Cylindrical Magnets and Coils

IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010

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Cylindrical Cylindrical Magnets and Coils: Fields, Forces, Forces, and Inductances Inductances R. Ravaud , G. Lemarquan Lemarquand d , S. Babic , V. Lemarquand , and C. Akyel LAUM, Universite du Maine, Av Olivier Messiaen, Le Mans 72085, France Departement Genie Physique, Ecole Polytechnique, Montreal, QC H3C 3A7, Canada 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. 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. Index Terms—Force, magnetic field, mechanical stiffness, mutual inductance, permanent magnet, self inductance.

I. INTRODUCTION AGNET AGNETOMEC OMECHANIC HANICAL AL devices devices used in electrical electrical engineering require permanent magnets or coils. However, some devices, such as magnetic couplings [1], are made of cylind cylindric rical al perman permanent ent magnet magnetss though though others others,, as the air transformers, use cylindrical air coils. These devices are very different from one point of view. Indeed, though coils require electrical electrical supplies, permanent magnets do not. Furthermore Furthermore,, some devices, such as sensors [2], actuators, and loudspeakers [3] are manufactured with magnets, coils, or both. However, all these devices can be modeled with similar approaches. Indeed, the magnetic field produced by a cylindrical permanent magnet can be determined determined with the same analytical analytical formulation formulation as the one used for a cylindrical thin coil. However, the magnetic field produced by a permanent magnet is not of the same order of magnitude as the one produced by a thin coil. Besides, two kinds of models are used: the coulombian approach and the amperian current one. We can say that the choice of the model used does not depend on the magnetic source nature. Indeed, in the coulom coulombia bian n approa approach, ch, a cylind cylindric rical al magnet magnet axiall axially y magnetized can be replaced by two charged planes which are located on the lower and upper faces of this cylinder. In the same same way way, a thin thin coil coil carryi carrying ng unifor uniform m curren currentt densit density y can also be represented by two charged planes. However, in the case of permanen permanentt magnet magnet whose whose polariza polarization tion is in Tesla, Tesla, its equivalent current must satisfy the following equation:

M

(1) Manuscript received January 26, 2010; revised March 09, 2010; accepted April 13, 2010. Date of publication May 03, 2010; date of current version August 20, 2010. Corresponding author: G. Lemarquand (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2049026 10.1109/TMAG.2010.2049026

Fig. 1. Representation of a cylindrical cylindrical permanent permanent magnet whose whose polarization is [T] and a thin coil having having loops in which a current circulates: these cylindrical topologies have the same radius and the same height .

wher wheree is the the norm normal al unit unit and and is the the perm permea eabi bili lity ty of the the vacuum. This implies that a cylindrical permanent magnet can be replaced by a thin coil whose current surface density is . In the case case of a thin thin coil coil of loops loops carryi carrying ng a curren currentt and whose height height is , its equivalent equivalent coulombia coulombian n model satisfy satisfy the following equation:

(2) Finally, these two previous equations lead to the following conclusion: the magnetic field produced by a permanent magnet or by a thin coil has the same topology. From this point of view, as shown in Fig. 1, we can say that a cylindrical cylindrical permanent magnet and a cylindrical thin coil own the same fictitious charge distribution. The authors generally use the amperian current model [4] or the coulombian approach [5] for calculating the magnetic field produced by permanent magnets. However, from a calculation point of view, these approaches do not lead to the same

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analytical expressions [6]. We assume, for this configuration, the amperian current model is more interesting. Therefore, the calculation of the magnetic field created by its constitutive elements, magnets or coils, is generally the first step for studying the characteristics of magnetomechanical devices [7]–[10]. Eventually, it is to be noted that various electromagnetic applications are composed of two coils that form a loosely coupled transformer. Basically, in such configurations, the first coil generates a magnetic field in all points in space. Then, this magnetic field is partly picked up by the secondary coil. As stated in [11], this a very efficient way of transferring power wirelessly. However, a decrease in power transfer efficiency can be caused by a lower mutual inductance between two coils [12], [13]. In other words, it is very useful to know the accurate value of the mutual inductance between two coils [14]. The calculation of the mutual inductance for cylindrical coils was studied by many authors [15]–[19]. These papers are generally based on the application of the fundamental laws of magnetostatics (Biot Savart law, Lorentz Force,Maxwell’s equations). By using these approaches, the mutual inductance of circular coils can be expressed in terms of analytical and semianalytical functions, as the elliptic integrals of the first, second, and third kind or the Bessel functions [20], [21]. This paper presents analytical formulations of every physical notion of interest when two coaxial cylindrical magnets or coils are studied: the created magnetic field, the axial force exerted between them, the mechanical stiffness, the self inductance and mutual inductance of the coils. Throughout this paper, we compare our semianalytical approach with the filament method [18] and Kajikawa calculations [22]. This comparison can be a way for us to verify the accuracy of our semianalytical model. The analytical and numerical simulations are in very good agreement.

Fig. 2. Geometry considered forthe restof this paper: twocoaxial cylinders radiallycentered: they canrepresent a thin coil or a cylindrical permanentmagnet.

gives the complete elliptic integral of the first kind (3) gives the elliptic integral of the first kind (4) gives the elliptic integral of the second kind (5)

II. NOTATION AND GEOMETRY

gives the complete elliptic integral of the second kind

We present first the notation and the geometry which will be (6) used throughout this paper. For this purpose, let us first consider Fig. 2 in which we have represented two cylinders axially gives the complete elliptic integral of the third kind decentered. As stated previously, these cylinders can represent either permanent magnets or thin coils. (7) The lower cylinder (1) has the following characteristics: radius, height,and the upper cylinder(2): radius, height. Moreover, gives the incomplete elliptic integral of Furthermore, for the third and fourth parts of this paper, the third kind the axial distance between the two cylinders is denoted . Therefore, the parameters defined in this section will be used (8) throughout this paper. polarization in Tesla, the cylinder 1, respectively 2, considered as a permanent magnet. current surface density [A/m], number of III. MAGNETIC FIELD PRODUCED BY PERMANENT MAGNETS loops, current [A], the cylinder 1, respectively 2, conAND THIN COILS sidered as a thin coil. We define here the special functions that are used throughout Let us consider the cylinder in Fig. 2. This cylinder is supthis paper. posed to be a thin coil having a current surface density that

RAVAUD et al.: CYLINDRICAL MAGNETS AND COILS: FIELDS, FORCES, AND INDUCTANCES

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Fig. 3. Representation of the radial and axial components of the magnetic induction field versus the radial and axial distances created by a thin coil in air; mm, mm, A/m; left representation: mm, right representation: mm.

equals . By using the Biot-Savart Law, the magnetic induction field created by this thin coil is expressed as follows: (9) where

is the Green’s function that is defined as follows: (10)

Furthermore, the parameters defined with the notation correspond to the points located on the lower cylinder whereas the parameters defined with the notation correspond to the points located on the upper one. In cylindrical coordinates, is reduced to the following form:

where the parameters are defined in Table III. We illustrate our previous analytical expressions in Fig. 3 where we have represented the radial and axial components of the magnetic induction field created by a thin coil. The parameter values are mm, mm, and A/m, this values correspond to a current of A and . Let us still consider the cylinder in Fig. 2. If this cylinder is a cylindrical permanent magnet having an axial polarization . By using the equivalence between the coulombian approach and the amperian current model, the magnetic induction field created by a cylindrical permanent magnet can be expressed as follows:

(15)

IV. FORCE EXERTED BETWEEN TWO PERMANENT MAGNETS OR TWO THIN COILS

(11) After having integrated with respect to and , we obtain the two following magnetic induction field components and :

This section presents analytical models for calculating the axial force exerted between thin coils or cylindrical permanent magnets. Let us consider the cylinders and in Fig. 2. These cylinders are supposed to be thin coils having current surface densities and . The axial force exerted between these two thin coils is derived with the following equation: (16) By inserting (9) in (16), we have

(12) where and source topology:

are scalar coefficients that depend on the

(17) Therefore, (17) becomes

(13)

(18) We obtain the final expression:

(14)

(19)

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where (20) It is useful to mention that we do not need to introduce another angle in the Green’s function because the magnetic field produced by the lower thin coil does not depend on angle . Moreover, it is to be noted that there is only the axial component of the force that is required to determine. By projecting and in c artesian coordinates, we find the following form of the axial force exerted between two toroidal conductors radially centered:

(21)

Fig. 4. Representation of the force exerted between two thin coils versus the axial distance . We take the following parameters: m, m, m, m, A/m, points filament method, line this work.

TABLE I COMPARISON OF THE AXIAL FORCE [MN] E XERTED BETWEEN TWO THIN COILS IN AIR WITH THE FILAMENT METHOD AND OUR ANALYTICAL M, M, M, APPROACH. M,

After integrating with respect to and , the final analytical expression of the force exerted between two thin coils in air is expressed as follows: (22) where is the fully analytical part and based on elliptic functions:

, the analytical part

(23) with

Fig. 5. Representation of the axial force (bold) and stiffness exerted between cylindrical permanent magnets in air versus the axial distance mm, mm, mm, mm, T.

exerted between two thin coils in air, we can directly write that the axial force exerted between two cylindrical permanent magnets (Fig. 5): (24) where verifies and the otherparameterscan depend on i and j (see Table III). We can illustrate and compare our analytical approach with the filament method [11]. For this purpose, we have represented in Fig. 4 the axial force exerted between two thin coils in air with both our analytical method and the filament method. We have given in Table I some numerical results. Fig. 4 and Table I show that the two approaches are in excellent agreement. Let us consider the cylinders and in Fig. 2. These cylinders are supposed now to be cylindrical permanent magnets axially magnetized. By using the analytical expression of the force

(25)

V. STIFFNESS EXERTED BETWEEN PERMANENT MAGNETS AND THIN COILS Let us consider the cylinders and in Fig. 2. These cylinders are supposed to be cylindrical permanent magnets axially magnetized. The axial stiffness can be deducted from the following equation: (26)

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TABLE II COMPARISON OF THE MUTUAL INDUCTANCE [MH] D ETERMINED WITH THE FILAMENT METHOD AND OUR ANALYTICAL APPROACH; M, M, M, M,

Fig. 6. Representation of the mutual inductance between two thin coils in air versus the axial distance m, m, m, m, .

mathematical manipulations, the reduced form of the mutual inductance can be expressed as follows (Fig. 6):

where is the axial force exerted between the two permanent magnets. The axial stiffness exerted between two cylindrical permanent magnets is expressed as follows (Fig. 5):

(32) where

is a scalar depending on the two source distributions

(27) where is a scalar coefficient that depends on the source distributions

(33) with

(28) It is emphasized here that it is simple to calculate the radial stiffness as it satisfies the following equation [23] for ironless structure:

(34) where

and

and

(29) Let us still consider the cylinders and in Fig. 2. These cylinders are supposed now to be thin coils in air. The axial stiffness exerted between two thin coils in air can be deducted directly from the equivalence between the coulombian approach and the amperian current model: (30) where

has been defined previously (28).

VI. MUTUAL INDUCTANCE BETWEEN TWO THIN COILS

(35)

A. Analytical Calculation

where

The mutual inductance between two thin coils can be derived from the following equation:

B. Comparison With the Filament Method

(31) where is the flux created by the thin coil in the thin coil . By using here previous integral formulations and after

.

We have compared our previous analytical model of the mutual inductance between two thin coils in air with the filament method, largely used by [11]. We represent in Table II the numerical results obtained for given axial distances . Table II clearly shows that all the methods are in excellent agreement.

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TABLE III DEFINITION OF PARAMETERS USED IN THIS PAPER

information for the design of magnetomechanical devices composed of thin coils or cylindrical permanent magnets. It is emphasized that all the results are in an excellent agreement with the finite element method or the filament method. REFERENCES

The analytical expressions presented in this paper are accurate whatever the coil or permanent magnet whereas the accuracy of the filament method depends greatly on the number of filaments used for modeling the topology of the magnetic source. These analytical formulations do not require the determination by trial and error methods of the adequate number of filaments to reach the intended accuracy. This is one of the real benefits. The consequence is that the computational cost of the analytical calculations is far lower as the one of the filament method. For example, it is smaller than 0.3 s with this analytical approach whereas it can reach several hundreds of second with the filament method if a great precision is required. VII. CONCLUSION This paper has presented a synthesis about analytical calculations of cylindrical magnetic sources. We have presented the analytical expressions of the magnetic induction field created by cylindrical permanent magnets axially magnetized and thin coils in air. These expressions have been confirmed with the finite element method. Then, we have presented analytical models allowing us to calculate the axial force exerted between two cylindrical permanent magnets axially magnetized and two thin coils in air. We have compared these models with the filament method. Moreover, we have presented the analytical calculation of the stiffness exerted between these magnetic source distributions. Eventually, we have presented an analytical calculation of the mutual inductance between two thin coils in air. Our model has been compared to the filament method. More generally, we can say that the equivalence between the coulombian model of a magnet and the amperian current model is a previous element of

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Cylindrical Magnets and Coils

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