IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 9, SEPTEMBER 2006 2159

A Comparison of Single-Layer Coaxial Coil MutualInductance Calculations Using Finite-Element

and Tabulated MethodsThomas G. Engel and Stacy N. Rohe

Electrical and Computer Engineering, University of Missouri-Columbia, Columbia, MO 65211 USAExelon Nuclear, Byron Generating Station, IL 61109 USA

Quick and accurate methods to calculate the mutual inductance of coaxial single layer coils remains important to this day in a largevariety of engineering and physical disciplines. While modern finite-element electromagnetic field codes can do this accurately, the en-gineer often requires only a first- or second-order estimate before proceeding to the numerical analysis stage. Grover’s tabular data,developed in the first half of the 20th century, remains the standard for manually calculating mutual inductance for a wide variety ofcoil and wire forms. This investigation reports the accuracy of mutual inductance calculations for single-layer coaxial coils based onGrover’s tables when compared to estimates obtained with a finite-element electromagnetic field code (FEEFC). Since it is impractical toconstruct and characterize the numerous coils needed for this type of investigation, the FEEFC results are treated as actual inductancemeasurements. Grover reported his tabular data to be accurate within five significant digits excluding the cases when the coils are looselycoupled and when the coils are short. This investigation found Grover’s tabular method to be inaccurate for loosely coupled and shortcoils, but also found that significant error for closely coupled coils as well. The maximum error between Grover’s tabular method andthe FEEFC results is 9.8%. Knowing the error associated with Grover’s method and the coil geometry for which the error occurs is animportant aid for the engineer and scientist.

Index Terms—Coils, error analysis, induction measurement, modeling.

I. INTRODUCTION

QUICK and accurate mutual inductance calculations ofcoaxial single-layer coils remains essential in manyfields of engineering and science despite the advent of

the digital computer [1]–[9]. While digital computers runningmodern codes can accurately calculate mutual inductance, thecoil geometry must first be programmed into the code andis often more time consuming than the calculation itself. Inaddition, the engineer often desires only a first- or second-orderestimate of the mutual inductance. The majority of work donein the area of inductance calculations occurred in the first halfof the 20th century. Starting with Neumann’s formulation,Nagaoka, Olshausen, and Terezawa derived absolute formulasfor the general case and Kirchhoff and Cohen for the concentriccase [10]. Generally, these formulas involved elliptical integralsof the first kind given by [11]

(1)

where is the elliptic integral of the first kind, is themodulus and equal to with , and is theamplitude. Since closed form analytic solutions of elliptical in-tegrals do not exist, series solutions were typically used to eval-uate the elliptic integral prior to the invention of the digitalcomputer.

In 1933, Grover introduced a set of three standard tables anda single formula that allowed one to calculate the mutual in-ductance of a wide variety of coil geometries with high accu-racy and without the use of a digital computer [10]. Grover

Digital Object Identifier 10.1109/TMAG.2006.880687

Fig. 1. Geometry for the mutual inductance of two coaxial single-layer coils.

used Clem’s series solution in combination with other series-form solutions to obtain the high degree of accuracy for the el-liptic integral values [12]. The error associated with Grover’stables is reportedly 10 to 10 , the higher error occurringwhen the coils are short or are loosely coupled [13]. Whenthe coils are loosely coupled, a situation is created where theterms in Grover’s formula nearly cancel making it difficult toobtain a high degree of accuracy. Grover’s tables are widelyused to manually calculate the mutual inductance of coaxialsingle-layer coils. Computer codes using Neumann’s formula-tion, electromagnetic field codes, or mathematical analysis soft-ware are more commonly used today when a high degree of ac-curacy is required.

As for hand, or manual calculations, the accuracy of Grover’stabular method is unknown. Anecdotal evidence suggests errorsas high as 40% and more. There are no comparative analyses ofGrover’s method in the open literature. One investigator founda more accurate method to evaluate the elliptic integral [13] of

0018-9464/$20.00 © 2006 IEEE

2160 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 9, SEPTEMBER 2006

Fig. 2. Percentage error for equal length coils.

Neumann’s formulation and compared this result to Grover’stabular method but did not comprehensively examine the pa-rameter space. Given the importance and common usage ofGrover’s tabular method in such fields as pulse power, electricmachinery, and electromagnetic compatibility, a comparativeanalysis is needed. This investigation summarizes a thesis onthis topic [14] and reports the accuracy of Grover’s mutualinductance calculation when compared to results obtainedwith finite-element electromagnetic field code (i.e., FEEFC)software. The FEEFC results in this investigation are used as asubstitute for physical inductance measurements. Constructingand accurately characterizing the large number of coils neededfor this type of investigation would be impractical, if notimpossible. This investigation shows that Grover’s claim ofaccuracy within five significant digits is not correct. With equallength concentric coils, the accuracy is three significant digits.Any separation of the coils causes the accuracy to drop to 2significant digits. With unequal length coils, Grover’s tabulardata is significant to 4 digits only. The accuracy of Grover’smethod decreases as the coil separation distance increasesand also as the radii ratio increases with the largest errorbeing 10%. Overall, the error is less for unequal length coilsthan for equal length coils.

II. THEORY AND MODELING

A. Grover’s Tabular Method

To find the mutual inductance of two coaxial single-layercoils, Grover started with the geometry of Fig. 1. In that figure,

and are the radial lengths, and are the axiallengths. The thickness of the coils is infinitesimally small. Themutual inductance is given by the sum of four elliptic integralswhich are functions of the four distances measured between thecoils. With reference to Fig. 1, these four distances are given as

(2)

TABLE ICOIL PARAMETER RANGES

where is the axial length of coil 1, is the axial lengthof coil 2, and is the separation of the two coil centers. It isto be noted that all dimensions in (2) are given in centimeters.Depending on , the coils could be partially inside, completelyinside, or completely outside each other. Four correspondingdiagonal distances can be calculated from (2) as

(3)

where is the smaller coil 1 radius and is the larger coil 2radius. The mutual inductance of coaxial single-layer coils isgiven by Grover’s general formula as

(4)

where is the mutual inductance given in is the coil 1winding density, and is the coil 2 winding density. Thevariables in (4) are the elliptic integral values and are given inGrover’s tables according to the two look-up constants

(5)

and

(6)

ENGEL AND ROHE: COMPARISON OF SINGLE-LAYER COAXIAL COIL MUTUAL INDUCTANCE CALCULATIONS 2161

TABLE IICOMPARISON BETWEEN GROVER’S EXAMPLE CALCULATIONS AND THE FEEFC RESULTS

MatLab mathematical analysis software [15] was used to calcu-late Grover’s mutual inductance in (4) over the coil parameterranges listed in Table I. Two coil parameter ranges are listed inTable I. The first parameter range was used for all analyses. Thesecond coil parameter range is a subset of the first range. WithGrover’s tables stored in Matlab, the look-up constants of (5)and (6) were calculated and used to find the values. If the

value needed interpolation, double interpolation was used.

B. Finite-Element Electromagnetic Field Code Method

Strictly speaking, experimental verification of Grover’stabular method requires the construction of a large numberof coils with closely controlled tolerances. Stringent controlswould have to be used to in order to accurately characterizethe coils since any deviation in winding density or coil lengthcould affect the measurements. Obviously, constructing sucha large number of coils and accurately characterizing themis very impractical, if not impossible, and probably explainswhy an investigation of this nature has not been attemptedbefore. A more practical method would be to use FEEFCsoftware, treating the results obtained as experimentally mea-sured values. There is a general consensus amongst usersthat well-known commercially available FEEFC software isaccurate. The FEEFC software used in this investigation isMagNet by Infolytica [16]. The FEEFC solution assumes atwo-dimensional (2-D) rotationally symmetric geometry and issolved using a magnetic vector potential formulation.

Once the coil and winding parameter information is enteredinto the FEEFC, MagNet creates a 2-D rotationally symmetricconstruct of the two coils and an “air space” to enclose them.The air space boundary locates the infinitely distant point inspace where the electromagnetic field is zero and serves to limitthe extent of the computational domain. In the numerical model,the air box length is 300 times the larger coil length with a height75 times the larger coil radius. The MagNet solution consists ofstored energy, force, flux linkage, power loss, and current foreach coil. The mutual inductance is solved using

(7)

where is the mutual inductance, is the total flux linkagein coil 1 due to flux produced by in coil 2 and is the totalflux linkage in coil 2 due to flux produced by in coil 1. Total

flux linkage is the sum of the flux linkage for each coil turn.In this investigation, the turn-to-turn spacing is zero, while thecoil thickness is 0.01 cm, in accordance with Grover’s use ofinfinitesimally thin current sheets. Further reductions in the coilthickness had no detectable affect on the FEEFC results. The“stranded” type of coil is used in MagNet to keep the currentand coil turns separate.

To achieve high accuracy in the FEEFC results, the h-adap-tation in MagNet (i.e., finite-element grid) is set to refine 25%of the elements to a tolerance of 0.01%. The convergence tol-erance for the FEEFC solution is 0.01% with a second-degreepolynomial fit.

III. RESULTS

Example calculations contained in Grover’s published reports[10], [12] are generally accepted as accurate and are, therefore,used as to verify the accuracy of the FEEFC results. Table IIshows the error between Grover’s example calculations and theFEEFC results. The smallest error in this comparison is 0.01%occurring when coil 1 is 4 cm long, coil 2 is 50 cm long, and theseparation between the coil centers is 0 cm while largest erroris 1.07% occurring when coil 1 is 10 cm long, coil 2 is 6 cm long,and the separation between the coil centers is 18 cm. While 1%error is acceptable for manual calculations, one would expecta lower error value given Grover’s statement of agreement towithin 5 significant digits.

The next part of the experimental results consists of com-paring Grover’s and FEEFC results over a wide range of coilgeometries. The coil geometries include those of practical in-terest such as equal radii, very different radii, equal lengths, dif-ferent lengths, and very short coils. This analysis is performedby fixing the coil lengths, turn densities, and radius of the largercoil while sweeping the separation distance and the radii ratio

. It is noted that the smaller coil radius is affected by changesin .

The results of this analysis can be seen in Figs. 3 and 4,which show the error between Grover’s method and the FEEFCmethod to be %. In general and for all the all cases consid-ered in this investigation, the error increases as increases, i.e.,as the small coil radius approaches the large coil radius, whichappears to contradict Grover’s statement that error in his tabulardata is largest for poorly coupled coils.

2162 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 9, SEPTEMBER 2006

Fig. 3. Percentage error for equal length coils for large �.

Fig. 4. Percentage error for unequal length coils.

Figs. 2 and 3 show the error between the Grover’s method andthe FEEFC method when the coil lengths are equal. In thosefigures, the error is very small for . As increasesabove 0.6 the error becomes larger, then when , theerror increases even more and is greater than 5%. The maximumerror in Fig. 2 is 9.8% occurring when cm and

. The minimum error in Fig. 2 is 0.00095% occurring whencm and . One of the most remarkable trends

in Fig. 3 is a significant decrease in the error to 1% or less forand . Fig. 3 is an enlarged

portion of the Fig. 4 data when and.

Fig. 4 data illustrates the error between Grover’s method andthe FEEFC method when the coil lengths are not equal. For

, there is approximately 0.2% or less error be-tween the two methods. The majority of the error occurs for

and for , again contradictory to Grover’sclaims. The minimum error in the data of Fig. 4 is 0.00171%when and while the maximum error is 5.7%when and .

The results show that the claim of Grover that his tables areaccurate to four or five significant digits does not appear tobe correct. With equal length concentric coils, the accuracy is

three significant digits. Any separation of the coils causes theaccuracy to drop to 2 significant digits. With unequal lengthcoils, Grover’s tabular data is significant to 4 digits only when

and . Otherwise, the accuracy decreasesto only 1 digit. The loss in accuracy occurs when the fourvalues are nearly equal, thereby canceling each other. From ageometric viewpoint, this investigation did find agreement withGrover’s claim that higher error is associated with very short orare loosely coupled coils.

IV. CONCLUSION

Mutual inductance calculations comparing Grover’s tabulardata method and an FEEFC method are performed. The majorityof the results do not agree with the four to five significant digitaccuracy reported by Grover. The results show the accuracy ofGrover’s method decreases as the coil separation distance in-creases and also as the radii ratio increases with the largesterror being 10%. Overall, the error is less for unequal lengthcoils than for equal length coils.

While Grover’s tabular method was not as accurate asclaimed in comparison to calculations done with FEEFC, thetabular method remains an important and accurate means to

ENGEL AND ROHE: COMPARISON OF SINGLE-LAYER COAXIAL COIL MUTUAL INDUCTANCE CALCULATIONS 2163

manually calculate mutual inductance. The exact cause forthe larger than expected error in Grover’s tabular data is un-known, but [13] speculates the error is caused by the use ofless-than-optimal series solution for the elliptic integrals.

Future investigations should explore the use of other series-form solutions that can potentially increase the accuracy of el-liptic integral evaluation, as is done in [13], or compare Grover’stabular method directly to Neumann’s formulation in the hopesof more accurate tabular data.

ACKNOWLEDGMENT

This work was supported by the Air Force Office of ScientificResearch under Contract F49620-03-1-0350.

REFERENCES

[1] S. Lee et al., “Reduced modeling of eddy current-driven electromechan-ical system using conductor segmentation and circuit parameters ex-tracted by FEA,” IEEE Trans. Magn., vol. 41, no. 5, pp. 1448–1451,May 2005.

[2] O. A. Mohammed et al., “High frequency PM synchronous motor modeldetermined by FEA analysis,” IEEE Trans. Magn., vol. 42, no. 4, pp.1291–1294, Apr. 2006.

[3] S. I. Babic and C. Akyel, “New analytical-numerical solutions for themutual inductance of two coaxial circular coils with rectangular crosssection in air,” IEEE Trans. Magn., vol. 42, no. 6, pp. 1661–1669, Jun.2006.

[4] D. Ban et al., “Turbogenerator end-winding leakage inductance calcula-tion using a 3-D analytical approach based on the solution of Neumannintegrals,” IEEE Trans. Energy Convers., vol. 20, no. 1, pp. 98–105, Mar.2005.

[5] K. Agarwal et al., “Modeling and analysis of crosstalk noise in coupledRLC networks,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.,vol. 25, no. 5, pp. 892–901, May 2006.

[6] T. Tera et al., “Performances of bearingless and sensorless inductionmotor drive based on mutual inductances and rotor displacements es-timation,” IEEE Trans. Ind. Electron., vol. 53, no. 1, pp. 187–194, Feb.2006.

[7] S. Yu et al., “Loop-based inductance extraction and modeling for mul-ticonductor on-chip interconnects,” IEEE Trans. Electron. Devices, vol.53, no. 1, pp. 135–145, Jan. 2006.

[8] S. Wang et al., “Improvement of EMI filter performance with parasiticcoupling cancellation,” IEEE Trans. Power Electron., vol. 20, no. 5, pp.1221–1228, Sep. 2005.

[9] T. M. Zeeff et al., “Traces in proximity to gaps in return planes,” IEEETrans. Electromagn. Compat., vol. 47, no. 2, pp. 388–392, May 2005.

[10] F. W. Grover, “Tables for calculation of the mutual inductance of any twocoaxial single layer coils,” in Proc. IRE, vol. 21, 1933, pp. 1039–1049.

[11] M. L. Boas, Mathematical Methods in the Physical Sciences. NewYork: Wiley, 1983, pp. 474–479.

[12] F. Grover, Inductance Calculations; Working Formulas and Ta-bles.. New York: Van Nostrand, 1946.

[13] T. H. Fawzi and P. E. Burke, “The accurate computation of self and mu-tual inductances of circular coils,” IEEE Trans Power App. Syst., vol.PAS-97, no. 2, pp. 464–468, 1978.

[14] S. N. Rohe, “Investigation of the accuracy of Grover’s method whensolving for the mutual inductance of two single-layer coaxial coils,”M.S. thesis, Univ. Missouri-Columbia, Dec. 2005.

[15] MatLab, ver. 7 student. The MathWorks, Inc., 3 Apple Hill Drive, Natick,MA 01760-2098.

[16] MagNet, ver. 6.20. Infolytica Corporation, P.O. Box 1144, Station Placedu Parc, Montreal, QC H2W 2P4, Canada.

Manuscript received March 21, 2006; revised June 27, 2006. Correspondingauthor: T. Engel (e-mail: engelt@missouri.edu).