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Rotational hysteresis and eddy current losses in electrical motor stators under non-conventional supply
Journal of Magnetism and Magnetic Materials 254–255 (2003) 241–243
Rotational hysteresis and eddy current losses in electrical motor stators under non-conventional supply Oriano Bottauscioa,*, Aldo Canovab, Mario Chiampib, Maurizio Repettob a
Istituto Elettrotecnico Nazionale Galileo Ferraris, Corso M.d’Azeglio, 42 10125 Torino, Italy b Dipartimento di Ingegneria Elettrica Industriale, Politecnico di Torino, Italy
Abstract The magnetic analysis of stators of electrical motors is performed through an innovative 2D finite element formulation that takes into account the effects of eddy currents within the laminations by means of a generalized constitutive relationship also including vector hysteresis. This approach is applied to a deep estimation of magnetic flux distribution and magnetic losses in stator of induction motors supplied by high-frequency sinusoidal or six-step voltage sources. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Magnetic loss; Induction motor; Numerical simulation
The analysis of the stator of three phase electrical motors at power frequency (50/60 Hz) shows that the role of eddy currents induced in the lamination is not relevant neither in the definition of the flux patterns inside the magnetic core nor in the increase of magnetic losses. Thus the use of highly sophisticated models of the diffusion phenomena is not always justified. On the contrary, the question of using accurate magnetic material models, which include skin effect, still remains an open problem when analysing high-speed electrical motors or motors driven by electronic static converters, which generate significant higher-order harmonics. The field analysis under these working conditions is further complicated by the presence of rotational fluxes produced by multi-phase supply. The aim of this paper is to clarify the role of skin effect and vector hysteresis on the phenomena occurring in electrical motor stators under non-conventional supply. The analysis employs an innovative finite element model including vector hysteresis and skin effects. The rotational hysteresis is described through the Preisach approach [1], while the skin effect in the laminated core is accounted for by the procedure *Corresponding author. Tel.: +39-011-3919-776; fax: +39011-6509-471. E-mail address: [email protected] (O. Bottauscio).
proposed in Ref. [2]. This approach is based on the idea of performing a 2D analysis of the magnetic structure without eddy currents in the lamination depth and of including their effects by means of a constitutive model. Thus the 2D field problem, expressed in terms of magnetic vector potential, leads to the well-known standard periodic formulation, which is developed in the harmonic domain and linearized by fixed point technique. The constitutive relationship is obtained, for each finite element, by the solution of two 1D fluxdriven electromagnetic field problems in the lamination depth, expressed in terms of magnetic vector potential a: n1D
ðk1Þ qS y;m q2 aðkÞ % % x;m ; ¼ iomsaðkÞ x;m 2 @z qz %
n1D
q2 aðkÞ qS ðk1Þ x;m % y;m : ¼ iomsaðkÞ y;m þ % 2 qz qz %
ð1Þ
In Eq. (1), n1D and S are, respectively, the FP linear term and residual, s is the electrical conductivity of the lamination, o is the fundamental angular frequency and i is the imaginary unit. Non-homogenous Dirichlet boundary conditions impose the flux UðtÞ through the sheet, obtained by the 2D field problem. The iterative solution of problem (1) provides the time waveform of the magnetic field on the lamination surface hS ðtÞ;
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 7 7 8 - 3
O. Bottauscio et al. / Journal of Magnetism and Magnetic Materials 254–255 (2003) 241–243
Fig. 1. Domain under study and subdivision of the core into three regions.
Hysteresis and skin effect Hysteresis First Magn.curve 0.50
0.25
By (T)
enforcing the flux UðtÞ; which is used to update the FP residual for the iterative 2D problem. The model has been applied to the analysis of the stator of a four poles 7.5 kW induction motor, under sinusoidal (400 and 1000 Hz) and 50 Hz six-step supply. The domain under study is limited to a phase (three slots) imposing periodicity conditions on the lateral boundaries (Fig. 1). The stator has been subdivided into three different regions in order to evaluate the distribution of magnetic losses inside the core. The analysis has been developed considering three models of the magnetic material behaviour: (i) first magnetization curve without skin effect, (ii) hysteresis without skin effect and (iii) hysteresis with skin effect. The comparison between these approaches puts in evidence that the flux distribution inside the stator core is not sensibly affected by the use of refined material models. As an example, the loci of the magnetic flux density over a slot under 1000 Hz sinusoidal supply (Fig. 2a) show little discrepancies, while the waveforms of magnetic flux density in a tooth under six-step supply condition (Fig. 2b) are indistinguishable. As a consequence, also the loss estimations are found to be practically independent of the magnetic material model. Thus, the loss evaluation has been performed employing the most accurate model (iii), which makes available the hysteresis cycles and the eddy current distribution inside the sheets. The loss evaluation is performed following three approaches of increasing accuracy based on: (a) loss characteristic at industrial frequency provided by the material builder, corrected using empirical formula or table given by previous experience [3], (b) loss separation into three fundamental components on the basis of the Fiorillo–Novikov theory [4], (c) direct computation from the field distribution for classical and static losses and use of Fiorillo–Novikov theory for excess losses. Fig. 3 presents the computed losses for sinusoidal voltage supply at 400 and 1000 Hz. The amplitude of the
0.00
-0.25
-0.50 -0.4
-0.2
(a)
0.0
0.2
0.4
15
20
Bx (T)
2.0 Hysteresis and skin effect Hysteresis First Magn.curve
1.5 Magnetic flux density (T)
242
1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 0
(b)
5
10 Time (ms)
Fig. 2. Computed results under different supply conditions: (a) sinusoidal voltage at 1000 Hz; and (b) six-step voltage.
supply voltage has been kept constant so that the magnetic fluxes inside the core reduce, as usual in highspeed induction motors; this justifies the same magnitude of losses at 400 and 1000 Hz. Method (a) provides results which are not reliable, due to the uncertainty of the correction factor which does not take into account the different behaviour of loss components. The comparison between method (b) and (c) shows that a complex and heavy model is justified only for frequency higher that 1000 Hz, while at 400 Hz, the two methods only give a different separation between classical and static losses. Moreover, the use of different models for loss estimation does not modify the spatial distribution of specific losses in the three considered stator regions; in any case the teeth region is the most stressed one. These results are also confirmed under distorted supply conditions; as an example, Fig. 4 shows the loss estimation under 50 Hz six-step voltage supply. The
O. Bottauscio et al. / Journal of Magnetism and Magnetic Materials 254–255 (2003) 241–243
Method (a) Method (b) Method (c)
2000
Sinusoidal (50 Hz): 90 Six-Step (50 Hz): 80
Method (b) Method (b) Method (c)
70
1500
60 Losses (W)
Losses (W)
243
1000
50 40 30
500
20 10 0 Clas. Stat. Exc. Tot. Clas. Stat. Exc. Tot.
400 Hz
1000 Hz
Fig. 3. Loss components (classical, static and excess) under sinusoidal supply at two different frequencies.
increase of losses in the stator core with respect to the sinusoidal supply is not appreciable; this result entitles us to believe that higher losses, which are usually experienced, are probably due to an increase of losses in the cage of the rotor [5].
References [1] C. Ragusa, M. Repetto, Physica B 275 (1–3) (2000) 92.
0 Classical
Static
Excess
Total
Fig. 4. Loss components (classical, static and excess) under 50 Hz six-step and sinusoidal supplies.
[2] O. Bottauscio, M. Chiampi, J. Appl. Phys. 89 (2001) 6728. [3] A. Boglietti, M. Lazzari, M. Pastorelli, A simplified method for the iron losses prediction in soft magnetic materials with arbitrary voltage supply, IEEE IAS Annual Meeting 2000, 8–12 October, Rome, Italy. [4] F. Fiorillo, A. Novikov, IEEE Trans. Magn. 26 (1990) 2904. [5] A. Boglietti, O. Bottauscio, M. Chiampi, M. Lazzari, Modelling high frequency phenomena in the rotor of induction motors under no-load test conditions, Presented at SMM15, Bilbao, Spain.
Rotational hysteresis and eddy current losses in electrical motor stators under non-conventional supply Oriano Bottauscioa,*, Aldo Canovab, Mario Chiampib, Maurizio Repettob a
Istituto Elettrotecnico Nazionale Galileo Ferraris, Corso M.d’Azeglio, 42 10125 Torino, Italy b Dipartimento di Ingegneria Elettrica Industriale, Politecnico di Torino, Italy
Abstract The magnetic analysis of stators of electrical motors is performed through an innovative 2D finite element formulation that takes into account the effects of eddy currents within the laminations by means of a generalized constitutive relationship also including vector hysteresis. This approach is applied to a deep estimation of magnetic flux distribution and magnetic losses in stator of induction motors supplied by high-frequency sinusoidal or six-step voltage sources. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Magnetic loss; Induction motor; Numerical simulation
The analysis of the stator of three phase electrical motors at power frequency (50/60 Hz) shows that the role of eddy currents induced in the lamination is not relevant neither in the definition of the flux patterns inside the magnetic core nor in the increase of magnetic losses. Thus the use of highly sophisticated models of the diffusion phenomena is not always justified. On the contrary, the question of using accurate magnetic material models, which include skin effect, still remains an open problem when analysing high-speed electrical motors or motors driven by electronic static converters, which generate significant higher-order harmonics. The field analysis under these working conditions is further complicated by the presence of rotational fluxes produced by multi-phase supply. The aim of this paper is to clarify the role of skin effect and vector hysteresis on the phenomena occurring in electrical motor stators under non-conventional supply. The analysis employs an innovative finite element model including vector hysteresis and skin effects. The rotational hysteresis is described through the Preisach approach [1], while the skin effect in the laminated core is accounted for by the procedure *Corresponding author. Tel.: +39-011-3919-776; fax: +39011-6509-471. E-mail address: [email protected] (O. Bottauscio).
proposed in Ref. [2]. This approach is based on the idea of performing a 2D analysis of the magnetic structure without eddy currents in the lamination depth and of including their effects by means of a constitutive model. Thus the 2D field problem, expressed in terms of magnetic vector potential, leads to the well-known standard periodic formulation, which is developed in the harmonic domain and linearized by fixed point technique. The constitutive relationship is obtained, for each finite element, by the solution of two 1D fluxdriven electromagnetic field problems in the lamination depth, expressed in terms of magnetic vector potential a: n1D
ðk1Þ qS y;m q2 aðkÞ % % x;m ; ¼ iomsaðkÞ x;m 2 @z qz %
n1D
q2 aðkÞ qS ðk1Þ x;m % y;m : ¼ iomsaðkÞ y;m þ % 2 qz qz %
ð1Þ
In Eq. (1), n1D and S are, respectively, the FP linear term and residual, s is the electrical conductivity of the lamination, o is the fundamental angular frequency and i is the imaginary unit. Non-homogenous Dirichlet boundary conditions impose the flux UðtÞ through the sheet, obtained by the 2D field problem. The iterative solution of problem (1) provides the time waveform of the magnetic field on the lamination surface hS ðtÞ;
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 7 7 8 - 3
O. Bottauscio et al. / Journal of Magnetism and Magnetic Materials 254–255 (2003) 241–243
Fig. 1. Domain under study and subdivision of the core into three regions.
Hysteresis and skin effect Hysteresis First Magn.curve 0.50
0.25
By (T)
enforcing the flux UðtÞ; which is used to update the FP residual for the iterative 2D problem. The model has been applied to the analysis of the stator of a four poles 7.5 kW induction motor, under sinusoidal (400 and 1000 Hz) and 50 Hz six-step supply. The domain under study is limited to a phase (three slots) imposing periodicity conditions on the lateral boundaries (Fig. 1). The stator has been subdivided into three different regions in order to evaluate the distribution of magnetic losses inside the core. The analysis has been developed considering three models of the magnetic material behaviour: (i) first magnetization curve without skin effect, (ii) hysteresis without skin effect and (iii) hysteresis with skin effect. The comparison between these approaches puts in evidence that the flux distribution inside the stator core is not sensibly affected by the use of refined material models. As an example, the loci of the magnetic flux density over a slot under 1000 Hz sinusoidal supply (Fig. 2a) show little discrepancies, while the waveforms of magnetic flux density in a tooth under six-step supply condition (Fig. 2b) are indistinguishable. As a consequence, also the loss estimations are found to be practically independent of the magnetic material model. Thus, the loss evaluation has been performed employing the most accurate model (iii), which makes available the hysteresis cycles and the eddy current distribution inside the sheets. The loss evaluation is performed following three approaches of increasing accuracy based on: (a) loss characteristic at industrial frequency provided by the material builder, corrected using empirical formula or table given by previous experience [3], (b) loss separation into three fundamental components on the basis of the Fiorillo–Novikov theory [4], (c) direct computation from the field distribution for classical and static losses and use of Fiorillo–Novikov theory for excess losses. Fig. 3 presents the computed losses for sinusoidal voltage supply at 400 and 1000 Hz. The amplitude of the
0.00
-0.25
-0.50 -0.4
-0.2
(a)
0.0
0.2
0.4
15
20
Bx (T)
2.0 Hysteresis and skin effect Hysteresis First Magn.curve
1.5 Magnetic flux density (T)
242
1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 0
(b)
5
10 Time (ms)
Fig. 2. Computed results under different supply conditions: (a) sinusoidal voltage at 1000 Hz; and (b) six-step voltage.
supply voltage has been kept constant so that the magnetic fluxes inside the core reduce, as usual in highspeed induction motors; this justifies the same magnitude of losses at 400 and 1000 Hz. Method (a) provides results which are not reliable, due to the uncertainty of the correction factor which does not take into account the different behaviour of loss components. The comparison between method (b) and (c) shows that a complex and heavy model is justified only for frequency higher that 1000 Hz, while at 400 Hz, the two methods only give a different separation between classical and static losses. Moreover, the use of different models for loss estimation does not modify the spatial distribution of specific losses in the three considered stator regions; in any case the teeth region is the most stressed one. These results are also confirmed under distorted supply conditions; as an example, Fig. 4 shows the loss estimation under 50 Hz six-step voltage supply. The
O. Bottauscio et al. / Journal of Magnetism and Magnetic Materials 254–255 (2003) 241–243
Method (a) Method (b) Method (c)
2000
Sinusoidal (50 Hz): 90 Six-Step (50 Hz): 80
Method (b) Method (b) Method (c)
70
1500
60 Losses (W)
Losses (W)
243
1000
50 40 30
500
20 10 0 Clas. Stat. Exc. Tot. Clas. Stat. Exc. Tot.
400 Hz
1000 Hz
Fig. 3. Loss components (classical, static and excess) under sinusoidal supply at two different frequencies.
increase of losses in the stator core with respect to the sinusoidal supply is not appreciable; this result entitles us to believe that higher losses, which are usually experienced, are probably due to an increase of losses in the cage of the rotor [5].
References [1] C. Ragusa, M. Repetto, Physica B 275 (1–3) (2000) 92.
0 Classical
Static
Excess
Total
Fig. 4. Loss components (classical, static and excess) under 50 Hz six-step and sinusoidal supplies.
[2] O. Bottauscio, M. Chiampi, J. Appl. Phys. 89 (2001) 6728. [3] A. Boglietti, M. Lazzari, M. Pastorelli, A simplified method for the iron losses prediction in soft magnetic materials with arbitrary voltage supply, IEEE IAS Annual Meeting 2000, 8–12 October, Rome, Italy. [4] F. Fiorillo, A. Novikov, IEEE Trans. Magn. 26 (1990) 2904. [5] A. Boglietti, O. Bottauscio, M. Chiampi, M. Lazzari, Modelling high frequency phenomena in the rotor of induction motors under no-load test conditions, Presented at SMM15, Bilbao, Spain.