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..doc》由会员分享,可在线阅读,更多相关《An improvement in the calculation of the self-inductance of thin disk coils with air-core..doc(6页珍藏版)》请在三一文库上搜索。
1、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
2、-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 wi
3、th 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
4、 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
5、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 ru
6、le 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)
7、, 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 c
8、ompare 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
9、 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
10、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 con
11、sidered 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
12、 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 s
13、ingularities 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 =
14、 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 mem
15、bers. 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 Ga
16、ussian 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 ev
17、en 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 i
18、nfluence 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 sav
19、e 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
20、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 no
21、t 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 c
22、onclusion 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 0.0004
23、0 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 t
24、o 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 pres
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- An improvement in the calculation of self-inductance thin disk coils with air-core. self inductance air
链接地址:https://www.31doc.com/p-8865116.html