Magnetic and magnetostrictive forces The forces, which are the load for the elastic field, are separated into magnetic force also called reluctance forces in some works [4] and magnetost
Trang 1Magnetic and elastic fields
The A −φ formulation of the magnetic field in two dimensions and the displacement based
formulation of the elastic field are used
⎡
⎢
⎢
⎢
⎢
⎣
S( A k+1, U k+1)+
∂ S(A k+1, U k+1)
∂ A A k+1
[Dr]T [L Ds]T ∂ S(A k+1, U k+1)
−∂ ˜ F k+1
∂ ˜ F k+1
∂U
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
A n
k+1
u r n
k+1
i s n
k+1
U n
k+1
⎤
⎥
⎥
⎥
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
S( A
k ) A k + [Dr]Tur
k + [Ds]TKTis
k−
(S( A n k+1) A n k+1+ [Dr]Tu r n
k+1+ [Ds]TKTi s n k+1)
L DsA k − Hsisk − Cs(Vsk+1+ Vs
k)−
(L DsA n k+1− Hsi s n k+1− Cs(Vsk+1+ Vs
k))
DrA k − Crur
k − Grir
k−
(DrA n k+1− Cru r n k+1− Gri r n k+1)
˜
F n k+1− ˜KU n
k+1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(1)
The 2D finite element (FE) equations for the magnetic, and elastic field are coupled through the displacements and the forces These equations are solved together with the circuit equations of the windings of the machine as described in [9] The system of equations to
be solved at each iteration is written as (1), where A is the magnetic vector potential; C, D,
G, and H are the coupling matrices between the magnetic vector potential and the electric
parameters in the windings parts of the machine; L is a matrix for the connections of the stator windings; S and K are the magnetic and mechanical stiffness matrices; u and i are respectively voltages and currents and U is the nodal displacements vector.
The superscripts r and s refer respectively to rotor and stator The subscripts k and k+ 1 refer respectively to previous and present steps in the time stepping method The sign∼ over
the matrix K and the force F means that they are replaced by their dynamic counterparts.
A full description of the above matrices is given in [9,11]
Magnetic and magnetostrictive forces The forces, which are the load for the elastic field, are separated into magnetic force (also called reluctance forces in some works [4]) and magnetostriction forces
The magnetic forces are calculated, at any iteration, from the calculated magnetic vector potential, based on the local application of the virtual work principle:
FT= −∂W ∂U
Trang 2With the energy W per element Segiven by:
Se
B
0
we obtain the contribution of one element to the nodal magnetic forces calculated on the reference element ˆSeas:
FT= −
ˆ
Se
1
μe
∇ A · ∂U ∂ (∇ A)|J| +
B
0
H · dB ∂U ∂ (|J|)
where,|J| is the determinant of the Jacobian matrix for the transformation from the reference
element to the actual one These individual contributions from elements to nodal forces are added to each other to obtain the global nodal magnetic forces The nodal magnetostriction forces are calculated also at element level and assembled in the same manner as the magnetic forces
The calculation of the magnetostrictive forces is based on an original method called the
method of magnetostrictive stress This method is explained hereafter.
Let’s consider an element of iron in a magnetic field H Due to magnetostriction of iron,
this element will shrink or stretch depending on the sign of its magnetostriction This change
in dimensions is described by a magnetostrictive strain tensor{ε ms } Corresponding to this
strain, a magnetostrictive stress tensor{σ ms} can be calculated using Hook’s law The nodal magnetostrictive forces are the set of nodal forces due to this stress
The measurements presented in [10] give the component of magnetostrictive stressσ ms
in a direction parallel to that of the magnetic field The other component of
magnetostric-tive stress orthogonal to the direction of the magnetic field H can be calculated within two
assumptions First, there is no magnetostrictive shear stress in the frame defined by the
direc-tion parallel to H and the one orthogonal to it Second, there is no volume magnetostricdirec-tion;
which is a good assumption in the range of flux density occurring in electrical machines The latter assumption means that the magnetostriction strain in the direction orthogonal
to that of the magnetic field is opposite and has half the amplitude compared to the strain parallel to the direction of the magnetic field
Ifσ ms⊥ is the magnetostrictive stress in the direction orthogonal to the magnetic field,
using the first assumption, we can write:
⎡
⎣σ σ ms ms⊥ 0
⎤
⎦ = E
⎡
⎣ε ε ms ms⊥ 0
⎤
where E is the stress-strain matrix In the case of plane stress, use of the second assumption
leads to:
σ ms⊥ =2v − 1
The magnetostrictive nodal forces are calculated for each element as follows Letθ be the angle defined by the direction of the magnetic field and the x-axis The projections of
each edge of the element on the directions parallel and orthogonal to the magnetic field are respectively
Trang 3where s x and s yare respectively the projection of the considered edge of the element on the
x- and y-axis The forces, per unit length, parallel and orthogonal to the direction of the
magnetic field are respectively
and
These forces are distributed equally between the two nodes of the given edge The forces
in the original Cartesian coordinate system are obtained as the projection of F msand F ms⊥
on the axis of that system:
F msx = cos(θ)F ms− sin(θ)F ms⊥ (11) and
F msy = sin(θ)F ms+ cos(θ)F ms⊥ (12)
When the stress dependency of magnetostriction is taken into account, only the data of magnetostrictive stress are changed All the rest is the same
Vibrations The solution of the magnetoelastic FE analysis produces among others the nodal displace-ments as a function of time The displacedisplace-ments of a node on the outer surface of the stator core are transformed with Discrete Fourier Transform (DFT) and numerically differentiated
to obtain the frequency components of the velocity of vibrations of the node considered
In the following calculations, a total of 3,000 time steps are calculated with 300 time steps per period of the line voltage (20 ms) This leads to a sampling frequency of 15 kHz,
a frequency resolution of 5 Hz and a maximum frequency of 7.5 kHz The amplitudes of these vibrations are the quantities under consideration in the result section
Results
Validation
An induction machine-like test device has been built to verify the presented model for magnetostriction The test device is shown in Fig 1
The test device is constructed in a way that simulates the flux path in an induction machine, meanwhile it minimizes the reluctance forces The latter are due to the presence
of the air gap while the test device has no air gap The only magnetic forces in the test device are the magnetostrictive forces and the Lorentz forces The effect of the latter ones on the vibrations of the test device can be neglected due to the low currents in the windings and also due to the high relative mass-ratio between the iron core of the device and its windings Thus, the only cause of vibrations in the test device is the magnetostriction
Computations and measurements have been made for the test device The simulated flux lines in the cross section of the device are shown in Fig 2 The flux density in the back
Trang 4Figure 1 Picture of the test device The search coil for measurement of the back iron magnetic flux
density can be seen Rubber tubes separate both mechanically and electrically the windings from the iron core
iron core of the test device has also been measured by a search coil This flux density is compared to the simulated one in Fig 3 The vibrations of the outer surface of the test device have been measured with a laser vibrometer at different points The same values have been calculated The simulated and measured displacements at a point on the outer surface are shown in Fig 4
Application The method developed in this work is applied to a small size (37 kW) induction machine Dif-ferent computational approaches are used to establish both the effect of stress-independent and stress-dependent magnetostriction on the vibrations of such a machine The parameters
of the simulation machine are given in Table 1
Figure 2 Plot of the calculated flux lines in the test device.
Trang 5Table 1 Parameters of the induction machine
Rated power 37 kW Rated voltage 380 V
Number of phases 3 Number of poles 4 Stator outer diameter 310 mm Stator inner diameter 200 mm Stack length 289 mm Number of stator slots 48 Rotor outer diameter 198.4 mm Number of rotor slots 40
1.5
1
0.5
0
–0.5
–1
–1.5
1.5
1
0.5
0
–0.5
–1
–1.5
0.1 0.12 0.14 0.16 0.18 0.2
Figure 3 Measured and simulated flux density in the back iron core of the test device.
(a) Measured (b) Simulated
0.3 0.32 0.34 0.36 0.38 0.4
− 2
− 1.5
− 1
− 0.5
0
0.5
1
1.5
2
2.5x 10
Time (s)
0.1 0.12 0.14 0.16 0.18 0.2 3
3.5 4 4.5 5 5.5 6 6.5 7 7.5x 10
Time (s)
Figure 4 Measured and simulated displacements at the surface of the test device (in measurement
the DC-component is omitted) (a) Measured (b) Simulated
Trang 6− 0.2 − 0.1 0 0.1 0.2
0
0.1
0.2
X-coordinate (m)
0 0.1 0.2
X-coordinate (m)
original deformed
Figure 5 Calculated magnetostrictive force (normalized to 50,870 N/m) and deformation (magnified
20,000 times) of the stator core of the induction machine (a) Forces (b) Deformation
The calculated magnetostrictive forces acting on the stator core of the induction machine
at the last time step and the corresponding deformation are shown in Fig 5(a,b) respectively These are forces and deformations calculated with stress-independent magnetostriction The reluctance forces are not shown
Calculations with no magnetostriction and these with stress-dependent magnetostriction have also been undertaken The velocity of vibrations of a node on the outer surface of the stator core of the machine (point P in Fig 5(b)) calculated with different approaches are compared
Fig 6 shows the relative difference in the amplitudes of velocity between the cases
no magnetostriction and stress-independent magnetostriction Fig 7 shows the relative
0 1000 2000 3000 4000 5000 6000 7000 8000
− 10
− 8
− 6
− 4
− 2
0
2
Frequency (Hz)
Figure 6 Calculated relative difference in the amplitude of velocities of point P (Fig 5b) Differences
between the cases no magnetostriction and stress-independent magnetostriction
Trang 70 200 400 600 800 1000
− 70
− 60
− 50
− 40
− 30
− 20
− 10
0
10
Frequency (Hz)
Figure 7 Calculated relative difference in the amplitude of velocities of point P (Fig 5b) Differences
between the cases stress-independent and stress-dependent magnetostriction
difference in the amplitudes of velocity between the cases independent and stress-dependent magnetostriction The relative differences are calculated as (|v1| − |v2|)/|v1|
Analysis and discussion
Validation The measured displacements from the test device are slightly higher than the simulated ones This difference can be seen also in the measured and simulated flux densities They are due mainly to differences in the magnetic properties of the materials used in simulations Indeed, the manufacturing process of the test device slightly deteriorated the magnetic properties of the iron sheets However, the correspondences between measured quantities and these simulated with the presented model for magnetostriction are rather good from both the amplitudes and wave forms points of view In the future, better magnetic properties
of the manufactured machine can be introduced into the simulation software for better results
Simulations The vibrations of the induction machine are affect by the magnetostriction The amplitudes of most of the frequency components are increased The most increased frequency components are these at 490 Hz (840%), 40 Hz (400%), 60 Hz (270%), and 50 Hz (320%) Some other frequency components are damped due to the magnetostriction Among these the ones at 1,190 and 1,470 Hz damped respectively 80% and 70% The 100 Hz component is damped only by 7%
Trang 8The stress dependency of magnetostriction adds to the effect of stress-independent mag-netostriction so that the amplitudes of almost all the frequencies are increased The increase reaches some 7,000% for the frequency component at 1,595 Hz e.g However, the accuracy
of the simulations with stress-dependent magnetostriction cannot be established due to the effect of magnetostrictive stress Indeed, in the FE iteration process, the stress from mag-netostrictive forces cannot be separated from the stress due to other forces (reluctance and Lorentz forces) Thus the stress state of the material is not accurately estimated, leading to inaccuracies in calculation of the stress-dependent magnetostriction Although, we can say that both stress-independent and stress-dependent affect the vibrations of rotating electrical machines
Conclusions
A model for the magnetoelastic coupling is presented and used in the simulations of an induction machine The goal of these simulations is to establish the effect of the magne-tostriction on the vibrations of rotating electrical machines For this purpose an original method for the calculation of magnetostrictive forces is presented
It is shown that the magnetostriction affects the vibrations of rotating electrical machines
by increasing or decreasing the amplitudes of velocities measured at the outer surface of the stator core of the machine These velocity are the ones responsible for acoustic noise Furthermore, The stress dependency of the magnetostriction adds to the increase of the above amplitudes
The modeling of vibrations and noise of electrical machines should take into account the effect of magnetostriction and its stress dependency
References
[1] P Witczak, Calculation of force densities distribution in electrical machinery by means of magnetic stress tensor, Arch Electr Eng., Vol XLV, No 1, pp 67–81, 1996
[2] L L˚aftman, “The Contribution to Noise from Magnetostriction and PWM Inverter in an Induction Machine”, Doctoral thesis, IEA Lund Institute of Technology, Sweden, 94 p, 1995 [3] K Delaere, “Computational and Experimental Analysis of Electrical Machine Vibrations Caused By Magnetic Forces and Magnetostriction”, Doctoral thesis, Katholieke Universiteit Leuven, Belgium, 224 p, 2002
[4] Z Ren, B Ionescu, M Besbes, A Razek, Calculation of mechanical deformation of mag-netic material in electromagmag-netic devices, IEEE Trans Magn Vol 31, No 3, pp 1873–1876, 1995
[5] O Mohammed, T Calvert, R McConnell, “A Model for Magnetostriction in Coupled Nonlinear Finite Element Magneto-elastic Problems in Electrical Machines”, International Conference
on Electric Machines and Drives IEMD ’99, Seattle, Washington, USA, pp 728–735, May 1999
[6] F Ishibashi, S Noda, M Mochizuki, “Numerical simulation of electromagnetic vibration of small induction motor”, IEE Proc Electr Power Appl., Vol 145, No 6, pp 528–534, November 1998
[7] G.H Jang, D.K Lieu, “The effect of magnetic geometry on electric motor vibration”, IEEE Trans Magn., Vol 27, No 6, pp 5202–5204, November 1991
Trang 9[8] C.G Neves, R Carlson, N Sadowski, J.P.A Bastos, N.S Soeiro, “Forced Vibrations Calcu-lation in Switched Reluctance Motor Taking into Account Viscous Damping”, International Conference on Electric Machines and Drives IEMD’99, May 1999
[9] A Belahcen, “Magnetoelastic Coupling in Rotating Electrical Machines”, IEEE Tran Mag., Vol 41, No 5, pp 1624–1627, May 2005
[10] A Belahcen, M El Amri, “Measurement of Stress-Dependent Magnetisation and Magne-tostriction of Electrical Steel Sheets”, Internation Conference on Electrical Machines, Cracow, Poland, CD-ROM Paper No 258, September 5–8, 2004
[11] A Arkkio, “Analysis of Induction Motors Based on the Numerical Solution of the Mag-netic Field and Circuit Equations”, Doctoral thesis, Acta Polytechnica Scandinavica, Electrical Engineering Series No 59, 97 p Available at http://lib.hut.fi/Diss/198X/isbn951226076X/
Trang 10II-7 COMPARISON OF STATOR- AND ROTOR-FORCE EXCITATION FOR THE ACOUSTIC SIMULATION OF AN
INDUCTION MACHINE WITH
SQUIRREL-CAGE ROTOR
C Schlensok and G Henneberger
Institute of Electrical Machines (IEM), RWTH Aachen University, Schinkelstraße 4,
D-52056 Aachen, Germany
christoph.schlensok@iem.rwth-aachen.de, henneberger@iem.rwth-aachen.de
Abstract In this paper the structure- and air-borne noise of an induction machine with squirrel-cage
rotor are estimated For these, different types of surface-force excitations and rotational directions are regarded for the first time The comparison of the different excitations shows, that it is necessary to take the rotor excitation into account, and that the direction of the rotation has a significant effect on the noise generation
Introduction
The drivers of passenger cars nowadays make great demands on the acoustics of the technical equipment such as the electrical power steering Therefore, it is of high interest to estimate the audible noise radiation of these components The induction machine with squirrel-cage rotor used as power-steering drive is computed in three steps: coupled to the casing caps by the bearings For this, the rotor excitation has to be taken into account as well for comparison reasons
1 electromagnetic simulation,
2 structural-dynamic computation, and
3 acoustic estimation
The theory is briefly described in [1] and therefore not repeated In the case of an induction machine with skewed squirrel-cage rotor the location of the maximum force excitation of the stator teeth depends on the rotational direction So far, only stator-teeth excitation has been regarded in literature [2–4] Further on the impact of the force exciting the rotor is taken into account Therefore, four different cases of electromagnetic surface-force excitation are compared and discussed in this paper as listed in Table 1
Since the rotor of the induction machine is skewed (skewing angle = 10◦) the stator
teeth are excited very asymmetrically The location of the maximal tooth excitation de-pends on the direction of rotation In case of right-hand rotation the highest excitation
2006 Springer.