Slobodan Babic

Calculating Mutual Inductance Between Circular Coils With Inclined Axes in Air

In this paper we present a lucid, easy, and accurate approach for calculation of the mutual inductance between all inclined circular coils with either rectangular cross section or negligible section. We use Grovers formula for the mutual inductance between two filamentary circular coils with inclined axes that lie in the same plane. Their centers are either displaced along the axis of one coil or displaced along one axis of the first coil and then displaced sideways in addition. We apply the filament method for coil combinations comprising circular coils of rectangular cross section, thin wall solenoids, thin disk coils (pancakes), and filamentary circular coils. In this approach we clarify how Grovers formulas have to be used for different coil combinations in the filament treatment. Thus, two well-known methods (Grovers formulas and the filament method) can be easily used to calculate the mutual inductance between all inclined circular coils, even though the problem is purely three-dimensional.

Improvement in calculation of the self- and mutual inductance of thin-wall solenoids and disk coils

The self-inductance expressions given by Yu and Han (1987) for air-core circular coils with rectangular cross sections, thin solenoids, and disk coils can be solved only by the numerical integration methods. We propose as an alternative a combined analytic and numeric approach. The approach brings some improvement in the calculations of self-inductance of thin-wall solenoids and disk coils that can be encountered in superconducting magnetic energy storage (SMES) problems. We also give a method for the calculation of mutual inductance of disk coils and of thin-wall solenoids. The results are obtained in an analytical form over the complete elliptic integrals of the first and second kind and Heumans Lambda function. It is important to mention that the kernels of these integrals are always continuous functions on intervals of integration, so singularities are avoided. The results enable one to calculate the self-inductance and the mutual inductance of any thin air-core coil precisely and fast. For practical applications, the results are so simple to use that we recommend them to avoid the problems of solving the singular cases.

Cylindrical Magnets and Coils: Fields, Forces, and Inductances

This paper presents a synthesis of analytical calculations of magnetic parameters (field, force, torque, stiffness) in cylindrical magnets and coils. By using the equivalence between the amperian current model and the coulombian model of a magnet, we show that a thin coil or a cylindrical magnet axially magnetized have the same mathematical model. Consequently, we present first the analytical expressions of the magnetic field produced by either a thin coil or a ring permanent magnet whose polarization is axial, thus completing similar calculations already published in the scientific literature. Then, this paper deals with the analytical calculation of the force and the stiffness between thin coils or ring permanent magnets axially magnetized. Such configurations can also be modeled with the same mathematical approach. Finally, this paper presents an analytical model of the mutual inductance between two thin coils in air. Throughout this paper, we emphasize why the equivalence between the coulombian and the amperian current models is useful for studying thin coils or ring permanent magnets. All our analytical expressions are based on elliptic integrals but do not require further numerical treatments. These expressions can be implemented in Mathematica or Matlab and are available online. All our models have been compared to previous analytical and semianalytical models. In addition, these models have been compared to the finite-element method. The computational cost of our analytical model is very low, and we find a very good agreement between our analytical model and the other approaches presented in this paper.

MUTUAL INDUCTANCE CALCULATION FOR NON- COAXIAL CIRCULAR AIR COILS WITH PARALLEL AXES

We present a practical and simple method for calculating the mutual inductance between two non-coaxial circular coils with parallel axes. All possible circular coils such as coils of rectangular cross section, thin wall solenoids, thin disk coils (pancakes) and circular filamentary coils are taken into consideration. We use Grover’s formula for the mutual inductance between two filamentary circular coils with parallel axes. The filament method is applied for all coil combinations, for coils of the rectangular cross section and for thin coils. We consider that the proposed method is very simple, accurate and practical for engineering applications. Computed mutual inductance values obtained by the proposed method have been verified by previously published data and the software Fast-Henry. All results are in a very good agreement. This method can be used in various electromagnetic applications such as coil guns, tubular linear motors, transducers, actuators and biomedical implanted sensors.

Magnetic Force Calculation Between Thin Coaxial Circular Coils in Air

We present new and fast procedures for calculating magnetic forces between thin coaxial circular coaxial coils in air. The results are expressed in semianalytical form in terms of the complete elliptical integrals of the first and second kind, Heumans Lambda function, and a term that must be solved numerically. These expressions are accurate and simple to use for several practical applications. We also describe a comparative method based on the filament technique. We discuss the computational cost and the accuracy of two methods and compare them with already published data. Results obtained by our two approaches are in excellent agreement with each other. They can be used in industrial electromagnetic applications such as electrodynamic levitation systems, linear induction launchers, linear actuators, and coil guns.

Mutual Inductance Calculation Between Circular Filaments Arbitrarily Positioned in Space: Alternative to Grover's Formula

In this paper, we present the full derivation of a new formula for calculating the mutual inductance between inclined circular filaments arbitrarily positioned with respect to each other. Although such a formula was already proposed by Grover more than 50 years ago, the formula presented here is slightly more general and simpler to use, i.e., it involves only a sequential evaluation of expressions and the numerical resolution of a simple numerical integration. We derived the new formula using the method of vector potential, as opposed to Grovers approach, which was based on the Neumann formula. We validated the new formula through a series of examples, which are presented here. Finally, we present the relationship between the two general formulas (i.e., Grovers and our new one) explicitly (Example 12).

The mutual inductance of two thin coaxial disk coils in air

This paper presents an efficient method for computing the mutual inductance between two thin coaxial disk coils in air. The derived principal semi-analytical expressions involve complete elliptic integrals of the first and second kind, Heumans Lambda function and three terms that have to be computed numerically. The presented method is compared to the filament method where both conductors are approximated by Maxwells filamentary coils. The mutual inductance values (for a chosen case) computed by both methods are in good agreement. However, the method presented in this work exceeds by far the filament method in its precision and computational efficiency. In another example, the computed mutual inductance is validated by measurement.

MUTUAL INDUCTANCE AND FORCE EXERTED BETWEEN THICK COILS

We present exact three-dimensional semi-analytical ex- pressions of the force exerted between two coaxial thick coils with rect- angular cross-sections. Then, we present a semi-analytical formulation of their mutual inductance. For this purpose, we have to calculate six and seven integrations for calculating the force and the mutual induc- tance respectively. After mathematical manipulations, we can obtain semi-analytical formulations based on only two integrations. It is to be noted that such integrals can be evaluated numerically as they are smooth and derivable. Then, we compare our results with the flla- ment and the flnite element methods. All the results are in excellent agreement.

New and fast procedures for calculating the mutual inductance of coaxial circular coils (circular coil-disk coil)

This paper deals with two efficient approaches for determining the mutual inductance between thin circular coils and disk coils in air. The first approach gives new expressions for calculating the mutual inductance of treated configurations. These results are expressed over the complete elliptical integrals of the first and second kind, Heumans Lambda function, and one term that must be solved numerically. Another approach is based on the filament method where conductors are approximated by the set of Maxwells coils. The obtained expressions are expressed over the complete elliptical integrals of the first and second kind and permit also fast calculation of the mutual inductance for mentioned systems. These new expressions are accurate and simple for useful applications. All results obtained by the two approaches are in excellent agreement.

New procedures for calculating the mutual inductance of the system: filamentary circular coil-massive circular solenoid

New and fast procedures for calculating the mutual inductance between a thin circular filament coil and a massive circular solenoid in air are presented. The coils are axisymmetric. These results are expressed over the complete elliptical integrals of the first and second kind, Heumanns Lambda function, and one term that has to be solved numerically. The numerical integration of this term is based on some numerical integrations to get the satisfactory accuracy and computational cost. These new expressions are accurate and simple to apply for useful applications. Also, another comparative method based on an approximation of the massive solenoid by Maxwells coils is given. The paper discusses the computational cost and the accuracy. Results obtained by the two approaches are in excellent agreement.