Magnetic Field Slot Machine

1. Analytical and Numerical Calculation of Magnetic Field Distribution in the Slotted. 22 wb p wa − = −, CR=ln js +θ2, ln s r R g R ⎛⎞ ′= ⎜⎟ ⎝⎠, 2 2 001 22 bb b gg ⎡ ′′⎛⎞⎤ =+ +⎢ ⎜⎟⎥ ⎢ ′′⎝⎠⎥ ⎣ ⎦, 1 a b =, b02 1′=θ−θ Fig. 3 – Single infinitely deep slot opening in the S plane. 4 – Slot opening in the K plane. The value of s is known since it is a coordinate in the slotted air gap where.
2. The illustrate of the slots effects on magnetic field distribution is the most important in this study The slots are simplified to a rectangular shape it based on the modified of the slots shape in order to minimize their harmhere onic, wthe slot takes the form of a rectangle, as shown in the following Figure, it named modified slots type. The magnetic field is calculated in the proposed models.

Increases machine output though precise switching at the first attempt. Saves time on initial installation and when replacing devices as the sensor can be easily inserted into the slot from above. The end caps of the cylinder do not have to be removed. Magnetic field and welding field resistance (interference fields) PTFE hose.

In electrical engineering, electric machine is a general term for machines using electromagnetic forces, such as electric motors, electric generators, and others. They are electromechanical energy converters: an electric motor converts electricity to mechanical power while an electric generator converts mechanical power to electricity. The moving parts in a machine can be rotating (rotating machines) or linear (linear machines). Besides motors and generators, a third category often included is transformers, which although they do not have any moving parts are also energy converters, changing the voltage level of an alternating current.^[1]

Electric machines, in the form of generators, produce virtually all electric power on Earth, and in the form of electric motors consume approximately 60% of all electric power produced. Electric machines were developed beginning in the mid 19th century and since that time have been a ubiquitous component of the infrastructure. Developing more efficient electric machine technology is crucial to any global conservation, green energy, or alternative energy strategy.

Generator[edit]

An electric generator is a device that converts mechanical energy to electrical energy. A generator forces electrons to flow through an external electrical circuit. It is somewhat analogous to a water pump, which creates a flow of water but does not create the water inside. The source of mechanical energy, the prime mover, may be a reciprocating or turbine steam engine, water falling through a turbine or waterwheel, an internal combustion engine, a wind turbine, a hand crank, compressed air or any other source of mechanical energy.

The two main parts of an electrical machine can be described in either mechanical or electrical terms. In mechanical terms, the rotor is the rotating part, and the stator is the stationary part of an electrical machine. In electrical terms, the armature is the power-producing component and the field is the magnetic field component of an electrical machine. The armature can be on either the rotor or the stator. The magnetic field can be provided by either electromagnets or permanent magnets mounted on either the rotor or the stator. Generators are classified into two types, AC generators and DC generators.

AC generator[edit]

An AC generator converts mechanical energy into alternating current electricity. Because power transferred into the field circuit is much less than power transferred into the armature circuit, AC generators nearly always have the field winding on the rotor and the armature winding on the stator.

AC generators are classified into several types.

• In an induction generator, the stator magnetic flux induces currents in the rotor. The prime mover then drives the rotor above the synchronous speed, causing the opposing rotor flux to cut the stater coils producing active current in the stater coils, thus sending power back to the electrical grid. An induction generator draws reactive power from the connected system and so cannot be an isolated source of power.
• In a Synchronous generator (alternator), the current for the magnetic field is provided by a separate DC current source.

DC generator[edit]

A DC generator is a machine that converts mechanical energy into Direct Current electrical energy. A DC generator generally has a commutator with split ring to produce a direct current instead of an alternating current.

Motor[edit]

An electric motor converts electrical energy into mechanical energy. The reverse process of electrical generators, most electric motors operate through interacting magnetic fields and current-carrying conductors to generate rotational force. Motors and generators have many similarities and many types of electric motors can be run as generators, and vice versa.Electric motors are found in applications as diverse as industrial fans, blowers and pumps, machine tools, household appliances, power tools, and disk drives. They may be powered by direct current or by alternating current which leads to the two main classifications: AC motors and DC motors.

AC motor[edit]

An AC motor converts alternating current into mechanical energy. It commonly consists of two basic parts, an outside stationary stator having coils supplied with alternating current to produce a rotating magnetic field, and an inside rotor attached to the output shaft that is given a torque by the rotating field.The two main types of AC motors are distinguished by the type of rotor used.

• Induction (asynchronous) motor, the rotor magnetic field is created by an induced current. The rotor must turn slightly slower (or faster) than the stator magnetic field to provide the induced current. There are three types of induction motor rotors, which are squirrel-cage rotor, wound rotor and solid core rotor.
• Synchronous motor, it does not rely on induction and so can rotate exactly at the supply frequency or sub-multiple. The magnetic field of the rotor is either generated by direct current delivered through slip rings (exciter) or by a permanent magnet.

DC motor[edit]

The brushed DC electric motor generates torque directly from DC power supplied to the motor by using internal commutation, stationary permanent magnets, and rotating electrical magnets. Brushes and springs carry the electric current from the commutator to the spinning wire windings of the rotor inside the motor. Brushless DC motors use a rotating permanent magnet in the rotor, and stationary electrical magnets on the motor housing. A motor controller converts DC to AC. This design is simpler than that of brushed motors because it eliminates the complication of transferring power from outside the motor to the spinning rotor.An example of a brushless, synchronous DC motor is a stepper motor which can divide a full rotation into a large number of steps.

Other electromagnetic machines[edit]

Other electromagnetic machines include the Amplidyne, Synchro, Metadyne, Eddy current clutch, Eddy current brake, Eddy current dynamometer, Hysteresis dynamometer, Rotary converter, and Ward Leonard set. A rotary converter is a combination of machines that act as a mechanical rectifier, inverter or frequency converter. The Ward Leonard set is a combination of machines used to provide speed control. Other machine combinations include the Kraemer and Scherbius systems.

Transformer[edit]

A transformer is a static device that converts alternating current from one voltage level to another level (higher or lower), or to the same level, without changing the frequency. A transformer transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying electric current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field through the secondary winding. This varying magnetic field induces a varying electromotive force (emf) or 'voltage' in the secondary winding. This effect is called mutual induction.

There are three types of transformers

1. Step-up transformer
2. Step-down transformer

There are four types of transformers based on structure

1. core type
2. shell type
3. power type
4. instrument type

Electromagnetic-rotor machines[edit]

Electromagnetic-rotor machines are machines having some kind of electric current in the rotor which creates a magnetic field which interacts with the stator windings. The rotor current can be the internal current in a permanent magnet (PM machine), a current supplied to the rotor through brushes (Brushed machine) or a current set up in closed rotor windings by a varying magnetic field (Induction machine).

Permanent magnet machines[edit]

PM machines have permanent magnets in the rotor which set up a magnetic field. The magnetomotive force in a PM (caused by orbiting electrons with aligned spin) is generally much higher than what is possible in a copper coil. The copper coil can, however, be filled with a ferromagnetic material, which gives the coil much lower magnetic reluctance. Still the magnetic field created by modern PMs (Neodymium magnets) is stronger, which means that PM machines have a better torque/volume and torque/weight ratio than machines with rotor coils under continuous operation. This may change with introduction of superconductors in rotor.

Since the permanent magnets in a PM machine already introduce considerable magnetic reluctance, then the reluctance in the air gap and coils are less important. This gives considerable freedom when designing PM machines.

It is usually possible to overload electric machines for a short time until the current in the coils heats parts of the machine to a temperature which cause damage. PM machines can in less degree be subjected to such overload because too high current in the coils can create a magnetic field strong enough to demagnetise the magnets.

Brushed machines[edit]

Brushed machines are machines where the rotor coil is supplied with current through brushes in much the same way as current is supplied to the car in an electric slot car track. More durable brushes can be made of graphite or liquid metal. It is even possible to eliminate the brushes in a 'brushed machine' by using a part of rotor and stator as a transformer which transfer current without creating torque. Brushes must not be confused with a commutator. The difference is that the brushes only transfer electric current to a moving rotor while a commutator also provide switching of the current direction.

There is iron (usually laminated steel cores made of sheet metal) between the rotor coils and teeth of iron between the stator coils in addition to black iron behind the stator coils. The gap between rotor and stator is also made as small as possible. All this is done to minimize magnetic reluctance of the magnetic circuit which the magnetic field created by the rotor coils travels through, something which is important for optimizing these machines.

Large brushed machines which are run with DC to the stator windings at synchronous speed are the most common generator in power plants, because they also supply reactive power to the grid, because they can be started by the turbine and because the machine in this system can generate power at constant speed without a controller. This type of machine is often referred to in the literature as a synchronous machine.

This machine can also be run by connecting the stator coils to the grid, and supplying the rotor coils with AC from an inverter. The advantage is that it is possible to control rotating speed of the machine with a fractionally rated inverter. When run this way the machine is known as a brushed double feed 'induction' machine. 'Induction' is misleading because there is no useful current in the machine which is set up by induction.

Induction machines[edit]

Induction machines have short circuited rotor coils where a current is set up and maintained by induction. This requires that the rotor rotates at other than synchronous speed, so that the rotor coils are subjected to a varying magnetic field created by the stator coils. An induction machine is an asynchronous machine.

Induction eliminates the need for brushes which is usually a weak part in an electric machine. It also allows designs which make it very easy to manufacture the rotor. A metal cylinder will work as rotor, but to improve efficiency a 'squirrel cage' rotor or a rotor with closed windings is usually used. The speed of asynchronous induction machines will decrease with increased load because a larger speed difference between stator and rotor is necessary to set up sufficient rotor current and rotor magnetic field. Asynchronous induction machines can be made so they start and run without any means of control if connected to an AC grid, but the starting torque is low.

A special case would be an induction machine with superconductors in the rotor. The current in the superconductors will be set up by induction, but the rotor will run at synchronous speed because there will be no need for a speed difference between the magnetic field in stator and speed of rotor to maintain the rotor current.

Another special case would be the brushless double fed induction machine, which has a double set of coils in the stator. Since it has two moving magnetic fields in the stator, it gives no meaning to talk about synchronous or asynchronous speed.

Reluctance machines[edit]

Reluctance machines have no windings on the rotor, only a ferromagnetic material shaped so that 'electromagnets' in stator can 'grab' the teeth in rotor and advance it a little. The electromagnets are then turned off, while another set of electromagnets is turned on to move rotor further. Another name is step motor, and it is suited for low speed and accurate position control. Reluctance machines can be supplied with permanent magnets in the stator to improve performance. The “electromagnet” is then “turned off” by sending a negative current in the coil. When the current is positive the magnet and the current cooperate to create a stronger magnetic field which will improve the reluctance machine's maximum torque without increasing the currents maximum absolute value.

Electrostatic machines[edit]

In electrostatic machines, torque is created by attraction or repulsion of electric charge in rotor and stator.

Electrostatic generators generate electricity by building up electric charge. Early types were friction machines, later ones were influence machines that worked by electrostatic induction. The Van de Graaff generator is an electrostatic generator still used in research today.

Homopolar machines[edit]

Homopolar machines are true DC machines where current is supplied to a spinning wheel through brushes. The wheel is inserted in a magnetic field, and torque is created as the current travels from the edge to the centre of the wheel through the magnetic field.

Electric machine systems[edit]

For optimized or practical operation of electric machines, today's electric machine systems are complemented with electronic control.

References[edit]

• Chapman, Stephen J. 2005. Electrical Machinery Fundamentals. 4th Ed. New York: McGraw Hill.

1. ^Flanagan. Handbook of Transformer Design and Applications, Chap. 1 p1.

Further reading[edit]

• Chisholm, Hugh, ed. (1911). 'Electrical Machine' . Encyclopædia Britannica. 9 (11th ed.). Cambridge University Press. pp. 176–179. This has a detailed survey of the contemporaneous history and state of electric machines.

Electromagnetically induced acoustic noise (and vibration), electromagnetically excited acoustic noise, or more commonly known as coil whine, is audible sound directly produced by materials vibrating under the excitation of electromagnetic forces. Some examples of this noise include the mains hum, hum of transformers, the whine of some rotating electric machines, or the buzz of fluorescent lamps. The hissing of high voltage transmission lines is due to corona discharge, not magnetism.

The phenomenon is also called audible magnetic noise,^[1] electromagnetic acoustic noise, or electromagnetically-induced acoustic noise,^[2] or more rarely, electrical noise,^[3] or 'coil noise', depending on the application. The term electromagnetic noise is generally avoided as the term is used in the field of electromagnetic compatibility, dealing with radio frequencies. The term electrical noise describes electrical perturbations occurring in electronic circuits, not sound. For the latter use, the terms electromagnetic vibrations^[4] or magnetic vibrations,^[5] focusing on the structural phenomenon are less ambiguous.

Acoustic noise and vibrations due to electromagnetic forces can be seen as the reciprocal of microphonics, which describes how a mechanical vibration or acoustic noise can induce an undesired electrical perturbation.

General explanation[edit]

Electromagnetic forces can be defined as forces arising from the presence of an electromagnetic field (electrical field only, magnetic field only, or both).

Electromagnetic forces in the presence of a magnetic field include equivalent forces due to Maxwell stress tensor, magnetostriction and Lorentz force (also called Laplace force).^[6] Maxwell forces, also called reluctances forces, are concentrated at the interface of high magnetic reluctivity changes, e.g. between air and a ferromagnetic material in electric machines; they are also responsible of the attraction or repulsion of two magnets facing each other. Magnetostriction forces are concentrated inside the ferromagnetic material itself. Lorentz or Laplace forces act on conductors plunged in an external magnetic field.

Equivalent electromagnetic forces due to the presence of an electrical field can involve electrostatic, electrostrictive and reverse piezoelectric effects.

These phenomena can potentially generate vibrations of the ferromagnetic, conductive parts, coils and permanent magnets of electrical, magnetic and electromechanical device, resulting in an audible sound if the frequency of vibrations lies between 20 Hz and 20 kHz, and if the sound level is high enough to be heard (e.g. large surface of radiation and large vibration levels). Vibration level is increased in case of a mechanical resonance, when electromagnetic forces match with a structural mode natural frequency of the active component (magnetic circuit, electromagnetic coil or electrical circuit) or of its enclosure.

The frequency of the noise depends on the nature of electromagnetic forces (quadratic or linear function of electrical field or magnetic field) and on the frequency content of the electromagnetic field (in particular if a DC component is present or not).

Electromagnetic noise and vibrations in electric machines[edit]

Electromagnetic torque, which can be calculated as the average value of the Maxwell stress tensor along the airgap, is one consequence of electromagnetic forces in electric machines. As a static force, it does not create vibrations nor acoustic noise. However torque ripple (also called cogging torque for permanent magnet synchronous machines in open circuit), which represents the harmonic variations of electromagnetic torque, is a dynamic force creating torsional vibrations of both rotor and stator. The torsional deflection of a simple cylinder cannot radiate efficiently acoustic noise, but with particular boundary conditions the stator can radiate acoustic noise under torque ripple excitation.^[7] Structure-borne noise can also be generated by torque ripple when rotor shaft line vibrations propagate to the frame^[8] and shaft line.

Some tangential magnetic force harmonics can directly create magnetic vibrations and acoustic noise when applied to the stator teeth: tangential forces create a bending moment of the stator teeth, resulting in radial vibrations of the yoke.^[9]

Besides tangential force harmonics, Maxwell stress also includes radial force harmonics responsible for radial vibrations of the yoke, which in turn can radiate acoustic noise.

Electromagnetic noise and vibrations in passive components[edit]

Inductors[edit]

In inductors, also called reactors or chokes, magnetic energy is stored in the airgap of the magnetic circuit, where large Maxwell forces apply. Resulting noise and vibrations depend on airgap material and magnetic circuit geometry.^[10]

Transformers[edit]

In transformers magnetic noise and vibrations are generated by several phenomena depending on the load case which include Lorentz force on the windings,^[11] Maxwell forces in the joints of the laminations, and magnetostriction inside the laminated core.

Capacitors[edit]

Capacitors are also subject to large electrostatic forces. When the capacitor voltage/current waveform is not constant and contains time harmonics, some harmonic electric forces appear and acoustic noise can be generated.^[12] Ferroelectric capacitors also exhibit a piezoelectric effect that can be source of audible noise. This phenomenon is known as the 'singing capacitor' effect.^[13]

Resonance effect in electrical machines[edit]

In radial flux rotating electric machines, resonance due to electromagnetic forces is particular as it occurs at two conditions: there must be a match between the exciting Maxwell force and the stator or rotor natural frequency, and between the stator or rotor modal shape and the exciting Maxwell harmonic wavenumber (periodicity of the force along the airgap).^[14]

As an example a resonance with the elliptical modal shape of the stator can occur if the force wavenumber is 2. Under resonance conditions, the maxima of the electromagnetic excitation along the airgap and the maxima of the modal shape displacement are in phase.

Numerical simulation[edit]

Methodology[edit]

The simulation of electromagnetically induced noise and vibrations is a multiphysic modeling process carried in three steps:

• calculation of the electromagnetic forces
• calculation of the resulting magnetic vibrations
• calculation of the resulting magnetic noise

It is generally considered as a weakly coupled problem: the deformation of the structure under electromagnetic forces is assumed not to change significantly the electromagnetic field distribution and the resulting electromagnetic stress.

Application to electric machines[edit]

The assessment of audible magnetic noise in electrical machines can be done using three methods:

• using dedicated electromagnetic and vibro-acoustic simulation software (e.g. MANATEE ^[15])
• using electromagnetic (e.g. Flux,^[16] Jmag,^[17] Maxwell,^[18] Opera^[19]), structural (e.g. Ansys Mechanical, Nastran, Optistruct) and acoustic (e.g. Actran, LMS, Sysnoise) numerical software together with dedicated coupling methods
• using multiphysics numerical simulation software environment (e.g. Comsol Multiphysics,^[20] Ansys Workbench^[21])

Examples of device subject to electromagnetic noise and vibrations[edit]

Static devices[edit]

Static devices include electrical systems and components used in electric power storage or power conversion such as

Slot Machine Games

• transformers^[22]
• resistors: the braking resistors of electric trains, used to dissipate electrical power when the catenary is not receptive during braking, can make electromagnetically induced acoustic noise
• coils: in magnetic resonance imaging, 'coil noise' is that part of total system noise attributed to the receiving coil, due to its non-zero temperature.

Rotating devices[edit]

Free Slot Machine

Rotating devices include radial and axial flux rotating electric machines used for electrical to mechanical power conversion such as

• induction motors^[23]
• synchronous motors with permanent magnets or DC wound rotor
• switched reluctance motors

In such device, dynamic electromagnetic forces come from variations of magnetic field, which either comes from a steady AC winding or a rotating DC field source (permanent magnet or DC winding).

Sources of magnetic noise and vibrations in electric machines[edit]

The harmonic electromagnetic forces responsible for magnetic noise and vibrations in a healthy machine can come from

• Pulse-width modulation supply of the machine^[24]
• slotting effects^[25][26][27]
• magnetic saturation^[28]

In a faulty machine, additional noise and vibrations due to electromagnetic forces can come from

• mechanical static and dynamic eccentricities^[29]
• uneven air-gap^[30]
• demagnetization
• short circuits
• missing magnetic wedges

Unbalanced Magnetic Pull (UMP) describes the electromagnetic equivalence of mechanical rotating unbalance: if electromagnetic forces are not balanced, a non-zero net magnetic force appears on stator and rotor. This force can excite the bending mode of the rotor and create additional vibration and noise.

Reduction of electromagnetic noise and vibrations[edit]

Reduction of magnetic noise and vibrations in electric machines[edit]

NVH mitigation techniques in electrical machines include^[31]

• reducing the magnitude of electromagnetic excitations, independently of the structural response of the electrical machine
• reducing the magnitude of the structural response, independently of the electromagnetic excitations
• reducing the resonances occurring between electromagnetic excitations and structural modes

Electromagnetic noise and vibration mitigation techniques in electrical machines include:

• choosing the right slot/pole combination and winding design
• avoiding resonances match between stator and electromagnetic excitations
• skewing the stator or the rotor
• implementing pole shaping / pole shifting / pole pairing techniques
• implementing harmonic current injection or spread spectrum PWM strategies
• using notches / flux barriers on the stator or the rotor
• increasing damping

Reduction of 'coil noise'[edit]

Coil noise mitigation actions include:

• add some glue (e.g. a layer of glue is often added on the top of television coils ; over the years, this glue degrades and the sound level increases)
• change the shape of the coil (e.g. change coil shape to a figure eight rather than a traditional coil shape)
• isolate the coil from the rest of the device to minimize structure-borne noise
• increase damping

Experimental illustrations[edit]

A varying electromagnetic force can be produced either by a moving source of DC magnetic field (e.g. rotating permanent magnet or rotating coil supplied with DC current), or by a steady source of AC magnetic field (e.g. a coil fed by a variable current).

Forced vibration by a rotating permanent magnet[edit]

This animation illustrates how a ferromagnetic sheet can be deformed due to the magnetic field of a rotating magnet. It corresponds to an ideal one pole pair permanent magnet synchronous machine with a slotless stator.

Acoustic resonance by a variable frequency coil[edit]

The resonance effect of magnetic vibration with a structural mode can be illustrated using a tuning fork made of iron. A prong of the tuning fork is wound with a coil fed by a variable frequency power supply. A variable flux density circulates between the two prongs and some dynamic magnetic forces appear between the two prongs at twice the supply frequency. When the exciting force frequency matches the fundamental mode of the tuning fork close to 400 Hz, a strong acoustic resonance occurs.

Examples of audio files[edit]

PMSM motor (traction application)[edit]

External links[edit]

• Video of a resonating tuning fork magnetically excited by a variable frequency current on YouTube
• Video of a tuning fork magnetically excited by a fixed frequency current on YouTube
• Video of a ferromagnetic cylinder deformed by a rotating magnet on YouTube

References[edit]

1. ^Le Besnerais, J., Lanfranchi, V., Hecquet, M., & Brochet, P. (2010). Characterization and Reduction of Audible Magnetic Noise Due to PWM Supply in Induction Machines. IEEE Transactions on Industrial Electronics. http://doi.org/10.1109/tie.2009.2029529
2. ^van der Giet, M., (2011). Analysis of electromagnetic acoustic noise excitations – a contribution to low-noise design and to the auralization of electrical machines, RWTH Aachen University, Shaker Verlag.
3. ^Finley, W. R., Hodowanec, M. M., & Holter, W. G. (1999). An Analytical Approach to Solving Motor Vibration Problems, 36(5), 1–16.
4. ^Carmeli, M. S., Castelli Dezza, F., & Mauri, M. (2006). Electromagnetic vibration and noise analysis of an external rotor permanent magnet motor. International Symposium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), 1028–33. http://doi.org/10.1109/SPEEDAM.2006.1649919
5. ^Le Besnerais, J. (2015). Effect of lamination asymmetries on magnetic vibrations and acoustic noise in synchronous machines. In 2015 18th International Conference on Electrical Machines and Systems (ICEMS). http://doi.org/10.1109/icems.2015.7385319
6. ^Belahcen, A. (2004). Magnetoelasticity, magnetic forces and magnetostriction in electrical machines. PhD thesis, Helsinki University of Technology, Finland.
7. ^Tan Kim A. (2013). Contribution à l'étude du bruit acoustique d'origine magnétique en vue de la conception optimale de machines synchrones à griffes pour application automobile. PhD thesis, Université de Technologie de Compiègne, France.
8. ^De Madinabeitia I. G, (2016).Analysis of force and torque harmonics spectrum in an induction machine for automotive NVH Purposes. Master's thesis, University of Technology of Chalmers, Sweden.
9. ^Devillers E., Le Besnerais J., Regniez M. and Hecquet M., (2017). Tangential effects on magnetic vibrations of induction machines using subdomain method and electromagnetic vibration synthesis, Proceedings of IEMDC 2017 Conference, Miami, USA. https://eomys.com/recherche/publications/article/tangential-effects-on-magnetic-vibrations-and-acoustic-noise-of-induction
10. ^M. Rossi and J. Le Besnerais, Vibration Reduction of Inductors Under Magnetostrictive and Maxwell Forces Excitation, in IEEE Transactions on Magnetics, vol. 51, no. 12, pp. 1–6, Dec. 2015. https://doi.org/10.1109/TMAG.2015.2469643
11. ^Arturi, C.M., 1992. Force calculation in transformer windings under unbalanced MMFs by a non-linear finite element code. IEEE transactions on magnetics, 28(2), pp.1363-1366.
12. ^M. Hurkala, Noise analysis of high voltage capacitors and dry-type air-core reactors. Doctoral dissertation, Aalto University, Finland, 2013
13. ^https://product.tdk.com/en/contact/faq/31_singing_capacitors_piezoelectric_effect.pdf
14. ^Le Besnerais, J. (2008). Reduction of magnetic noise in PWM-supplied induction machines − low-noise design rules and multi-objective optimization. PhD Thesis, Ecole Centrale de Lille, Lille, France. https://hal.archives-ouvertes.fr/tel-00348730/
15. ^'MANATEE software (Magnetic Acoustic Noise Analysis Tool for Electrical Engineering), official website'. Retrieved September 15, 2017.
16. ^'Flux software official website'.
17. ^'Jmag software official website'.
18. ^'Maxwell software official website'.
19. ^'Opera software official website'.
20. ^'Comsol software official website'.
21. ^'Ansys software official website'.
22. ^Weiser, B., Pfützner, H., & Anger, J. (2000). Relevance of Magnetostriction and Forces for the Generation of Audible Noise of Transformer Cores, 36(5), 3759–3777.
23. ^Le Besnerais, J. (2008). Reduction of magnetic noise in PWM-supplied induction machines − low-noise design rules and multi-objective optimization. PhD Thesis, Ecole Centrale de Lille, Lille, France. https://hal.archives-ouvertes.fr/tel-00348730/
24. ^Le Besnerais, J., Lanfranchi, V., Hecquet, M., & Brochet, P. (2010). Characterization and Reduction of Audible Magnetic Noise Due to PWM Supply in Induction Machines. IEEE Transactions on Industrial Electronics. http://doi.org/10.1109/tie.2009.2029529
25. ^Le Besnerais, J., Lanfranchi, V., Hecquet, M., & Brochet, P. (2009). Optimal Slot Numbers for Magnetic Noise Reduction in Variable-Speed Induction Motors. IEEE Transactions on Magnetics. http://doi.org/10.1109/tmag.2009.2020736
26. ^Verez, G., Barakat, G., Amara, Y., Bennouna, O., & Hoblos, G. (n.d.). Impact of Pole and Slot Combination on Noise and Vibrations of Flux-Switching PM Machines, (1).
27. ^Zhu, Z. Q., Xia, Z. P., Wu, L. J., & Jewell, G. W. (2009). Influence of slot and pole number combination on radial force and vibration modes in fractional slot PM brushless machines having single- and double-layer windings. 2009 IEEE Energy Conversion Congress and Exposition, ECCE 2009, 3443–3450. http://doi.org/10.1109/ECCE.2009.5316553
28. ^Le Besnerais, J., Lanfranchi, V., Hecquet, M., Lemaire, G., Augis, E., & Brochet, P. (2009). Characterization and Reduction of Magnetic Noise Due to Saturation in Induction Machines. IEEE Transactions on Magnetics. http://doi.org/10.1109/tmag.2008.2012112
29. ^Torregrossa, D., Khoobroo, A., & Fahimi, B. (2012). Prediction of acoustic noise and torque pulsation in PM synchronous machines with static eccentricity and partial demagnetization using field reconstruction method. IEEE Transactions on Industrial Electronics, 59(2), 934–944. http://doi.org/10.1109/TIE.2011.2151810
30. ^Le Besnerais, J. (2015). Effect of lamination asymmetries on magnetic vibrations and acoustic noise in synchronous machines. In 2015 18th International Conference on Electrical Machines and Systems (ICEMS). http://doi.org/10.1109/icems.2015.7385319
31. ^'Noise mitigation techniques in electric machines'. www.eomys.com. EOMYS ENGINEERING. Retrieved September 15, 2017.