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. F