Electromagnetic and vibro-acoustic models 

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State of the art and orientation of the work

In this chapter, an overview of the most important works seen during the bibliographic study are presented. As the thesis scope involves several fields of physics (electromagnetism, mechanics, acoustics) and mathematics (optimisation), an exhaustive list of all the references studied during these three years would be inappropriate: the more distinctive publications have been selected in terms of clearness, scientific rigour, synoptic view and pioneering work.
Modelling magnetic noise generation requires to model both the electromagnetic exciting force and the mechanical response of the excited structure (magnetic force / structure vibration transfer). Once the vibrations of this structure are known, an acoustic model is necessary to compute the sound power level radiated by the machine (air vibrations / SWL transfer). This first part details the different analytical models that were found in literature. The most appropriate ones to fulfil our goals (cf. section 1.2) are chosen, and all their assumptions are clarified.

Electromagnetic and vibro-acoustic modelling

Electromagnetic forces modelling

Magnetic forces occurring in electrical machines are traditionally classified in three types:
1. Maxwell forces (sometimes called reluctance forces)
2. Laplace forces
3. magnetostrictive forces
This classification, which is often reported in literature, is as ill-defined as the previous one on noise sources: indeed, as pointed out in (15), one must distinguish force distributions or densities, acting on differential vol-umes or surfaces of the machine, from total forces, which are integrals of force distributions on some parts of the machine (e.g. conductors for Laplace forces). Moreover, Maxwell forces are often expressed from the definition of the Maxwell tensor, whose general definition already contains the Lorentz force.
Neglecting the electrical fields, the i-th component of the force density applying on a given volume element of the motor can be written under the form (136) :
where fi stands for the i-th (i=1,2,3 being the Cartesian coordinates in basis (x1, x2, x3)) component of force vector f, and jk is the magnetic permeability second order tensor (matrix) assuming the medium is anisotropic. The meaning of these six components is the following:
1. Y = C(E) is the mechanical stress tensor, C is the elasticity tensor and E the strain tensor. In the linear case we have yij = 3 k,l=1 cijklǫkl (Hooke’s law).
2. j × B is the Lorentz force per unit volume which applies to a coil flowed by a current j in an external flux density field B (also called Laplace force in its integral form).
3. −21 Hj Hk ∂i jk describes the force per unit volume caused by magnetic permeability inhomogeneities, assuming that it does not depend of the magnetic field intensity.
4. −21 [rot(H × B)]i represents a torque (136) which vanishes if H and B are colinear.
5. Φ is the magnetostriction tensor, related to the strain dependence of the magnetic permeability.
6. Ψ is the thermal stress tensor.
Using the conservation of magnetic flux, the magnetic force densities number 2, 3 and 4 can be expressed by a magnetic tensor called Maxwell tensor (96):
The extra-diagonal terms of this tensor stand for magnetic shear stresses, whereas the diagonal terms stand for magnetic normal stresses. All the magnetic stress present in an electrical machine parts, which vanishes when it is magnetic-field free (i.e. current-free in induction motors), either comes from Maxwell stress or magnetostrictive stress.
Focusing of these magnetic forces, the electromagnetic force density can be be written as f = div(T + Φ) (2.3)

Maxwell forces

Assuming that B = H (isotropic permeability), and taking a surface dS with normal direction x3 (Fig. 2.1), the magnetic stresses in its normal and tangential direction
Figure 2.1: Left: definition of dS surface and its normal and tangential directions. Right: definition of the local coordinate system defined by a magnetic field line (from (14)).
In an induction machine, where cylindrical coordinates (r, θ, z) are more adapted, σn becomes σr and σt becomes σθ . The modulus of this stress is constant and independent of the surface orientation.
which is the electromagnetic energy density (homogeneous to a pressure, which explains the use of the expression ”Maxwell pressure”) at that point. A geometrical interpretation of that magnetic stress is also reported in (14): the angle Δ between the magnetic force which applies to a given surface and its normal direction is twice the one between the local magnetic field line direction and the surface normal direction. As a consequence, when H is perpendicular to a surface (Δ = 0), the magnetic force seen by that surface is collinear to H; when H is parallel to a surface (Δ = π/2), the magnetic force is also perpendicular to the surface, but in opposite direction (see Fig. 2.2).
Figure 2.2: Direction of total Maxwell stress for different magnetic line directions.
If Maxwell tensor is expressed in the local coordinates defined by a given magnetic field line (cf. Fig. 2.1), it becomes
Two different stresses therefore occur in the perpendicular plane of the magnetic line, and in the direction of the magnetic line: the first one can be interpreted as an hydrostatic magnetic pressure, which tends to keep the magnetic field lines away from one another; the second one can be interpreted as a magnetic tension which tends to make magnetic field lines shorter (14). The first one also corresponds to the law of maximum flux, which ”inflates” a deformable solenoid, while the second one corresponds to the law of minimal reluctance, which tenses the solenoid.
Expression (2.4) can also be found using the Ostrogradsky theorem (or divergence theorem) (15), which shows that the volume integral of the Maxwell stress divergence can be reduced to a surface integral enclosing the volume under consideration:
F = f dτ = T.dS (2.7)
V S/V⊂S
If we want to calculate the total Maxwell force which applies on the volume V of the full stator, we can define a closed surface S composed of a cylinder in the air-gap, another cylinder outside the motor, and two rings in the motor end-regions. Assuming that the magnetic field vanishes outside the machine, the surface integration integration over S is reduced to a surface integration over the air-gap cylinder:
2π Ls 2π Ls 1 0(Hr2 − Hθ2)uθ )rdθdz
Fstator = z=0 T.urrdθdz = z=0( 0Hr Hθ ur + (2.8)
θ=0 θ=0 2
where i stands for the magnetic permeability of iron, H stands for magnetic field values in the iron, whereas H′ stand for magnetic field value in the air. Using the continuity laws of flux density and magnetic field, we
have Ht = H′t and Bn = B′n, so that
σn = H′t2 B′n2 1 1 1 ′2 ′2 1 1
( 0 − i) + ( − ) = ( 0 iH + B n)( − ) (2.11)
2 2 i 0 2 i 0
σt = 0 (2.12)
Maxwell stress at the interface between air-gap and stator iron (in front of stator teeth) is therefore a normal stress, in radial direction. Note that at this stage of the analysis, no assumption has been made on the fact that magnetic flux density lines radially enter in stator teeth. The property of a purely radial Maxwell stress at the interface is independent of the incidence angle of the flux density lines, and only comes from continuity laws.
This radial stress tends to pull stator towards rotor (law of minimal reluctance), it is a negative pressure which applies on the inner surface of the stator. As illustrated in Fig. 2.3, only a small amount of flux density lines enters in teeth sides. If the inner surface of stator slots had been included in the surface integration, we would have found some additional Maxwell forces acting on stator teeth sides in transverse direction. These tangential forces can play an important vibro-acoustic role in certain cases, e.g. in large turbines (66; 68; 127) where the teeth bending natural frequencies can be excited (their natural frequency computation is detailed for instance in (63)). In (137), some tangential tooth forces linked to PWM supply are also shown to be superior or equal to radial tooth forces in sinusoidal case. However, teeth radial compression is directly transmitted to the stator yoke and frame, and efficiently radiated in audible sound power level, whereas their bending is damped by windings and wedges, and badly transmitted to the stator frame: these tangential Maxwell forces will be neglected in this work.
Figure 2.3: Example of flux density lines distribution in an induction machine, in unsaturated case: the main part of the magnetic flux enters perpendicularly to tooth tips.
As 0 ≪ i, the radial Maxwell stress can finally be approximated by
σn ≈ − 1 ′2 ′2 1 i ′2 ′2
( 0 iH t + B n) = − ( B t + B n) (2.13)
2 0 2 0 0
In non-saturated case, magnetic flux lines enter almost perpendicularly into the iron (cf. Fig. 2.3), so that iB′t ≪ 0B′n, and a new approximation of σn can be made:
This negative radial pressure is here assumed to be the only significant source of audible magnetic noise in traction machines. In order to compute it analytically, the expression of the radial flux density B′n at the outer surface of stator teeth is needed. As the air-gap is generally very thin, B′n is approximated by the flux density value in the middle of the air-gap.
Expression (2.14) is only valid considering that stator iron has a linear characteristic. The effect of satura-tion will therefore have to be treated separately (see section 2.1.1.3.2).
Laplace forces are the integral expression of Maxwell forces on conductors, i.e. stator windings and rotor bars in an induction machine. They can produce high vibrations of the stator end-windings in both starting transient and steady state conditions (115). However, the stator slots flux density is too low to produce significant vibrations of the yoke, and stator conductors are wedged into slots. Acoustic noise due to Laplace forces is therefore neglected in this work.

Magnetostriction forces

The general phenomenon of magnetostriction in active materials comprise an isotropic phenomenon called volume magnetostriction, and an anisotropic phenomenon called Joule and transverse magnetostriction (15). Volume magnetostriction occurs in the iron at magnetic fields superior to 8 kA/m, but it can be neglected in the two-dimensional iron sheets of the stator stack: magnetostriction will therefore only refer to Joule and transverse magnetostriction in the following. Magnetostriction is a magnetomagnetic phenomenon which stretches or shrinks a material in the magnetic field lines direction (Fig. 2.4), keeping its volume constant (15; 69). The elongation value for an electrical iron sheet can reach from 1 to 10 m/m, which is also the order of magnitude of Maxwell relative displacements1.
Figure 2.4: Simplified representation of a material flux density lines (left) and the resulting magnetostrictive effect (right).
Magnetostrictive forces are defined as the force field which creates the same strain than the magnetostrictive stress. This stress is partly due to the overlapping of the dipoles electronic clouds that compose the iron ferromagnetic material, when they naturally align with an external magnetic field. Another part of this stress is produced at a larger scale between different domains of the crystalline structure (11; 99). As Maxwell forces, magnetostrictive forces depend on the observation direction and are therefore properly represented by a tensor.
Magnetostrictive vibrations occur at same frequencies than Maxwell vibrations (99): it is therefore impossible to distinguish them on a vibration spectrum. Moreover, it is hard to quantify how much Maxwell efforts are greater than magnetostrictive efforts. It was shown in particular that in can depend on the deflection shape of the stator (99): magnetostrictive deflections can either limit or reinforce Maxwell deflections at a given frequency. An example of Maxwell and magnetostrictive force distributions is displayed in Fig. 2.5.
The relative importance of magnetostrictive and Maxwell forces strongly depends on the air-gap width, on the intrinsic magnetic properties of the iron (for instance, silicium-enriched materials limit magnetostriction (92)), and on the frequency range. Many studies were carried on different motors with different assumptions, leading to different conclusions.
As an example, a first study was made by Belmans (19) who suggested that magnetostriction was negligible. Then, Garvey (67) proposed a method to compute magnetostriction effects, and its numerical application on a large machine showed that magnetostriction was negligible. Some other works (94) on a 2.2 kW motor showed that 10 to 30% of the magnitude of magnetic vibration lines were due to magnetostriction. The thesis work of Laftman (99) conluded that magnetostrictive forces magnitude could be as high as Maxwell forces magnitude 1 Taking the magnitude B02/(2 0 ) of the fundamental Maxwell force, which has a 2p spatial order and 2fs frequency, gives for B0 = 1.5 T on motor M5 an elongation of Y2spω ≈ 310−6 (see equation (3.39)). Note that kind of comparison can be misleading as magnetic noise does not come from the low frequency fundamental magnetic vibration, but from some of its harmonics of higher frequencies and smaller magnitude (the same remark applies to magnetostrictive and Laplace forces).
Figure 2.5: Maxwell (left) and magnetostrictive (right) force fields in a 37 kW induction machine (15). Note that different scales have been used in both figures: as discussed in (15), the FEA representation of nodal forces can be misleading because tooth tips mesh is generally finer: few but long arrows can stand for the same pressure as numerous but short arrows.
in mean power machines (50 kW), and suggested that magnetostriction could even make machines quieter. However, the comparison between Maxwell and magnetostriction is made on stator displacements magnitude, without considering the frequency factor: if magnetostriction is as high as Maxwell forces larger component occuring at two times the supply frequency, it can still be unconsequential on the A-weighted sound power level. This is precisely the conclusion of a FEM-based work of Delaere (51), who compared the vibration spectra induced by reluctance forces and magnetostriction on a 45 kW up to 2 kHz, and concluded that magnetostriction vibrations are considerably smaller than Maxwell ones, apart from the twice supply frequency component where both vibrations have same order of magnitude.
Given the complexity of the subject, we suggest to take the work of Belahcen (15), probably the most advanced in the domain, as a reference. His rigorous work suggested in particular that magnetostriction may only produce significant vibrations at frequencies inferior to 1500 Hz.
There exist different models of magnetostriction (15). The most commonly used model in FEM simulation (16; 49; 50; 51; 78; 79; 152) is the force-based one: magnetostrictive forces are assimilated as a thermal tensor whose characteristics are a function of magnetostriction coefficients. These coefficients are obtained by experiments made on iron sheets (7; 41).
Magnetostrictive effects will be neglected in this thesis. We will see thereafter that magnetostriction mod-elling is not necessary to satisfyingly explain the acoustic behaviour of ALSTOM traction machines studied along this thesis.

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Conclusion

The only electromagnetic forces that are assumed to play a significant role in acoustic noise radiation are air-gap radial Maxwell forces, which depend on the air-gap radial flux density.

Air-gap flux density modelling

There are two main analytical methods able to determine the air-gap radial flux density distribution. The first one consists in decomposing the flux density as the product of a permeance function and magnetomotive forces (mmf ), the latter being decomposed in current and winding functions (wf ) (34; 70). The second method consists in analytically solving the electromagnetic field equations in a simplified geometry by applying conformal transformations (128; 135). However, the latter does not take into account iron saturation yet, and some progress is needed to make it a viable design option to other analytical models and FEA.
The permeance/winding function decomposition is therefore adopted in this thesis.

Saturation modelling

Saturation of stator and rotor iron sheets (see B(H) curve in Fig. 2.6) has several influences on our electromag-netic model. Firstly, it decreases the magnetising inductance, which also increases stator and rotor currents. This effect can be modelled by introducing in the electrical circuit a dependence of the magnetising inductance with respect to the saturation level, quantified by the saturation factor (32; 110). Secondly, the top of the air-gap flux density distribution cannot increase proportionally to the applied current, given the saturation of the B(H) curve in the active parts and the magnetic flux conservation: the flux density distribution in the air-gap is therefore flattened. This way, saturation changes the electromagnetic force spectrum, and can have a strong influence on magnetic noise.

Table of contents :

Nomenclature
1 Introduction 
1.1 Background
1.1.1 Noise of railway transport systems
1.1.1.1 Sources of noise
1.1.1.2 Acoustic norms
1.1.2 PROSODIE Project
1.2 Objectives
1.3 General approach
2 State of the art and orientation of the work 
2.1 Electromagnetic and vibro-acoustic modelling
2.1.1 Electromagnetic forces modelling
2.1.1.1 Maxwell forces
2.1.1.2 Magnetostriction forces
2.1.1.3 Conclusion
2.1.1.3.1 Air-gap flux density modelling
2.1.1.3.2 Saturation modelling
2.1.2 Vibro-acoustic modelling
2.1.2.1 Static displacements
2.1.2.2 Natural frequencies and dynamic deflections
2.1.2.3 Radiation factor and sound power level
2.2 Noise reduction methods
2.2.1 Low-noise design rules
2.2.1.1 Design variables influence
2.2.1.1.1 Rotor and stator slot number combination
2.2.1.1.2 Rotor and stator slots shape
2.2.1.1.3 Skewing
2.2.1.1.4 Stator stack dimensions
2.2.1.1.5 Eccentricities
2.2.1.1.6 Manufacturing errors: tolerances, asymmetries
2.2.1.1.7 Materials
2.2.1.2 Mounting and coupling influences
2.2.1.3 Supply current influence
2.2.1.3.1 Supply frequency
2.2.1.3.2 PWM strategy
2.2.1.3.3 Switching frequency
2.2.1.3.4 Comparison between PWM noise and slotting noise
2.2.2 Active methods
2.2.2.1 Current injection methods
2.2.2.2 Piezo-electric methods
2.2.2.3 Other methods
2.3 Vibro-acoustic optimisation
2.4 Conclusion
2.4.1 Modelling assumptions
2.4.2 Position of the work
3 Electromagnetic and vibro-acoustic models 
3.1 Electromagnetic model
3.1.1 Currents computation
3.1.1.1 Voltage computation
3.1.1.1.1 Method
3.1.1.1.2 Validation
3.1.1.2 Extended single phase equivalent circuit
3.1.1.2.1 Fundamental case
3.1.1.2.1.1 Expression
3.1.1.2.1.2 Saturation factor computation
3.1.1.2.2 Harmonic extension
3.1.1.3 Validations
3.1.1.3.1 No-load saturated sinusoidal case
3.1.1.3.2 On-load unsaturated sinusoidal case
3.1.1.3.3 No-load unsaturated PWM case
3.1.2 Magnetomotive forces computation
3.1.2.1 Stator magnetomotive force
3.1.2.2 Rotor magnetomotive force
3.1.3 Permeance computation
3.1.3.1 Expression
3.1.3.2 Skewed case
3.1.3.3 Saturated case
3.1.3.4 Eccentric case
3.1.3.4.1 Static eccentricity
3.1.3.4.2 Dynamic eccentricity
3.1.4 Air-gap radial flux density computation
3.1.4.1 Offload sinusoidal validations
3.1.4.1.1 Unsaturated case
3.1.4.1.2 Saturated case
3.1.4.2 On-load sinusoidal validations
3.1.5 Traction characteristics computation
3.1.5.1 Torque, power factor, efficiency expressions
3.1.5.2 Variable-speed characteristics
3.1.5.3 Validation
3.1.6 Instantaneous electromagnetic torque
3.2 Vibro-acoustic model
3.2.1 Natural frequencies computation
3.2.1.1 Expression
3.2.1.2 Effect of magnetic stiffness
3.2.1.3 Validation
3.2.2 Vibration computation
3.2.2.1 Expression
3.2.2.2 Experimental validation
3.2.2.2.1 Sinusoidal case
3.2.2.2.2 PWM case
3.2.3 Radiation factor computation
3.2.3.1 Expression
3.2.3.2 Validation
3.2.4 Sound power level computation
3.2.4.1 Expression
3.2.4.2 Validation
3.2.4.2.1 FEM/BEM simulations
3.2.4.2.2 Experiments
3.2.4.2.2.1 No-load sinusoidal case
3.2.4.2.2.2 No-load PWM case
3.3 Numerical considerations
3.3.1 Fourier versus time/space domain modelling
3.3.2 Computing tricks
3.3.3 Discretisation quality, spectral range and resolution
3.3.3.1 Time and space discretisation
3.3.3.2 Speed discretisation
3.3.3.3 Spectral resolution
3.3.3.4 Spectral range
3.3.3.5 Conclusion
4 Characterisation and reduction of noise 
4.1 Analytical charaterisation of magnetic force lines
4.1.1 General method
4.1.2 Standing versus rotating waves
4.1.3 Expression of main magnetic lines orders and frequencies
4.1.3.1 General case
4.1.3.2 Sinusoidal case
4.1.3.2.1 Expression of main lines
4.1.3.2.2 Pure slotting force lines
4.1.3.2.2.1 Expression
4.1.3.2.2.2 Validation
4.1.3.2.2.3 Effect of current magnitude
4.1.3.2.3 Saturation force lines
4.1.3.2.3.1 Expression
4.1.3.2.3.2 Validation
4.1.3.2.3.3 Interaction with rotor skewing
4.1.3.2.4 Eccentricity force lines
4.1.3.2.5 Winding force lines
4.1.3.2.5.1 Expression
4.1.3.2.5.2 Validation
4.1.3.3 PWM case
4.1.3.3.1 Pure PWM force lines
4.1.3.3.1.1 Expression
4.1.3.3.1.2 Validation
4.1.3.3.1.3 Influence of current magnitude
4.1.3.3.1.4 Influence of phase angle
4.1.3.3.2 Slotting PWM force lines
4.1.3.3.2.1 Expression
4.1.3.3.2.2 Validation
4.1.4 Expression of main magnetic lines magnitude
4.1.5 Conclusion
4.2 Low-noise design rules
4.2.1 Slot combination
4.2.1.1 Exhaustive search
4.2.1.1.1 Realisation of the slot combination database
4.2.1.1.2 Application
4.2.1.2 Special slot numbers
4.2.2 Slot openings width
4.2.2.1 Optimal choice
4.2.2.2 Application
4.2.3 PWM supply
4.2.3.1 Psychoacoustic factors
4.2.3.2 Choice of the switching frequency
4.2.3.2.1 Pure PWM noise
4.2.3.2.2 Slotting PWM noise
4.2.3.3 Current injection method
4.3 Noise minimisation
4.3.1 Optimisation problem
4.3.1.1 Objectives
4.3.1.2 Design variables
4.3.1.3 Constraints
4.3.2 Optimisation method
4.3.3 Application
4.3.3.1 Rotor optimisation
4.3.3.2 PWM strategy optimisation
4.3.3.3 Experimental validation
4.3.3.3.1 Traction characteristics
4.3.3.3.2 No-load sinusoidal tests
4.3.3.3.3 On-load sinusoidal tests
4.3.3.3.4 No-load PWM tests
5 Conclusion, future work and prospects 
5.1 Conclusion
5.2 Future work
5.3 Prospects

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