An improvement in the calculation of the self-inductance of thin disk coils with air-core..doc

An Improvement in the Calculation of the Self -Inductance of Thin Disk Coils with Air-Core SLOBODAN I. BABIC and CEVDET AKYEL Dpartement de Gnie lectrique & de Gnie Informatique cole Polytechnique de Montral Montral, C.P. 6079, Succ. Centre Ville QUBEC, CANADA Abstract: - The calculation of the self-inductance and the mutual inductance of air-core circular coils as thin current cylinders, thin disks, and massive coils have existed since the time of Maxwell - but were laborious without computers. In this paper we propose an improvement in calculation of the self-inductance of thin disk coils with air-core that can be encountered in SMES problems. The paper focuses on the numerical integration to perform the computational cost and the accuracy. Key-Words: - Self-inductance, thin disk coils, Gaussian numerical integration 1 Introduction Several monographs and papers are devoted to calculate the self-inductance of thin disk coils with air-core 1-12. In this paper, the new improved expressions are derived and presented to compute the self-inductance of thin disk coils with air-core. The obtained results are expressed over the complete elliptic integral of the first kind and two members that should have been solved numerically because their analytical solutions do not exist. The Gaussian numerical integrations 13 and 14 are used to evaluate these integrals whose kernel functions are continuous functions on the interval of integration 0, /2 so that we do not need to use LHopitals rule 6. We already gave the expressions of this calculation, 7 but the new proposed calculation represents significant simplification regarding to the computational cost and the range of application. Also this formula presents a simplification and improvement of the formula for the thin disk (pancake), 6. According to our research, we considerably reduced and ameliorated these expressions comparing to those they are known in literature 5-7. In this calculation we point out on the accuracy and the computational cost as advantages and advantages of our formula for practical applications. We also compare our results with those obtained by the program INCA 15 for single layer coaxial solenoids, conical and flat coils of round wire with the cross section. Computed self- inductance values agree with those found in the literature. All proposed procedures are suitable for practical applications. 2 Basic Expressions The self-inductance as a fundamental electrical engineering parameter for a coil, which can be computed by applying the Biot - Savart law directly or using the alternate methods 1 - 9. In the case of a thin disk coil (Fig. 1) the self-inductance can be calculated by 6, (1) 0 cos2 dddcos )( = 2 1 2 121 2 2 2 1 2211 2 12 2 0 R R R Rrrrr rrrr RR N L where R1 and R2 are the radius of the thin disk coil and N is the number of turns of the winding. It is supposed that the coil is compactly wound and the insulation layer on the wire is thin so that the electrical current can be considered uniformly distributed over the thin cross section of the winding, whose density is J (A/m). Also, the shape factor is defined as: (2) 1 1 2 R R r z R 2 R 1 Fig 1. Schematic drawing of a disk coil 3 Calculation Method In (1) we integrate over r2 , r1 and respectively. The self-inductance of a disc coil is obtained as an analytical/numerical combination expressed over the complete elliptic function of the first kind and two members, which will be solved numerically. All kernels of these integrals are continuous functions on the intervals of integration because we smartly eliminated all singularities in the process of calculation. The self inductance of the thin disk coil can be calculated as (Appendix), (3)( ) 1(3 2 2 1 2 0 S RN L where )() 1()() 1()() 1( )2/ )(log() 12)(1(=)( 2 3 1 3 3 IIkE kGS 2 2 )1 ( 4 )( k 2/ 0 22 1 d )(sin)(1log1)( xxkI 2/ 0 222 2 d )(sin)(1)(1log)( xxkkI G = 0.9159655941772190.,Catalanas constant. E is complete elliptic integrals of the first kind, 13. We obtain relatively easy expression for the self inductance which is not complicate as those in 1, 6 and 7. In 1 Spielrein obtained the self inductance over convergent series expressed with a lot of members. In 6 and 7 the self inductance is expressed by several integrals and some numerous expressions with the elliptic integrals. 4 Examples To verify the proposed computing approach we will treat all cases for the different shape factors concerning the accuracy and the computational cost. We use Gaussian numerical integrations 13 and 14 to solve integrals I1 and I2 because it does not take in consideration limits of integrals that is important if for one or two limits the kernel function has singularities. It is the case of the integral I2 which has the singularity at the right end for = 1 even though this integral converges, 16. For all shape factors different from 1 integrals I1 and I2 are continuous functions without singularities on all interval of integration. Before calculating the self- inductance for different shape factors let us evaluate the integrals I1 and I2 to find their influence on the computational cost and the accuracy. The best way to numerically evaluate these integrals is to take the extreme case for which the shape factor is equal to 1. It will give us idea about the number of integration points to correctly evaluate these integrals for all from 0 to 1 to save the computational time. The integrals I1 and I2 became 16 32026369410.743138 )2log( 2 2d )cos(log1) 1( 2/ 0 1 GxxI 518010887930451 . 1 )2log( 2 d )cos(log) 1( 2/ 0 2 xxI These integrals will be evaluated numerically using Gaussians numerical integration, 13. TABLE I .EVALUATION OF THE INTEGRAL I1 FOR = 1, 13 N-Gauss Points I1Gauss Computational Time (Seconds) Ab. Error (%) 20 0.7431381432026374 0.05 6.73e-14 50 0.7431381432026366 0.06 4.04e-14 100 500 0.7431381432026364 0.7431381432026368 0.13 10.96 6.73e-14 1.35e-14 From Table I we can see that the increasing in the Gaussians points does not have considerable influence in the accuracy. It means that the integral I1( =1) converges rapidly to its exact value. It is enough to take about 20 Gaussians points to evaluate this integral with the high precision. The absolute error regarding the exact value is about 6.73e-14 %. It is the same conclusion for the integral I1 in which the factor takes values from 0 to 1. TABLE II. EVALUATION OF THE INTEGRAL I2 FOR = 1, 13 N Gauss points I2Gauss Computational Time (Seconds) Ab. Error (%) 20 -1.086432063660191 0.04 0.21684 100 -1.088694850468245 0.11 0.00902 500 -1.088789085986257 11.09 0.00040 1000 -1.088792054330082 89.88 0.00009 2500 -1.088792886652334 1401.12 1.456e-5 From Table II we can see that the integral I2( =1) slowly converges to its exact value because its kernel function is not smooth function nearby the right end and has the singularity at this extremity. It is necessary to evaluate this integral with at least 2500 Gaussians points to have the precision about 1.456e-5 %. The difference is on sixth decimal place and the computational time is about 1401 seconds. This way we need a lot of Gaussians points to evaluate this integral to have the high precision. In the presented approach the kernel function of the integral I2 is obtained for more efficient numerical integration. Undoubtedly the integral I2 is an integral that converges slowly to its exact value but for its evaluation we do not need more than 1000 Gaussians points to obtained satisfactory results if the shape factor is approximately equal to 1. In this case absolute error regarding the exact value is about 9.00e-5 %. Let us numerically evaluate integrals I1 and I2 using Gaussians numerical integration, 14. In Tables III and IV the evaluation of these integrals (accuracy, computational cost) has been made in the function of the tolerance that directly has the influence on the accuracy and the computational cost. We chose the arbitrary tolerance and this one proposed in MATLAB programming (Eps=2.220446049250313x10-16). TABLE III. EVALUATION OF THE INTEGRAL I1 FOR = 1, 14 Tol. I1Gauss Computational Time (Seconds) Ab. Error (%) 0.00 0.7431381432028009 0.05 2.21e-11 10 -5 0.7431381432028009 0.16 2.21e-11 10 -10 Eps 0.7431381432026368 0.7431381432026371 0.16 0.16 1.35e-14 1.35e-14 TABLE IV. EVALUATION OF THE INTEGRAL I2 FOR = 1, 14 Tol I2Gauss Computational Time (Seconds) Ab. Error (%) 0.00 -1.088554642085562 0.141 0.021896 10-5 -1.088792100194056 0.791 0.000087 10-10 -1.088792808837103 2.143 0.000022 10-14 -1.088792808837103 2.143 0.000022 Eps -1.088792808837103 2.143 0.000022 From Tables III and IV one can see the good agreement of all results obtained by Gaussians numerical integrations 13, 14 and analytically, but Gaussians numerical integration 14 gives the same accuracy as 13 with considerably reduced computational time. We will use Gaussians numerical integration 14 because of its speed (about 2.1 s. for integrals that have possible singularities and oscillations nearby their limits) and accuracy. TABLE V. SELF-INDUCTANCE OF DISK COILS L/(N 2) (H) Shape factor In 1 In 6 This work 50.00 36.2822050627 36.2822050627 3.6282205063 10.00 8.5558078657 8.5558078657 8.5558078657 3.00 4.1202478984 4.1202477709 4.1202477709 1.50 3.9375565536 3.9375569573 3.9375569573 1.10 5.1875898298 5.1875898299 5.1875898299 1.01 7.8169836166 7.8169836167 7.8169836166 1.001 10.6712873756 10.6712873807 10.6712873753 1.00001 16.4524421475 17.0989093176 16.4524654449 1.000001 19.3458776688 -4378442.499276 19.3242796394 1.0000001 22.2393823064 -14026882.60408 26.2845769703 To verify the proposed computing approach we will compare the results of this method to results obtained in 1 and 6. In all calculations we will keep then decimal place and the tolerance Eps. From Table V we can see very good agreement between results obtained by the proposed approach and 1 except in the last case ( = 1.0000001). Also results obtained by 6 are in good agreement with these previously mentioned except three last cases. Unexpectedly, they are erroneous in two last cases. It can be explained by the fact of the uncomfortable presentation of all integrals in 6 when the shape factor takes values in close proximity to 1. It means that integrals in 6 have to be rearranged to improve the accuracy in the case where is inferior of 1.00001. This problem is efficiently solved in this approach during all integrations in expression (1). To improve the accuracy of presented approach (for 1 1.0000001) and these in 6 (for 1 1.00001) it is possible to use Gaussians numerical integration 13 with numerous points of integration that will considerably increase the computational cost. In these cases we practically obtain the extreme case 1 (thin filamentary coil for which the self inductance is infinity) so that for practical engineering applications this range of applications is not important. In Table VI the computational time of three approaches are given. TABLE VI.COMPARISON OF COMPUTATIONAL EFFICIENCY Computational time In 1 In 6 This work seconds 0.010 2.321 2.203 Finally, by using the program INCA, 15 let us calculate the self inductance of thin disk pancake formed by N conductors whose wire diameter is d. The minimum and maximum radius of the disk are respectively R1 and R2. For R1 = 0.3 m, R2 = 0.4 m and d = 0.0002 m 15 permits the maximum number of turns N = 499 for which the self inductance 8 MMaxwell = 310.5707755040 mH and by 9 MWheeler = 307.9206793862 mH Applying Spielreins method, 1 the self inductance is MSpielrein = 310.8176300221 mH The presented approach gives, MThis work = 310.8176340377 mH From preceding results we can see that the presented method is in excellent agreement with 1 and both in very good agreement with 8. There is an insignificant difference between these results (about 0.08 %) that can be explained by the fact that presented approach and 1 do not take into consideration the wire cross section that is the case of the formula 8 in the program INCA. All results differ from 9 about 0.85 %. The self-inductances 8 and 9 are included in the program INCA. Even though the method 1 proposed by Spielrein (convergent series expressed with lot of members) is the fastest known method, the method proposed in this paper is suitable for practical applications because of its simplicity (only five members they are easy for the numerical treatments) and rapidity of the calculation. Finally, from these results we can validate the excellent thought that somebody says: The numerical integration is not the science it is the art. The comparative calculation was made in MATLAB programming on a personal computer with Pentium III 700 MHz processor. 5 Conclusion The accurate self-inductance expressions for thin circular disk coils are derived and presented in this paper. The results are obtained in an analytical/numerical form over the complete elliptic integral of the first kind and two members, which have to be solved numerically using the single integration. In these integrals the kernels are continuous functions on intervals of interest thus there are not any problems concerning the singularities which have been smartly avoided during the integration. The results are in very good agreement with already published data. References: 1 J. Spielrein, Arch. El. 3, pp. 187, 1915. 2 2 F. W. Grover, Inductance Calculations, New York: Dover, 1964, chs.2 and 13. 3 H. B. Dwight, Electrical Coils and Conductors, McGraw-Hill Book Company, INC. New York, 1945. 4 Chester Snow, Formulas for Computing Capacitance and Inductance, National Bureau of Standards Circular 544, Washington DC, December 1954. 5 P.L. Kalantarov et al., Inductance Calculations, Moscow, USSR: National Power Press, 1955, chs.1 and 6. 6 Dingan Yu and K. S. Han, Self-Inductance of Air-Core Circular Coils with Rectangular Cross-Section, IEEE Trans. on Mag., Vol. 23, No. 6, pp. 3916-3921, November 1987. 7 S. Babic and C. Akyel, An Improvement in Calculation of the Self- and Mutual Inductance of Thin-Wall Solenoids and Disk Coils, IEEE Trans. on Magnetics, Vol. 36, No.4, pp. 678- 684, July 2000. 8 James Clerk Maxwell, A Treatise on Electricity an Magnetism, Dover Publications Inc, New York,1954 (reprint from the original 1873). 9 H. A. Wheeler, Simple inductance formulas for radio coils, Proceedings of the IRE, Vol 16, No.10, October 1928. 10 C.Akyel, S.Babic and S. Kincic, New and Fast Procedures for Calculating the Mutual Inductance of Coaxial Circular Coils (Disk Coil-Circular Coil) , IEEE Transactions on Magnetics, Vol. 38, No. 5, Part 1,(2002), p