the modifiedformulas for the mutual inductance and the magnetic forcefor the treated configuration (see Figure 2) using the filamentmethod
Applying some modification in the mutual inductancecalculation given in [17] we deduced the mutual inductanceand the magnetic force between the Bitter coil and thefilamentary coil as follows
119872BF =1198731
(1198772
minus 1198771
)
(2119899 + 1) ln (1198772
1198771
)
119897=119899
sum
119897=minus119899
119872(119897)
119903119868
(119897)
119865BF =1198731
1198681
1198682
(1198772
minus 1198771
)
(2119899 + 1) ln (1198772
1198771
)
119897=119899
sum
119897=minus119899
119865 (119897)
119903119868
(119897)
(13)
where
119872(119897)
=
21205830
radic119903119868
(119897) 119877
119896 (119897)
[(1 minus
1198962
(119897)
2
)119870 (119896 (119897)) minus 119864 (119896 (119897))]
119865 (119897)
= minus
1205830
1198681
1198682
119888119896 (119897)
4radic119903119868
(119897) 119877
[
2 minus 1198962
(119897)
1 minus 1198962
(119897)
119864 (119896 (119897)) minus 2119870 (119896 (119897))]
z
c
R
cells
R4
R3
2n + 1
RII
hII
Figure 2 Configuration of mesh matrix Filamentary coil-thin diskcoil (pancake)
119903119868
(119897) =
1198772
+ 1198771
2
+
1198772
minus 1198771
2119899 + 1
119897 (119897 = minus119899 0 119899)
119888 = 1199112
minus 1199111
= 119911119876
1198962
(119897) =
4119903119868
(119897) 119877
(119903119868
(119897) + 119877)2
+ 1198882
(14)
Expressions (13) will be used to confirm the validity ofanalytical formulas (8) and (9)
5 Examples
To verify the validity of the new formulas we apply them tothe following set of examples
Example 1 Calculate the mutual inductance and the mag-netic force between the Bitter disk coil and a filamentarycircular coil with the following dimensions and the numberof turns Currents in coils are unit
Filamentary Circular Coil 119877 = 20mm 1199111
= 0mm
Bitter Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 50mm1198731
= 100
Applying (8) and (9) the mutual inductance and themagnetic force are respectively
119872119861
= 52147220251309271 nH
119865119861
= minus16016662833788816 120583N(15)
The computational time is about 0037677 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 5214722025131178 nH
119865BF = minus1601666283378895 120583N(16)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0362069 seconds
Example 2 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
4 Advances in Electrical Engineering
Filamentary Circular Coil 119877 = 20mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100Applying (8) and (9) the mutual inductance and the
magnetic force are respectively
119872119861
= 17442876932070914 120583H
119865119861
= 0N(17)
The computational time is about 0020657 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 1744287693207058 120583H
119865BF = 0N(18)
The number of subdivision was 119899 = 20000 and thecomputational time about 0030564 seconds
Example 3 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
Filamentary Circular Coil 119877 = 40mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100
This case is the singular case because 119877 = 1198771
Applying (8) (9) and (11) the mutual inductance and the
magnetic force are respectively
119872119861
= 10931677144024915 120583H
119865119861
= 0N(19)
The computational time is about 0020223 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 1093167597572536 120583H
119865BF = 0N(20)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0513017 seconds
Example 4 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
Filamentary Circular Coil 119877 = 60mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100
This case is also the singular case for which 119877 = 1198772
Applying (8) (9) and (11) the mutual inductance and the
magnetic force are respectively
119872119861
= 14062056002507577 120583H
119865119861
= 0N(21)
The computational time is about 003929 seconds
By using the modified filament (13) and (14) we obtain
119872BF = 140620549882661 120583H
119865BF = 0N(22)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0487595 seconds
In comparative calculations the significant figures whichagree are bolded
Thus with these examples we confirmed the validity ofthe present analytical method for calculating the mutualinductance and the magnetic force between the Bitter diskwith the nonlinear radial current distribution and an ordi-nary filamentary circular coil All calculations were executedin Matlab programming
6 Conclusion
The new accurate mutual inductance and magnetic forceformulas for the system of the Bitter disk with the nonlinearradial current and the filamentary circular with the azimuthalin air are derived and presented in this paper All expressionsfor either the mutual inductance or the magnetic force areobtained in the close form expressed over complete ellipticintegrals of the first and second kind and Heumanrsquos Lambdafunction Also we gave in this paper the improved formulasfor the mutual inductance and the magnetic force betweentreated coils by using the filament method All singular casesare obtained in a closed form The presented method can beused to calculate these important electrical quantities suchas the mutual inductance and the magnetic force for coilscombinations comprising Bitter coils of rectangular crosssection and the circular coils with uniform current densities
Appendix
A Mutual Inductance Integrals
The first integral in (5) is 11986810
which can be solved analytically[19]
11986810
= int
119877
2
119877
1
119889119903119868
radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
= aresh119903119868
+ 119877 cos 2120573
radic1198772sin22120573 + 1199112
119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(A1)
The last integral in (5) is
11986820
= int
1205872
0
cos 2120573 sdot aresh119877119899
+ 119877 cos 2120573
radic1198772sin22120573 + 1199112
119876
119889120573 (A2)
Advances in Electrical Engineering 5
which can be obtained in the following form
11986820
=
119896119899
2
radic119877119899
119877
int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
119896119899
2
radic119877119899
119877
1198680
(A3)
where
119911119876
= 1199112
minus 1199111
1198962
119899
=
4119877119877119899
(119877 + 119877119899
)2
+ 1199112
119876
le 1
Δ = radic1 minus 1198962
119899
sin22120573
(A4)
The integral 1198680
can be obtained in its final form expressed overcomplete elliptic integrals of the first and second kind 119870(119896)119864(119896) as well as Heumanrsquos Lambda function Λ
0
(120576 119896) [18 19]
1198680
= int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 = int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
Δ
119889120573
+
1199112
119876
1198772
int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
2119877radic1198772
+ 1199112
119876
minus 1198772
119899
minus 1198772
minus 1199112
119876
2119877119877119899
119870(119896119899
) +
(119877 + 119877119899
)2
+ 1199112
119876
2119877119877119899
119864 (119896119899
)
minus
1205871003816100381610038161003816119911119876
1003816100381610038161003816
2119896119899
radic119877119877119899
119881119899
(A5)
where
119881119899
= 1 minus Λ0
(1205791119899
119896119899
)
+ sgn (radic1198772 + 1199112119876
minus 119877119899
) [1 minus Λ0
(1205792119899
119896119899
)]
(A6)
Replacing 11986820
in expression (5) we obtain (8)
B Magnetic Force Integrals
The first integral in (6) is 11987110
which has the analytic solution[19]
11987110
= int
119877
2
119877
1
119889119903119868
(1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
)
32
=
119903119868
+ 119877 cos 2120573
(1198772sin22120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(B1)
The last integral in (6) is
11987120
= 2119877119911119876
int
1205872
0
cos 2120573 (119877119899
+ 119877 cos 2120573) 119889120573
(1198772 sin2 2120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
(B2)
which can be given in the following form
11987120
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+ int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573]
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+
119877119899
119877
1198680
]
= minus
119896119899
119911119876
radic119877119899
119877
[119870 (119896119899
) +
119877119899
119877
1198680
]
(B3)
Integral 1198680
appears in Appendix A so that replacing 11987120
in (6)we obtain (9)
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Akyel S I Babic and S Kincic ldquoNew and fast proceduresfor calculating the mutual inductance of coaxial circular coils(circular coil-disk coil)rdquo IEEE Transactions on Magnetics vol38 no 5 pp 2367ndash2369 2002
6 Advances in Electrical Engineering
[2] C Akyel S I Babic S Kincic and P J Lagace ldquoMagnetic forcecalculation between thin circular coils and thin filamentarycircular coil in airrdquo Journal of Electromagnetic Waves andApplications vol 21 no 9 pp 1273ndash1283 2007
[3] S Babic C Akyel Y Ren and W Chen ldquoMagnetic forcecalculation between circular coils of rectangular cross sectionwith parallel axes for superconducting magnetrdquo Progress InElectromagnetics Research B no 37 pp 275ndash288 2012
[4] S I Babic and C Akyel ldquoNew analytic-numerical solutions forthe mutual inductance of two coaxial circular coils with rectan-gular cross section in airrdquo IEEE Transactions on Magnetics vol42 no 6 pp 1661ndash1669 2006
[5] S I Babic and C Akyel ldquoMagnetic force calculation betweenthin coaxial circular coils in airrdquo IEEE Transactions on Magnet-ics vol 44 no 4 pp 445ndash452 2008
[6] J T Conway ldquoInductance calculations for noncoaxial coilsusing bessel functionsrdquo IEEE Transactions onMagnetics vol 43no 3 pp 1023ndash1034 2007
[7] R Ravaud G Lemarquand S Babic V Lemarquand andC Akyel ldquoCylindrical magnets and coils fields forces andinductancesrdquo IEEE Transactions onMagnetics vol 46 no 9 pp3585ndash3590 2010
[8] R Ravaud G Lemarquand V Lemarquand S Babic and CAkyel ldquoMutual inductance and force exerted between thickcoilsrdquo Progress in Electromagnetics Research vol 102 pp 367ndash380 2010
[9] A Shiri and A Shoulaie ldquoA new methodology for magneticforce calculations between planar spiral coilsrdquo Progress inElectromagnetics Research vol 95 pp 39ndash57 2009
[10] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets Part 1 axial magnetizationrdquo IEEE Transactions onMagnet-ics vol 45 no 7 pp 2996ndash3002 2009
[11] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets part 2 radial magnetizationrdquo IEEE Transactions on Mag-netics vol 45 no 9 pp 3334ndash3342 2009
[12] J Coulomb and G Meunier ldquoFinite element implementation ofvirtual work principle for magnetic or electric force and torquecomputationrdquo IEEETransactions onMagnetics vol 20 no 5 pp1894ndash1896 1984
[13] A Benhama A C Williamson and A B J Reece ldquoForceand torque computation from 2 D and 3-D finite elementfield solutionsrdquo IEE ProceedingsmdashElectric Power ApplicationsJournal vol 146 no 1 pp 25ndash31 1999
[14] E P Furlani ldquoA formula for the levitation force betweenmagnetic disksrdquo IEEE Transactions on Magnetics vol 29 no 6pp 4165ndash4169 1993
[15] E P Furlani ldquoFormulas for the force and torque of axialcouplingsrdquo IEEE Transactions on Magnetics vol 29 no 5 pp2295ndash2301 1993
[16] J T Conway ldquoNon coaxial force and inductance calculationsfor bitter coils and coils with uniform radial current distribu-tionsrdquo in Proceedings of the International Conference on AppliedSuperconductivity and Electromagnetic Devices (ASEMD rsquo11) pp61ndash64 Sydney Australia December 2011
[17] Y Ren F Wang G Kuang et al ldquoMutual inductance and forcecalculations between coaxial bitter coils and superconductingcoils with rectangular cross sectionrdquo Journal of Superconductiv-ity and Novel Magnetism vol 24 no 5 pp 1687ndash1691 2011
[18] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Series 55 National Bureau of Standards AppliedMathematics Washington DC USA 1972
[19] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press New York NY USA 1965
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Electrical and Computer Engineering
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Navigation and Observation
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DistributedSensor Networks
International Journal of
4 Advances in Electrical Engineering
Filamentary Circular Coil 119877 = 20mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100Applying (8) and (9) the mutual inductance and the
magnetic force are respectively
119872119861
= 17442876932070914 120583H
119865119861
= 0N(17)
The computational time is about 0020657 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 1744287693207058 120583H
119865BF = 0N(18)
The number of subdivision was 119899 = 20000 and thecomputational time about 0030564 seconds
Example 3 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
Filamentary Circular Coil 119877 = 40mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100
This case is the singular case because 119877 = 1198771
Applying (8) (9) and (11) the mutual inductance and the
magnetic force are respectively
119872119861
= 10931677144024915 120583H
119865119861
= 0N(19)
The computational time is about 0020223 secondsBy using the modified filament (13) and (14) we obtain
119872BF = 1093167597572536 120583H
119865BF = 0N(20)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0513017 seconds
Example 4 Calculate the mutual inductance between a diskand a filamentary circular coil with the following dimensionsand the number of turns
Filamentary Circular Coil 119877 = 60mm 1199111
= 0mm
Disk Coil 1198771
= 40mm 1198772
= 60mm 1199112
= 0mm1198731
= 100
This case is also the singular case for which 119877 = 1198772
Applying (8) (9) and (11) the mutual inductance and the
magnetic force are respectively
119872119861
= 14062056002507577 120583H
119865119861
= 0N(21)
The computational time is about 003929 seconds
By using the modified filament (13) and (14) we obtain
119872BF = 140620549882661 120583H
119865BF = 0N(22)
The number of subdivision was 119899 = 1000000 and thecomputational time about 0487595 seconds
In comparative calculations the significant figures whichagree are bolded
Thus with these examples we confirmed the validity ofthe present analytical method for calculating the mutualinductance and the magnetic force between the Bitter diskwith the nonlinear radial current distribution and an ordi-nary filamentary circular coil All calculations were executedin Matlab programming
6 Conclusion
The new accurate mutual inductance and magnetic forceformulas for the system of the Bitter disk with the nonlinearradial current and the filamentary circular with the azimuthalin air are derived and presented in this paper All expressionsfor either the mutual inductance or the magnetic force areobtained in the close form expressed over complete ellipticintegrals of the first and second kind and Heumanrsquos Lambdafunction Also we gave in this paper the improved formulasfor the mutual inductance and the magnetic force betweentreated coils by using the filament method All singular casesare obtained in a closed form The presented method can beused to calculate these important electrical quantities suchas the mutual inductance and the magnetic force for coilscombinations comprising Bitter coils of rectangular crosssection and the circular coils with uniform current densities
Appendix
A Mutual Inductance Integrals
The first integral in (5) is 11986810
which can be solved analytically[19]
11986810
= int
119877
2
119877
1
119889119903119868
radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
= aresh119903119868
+ 119877 cos 2120573
radic1198772sin22120573 + 1199112
119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(A1)
The last integral in (5) is
11986820
= int
1205872
0
cos 2120573 sdot aresh119877119899
+ 119877 cos 2120573
radic1198772sin22120573 + 1199112
119876
119889120573 (A2)
Advances in Electrical Engineering 5
which can be obtained in the following form
11986820
=
119896119899
2
radic119877119899
119877
int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
119896119899
2
radic119877119899
119877
1198680
(A3)
where
119911119876
= 1199112
minus 1199111
1198962
119899
=
4119877119877119899
(119877 + 119877119899
)2
+ 1199112
119876
le 1
Δ = radic1 minus 1198962
119899
sin22120573
(A4)
The integral 1198680
can be obtained in its final form expressed overcomplete elliptic integrals of the first and second kind 119870(119896)119864(119896) as well as Heumanrsquos Lambda function Λ
0
(120576 119896) [18 19]
1198680
= int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 = int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
Δ
119889120573
+
1199112
119876
1198772
int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
2119877radic1198772
+ 1199112
119876
minus 1198772
119899
minus 1198772
minus 1199112
119876
2119877119877119899
119870(119896119899
) +
(119877 + 119877119899
)2
+ 1199112
119876
2119877119877119899
119864 (119896119899
)
minus
1205871003816100381610038161003816119911119876
1003816100381610038161003816
2119896119899
radic119877119877119899
119881119899
(A5)
where
119881119899
= 1 minus Λ0
(1205791119899
119896119899
)
+ sgn (radic1198772 + 1199112119876
minus 119877119899
) [1 minus Λ0
(1205792119899
119896119899
)]
(A6)
Replacing 11986820
in expression (5) we obtain (8)
B Magnetic Force Integrals
The first integral in (6) is 11987110
which has the analytic solution[19]
11987110
= int
119877
2
119877
1
119889119903119868
(1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
)
32
=
119903119868
+ 119877 cos 2120573
(1198772sin22120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(B1)
The last integral in (6) is
11987120
= 2119877119911119876
int
1205872
0
cos 2120573 (119877119899
+ 119877 cos 2120573) 119889120573
(1198772 sin2 2120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
(B2)
which can be given in the following form
11987120
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+ int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573]
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+
119877119899
119877
1198680
]
= minus
119896119899
119911119876
radic119877119899
119877
[119870 (119896119899
) +
119877119899
119877
1198680
]
(B3)
Integral 1198680
appears in Appendix A so that replacing 11987120
in (6)we obtain (9)
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Akyel S I Babic and S Kincic ldquoNew and fast proceduresfor calculating the mutual inductance of coaxial circular coils(circular coil-disk coil)rdquo IEEE Transactions on Magnetics vol38 no 5 pp 2367ndash2369 2002
6 Advances in Electrical Engineering
[2] C Akyel S I Babic S Kincic and P J Lagace ldquoMagnetic forcecalculation between thin circular coils and thin filamentarycircular coil in airrdquo Journal of Electromagnetic Waves andApplications vol 21 no 9 pp 1273ndash1283 2007
[3] S Babic C Akyel Y Ren and W Chen ldquoMagnetic forcecalculation between circular coils of rectangular cross sectionwith parallel axes for superconducting magnetrdquo Progress InElectromagnetics Research B no 37 pp 275ndash288 2012
[4] S I Babic and C Akyel ldquoNew analytic-numerical solutions forthe mutual inductance of two coaxial circular coils with rectan-gular cross section in airrdquo IEEE Transactions on Magnetics vol42 no 6 pp 1661ndash1669 2006
[5] S I Babic and C Akyel ldquoMagnetic force calculation betweenthin coaxial circular coils in airrdquo IEEE Transactions on Magnet-ics vol 44 no 4 pp 445ndash452 2008
[6] J T Conway ldquoInductance calculations for noncoaxial coilsusing bessel functionsrdquo IEEE Transactions onMagnetics vol 43no 3 pp 1023ndash1034 2007
[7] R Ravaud G Lemarquand S Babic V Lemarquand andC Akyel ldquoCylindrical magnets and coils fields forces andinductancesrdquo IEEE Transactions onMagnetics vol 46 no 9 pp3585ndash3590 2010
[8] R Ravaud G Lemarquand V Lemarquand S Babic and CAkyel ldquoMutual inductance and force exerted between thickcoilsrdquo Progress in Electromagnetics Research vol 102 pp 367ndash380 2010
[9] A Shiri and A Shoulaie ldquoA new methodology for magneticforce calculations between planar spiral coilsrdquo Progress inElectromagnetics Research vol 95 pp 39ndash57 2009
[10] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets Part 1 axial magnetizationrdquo IEEE Transactions onMagnet-ics vol 45 no 7 pp 2996ndash3002 2009
[11] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets part 2 radial magnetizationrdquo IEEE Transactions on Mag-netics vol 45 no 9 pp 3334ndash3342 2009
[12] J Coulomb and G Meunier ldquoFinite element implementation ofvirtual work principle for magnetic or electric force and torquecomputationrdquo IEEETransactions onMagnetics vol 20 no 5 pp1894ndash1896 1984
[13] A Benhama A C Williamson and A B J Reece ldquoForceand torque computation from 2 D and 3-D finite elementfield solutionsrdquo IEE ProceedingsmdashElectric Power ApplicationsJournal vol 146 no 1 pp 25ndash31 1999
[14] E P Furlani ldquoA formula for the levitation force betweenmagnetic disksrdquo IEEE Transactions on Magnetics vol 29 no 6pp 4165ndash4169 1993
[15] E P Furlani ldquoFormulas for the force and torque of axialcouplingsrdquo IEEE Transactions on Magnetics vol 29 no 5 pp2295ndash2301 1993
[16] J T Conway ldquoNon coaxial force and inductance calculationsfor bitter coils and coils with uniform radial current distribu-tionsrdquo in Proceedings of the International Conference on AppliedSuperconductivity and Electromagnetic Devices (ASEMD rsquo11) pp61ndash64 Sydney Australia December 2011
[17] Y Ren F Wang G Kuang et al ldquoMutual inductance and forcecalculations between coaxial bitter coils and superconductingcoils with rectangular cross sectionrdquo Journal of Superconductiv-ity and Novel Magnetism vol 24 no 5 pp 1687ndash1691 2011
[18] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Series 55 National Bureau of Standards AppliedMathematics Washington DC USA 1972
[19] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press New York NY USA 1965
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Electrical Engineering 5
which can be obtained in the following form
11986820
=
119896119899
2
radic119877119899
119877
int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
119896119899
2
radic119877119899
119877
1198680
(A3)
where
119911119876
= 1199112
minus 1199111
1198962
119899
=
4119877119877119899
(119877 + 119877119899
)2
+ 1199112
119876
le 1
Δ = radic1 minus 1198962
119899
sin22120573
(A4)
The integral 1198680
can be obtained in its final form expressed overcomplete elliptic integrals of the first and second kind 119870(119896)119864(119896) as well as Heumanrsquos Lambda function Λ
0
(120576 119896) [18 19]
1198680
= int
1205872
0
cos32120573 + ((1198772 + 1199112119876
) 119877119877119899
) cos22120573 minus cos 2120573 minus (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 = int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
Δ
119889120573
+
1199112
119876
1198772
int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573 =
2119877radic1198772
+ 1199112
119876
minus 1198772
119899
minus 1198772
minus 1199112
119876
2119877119877119899
119870(119896119899
) +
(119877 + 119877119899
)2
+ 1199112
119876
2119877119877119899
119864 (119896119899
)
minus
1205871003816100381610038161003816119911119876
1003816100381610038161003816
2119896119899
radic119877119877119899
119881119899
(A5)
where
119881119899
= 1 minus Λ0
(1205791119899
119896119899
)
+ sgn (radic1198772 + 1199112119876
minus 119877119899
) [1 minus Λ0
(1205792119899
119896119899
)]
(A6)
Replacing 11986820
in expression (5) we obtain (8)
B Magnetic Force Integrals
The first integral in (6) is 11987110
which has the analytic solution[19]
11987110
= int
119877
2
119877
1
119889119903119868
(1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
)
32
=
119903119868
+ 119877 cos 2120573
(1198772sin22120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119877
2
119877
1
(B1)
The last integral in (6) is
11987120
= 2119877119911119876
int
1205872
0
cos 2120573 (119877119899
+ 119877 cos 2120573) 119889120573
(1198772 sin2 2120573 + 1199112
119876
)radic1199032
119868
+ 2119903119868
119877 cos 2120573 + 1198772 + 1199112119876
(B2)
which can be given in the following form
11987120
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+ int
1205872
0
cos 2120573 + (1198772 + 1199112119876
) 119877119877119899
(cos22120573 minus (1198772 + 1199112119876
) 1198772
) Δ
119889120573]
= minus
119896119899
119911119876
radic119877119899
119877
[int
1205872
0
119889120573
Δ
+
119877119899
119877
1198680
]
= minus
119896119899
119911119876
radic119877119899
119877
[119870 (119896119899
) +
119877119899
119877
1198680
]
(B3)
Integral 1198680
appears in Appendix A so that replacing 11987120
in (6)we obtain (9)
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Akyel S I Babic and S Kincic ldquoNew and fast proceduresfor calculating the mutual inductance of coaxial circular coils(circular coil-disk coil)rdquo IEEE Transactions on Magnetics vol38 no 5 pp 2367ndash2369 2002
6 Advances in Electrical Engineering
[2] C Akyel S I Babic S Kincic and P J Lagace ldquoMagnetic forcecalculation between thin circular coils and thin filamentarycircular coil in airrdquo Journal of Electromagnetic Waves andApplications vol 21 no 9 pp 1273ndash1283 2007
[3] S Babic C Akyel Y Ren and W Chen ldquoMagnetic forcecalculation between circular coils of rectangular cross sectionwith parallel axes for superconducting magnetrdquo Progress InElectromagnetics Research B no 37 pp 275ndash288 2012
[4] S I Babic and C Akyel ldquoNew analytic-numerical solutions forthe mutual inductance of two coaxial circular coils with rectan-gular cross section in airrdquo IEEE Transactions on Magnetics vol42 no 6 pp 1661ndash1669 2006
[5] S I Babic and C Akyel ldquoMagnetic force calculation betweenthin coaxial circular coils in airrdquo IEEE Transactions on Magnet-ics vol 44 no 4 pp 445ndash452 2008
[6] J T Conway ldquoInductance calculations for noncoaxial coilsusing bessel functionsrdquo IEEE Transactions onMagnetics vol 43no 3 pp 1023ndash1034 2007
[7] R Ravaud G Lemarquand S Babic V Lemarquand andC Akyel ldquoCylindrical magnets and coils fields forces andinductancesrdquo IEEE Transactions onMagnetics vol 46 no 9 pp3585ndash3590 2010
[8] R Ravaud G Lemarquand V Lemarquand S Babic and CAkyel ldquoMutual inductance and force exerted between thickcoilsrdquo Progress in Electromagnetics Research vol 102 pp 367ndash380 2010
[9] A Shiri and A Shoulaie ldquoA new methodology for magneticforce calculations between planar spiral coilsrdquo Progress inElectromagnetics Research vol 95 pp 39ndash57 2009
[10] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets Part 1 axial magnetizationrdquo IEEE Transactions onMagnet-ics vol 45 no 7 pp 2996ndash3002 2009
[11] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets part 2 radial magnetizationrdquo IEEE Transactions on Mag-netics vol 45 no 9 pp 3334ndash3342 2009
[12] J Coulomb and G Meunier ldquoFinite element implementation ofvirtual work principle for magnetic or electric force and torquecomputationrdquo IEEETransactions onMagnetics vol 20 no 5 pp1894ndash1896 1984
[13] A Benhama A C Williamson and A B J Reece ldquoForceand torque computation from 2 D and 3-D finite elementfield solutionsrdquo IEE ProceedingsmdashElectric Power ApplicationsJournal vol 146 no 1 pp 25ndash31 1999
[14] E P Furlani ldquoA formula for the levitation force betweenmagnetic disksrdquo IEEE Transactions on Magnetics vol 29 no 6pp 4165ndash4169 1993
[15] E P Furlani ldquoFormulas for the force and torque of axialcouplingsrdquo IEEE Transactions on Magnetics vol 29 no 5 pp2295ndash2301 1993
[16] J T Conway ldquoNon coaxial force and inductance calculationsfor bitter coils and coils with uniform radial current distribu-tionsrdquo in Proceedings of the International Conference on AppliedSuperconductivity and Electromagnetic Devices (ASEMD rsquo11) pp61ndash64 Sydney Australia December 2011
[17] Y Ren F Wang G Kuang et al ldquoMutual inductance and forcecalculations between coaxial bitter coils and superconductingcoils with rectangular cross sectionrdquo Journal of Superconductiv-ity and Novel Magnetism vol 24 no 5 pp 1687ndash1691 2011
[18] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Series 55 National Bureau of Standards AppliedMathematics Washington DC USA 1972
[19] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press New York NY USA 1965
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Advances in Electrical Engineering
[2] C Akyel S I Babic S Kincic and P J Lagace ldquoMagnetic forcecalculation between thin circular coils and thin filamentarycircular coil in airrdquo Journal of Electromagnetic Waves andApplications vol 21 no 9 pp 1273ndash1283 2007
[3] S Babic C Akyel Y Ren and W Chen ldquoMagnetic forcecalculation between circular coils of rectangular cross sectionwith parallel axes for superconducting magnetrdquo Progress InElectromagnetics Research B no 37 pp 275ndash288 2012
[4] S I Babic and C Akyel ldquoNew analytic-numerical solutions forthe mutual inductance of two coaxial circular coils with rectan-gular cross section in airrdquo IEEE Transactions on Magnetics vol42 no 6 pp 1661ndash1669 2006
[5] S I Babic and C Akyel ldquoMagnetic force calculation betweenthin coaxial circular coils in airrdquo IEEE Transactions on Magnet-ics vol 44 no 4 pp 445ndash452 2008
[6] J T Conway ldquoInductance calculations for noncoaxial coilsusing bessel functionsrdquo IEEE Transactions onMagnetics vol 43no 3 pp 1023ndash1034 2007
[7] R Ravaud G Lemarquand S Babic V Lemarquand andC Akyel ldquoCylindrical magnets and coils fields forces andinductancesrdquo IEEE Transactions onMagnetics vol 46 no 9 pp3585ndash3590 2010
[8] R Ravaud G Lemarquand V Lemarquand S Babic and CAkyel ldquoMutual inductance and force exerted between thickcoilsrdquo Progress in Electromagnetics Research vol 102 pp 367ndash380 2010
[9] A Shiri and A Shoulaie ldquoA new methodology for magneticforce calculations between planar spiral coilsrdquo Progress inElectromagnetics Research vol 95 pp 39ndash57 2009
[10] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets Part 1 axial magnetizationrdquo IEEE Transactions onMagnet-ics vol 45 no 7 pp 2996ndash3002 2009
[11] R Ravaud G Lemarquand and V Lemarquand ldquoForce andstiffness of passive magnetic bearings using permanent mag-nets part 2 radial magnetizationrdquo IEEE Transactions on Mag-netics vol 45 no 9 pp 3334ndash3342 2009
[12] J Coulomb and G Meunier ldquoFinite element implementation ofvirtual work principle for magnetic or electric force and torquecomputationrdquo IEEETransactions onMagnetics vol 20 no 5 pp1894ndash1896 1984
[13] A Benhama A C Williamson and A B J Reece ldquoForceand torque computation from 2 D and 3-D finite elementfield solutionsrdquo IEE ProceedingsmdashElectric Power ApplicationsJournal vol 146 no 1 pp 25ndash31 1999
[14] E P Furlani ldquoA formula for the levitation force betweenmagnetic disksrdquo IEEE Transactions on Magnetics vol 29 no 6pp 4165ndash4169 1993
[15] E P Furlani ldquoFormulas for the force and torque of axialcouplingsrdquo IEEE Transactions on Magnetics vol 29 no 5 pp2295ndash2301 1993
[16] J T Conway ldquoNon coaxial force and inductance calculationsfor bitter coils and coils with uniform radial current distribu-tionsrdquo in Proceedings of the International Conference on AppliedSuperconductivity and Electromagnetic Devices (ASEMD rsquo11) pp61ndash64 Sydney Australia December 2011
[17] Y Ren F Wang G Kuang et al ldquoMutual inductance and forcecalculations between coaxial bitter coils and superconductingcoils with rectangular cross sectionrdquo Journal of Superconductiv-ity and Novel Magnetism vol 24 no 5 pp 1687ndash1691 2011
[18] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Series 55 National Bureau of Standards AppliedMathematics Washington DC USA 1972
[19] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press New York NY USA 1965
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of