Permanent Magnet -- Math Description

Permanent magnets have been in use for a long time. They have been applied as components of technical devices for nearly 200 years. The acquaintance with the physics of magnetic materials and principles governing magnetization on the micro level has occupied the attention of scientists for the last 50 years while the technicalities of the process of production of up-to-date composite magnets has dated since 1980s. At present we are familiar with permanent magnets with stable magnetic properties on condition of not exceeding admissible temperatures with high value of unitary internal energy, magnetic induction under the magnet in excess of 1[T] and a broad magnetization loop. The basic characteristics of the permanent magnet.

For the description of the operation of permanent magnet we normally present the characteristics of magnetization merely in the II quadrant of the coordinate system since this is the operating range of a magnet. In a simple magnetic circuit consisting of a permanent magnet, air gaps and a small ferromagnetic core (µ = 8) ... that is used for closing the magnetic circuit, one can state that ...

In the above formula:

...are: magnetic field strength and induction in the magnet and, consecutively, the length and internal cross section of the permanent magnet

...are: magnetic field strength and induction in the air gap and consecutively, the length and cross section of the air gap:

- µ, µ0 - is magnetic permeability, relative one and that of the air.

- _M - denotes, in accordance with the unit magnetic conductance of the magnetic circuit also known as the inclination of the straight line of magnet load.

The inclination of the straight with the directional coefficient _M corresponds to the conductance of the air gap in a simple magnetic circuit and the cross section between the straight line and characteristics of the magnet determines the operating point H0,B0 of the permanent magnet in a given magnetic circuit. Concurrently, the product of H0,B0 determines the unitary energy of the magnet (per unit of volume) at a given operating point H0,B0. For a certain inclination of the straight line _M the rectangle with the sides marked as H0,B0 has the largest area for a given characteristic of operation and this specific operating point determines the maximum operating energy (H B)max for the magnetic material from which the magnet is formed. A given material is optimal in terms of magnetic properties when it has concurrently a large value of induction of the magnetic remnant B_r and intensity of the coercion of the magnetic field |-Hc|, as well as the large value of the maximum operating energy (H B)max. An ideal would involve a magnet with a nearly rectangular magnetization loop for large values of Br and |-Hc| since it ensures a large and nearly constant induction under a magnet with a wide range of loads. As a result of the wide application of rare earth elements in magnets, they are able to come closer to this specific requirements to a much larger degree.

+-+-+- A family of magnetizing curves of a rare earth permanent magnet, for different temperature values of operation.

As one can conclude from z, the increase of temperature has a considerable effect on the magnetization characteristics of up-to-date permanent magnets based on rare earth elements. There is a certain, small reduction of the value of the remnant induction B_r and very large decrease of the absolute value of |-Hc| that is the intensity of the magnetic coercion. Too high an ambience temperature of a magnet results in the deterioration of the range of adequate conditions for the mag net to operate. This comes as a consequence of the fact that following the change of |-Hc| the inclination of _M, i.e. the characteristic of magnet loading is limited, which means that the admissible air gap in the magnetic circuit is considerably smaller.

The fundamental parameters of major families of permanent magnets, i.e. ferrite magnets, alloy based ones with aluminum, nickel and cobalt (AlNiCo) and two major groups of magnets with rare earth elements, i.e. ones with samarium (Sr) and neodymium (Nd), are presented.

Table 1: Basic magnetic properties of the main PM materials. Family of a PM materials.

Table 2: Basic temperature parameters of the main PM materials. Family of a PM materials Maximum operating temperature.

The tables contain mean and approximated values of parameters taken from various references in a manner that does not reflect any particular magnetic material available in the market. One can note that the details of the materials summarized in the tables are offered commercially in various alloy combination, as composites or sinters, as it’s the case for ferrites. The particular materials de scribed in manufacturers' catalogues display various properties despite belonging to a single family. From the data one can conclude that neodymium magnets are suitable for operation with lower operating temperatures while the ones with samarium display much better properties in higher temperature ranges. There are couple of methods of modeling on the macroscopic scale of PMs applied in electromechanical devices. We mean here simplified modeling, such that makes it possible to present the operation of electromechanical transducers and enable their modeling and simulation of operation in drive systems. One of the methods involves the replacement of the magnet with a compact turn with zero resistance and an adequately adapted self-inductance and circulating current if0 in this turn. The reverse effect of the armature on the magnet occurs as a result of the armature current ia via the mutual inductance M. The product M if0 corresponds to the magnetic flux _f0 by means of which the permanent magnet affects the armature circuit. Another quite simple way involves the presentation of permanent magnet flux in the mathematical model in the form

The effect of the armature is modeled using the term -iaM/Lf, which reduces the conventional magnetizing current if0 originating from the permanent magnet. The simplest way of modeling the current originating from PM coupled with a given circuit is the adoption of its value _f as a constant. This involves disregarding armature currents during the operation of a machine for a small air gap in the magnetic circuit. It also corresponds to the operation of the magnet in the initial section of magnetization characteristic of a magnet produced from alloys of rare earth elements. None of the presented here PM modeling methods ac counts for the magnetization characteristics under the effect of the temperature rise. In order to present the discussed PM modeling methods, below is found an example of a servomechanism with a movable coil swinging above the magnet.

____ Pendulum coil over PM.

A simplified model of the electromechanical system in which a pendulum coil moves in the field of a immobile PM is presented.

+-+-+-Model of a pendulum coil over PM

The kinetic energy of the system is: ... while the potential energy: U = mgy

This system has three degrees of freedom and after the introduction of generalized coordinates:

The maximum value of the magnetic flux coupled with the coil is:

... where the first right hand side term denotes electromechanical torque braking the motion of the pendulum:

The mathematical model presented in is applicable with regard to the first, least simplified way of modeling PMs. A more simplified magnet model involves disregarding of the modeling of the magnet by means of a separate differential equation and the presentation of the effect of the armature in the form resulting from. In this case, Lagrange's equation takes the form:

+-+-+- Swinging motion of the pendulum coil, for _0 = 36º, computed by the model: a) if current of PM b) ia coil current c) ? _ angular velocity of the pendulum d) sway angle theta e) electromagnetic torque Te.

+-+-+Comparison of electromagnetic torque computed with different PM model simplifications: a) full model b) armature reaction model c) constant _f model.

+-+-+-Comparison of swinging motion - _angle, for different PM models: a) full model b) armature reaction model c) constant _f model.

+-+-+-Characteristics of movement for a very strong field linkage _f = 1.0 [Wb]: a) if current of PM b) ia coil current c) _ angular speed of the pendulum d) sway angle _ e) electromagnetic torque Te.

On the basis of the results presented one can conclude that the differences in terms of the curves for the variables characterizing the pendulum motion gained for various versions of the PM model simplifications are in considerable and the magnet model for ?f = const is acceptable for the modeling of motion parameters of the drive.