Permanent magnet motors are used in a variety of high-tech applications, but they have some design limitations. One such example is sensitivity to high temperatures, which can be caused by the release of heat from flowing currents, and in particular, eddy currents. Version 5.3 software COMSOL® includes a feature to account for eddy current losses in the permanent magnets of such motors. Engineers can use these results to fully understand the performance of permanent magnet motors and identify ways to optimize their performance.
The use of permanent magnet motors in high-tech devices.
Saving energy is a common goal that all manufacturers around the world strive for. For example, consider the transport sector. Just last year, China unveiled a new high-speed subway system that offers significant energy savings. Meanwhile, the oldest operating ferry in Finland has had its original ones replaced diesel engines for new electric ones. And on the streets of London, the famous luxury car brand presented for the first time an all-electric car.
These examples demonstrate the evolution of transport towards a greener future. Also, these examples are united by the fact that for this purpose they use permanent magnet (PM) motors. These types of motors, with magnets instead of windings in the rotor, typically find use in high-tech applications. The most important is their use in electric and hybrid vehicles.
Electric vehicles are one of the applications of permanent magnet motors. Image courtesy of Mariodo. Available under a Creative Commons 2.0 license from Wikimedia Commons.
PM motors are highly valued due to their efficiency, but at the same time there are some limitations in their design. For example, permanent magnets are very sensitive to high temperatures. Such temperatures can be reached when currents, particularly eddy currents, cause heat to be generated as they flow. Although laminating steel/iron rotor sections helps reduce eddy current losses in these areas, manufacturing limitations make the process difficult. Thus, the heating of permanent magnets can be quite significant.
Let's look at the new one training model, available in version 5.3 of COMSOL Multiphysics®, which takes into account eddy current losses in PM motors
Simulate eddy current losses in a permanent magnet motor using COMSOL Multiphysics®.
Let's start with the geometry of our model. In this example we use a 3D model of an 18-pole PM motor. To simultaneously reduce computational costs and take into account the entire 3D geometry of the model, we will model one pole using longitudinal and mirror symmetries.
You can see an animation of the entire engine running below. It shows a rotor and an iron stator ( gray), stator winding (made of copper) and permanent magnets (blue and red depending on the radial magnetization).
Permanent magnet motor design.
To model the conductive part of the rotor, we use the Ampère’s law node. For the non-conducting parts of the rotor and stator, we use the Magnetic flux conservation node relative to the scalar magnetic potential.
Using the built-in Rotating Machinery physics interface, it is easy to simulate motor rotation. In the model, we consider the central upper pole, in which the rotor is located along with the air gap section, rotating relative to the stator coordinate system. Please note that in in this case assembly is required when completing the geometry, since the rotor and stator are two in separate parts designs.
To calculate and further use the value of eddy current losses in magnets over time, we introduce an additional variable. Although not required for this model, the variable can be used in subsequent heat transfer analyzes as a time-averaged and distributed heat source. Since thermal processes take much longer to establish than the change in direction of eddy currents and the losses caused by them, it is necessary to separate electromechanical and thermal calculations for greater calculation efficiency.
Analysis of simulation results.
Based on the simulation results in the first figure, we can see the distribution of magnetic induction in the motor at a standstill stationary state, in other words, the graph shows initial conditions for non-stationary research. The coil current in the initial state is zero. The figure on the right shows the distribution of magnetic induction after the motor has turned one sector. For better clarity, you can exclude the areas of air and coils in the figure.
Left: Distribution of magnetic induction in a stationary initial state. Right: Distribution of magnetic induction in the motor after turning one sector.
In the graph below we can see how eddy current losses in magnets change over time. The animation on the right shows the change in eddy current losses as the stator rotates one sector. Eddy currents are depicted by arrows.
Left: Eddy current loss plotted against time. Right: Change in eddy current loss density when rotated by one sector.
The above examples give a more complete picture of the characteristics of PM motors, taking into account eddy current losses in permanent magnets. This information will be useful for improving the design of PM motors and therefore the technology in which they are used.
Let's place a coil of wire in an alternating magnetic field. The coil is closed, and there is no galvanometer in the circuit, which could show the presence of induction current in our circuit. But the current can be detected because the conductor will heat up when current passes through it. If, without changing the remaining dimensions of the coil, we increase only the thickness of the wire from which the circuit is made, then induced emf($\varepsilon_i\sim \frac(\Delta Ф)(\Delta t)$) will not change, since the rate of change of the magnetic flux will remain the same. However, the coil resistance ($R\sim \frac(1)(S)$) will decrease. As a result, the induction current will increase ($I_i$). The power that is released in the circuit in the form of heat is directly proportional to $I_i\varepsilon_i$, therefore, the temperature of the conductor will increase. And so, experience shows that a piece of metal, when placed in a magnetic field, heats up, which indicates the occurrence of induced currents in massive conductors when the magnetic flux changes. Such currents are called eddy currents or Foucault currents.
Definition of Foucault currents
Definition
Tokami Fuko called vortex induction volumetric electric currents, which appear in conductors when the conductors are placed in an alternating magnetic field.
Properties of Foucault currents
By their nature, eddy currents are no different from induction currents that arise in wires.
The direction and strength of Foucault currents depend on the shape of the metal conductor, the direction of the alternating magnetic flux, the properties of the metal, and the rate of change of the magnetic flux. The distribution of Foucault currents in metal can be very complex.
In conductors that are large in the direction perpendicular to the direction of the induction current, eddy currents can be very large, which leads to a significant increase in body temperature.
The properties of eddy currents to heat a conductor are used in induction furnaces for melting metals.
Foucault currents, like other induction currents, obey Lenz’s rule, that is, they have such a direction that their interaction with the primary magnetic field inhibits the movement that causes induction.
Examples of problems with solutions
Example 1
Exercise. What is “magnetic damping”, which is used in electrical measuring instruments?
Solution. Consider the following experiment. We hang a light magnetic needle from a thread (Fig. 1).
If this arrow is left to itself, it is in a position of equilibrium set in the direction from north to south. When it is deviated from its equilibrium position, it will oscillate for a long time if the friction in the suspension is small. Let us place a large copper plate of significant mass under the arrow at a small distance from it. The damping of the arrow's oscillations in this case will occur very quickly, after making one or two swings the arrow will reach the equilibrium position. The reason is that when the magnetic needle moves, Foucault currents are induced in the copper conductor, the interaction of which with the magnetic field, in accordance with Lenz’s rule, inhibits the movement of the magnet. Kinetic energy, which was communicated to the magnetic needle at the moment of the push, thanks to eddy currents, turns into the internal energy of copper, increasing its temperature. This phenomenon is called "magnetic quieting".
Example 2
Exercise. A metal coin falls between the poles of an electromagnet. The first time the magnet is off, the second time the magnet is on. In which case will the coin fall at a slower rate?
Solution. If there is a magnetic field between the poles of an electromagnet, then the coin will slowly fall down, as if it were moving in a viscous liquid, and not in atmospheric air. The coin is decelerated by forces that act from the magnetic field on the eddy currents induced in the coin when it falls in the magnetic field. The speed of its movement will be significantly less than when the magnetic field is turned off.
Answer. The fall speed is slower when the magnet is on.
The winding of a laboratory regulating autotransformer (LATR) is wound on an iron core shaped like a rectangular toroid (Fig.). To protect against Foucault eddy currents, the core is made of thin iron plates coated with an insulating layer of varnish. Such a core can be made in different ways:
a) collecting it from thin rings placed in a stack one on top of the other;
b) rolling a thin long strip of width h;
c) assembling from rectangular plates of size l×h, placing them along the radii of the cylinder.
Experiment.
You can observe the occurrence of Foucault currents using the following setup. A pendulum, consisting of a piece of metal suspended on a thread between the poles of an electromagnet, removed from the equilibrium position in the absence of current in the electromagnet, makes a weak damped oscillations. When the current is turned on, the oscillations die out almost instantly, and the movement of the pendulum until it stops resembles movement in a viscous medium. This is explained by the fact that the Foucault currents that arise when the pendulum moves in a magnetic field have such a direction that the forces acting on them from the magnetic field inhibit the movement of the pendulum. 
If the solid sector of the pendulum is replaced by a comb with long teeth, then the excitation of Foucault currents will be very difficult. The pendulum will oscillate in a magnetic field with almost no damping. This experience explains why the cores of electromagnets and the frames of transformers are made not from a solid piece of iron, but from many sheets superimposed on each other. As a result currents Foucault are excited weakly and the harmful influence of the Joule heat generated by them is greatly reduced.
Theory.
Toki Fuko− induction currents arising in massive conductors
in an alternating magnetic field are called Foucault currents. Sometimes they play a beneficial role, and sometimes they play a harmful role.
Foucault currents play a useful role in the rotor asynchronous motor, driven by a rotating magnetic field, since the very implementation of the principle of operation of an asynchronous motor requires the occurrence of Foucault currents. Being conduction currents, Foucault currents dissipate part of the energy to release Joule heat. This loss of energy in the rotor of an induction motor is useless, but you have to put up with it, only avoiding excessive overheating of the rotor. But at the same time, in the cores of the electromagnets of an asynchronous motor, usually made of ferromagnets that are conductors, Foucault currents also arise, which have no significance for the principle of operation of electromagnets, but heat these cores, thereby deteriorating their characteristics. They must be dealt with as a harmful factor. The struggle lies in the fact that the cores are made of thin plates, separated from one another by layers of insulator, and they are installed so that the Foucault currents are directed across the plates. Due to this, with a sufficiently small thickness of the plates, Foucault currents cannot develop and have an insignificant volume density.
The Joule heat generated by Foucault currents is usefully used in processes of heating or even melting metals, when this turns out to be more profitable or expedient compared to other heating methods. If you heat the metal with currents very high frequency, then as a result of the skin effect, only the surface layer of the conductor is heated.
(b, c) Solid piece of metal, located in an alternating magnetic field, is a resistance conductor, as a result of which the strength of the induction currents in it reaches large values.
Since the induced emf is proportional to the rate of change in the flux of magnetic induction, the magnitude of the Foucault currents is greater, the faster the magnetic field into which a given conductor is inserted changes. Therefore, the occurrence of Foucault currents is easier to observe if a conductor is inserted into the cavity of the solenoid, through the winding of which a rapidly alternating current is passed, causing a magnetic field that also changes rapidly in magnitude. In this case, Foucault currents in massive, well-conducting bodies reach such strength that the generated heat is enough to heat the body. This method is widely used in vacuum technology for heating inside the pumped device. metal parts for their degassing. The same method is used for melting metals under vacuum.
In pieces quite thick, i.e. having large sizes in the direction, perpendicular to the direction of the induction current, eddy currents due to the low resistance can be very large and cause very significant heating. If, for example, you place inside a coil massive metal core and pass an alternating current through the coil, which changes its direction and strength 100 times per second, reaching zero and increasing again, then this core will heat up very much. This heating is caused by induction (eddy) currents arising as a result of a continuous change in the magnetic flux passing through the core. If this core is made of separate thin wires, isolated from each other by a layer of varnish or oxides, then the resistance of the core in the direction perpendicular to its axis, that is, the resistance for eddy currents, will increase, and the heating will decrease significantly. This technique - dividing solid pieces of iron into thin layers isolated from each other - is constantly used in all electrical machines to reduce their heating by induction currents arising in an alternating magnetic field. On the other hand, Foucault currents are sometimes used in so-called induction furnaces to strongly heat or even melt metals.
Transformers.
However, in many cases the heating caused by Foucault currents is harmful. Such cases include heating of transformer cores and, in general, metal cores of all kinds of windings through which alternating current flows. To avoid such heating, the cores are made layered, separating the layers from each other with a thin layer of insulation located perpendicular to the direction of the Foucault currents.
The appearance of ferrites (magnetic materials with high electrical resistance) made possible production solid cores.
(c) In low-power transformers, the magnetic circuit is assembled from plates P-, Sh- And ABOUT- shaped (Fig. a, b, c). 
Magnetic cores wound from a narrow strip of electrical steel or from special iron-nickel alloys such as permalloy are widely used. They can be used for rod, armored, toroidal and three-phase transformers (d, e, f, g).
Skin effect.
Foucault currents can also arise in the conductor itself through which alternating current flows. The appearance of such currents leads to a special surface effect (also called the skin effect from English word skin, which means leather). If alternating current flows through a cylindrical conductor, then at moments when the current increases, the Foucault induction currents will be directed as shown in the figure. 
These currents are directed at the surface of the conductor in the direction of the primary electric current, and at the axis of the conductor - towards the current. As a result, the current inside the conductor will weaken and increase near the surface. Thus, due to the occurrence of Foucault induction currents, the current will be distributed unevenly over the cross-section of the conductor.
With rapidly alternating currents, the current density near the axis of the conductor is practically equal to zero, and all the current flows along the surface of the conductor. As a result, the magnetic field inside the conductor becomes equal to zero. This phenomenon causes an increase in the resistance of the conductor, since current does not flow through the internal parts of the conductor. Since these internal parts turn out to be useless, in order to save metal, the wires for rapidly alternating currents are made hollow. Foucault currents also lead to a decrease in the self-inductance coefficient of the conductor. This can be illustrated using the example of a cylindrical conductor.
Due to the skin effect, it makes no sense to make conductors in high-frequency circuits solid. To reduce resistance, you need to increase their surface, and not their cross-section, i.e., make tube-shaped conductors. In electric furnaces, this circumstance is used by cooling the coil tubes, through which a high-frequency current flows, with the help of water circulating inside the tubes.
Generators.
Generators are usually driven by relatively low-speed water turbines or engines internal combustion. When working with steam turbines rotating at a frequency 1500 − 3000
revolutions per minute, a slightly different rotor (inductor) design is used. The rotor has no protrusions, but is a smooth cylinder, outer surface in which the winding is laid in the grooves. At high rotation speeds, this is more advantageous, because the protrusions on the rotor create air vortices and increase mechanical losses.
The shape of the pole pieces on the rotor protrusions is specially calculated so that the EMF induced in the winding changes over time according to the sine law, that is, so that the shape of the voltage and current given by the generator is sinusoidal.
The generator stator - its stationary part - is an iron ring in the grooves of which the armature windings are laid. To reduce losses due to Foucault currents, this ring is made not continuous, but consisting of individual thin sheets iron isolated from each other
friend.
MINISTRY OF EDUCATION AND SCIENCE
RUSSIAN FEDERATION
FEDERAL STATE BUDGET EDUCATIONAL INSTITUTION OF HIGHER PROFESSIONAL EDUCATION
"KURGAN STATE UNIVERSITY"
Abstract On the subject "Physics" Topic: "Foucault's currents and their application"
Completed by: Student of group T-10915 Logunova M.V.
Teacher Vorontsov B.S.
Kurgan 2016
Introduction 3
1. Toki Fuko 4
2.Whirls and skin effect 7
3. Practical application of Foucault currents 8
4.Derivation of formulas 10
4.1. Eddy current strength according to Ohm's law 10
4.2. Formulas for calculating losses due to Foucault currents 10
Conclusion 11
References 12
Introduction
Induction current can occur not only in linear circuits, that is, in conductors whose transverse dimensions are negligible compared to their length. Induction current also occurs in massive conductors. In this case, the conductor does not have to be included in a closed circuit. A closed circuit of induction current is formed in the thickness of the conductor itself. Such induced currents are called vortex or currentsFoucault.
Eddy currents, or Foucault currents (in honor of J. B. L. Foucault) are eddy induction currents that arise in conductors either due to a change in time of the magnetic field in which the body is located, or due to the movement of the body in a magnetic field, leading to a change in the magnetic flux through body or any part thereof.
The faster the magnetic flux changes, the greater the magnitude of Foucault currents.
Toki Fuko
Eddy currents were first discovered by the French scientist D. F. Arago (1786-1853) in 1824 in a copper disk located on an axis under a rotating magnetic needle. Due to eddy currents, the disk began to rotate. This phenomenon, called the Arago phenomenon, was explained several years later by M. Faradays from the position of the law he discovered electromagnetic induction: A rotating magnetic field induces eddy currents in the copper disk, which interact with the magnetic needle. Eddy currents were studied in detail by the French physicist Foucault (1819-1868) and named after him. He discovered the phenomenon of heating metal bodies rotated in a magnetic field by eddy currents.
Foucault currents arise under the influence of alternating electromagnetic field and by their physical nature they are no different from induction currents arising in linear wires.
But, unlike electric current in wires, flowing along precisely defined paths, eddy currents are closed directly in the conducting mass, forming vortex-like circuits. These current circuits interact with the magnetic flux that generated them. The electrical resistance of a massive conductor is low, so Foucault currents reach very high strength. According to Lenz's rule, the magnetic field of eddy currents is directed so as to counteract the change in magnetic flux that induces these eddy currents.
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For example, if a copper plate is tilted from its equilibrium position and released so that it enters at a speed υ into the space between the magnet strips, then the plate will practically stop at the moment it enters the magnetic field (Fig. 1).
The slowdown in motion is associated with the excitation of eddy currents in the plate, which prevent the flux of the magnetic induction vector from changing. Since the plate has a finite resistance, the induction currents gradually die out and the plate moves slowly in the magnetic field. If the electromagnet is turned off, the copper plate will perform the usual oscillations characteristic of a pendulum.
Eddy currents also lead to an uneven distribution of magnetic flux over the cross section of the magnetic core. This is explained by the fact that in the center of the cross-section of the magnetic core, the magnetizing force of eddy currents directed towards the main flow is greatest, since this part of the cross-section is covered by the largest number of eddy current circuits. This “displacement” of the flux from the middle section of the magnetic circuit is expressed more sharply, the higher the frequency alternating current and the greater the magnetic permeability of the ferromagnet. At high frequencies, the flow passes only in a thin surface layer of the core. This causes a decrease in the apparent (averaged over the cross section) magnetic permeability. The phenomenon of displacement of a magnetic flux changing with a high frequency from a ferromagnet is similar to the electrical skin effect and is called the magnetic skin effect.
In accordance with the Joule-Lenz law, eddy currents heat the conductors in which they arise. Therefore, eddy currents lead to energy losses (eddy current losses) in magnetic circuits (in the cores of transformers and AC coils, in the magnetic circuits of machines).
To reduce energy losses due to eddy currents (and harmful heating of magnetic circuits) and reduce the effect of “displacement” of magnetic flux from ferromagnets, the magnetic circuits of machines and alternating current devices are made not from a solid piece of ferromagnetic material (electrical steel), but from separate plates isolated from each other. This division into plates located perpendicular to the direction of the eddy currents limits the possible contours of the eddy current paths, which greatly reduces the magnitude of these currents. At very high frequencies, the use of ferromagnets for magnetic circuits is impractical; in these cases, they are made from magnetodielectrics, in which eddy currents practically do not arise due to the very high resistance of these materials.
When a conducting body moves in a magnetic field, the induced eddy currents cause a noticeable mechanical interaction of the body with the field. This principle is based, for example, on the braking of the moving system in electric energy meters, in which an aluminum disk rotates in the field of a permanent magnet. In alternating current machines with a rotating field, a solid metal rotor is carried away by the field due to the eddy currents arising in it. The interaction of eddy current with an alternating magnetic field is the basis of various types of pumps for pumping molten metal.
Eddy currents also arise in the conductor itself, through which alternating current flows, which leads to an uneven distribution of the current over the cross-section of the conductor. At moments of increasing current in the conductor, induction eddy currents are directed at the surface of the conductor along the primary electric current, and at the axis of the conductor - towards the current. As a result, the current inside the conductor will decrease, and at the surface it will increase. High frequency currents practically flow into thin layer near the surface of the conductor, but inside the conductor there is no current. This phenomenon is called the electrical skin effect. To reduce energy losses due to eddy currents, large-gauge AC wires are made from separate strands insulated from each other.
In electrical devices, instruments and machines, metal parts sometimes move in a magnetic field or stationary metal parts intersect power lines changing magnetic field strength. These metal parts are inducted.
Under the influence of these e. d.s. leaks in the mass of the metal part eddy currents (Foucault currents), which close in the mass, forming eddy current circuits.
Eddy currents (also Foucault currents) are electric currents that arise as a result of electromagnetic induction in a conducting medium (usually a metal) when the magnetic flux penetrating it changes.
Eddy currents generate their own magnetic fluxes, which counteract the magnetic flux of the coil and weaken it. They also cause the core to heat up, which is a waste of energy.
Let there be a core of metal material. Let's place a coil on this core and pass it through it. Around the coil there will be an alternating magnetic current crossing the core. In this case, an induced EMF will be induced in the core, which, in turn, causes currents called eddy currents in the core. These eddy currents heat the core. Since the electrical resistance of the core is small, the induced currents induced in the cores can be quite large, and the heating of the core can be significant.
Eddy currents were first discovered by the French scientist D.F. Arago (1786 - 1853) in 1824 in a copper disk located on an axis under a rotating magnetic needle. Due to eddy currents, the disk began to rotate. This phenomenon, called the Arago phenomenon, was explained several years later by M. Faraday from the standpoint of what he discovered.
Eddy currents were studied in detail by the French physicist Foucault (1819-1868) and named after him. He called the phenomenon of heating of metal bodies rotated in a magnetic field eddy currents.
As an example, the figure shows eddy currents induced in a massive core placed in a coil flowing around an alternating current. An alternating magnetic field induces currents that close along paths lying in planes perpendicular to the direction of the field.
Eddy currents: a - in a massive core, b - in a plate core
Ways to reduce Foucault currents
The power expended on heating the core by eddy currents needlessly reduces efficiency technical devices electromagnetic type.
To reduce the power of eddy currents, the electrical resistance of the magnetic circuit is increased; for this, the cores are assembled from separate thin (0.1-0.5 mm) plates, isolated from each other using a special varnish or scale.
Magnetic cores of all machines and alternating current devices and machine armature cores direct current assembled from plates insulated from each other with varnish or a surface non-conductive film (phosphated), stamped from sheet electrical steel. The plane of the plates must be parallel to the direction of the magnetic flux.
With such a division of the cross-section of the magnetic core, the eddy currents are significantly weakened, since the magnetic fluxes that interlock the eddy current contours are reduced, and consequently, the emission induced by these fluxes is also reduced. d.s., creating eddy currents.
Special additives are also introduced into the core material, which also increase it.For increase electrical resistance ferromagnetic electrical steel is prepared with a silicon additive.
The cores of some coils (bobbins) are made from pieces of annealed iron wire. The iron strips are placed parallel to the magnetic flux lines. Eddy currents flowing in planes perpendicular to the direction of the magnetic flux are limited by insulating gaskets. Magnetodielectrics are used for magnetic circuits of devices and devices operating at high frequencies. To reduce eddy currents in wires, the latter are made in the form of a bundle of individual conductors, insulated from each other.
Application of Foucault currents
Eddy currents have found useful application in the device of a magnetic disk brake electric meter. Rotating, the disk crosses . Eddy currents arise in the plane of the disk, which, in turn, create their own magnetic fluxes in the form of tubes around the eddy current. Interacting with the main field of the magnet, these flows slow down the disk.
In some cases, using eddy currents, it is possible to use technological operations that cannot be used without high-frequency currents. For example, when manufacturing vacuum instruments and devices, it is necessary to carefully pump out air and other gases from the cylinder. However, in metal fittings, located inside the cylinder, there are gas residues that can be removed only after brewing the cylinder. To completely degass the fittings, a vacuum device is placed in the field of a high-frequency generator; as a result of the action of eddy currents, the fittings are heated to hundreds of degrees, and the remaining gas is neutralized.
Eddy currents are found useful application also during surface hardening with high frequency currents.
The use of eddy currents in induction hardening of metals