3-D finite element analysis of interbar current of skewed squirrel-cage induction motor taking into account of contact resistance

Abstract

In this paper, we analyze the interbar current taking into account of the contact resistance between the secondary conductor and the steel sheets of squirrel-cage induction motors using the 3-D parallel finite element method. As a result, the effects of the contact resistance on the interbar current are clarified.

Keywords

3-D parallel finite element method squirrel-cage induction motor interbar current contact resistance skew

1. Introduction

The interbar current of the squirrel-cage induction motor (IM) flows in the steel sheets when the secondary conductor is not insulated from the steel sheets. The interbar current is influenced by the contact resistance between the secondary conductor and the steel sheets. However, it has not been considered about the contact resistance [1]. In this paper, a squirrel-cage IM taking into account of the interbar current is analyzed using the 3-D parallel finite element method [2, 3]. As a result, the effects of the contact resistance on the interbar current are clarified.

2. Analysis method

2.1 Magnetic field analysis

The fundamental equation of the magnetic field can be written using the magnetic vector potential A and the electric scalar potential $\phi$ as follows [4]:

$\displaystyle\text{rot}(\nu\text{ rot }{\bm{A}})={\bm{J}}_{0}+{\bm{J}}_{e}$ (1) $\displaystyle{\bm{J}}_{e}=-\sigma\left({\frac{\partial{\bm{A}}}{\partial t}+% \text{grad}\phi}\right)$ (2) $\displaystyle\text{div}{\bm{J}}_{e}=0$ (3)

where $\nu$ is the reluctivity, J ${}_{0}$ is the exciting current density, J ${}_{e}$ is the eddy current density, and $\sigma$ is the conductivity.

2.2 Loss calculation

In order to calculate the interbar current loss, we have assumed that the eddy current flows in the several steel sheets. In those steel sheets, the interbar current loss $W$ is given as follows [4]:

$\displaystyle W=\frac{1}{T}\int_{0}^{T}{\left\{{\int_{V_{e}}{\frac{(J_{e})^{2}% }{\sigma_{//}}dv}}\right\}dt}$ (4)

where $T$ is the period of the current waveform, $V_{e}$ is the region of the steel sheet with the interbar current, J ${}_{e}$ is the eddy current density, and $\sigma_{//}$ is the conductivity of the steel sheet, which the component perpendicular to the steel sheet is set to 0.

The hysteresis loss and the eddy current loss occurred in the steel sheet are approximately estimated from the calculated flux density B [5].

Figure 1.

Analyzed model.

3. Analyzed model and conditions

Figure 1 shows the analyzed model of a squirrel-cage IM. There are the contact parts between the secondary conductor and the steel sheets. The conductivity of the contact part is calculated from the contact resistance. When the contact resistance is zero, the conductivity of the contact parts is given as the conductivity of the steel sheets. Based on the periodicity of the model, the analysis region in the circumferential direction is 1/3 in no-skewed IM and skewed IM. Because of the symmetry of the analyzed model, the analyzed region is 1/2 of the whole in the axial direction in the no-skewed IM. On the other hand, that is 1/1 in the skewed IM.

Figure 2 shows the 3-D finite element mesh. If all laminated steel sheets are divided by the mesh with the actual thickness, the huge calculation time is required. Therefore, only 23 steel sheets are divided by the mesh with the actual thickness and some steel sheets are assumed to the steel lump between them. The steel sheets A, B shown in Fig. 2b are the steel sheet that the analyzed interbar currents show in Fig. 3.

Table 1 shows the analysis conditions. The three-phase sine wave AC of 640AT is excited to the coils.

Table 1
Analysis conditions

Skew type		No-skewed IM
Exciting current (AT)		640
Conductivity (S/m)	Secondary conductor	3.14 $\times$ 10 ${}^{7}$
	Steel sheet	6.67 $\times$ 10 ${}^{6}$
Contact resistance		0, $R$ , 10 $R$ , 100 $R$ , 1000 $R$ , $\infty$ ( $=$ insulated)

Table 2

Average torque and torque ripple

normalized by $T_{\textit{ave}}$
Contact resistance	0	$R$	10 $R$	100 $R$	1000 $R$	$\infty$
(a) no-skewed IM
Average torque	1.000	1.002	1.005	1.007	1.008	1.008
Torque ripple	0.163	0.164	0.164	0.164	0.165	0.165
normalized by $T_{\textit{ave}}$
(b) skewed IM
Average torque	0.905	0.900	0.891	0.895	0.900	0.901
Torque ripple	0.023	0.023	0.023	0.022	0.021	0.023

Figure 2.

3-D finite element mesh. (no-skewed IM).

Figure 3.

Distributions of interbar current density vectors.

Figure 4.

Distributions of interbar current loss.

Figure 5.

Electrical loss characteristics.

Figure 6.

Interbar current loss characteristics.

4. Results calculated and discussion

Table 2 shows the average torque and the torque ripple characteristics. Those are normalized by the average torque ( $T_{\textit{ave}}$ ) when the contact resistance is zero of the no-skewed IM. In the no-skewed IM, as the contact resistance increases, the average torque slightly increases. In the skewed IM, the average torque becomes the minimum when contact resistance is 10 $R$ . When the contact resistance is higher than 100 $R$ , the average torque decreases as the contact resistance increases. It is found that contact resistance has little effect on torque ripple regardless of the skew.

Figure 3 shows the distributions of the interbar current density vectors. Figure 3a shows the steel sheet A in the no-skewed IM because a lot of the interbar current flows on the steel sheet at the end of the rotor. Figure 3b shows the steel sheet B in the skewed IM because a lot of the interbar current flows on the steel sheet at the center of the rotor. We can see that the interbar current decreases owing to the increase of the contact resistance. In the skewed IM, there is the eddy current along the contact parts in the steel sheet when the contact resistance is large. It is because skew generated magnetic flux penetrating the steel sheet vertically.

Figure 4 shows the distributions of interbar current loss. Those are normalized by the summation of the interbar current loss ( $\Sigma\textit{loss}$ ) when the contact resistance is zero of the no-skewed IM. The interbar current loss of each steel plate includes the in-plane eddy current loss of the steel plate. In the no-skewed IM, the interbar current loss of each steel sheet decreases as the contact resistance increases. When the contact resistance is $R$ and 10 $R$ in the skewed IM, the interbar current loss is a larger than when the contact resistance is zero.

Figure 5 shows the calculated electrical loss characteristics. Those are normalized by the total of the electrical loss (Total Loss) when the contact resistance is zero of the no-skewed IM. It is found that the contact resistance has little effect on the loss except for interbar current loss regardless of the skew. In the no-skewed IM, the total of electrical loss decreases as the contact resistance increases. In the skewed IM the total of the electrical loss becomes the maximum at 10 $R$ . When the contact resistance is higher than 100 $R$ , the total of the electrical loss decrease as the contact resistance increases. The skewed IM has larger interbar current loss than the no-skewed IM. Therefore, the decrease in interbar current loss is large in the case where the contact resistance is large and in the case of insulation.

Figure 6 shows the calculated the interbar current loss characteristics. Those are normalized by the total of the electrical loss (Total Loss) when the contact resistance is zero of the no-skewed IM. In the no-skewed IM, total of the interbar current loss decreases as the contact resistance increases. In the skewed IM, The total of the interbar current loss reaches the maximum in 10 $R$ . When the contact resistance is higher than 100 $R$ , total of the interbar current loss decreases as the contact resistance increases. The total of the interbar current loss must consider two factors: a decrease in the interbar current loss of the steel sheets and an increase in the contact parts. We consider that the interbar current loss is the loss due to the in-plane eddy current when the contact part is insulated. There is almost no in-plane eddy current loss in the no-skewed IM. In the skewed IM, the in-plane eddy current loss is approximately 11% compared to the total interbar current loss of the model with the contact resistance value of zero when the contact part is insulated. The in-plane eddy current loss of the skewed IM cannot be ignored.

Table 3 shows the discretization data and elapsed time.

Table 3
Discretization data and elapsed time

${}^{*2}$ Intel Xeon (3.50 GHz) $\times$ 16
Computer used: ${}^{*1}$ Intel Xeon (3.40 GHz) $\times$ 16
Contact resistance	0	$R$	10 $R$	100 $R$	1000 $R$	$\infty$
(a) no-skewed IM
Number of elements	3,125,732
Number of nodes	0,538,428
Number of edges	3,705,519
Total number of time steps	1,067
Elapsed time (hours)	169.0 ${}^{*1}$	134.8 ${}^{*1}$	168.0 ${}^{*1}$	199.0 ${}^{*2}$	154.3 ${}^{*1}$	202.2 ${}^{*1}$
Computer used: ${}^{*1}$ Intel Xeon (3.40 GHz) $\times$ 16
${}^{*2}$ Intel Xeon (3.50 GHz) $\times$ 16
(b) skewed IM
Number of elements	6,344,388
Number of nodes	1,076,856
Number of edges	7,459,819
Total number of time steps	1,600	1,067	1,600
Elapsed time (hours)	555.3 ${}^{*1}$	344.0 ${}^{*1}$	606.3 ${}^{*1}$	740.0 ${}^{*2}$	1,011.3 ${}^{*1}$	1,356.4 ${}^{*2}$

5. Conclusion

In this paper, a squirrel-cage IM taking into account of the interbar current is analyzed using the 3-D parallel finite element method. As a result, the effects of the contact resistance on the interbar current are clarified. As a result, the following knowledge was obtained.

In the skewed IM, the in-plate eddy current along the contact parts in the steel sheet is generated. The in-plane eddy current loss of the skewed IM cannot be ignored.

As the contact resistance increases, the interbar current loss of the steel sheet decreases but the interbar current loss at the contact part increases and then decreases. The total of the interbar current loss must consider two factors: a decrease in the interbar current loss of the steel sheets and an increase in the contact parts.

Skewed IM has larger interbar current loss than no-skewed IM. Therefore, the decrease in interbar current loss is large in the case where the contact resistance is large and in the case of insulation.

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