New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air

IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 6, JUNE 2006 1661

New Analytic-Numerical Solutions for the MutualInductance of Two Coaxial Circular Coils

With Rectangular Cross Section in AirSlobodan I. Babic1 and Cevdet Akyel2

Département de Génie Électrique and Département de Génie Physique, École Polytechnique, Montréal, QC H3C 3A7, CanadaDépartement de Génie Électrique, École Polytechnique, Montréal, QC H3C 3A7, Canada

In this paper, we present analytic-numerical expressions for the calculation of the mutual inductance of two axisymetric circular coilswith rectangular cross section in air. This original and new method may seem complicated but it is explicit, accurate, and fast, eventhough all expressions are obtained by the complete elliptic integrals of the first and second kind, Heuman’s lambda function, and threeterms that must be solved numerically. We confirm the validity of this approach by comparing it with other approaches (filament methodand previously published data). We also compare the accuracy and the computational cost of this approach and that of the filamentmethod. All results obtained by the various approaches are in excellent agreement.

Index Terms—Filament method, Maxwell’s coils, mutual inductance, Neumann’s formula.

I. INTRODUCTION

THE mutual inductance as a fundamental electrical engi-neering parameter of practical importance in numerous

electrical applications (current reactors, superconducting mag-netic energy storage problems (SMES), magnetic resonanceapplications, coil guns, tubular linear motors, transmissionlines, antennas, very large scale integration systems, navaland spacecraft magnetics, implantable electronic devices,instrumented orthopedic implants, recording devices, tele-metric systems in biomedical engineering) can be computedby applying the Neumann’s formula directly or using alternatemethods [1]–[15]. Exact methods based on elliptic integral so-lutions for current loops, thin current cylinders, thin disks, andmassive coils have existed since at least the time of Maxwell butwere laborious without computers. The purpose of this paper isto present an elliptic integral-based solution for circular coilswith rectangular cross section in air. This calculation leads tovery accurate expressions obtained over the complete elliptic in-tegrals of the first and second kind, Heuman’s lambda function,and some members that have to be solved numerically usingnumerical integration. These members have been solved nu-merically because their analytical solutions do not exist. In thispaper, we used Gaussian numerical integration. All results willbe compared with an approach based on the filament method[7], which leads to relatively simple procedures for calculatingthe mutual inductance of two circular coils with rectangularcross section using the well-known formula for Maxwell’scoils [1]–[3]. It will be useful to compare the accuracy and thecomputational cost between new analytic-numerical approachand the filament method to confirm the rapidity and the pre-cision of the presented method because its analytical solutiondoes not exist. Also, the accuracy of the presented method willbe compared with other methods, which are known in the liter-ature. Computed mutual-inductance values obtained from the

Digital Object Identifier 10.1109/TMAG.2006.872626

proposed method are in excellent agreement with well-knownresults in the literature.

II. BASIC EXPRESSIONS

The mutual inductance of circuit elements not associated withmagnetic materials is independent of the current and dependantonly on geometry of the system. At relatively low frequency, thecurrent distribution varies very little in the cross sections and canbe assumed to be constant through the conductors. The mutualinductance can be calculated by the following expression [4]:

(1)

where and are the cross-sectional areas of two mas-sive conductors with current densities and whose corre-sponding currents are and and is the mutual induc-tance of two filaments and that belong to correspondingconductors each. This mutual inductance can be calculated byNeumann’s formula [4]

(2)

where is distance between and , and is the magneticpermeability in air.

Let us consider the system of two coaxial circular coils inair as shown in Fig. 1. This figure shows two air-core circuitsof rectangular cross section whose inner radii are , outerradii , with corresponding lengths and .

In a cylindrical coordinate system as shown in Fig. 1, twoelementary current loops, namely loop-I at and loop-IIat carry current and , respectively

(3)

0018-9464/$20.00 © 2006 IEEE

1662 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 6, JUNE 2006

Fig. 1. Two coaxial coils with rectangular cross section.

Respecting the cylindrical coordinate system and (1)–(3), themutual inductance of two coaxial coils with rectangular crosssection in air can be obtained by the following expression:

(4)

where

Equation (4) can be used to calculate the mutual inductance oftwo coaxial coils with rectangular cross section that are createdby and turns, respectively. If coils are compactly woundand the insulation thickness on the wire is thin, the electricalcurrents in these coils can be considered uniformly distributedover the whole cross sections on the winding with densitiesand , respectively, where and are the number of turnsin winding and are the electrical currents in the coils theirmutual inductance can be calculated by the expression

(5)

III. CALCULATION METHOD

In (4) or (5), integrations are made over and ,respectively (see the Appendix). The mutual inductance of twotreated coils is

(6)

(7)

where

BABIC AND AKYEL: ANALYTIC-NUMERICAL SOLUTIONS FOR THE MUTUAL INDUCTANCE OF TWO COAXIAL CIRCULAR COILS 1663

are complete elliptic integrals of the first and secondkind and is Heuman’s lambda function [16].

It is important to mention that the function is not asmooth function at one of its ends; it has a singularity at theleft end which can be eliminated by L’Hopital’s rule, [11]. In thecase of two solenoids of the same length ( and ),the expressions (6) or (7) are not directly applicable because pre-vious integrals have singularities at their ends. By performingsome transformations on them, it is possible to eliminate all sin-gularities without using the L’Hopital’s rule to obtain integralswhose kernel functions are continuous functions on all their in-terval of integration. The expression (6) or (7) with the followingchanges can calculate the mutual inductance

For

the expression does not change.

For

the expression becomes

(8)

where

Fig. 2. Maxwell’s coils.

The previous expressions can also be used in four possiblesingular cases and .

Expressions (6)–(8) have a wide range of numerical applica-tions. In the case when the distance between coils is very large

, more suitable expressions could be deduced usingrapidly convergent series which leads to relatively simple an-alytical expressions. Applying l’Hopital’s rule in the limit weobtain the mutual inductance .

IV. FILAMENT METHOD

In order to verify the accuracy and the computational costof expressions for the mutual inductance obtained by using theintegration of (4), we present another efficient method to calcu-late the mutual inductance of two circular coils with rectangularcross section. The main idea of this method, called the filamentmethod, is the using of Maxwell’s coils where coils are dividedinto coaxial filamentary circular coils of the negligible cross sec-tion, [7], Fig. 2 for which the mutual inductance is given by theexpression

(9)

Even though these procedures are given in [7], we briefly reca-pitulate this approach.

In order to account for the finite dimensions of the coils, cir-cular coils with rectangular cross section are considered to besubdivided into meshes of filamentary coils (Maxwell’s coils)as shown in Fig. 3. The cross-sectional area of coil I is di-vided into cells, and that of coil II into

cells. Each cell in each coil contains onefilament, and the current density in the coil cross section is as-sumed to be uniform, so that the filament currents are equal foreach coil. This means that it is possible to take into considera-tion the pair of filamentary unit turn coils for which the mutual


Fig. 3. Configuration of mesh coils.

inductance is given by (9), where and are the correspondingthin coils of the first and second coil, respectively.

Using the same procedures given in [7] the mutual inductancecan be expressed in the following form:

(10)where

V. EXAMPLES

To verify the validity of the expressions presented, let us solvesome problems.

Example 1: The coil dimensions and the number of turns areas follows:

Presented method:cm, cm, cm, cm,

cm, cm, cm, cm,.

TABLE ICOMPARISON OF COMPUTATIONAL EFFICIENCY (FILAMENT METHOD)

Filament method:Dimensions of the same coils given in the terms of suitable

for filament method are: cm, cm,cm, cm, .

The presented method (7) gives the mutual inductance

Execution time was 0.61 s. The mutual inductance is obtainedby using Gaussian numerical integration with 20 integrationpoints for solving integrals and . In [11], it wasshown that the evaluation of preceding integrals with 20 inte-gration points gives satisfactory results even though the kernelfunction of the integral has not defined for and hasfast oscillations nearby this point.

In Table I we show values of the mutual inductance usingthe filament method, expressions (10). Also, the correspondingcomputational time and the absolute error of calculation re-garding the obtained value by (7) are given. From Table I wesee that results obtained using (7) or (10) are in good agreement.But, to have approximately the same value of the mutual induc-tance obtained using the proposed method the filament methodrequires many subdivisions that considerably increases the com-putational cost.

Example 2: Consider two coaxial solenoids of inner radius0.5 m, outer radius 1.5 m, and height 1 m. Solenoids are adjacentwith spacing of 1 mm and , [12].

From [1], the mutual inductance is

From [12] the mutual inductance is

The presented method (6) gives the mutual inductance

All results are in good agreement.Example 3: Consider two coils of 200 turns each (Brooks

coils), mean radii cm, separation between theirmedian planes cm, and side of the square cross section

cm, [1].


The coil dimensions and number of turns are the following.Presented method:

Filament method:

From [1], the mutual inductance is

By the presented method (7), the mutual inductance is

By the filament method (10), the mutual inductance is

where the number of subdivisions was .From [13], the mutual inductance is

We can see that all results are in very good agreement.Example 4: Let us consider two coaxial solenoids of the same

length. Dimensions and the number of turns are the following.Presented method:

Filament method:

In this case, we use (8). The mutual inductance is

and the execution time was 0.61 s.The values of the mutual inductance obtained by the filament

method are given in Table II. From Table II, one can con-

TABLE IICOMPARISON OF COMPUTATIONAL EFFICIENCY (FILAMENT METHOD)

Fig. 4. Schematic of the linear induction launcher.

TABLE IIIMAIN DIMENSIONS OF MODEL LAUNCHER

TABLE IVMUTUAL INDUCTANCE M(�H)

clude that all results obtained using both methods are in goodagreement.

Example 5: In [17], three kinds of numerical methods (Fawzi/Burka’s, Williamson/Leonard’s, and Grover’s) for computingthe mutual inductance of circular coils are discussed and com-pared with experimental results. The linear induction launcher[18] (Fig. 4) was chosen for numerical calculation and experi-mental verification. Its main dimensions are shown in Table III.The number of turns of each coil was 10.

The mutual inductance for several separations were calcu-lated with the above three methods, measured results, and thepresented method. These results are shown in Table IV. FromTable IV, one can see that all results are in good agreement butthe best agreement is between experimental results and thoseproposed in this paper.

Example 6: In [19] the mutual inductance of Brooks coilswith a common axis has been calculated by two methods, onebased on the flux linkage and other based on the stored energy.


Fig. 5. Half cross section of a pair of Brooks coils.

Fig. 6. Transversal cross section of the system.

The results were obtained with MagNet version 6.4.1 (InfolyticaCorporation). The coils’ dimensions are shown in Fig. 5 and thenumber of turns is . The spacing is m, which isequal of the total height of one coil.

The value of the mutual inductance [1] is

By [19], the mutual inductance is

and

By the presented method (7), the mutual inductance is

Example 7: Finally, let us solve the problem proposed in [20]and [21]. They calculated the mutual inductance between twosteady currents flowing in two massive coaxial circular coilswith rectangular cross section in free space, (Fig. 6) by usingthe toroidal multipole expansion method, [20]. The distance be-tween the median planes of the coils is 0.5 m. The coil dimen-sions are given in Table V.

Presented method:

TABLE VDIMENSIONS OF THE COILS

From [20] and [21], the mutual inductance is

Applying (6), the mutual inductance is

All results are in very good agreement. The calculation of (6),(7), (8), and (10) was made on a personal computer with aPentium III 700-MHz processor.

VI. CONCLUSION

New accurate mutual-inductance expressions for two axisy-metric circular coils with rectangular cross section in air arederived and presented in this paper. The accuracy of the pro-posed approach has been shown by comparison with results ofthe filament method and other known approaches. The resultsare obtained in an analytical/numerical form over complete el-liptic integrals of the first and second kind, Heuman’s lambdafunction, plus three terms that have to be solved numericallyusing single integration (Gaussian numerical integration). Theresults are in very good agreement with published data. In pre-ceding derivations, we assumed that the current distribution isuniform in the cross section of the conductors. It means that weignored the skin effect. For cases where the skin effect cannotbe ignored, we can divide the cross section of conductors intoa mesh, similar as the filament method, and then apply the for-mula (7) for each pair of filaments in those meshes. The general-ized approach is accompanied by efficient software in MATLABprogramming language. Thus, the mutual inductance can be ef-ficiently calculated with only a personal computer.

APPENDIX AEXPLICIT INTEGRAL EXPRESSIONS OF FORMULA (4)

The first integral in (4) is

The solution of this integral, [22] is


The second integral in (2) is

The solution of this integral, [22] is

where

The integral in the last expression is very complicated to solve.Using the transformations 2.281, 2.282, and 2.283 in [22] weobtained

Finally, the integral is

The third integration is made over the variable using thesubstitution

It is important to mention that the factor in is incorpo-rated in the final formula for the mutual induction. After somecomplicated transformations, the integral is obtained in thefollowing form:

where

The next integration is regarding to the variable .Before making this integration, let us introduce the next sub-

stitution . The integral is

The factor in and the factor 2 from the last substitution areincorporated in the final formula for the mutual induction. Letus introduce next substitutions, and .It is possible to get this integration over only one variable either

or .


Even though the last integral is tedious, we will give its solu-tion step by step for each term

Finally, the last integral is

where and are given in (6) or (7).

ACKNOWLEDGMENT

This work was supported by Natural Science and Engi-neering Research Council of Canada (NSERC) under GrantRGPIN 4476-05 NSERC NIP 11963.

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Manuscript received December 4, 2005; revised February 22, 2006. Corre-sponding author: S. I. Babic (e-mail: [email protected]).

Slobodan I. Babic was born in Tuzla, Bosnia and Herzegovina. He receivedthe Dipl. Ing. degree from the Faculty of Electrical Engineering, University ofSarajevo, the M.Sc. degree from the Faculty of Electrical Engineering, Univer-sity of Zagreb, Croatia, and the Ph.D. degree from the Faculty of Electrical En-gineering, University of Sarajevo, Bosnia and Herzegovina, in 1975, 1992 and1980, respectively.

From 1975, he was with the Electrical Engineering Faculty of the Universityof Sarajevo, where he held an Associate Professor position until 1994. Since1997 he has been a Lecturer at École Polytechnique de Montréal, Montréal,QC, Canada. His major interests are in the mathematical modeling of stationaryand quasi-stationary fields, electromagnetic fields in machines, transformers,computational electromagnetics, magnetic materials, and field theory. He haspublished over 70 papers in these fields.

Dr. Babic is a member of the International Compumag Society.

Cevdet Akyel (M’81) was born in Samsun, Turkey. He received the Sup. Ing.degree from the Technical University of Istanbul in 1971 and the M.Sc.A. andD.Sc.A. degrees from École Polytechnique de Montréal, Montréal, QC, Canada,in 1975 and 1980, respectively.

He had engineering positions in 1972 and 1976 at Northern Telecom ofCanada as a System Engineer in radio telecommunications. Since 1986, hehas been a Professor of Electrical Engineering at École Polytechnique deMontréal, where he teaches electromagnetic theory and automated microwaveinstrumentation. In 1991, he joined the Group of Poly-Grames involved inspace electronics and microwave advanced technologies at the same university.His main research interests are the permittivity measurement with microwaveactive cavity methods, the characterization of materials (conductive polymers,superconductivity ceramics, ferromagnetic materials, etc.), and high powermicrowave measurement systems and applications.

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New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air