1 Introduction
The calculation of losses in electrical machines has attracted a great deal of attention in recent years since reducing losses may facilitate energy saving and environmental pollution reduction. However, an accurate calculation of rotor eddy current losses is still a challenge especially for special induction motors [1].
Several papers have been dealing with the calculation of rotor eddy current losses of rotating machines. It is confirmed that the rotor surface losses (eddy current losses on the rotor surface) are caused by the slotting effect and the harmonic currents [2,3]. The rotor eddy current losses in permanent magnet motors were widely studied and they can be obtained by analytical method, finite element method or experiment method [4,5]. Y.Amara et al. presented an analytical method for calculating the eddy current losses in the moving armature of a tubular permanent magnet machine [6]. Dahaman et al. developed an analytical model for predicting the eddy current losseses in the rotor magnets of permanent magnet brushless machines which have a fractional number of slots per pole [7]. It can account for space-harmonic magnetomotive forces (MMF) resulting from the winding distribution and time-harmonic MMF due to non-sinusoidal phase currents. Jiabin Wang et al. proposed an analytical formula for predicting eddy current losses in each segment of permanent magnet brushless machines and concluded that forward and backward rotating space harmonics of different orders may result in the same frequency seen by the rotor [8]. However, the presented analytical techniques are not suitable for predicting the eddy-current losses in individual segments when more than two segments per pole are employed. In the case of permanent magnet synchronous motors, two-dimensional analytical method was first proposed by Frederic Dubas and the effects of the rotor speed and the initial rotor mechanical angular position were also investigated [9]. The analytical method can give a good prediction for evaluation rotor eddy current losses, but the accuracy is limited since many assumptions are made. Finite element methods including post1-D, 2-D and 3-D were widely employed in many papers for calculating the rotor eddy current losses [10,11,12,13,14]. Apart from calculating the rotor eddy current losses, many papers focused on reduction of rotor eddy current losseses in permanent magnet motors [15,16,17,18]. Shuangxia Niu et al. proposed a method called axial segmentation of the permanent magnets to cut off the eddy-current axial paths [15]. In the paper, the steady-state and dynamic characteristics of the permanent magnet motors were performed by a nodal method based on network-field coupled multislice time stepping finite element method. Jianxin Shen et al. presented grooving retaining sleeve method for reduction of rotor eddy current losses in high speed permanent magnet machines and concluded that the circumferential grooving is the most effective one to reduce the rotor eddy current losses [16]. In addition, the optimized design of rotor shape is an effective method for reducing rotor surface loss in the case of interior permanent magnet synchronous motor [17,18].
Although many papers investigated the rotor eddy current losses of the permanent magnet motors, few concentrated on the induction machines. High-frequency loss calculation in smooth rotor induction motor was presented in Ref.[19], and the approach was useful for the prediction of high-frequency losses at the design stage. Omar Laldin et al. employed the circuit models for predicting the core losses in a caged induction machine but the mechanism of rotor eddy current losses was not presented [20]. In Ref.[21] 3-D finite element analysis of additional eddy current losses in induction motors was performed but the aim was to extend the loss computation to the end-ring of the squirrel cage induction machine. In Ref.[22] stray load losses in cores and secondary conductors of induction motor without skew was evaluated by 2-D finite element method. However, the causes of the stray load losses were not investigated and the reduction was not made. Paul Handgruber et al. studied the effects of skewing on the loss behavior of a slip-ring induction machine [23]. However, the paper concentrated on the stator sheet model and gave no presentations about rotor eddy current losses.
It is confirmed that the air gap permeance harmonics produced by stator slotting can cause rotor surface losses. However, the permeance harmonics are rather difficult to calculate because of saturation and non-linear of the material. Because of this problem, Steve et al. proposed an analytical method for determining the slot-harmonic fields in the air gap which are produced by the currents flowing in closed rotor slots [24]. However, the stator slotting effects were not considered. As mentioned before, no one puts forward a model of electrical submersible motor with different stator slot types for the calculation of rotor surface losses at no-load condition.
In this paper, the Carter Factor of the submersible motor with different stator slot types is calculated. Since the stator tooth harmonic fields caused by stator slotting are the primary cause of the rotor surface loss at no-load condition, the harmonic air-gap permeance distributions produced by stator slotting were analyzed and the tooth harmonic fields caused by stator slotting were presented by analytical expressions. Then the rotor surface losses were calculated and the penetration depth was considered. Moreover, the flux density of the tooth harmonic caused by the stator slotting was separated from the total tooth harmonic. Finally, the rotor surface losses of the submersible with open stator slots and closed stator slots were analyzed and an improved stator slot was proposed for reduction of rotor surface losses in submersible motor.
2 Analytical Model of Closed-Slot Machine
Fig.1
3 Rotor Surface Losses in Closed-Slot Submersible Motor
3.1 Carter Factor
The Carter Factor for stator slotting can be expressed as
where ts is the stator slot pitch; b0 is the stator slot opening.
When both the stator and the rotor surface are split with slots, we first calculate Kcs by assuming the rotor surface to be smooth. Then calculations are repeated by assuming the stator surface to be smooth and the total Carter Factor can be written as [25]
In this paper, four stator slot types including full-open slot, semi-open slot, original slot and improved slot are introduced, as shown in Fig.2. According to the analytical method, the Carter Factors of the models with the full-open slot and the semi-open slot are 1.45 and 1.14, respectively. Both Carter Factors of the models with closed slot types are zero. The calculation results of the Carter Factor varied with the stator slot opening are shown in Fig.3.
Fig.2
Fig.3
Fig.3
Calculation results of the Carter Factor varied with the stator slot opening
3.2 Analysis of Magnetic Field
The rotor surface experiences alternating flux components because of a changing permeance caused by the slots. The harmonic air-gap permeance distribution produced by stator slotting has the general forms.
where g is the air gap length; Kcs is the Carter factor for the stator slotting; Kc is the Carter factor for both the stator slotting and rotor slotting; μ0 is the vacuum permeability; 
The harmonic air-gap permeance distributions of full-open slot model and semi-open slot model are shown in Fig.4. It can be seen from Fig.4 that the invariable part of the air gap permeance of the full-open slot model is smaller than that of the semi-open slot model. However, the first order harmonic air gap permeance of the full-open slot model is much larger than that of the semi-open slot model. In the case of the full-open slot model, the air gap permeance from big to small are 1st, 2nd, 5th, 4th and 3rd, respectively. However, in the case of the semi-open slot, the air gap permeance from big to small are 1st, 2nd, 3rd, 4th and 5th, respectively. This is because the amplitude of the air gap permeance caused by slotting effect is affected by the Carter Factor and the harmonic order.
Fig.4
Then the tooth harmonic fields caused by stator slotting can be expressed as
where F0 is the amplitude of synthetic magnetic potential; Ks is saturation coefficient; φ0k1 is the initial phase angle; νz = ±k1Z1 + p, k1 = 1, 2, ….
It should be noted that the first order forward-rotating and backward-rotating magnetic fields produced by stator slotting can be written as
The results of the first order magnetic field (including the forward-rotating magnetic field and the backward-rotating magnetic field) and the total magnetic fields caused by stator slotting are shown in Fig.5. It can be concluded that the distribution of the first order magnetic field of the full-open slot model has a similar trend with that of the semi-open slot model. However, the amplitude of the first order magnetic fields of the full-open slot is much larger than that of the semi-open slot. In addition, the amplitude of the total magnetic field of the full-open slot model is approximately equal to that of the semi-open slot. The amplitudes of the harmonic fields of the full-open slot model and the semi-open slot model are concluded as shown in Tab.1.
Fig.5
Fig.5
Results of the first order magnetic field and the total magnetic field caused by stator slotting
Tab.1 Amplitudes of the harmonic fields
| Harmonic order | Amplitude of forward-rotating tooth harmonic field/T | |
|---|---|---|
| Full-open slot | Semi-open slot | |
| 1st order | 0.151 | 0.070 |
| 2nd order | 0.085 | 0.060 |
| 3rd order | 0.014 | 0.056 |
| 4th order | 0.031 | 0.046 |
| 5th order | 0.035 | 0.034 |
| Total | 0.188 | 0.186 |
3.3 Analytical Calculation of Rotor Surface Losses
Although the magnetic cores are made of sheets, a thin sheet also enables eddy currents to occur when the flux alternates. According to Lenz’s law, the eddy currents try to prohibit the flux from penetrating the laminations, as shown in Fig.6. In the figure, d is the penetration depth of the eddy current induced in rotor, l is the thickness of the lamination, and lrt is the average eddy current path width. More details of eddy current path can be found in Ref.[19].
Fig.6
The governing equations in rotor can be written as
where ωz is the angular frequency of eddy current; ρ is the resistivity of the rotor; μ is the permeability of the rotor; j is the imaginary unit; Jz is the eddy current density.
By assuming the distribution of the tooth harmonic is sinusoidal, we can obtain the rotor surface loss density caused by the air gap permeance harmonic magnetic field as following formula
where Z1 is the number of stator slots; t1 is the stator tooth pitch; n is the rotating speed which is approximately equal to synchronous speed at no-load condition; k0 is a coefficient which is related to the resistivity and the permeability of the rotor; B0 is the amplitude of the tooth harmonic flux density caused by the stator slotting.
Then the total rotor surface loss can be written as
Fig.7 shows the analytical calculation results of the rotor surface losses. It is concluded that the first order of the tooth harmonic can produce the largest rotor surface loss in case of both the full-open slot model and the semi-open slot model. It should be noted that the third order tooth harmonic of the full-open slot model is much smaller than others since the tooth harmonic field is the smallest among the five orders in case of the full-open slot model.
Fig.7
3.4 Finite Element Analysis of Rotor Surface Losses
3.4.1 Finite Element Model
In this paper, models with four different stator slot types are solved by 2-D transient magnetic field analysis as space-harmonic magnetic problems. Maxwell’s equations for each region can be written as [3]
Where 
For all practical purposes, it is sufficient for us to consider a 2-D model by assuming symmetry along the axial direction. Then the eddy currents would appear to be flowing in the axial direction [19]. Eddy current density induced from the time-varying magnetic field in case of 2D problem can be expressed as
Then the rotor surface loss can be written as
where l is the depth of model; S is the area of the rotor region considering the penetration depth of the eddy current.
The penetration depth of the eddy current induced in rotor can be expressed as [14]
where d is the penetration depth of the eddy current; fz is the frequency of the eddy current; μr is the relative magnetic permeability of rotor; μ0 is the vacuum permeability; σ is the electrical conductivity of rotor.
Hence, the rotor surface loss is
where l is approximately equal to the outer perimeter of the rotor.
The penetration depth of the eddy current is calculated to be 0.2mm. This means that the mesh of the rotor surface is of great significance due to the skin depth. In order to improve the calculation accuracy, the maximum normal deviation is set to be 1deg. The skin depth is set to be 0.2mm, and the number of layers of elements is set to be 5. The finite element model and meshes are shown in Fig.8.
Fig.8
3.4.2 Analysis of Magnetic Field
In this section, the magnetic field is analyzed, as shown in Fig.9, and the mechanism of the rotor surface loss is investigated by using the spatial distributions of the flux densities in the air gap, as shown in Fig.10. The Fast Fourier Transformation (FFT) of the air gap flux densities of the four slot types are shown in Fig.11. It can be seen from Fig.10 and Fig.11 that the flux density of the full-open slot has serious wave distortion, and the flux density of the improved slot is in good agreement with sinusoidal wave. This is because the stator slotting has a significant influence on the air gap permeance which affects the flux density distribution.
Fig.9
Fig.10
Fig.11
The amplitude of the tooth harmonic flux density can be obtained by FFT analysis of radial air gap flux density, as shown in Fig.12. It can be seen from Fig.12 that spatial harmonics of the open slot models including full-open slot model and semi-open slot model are larger than the spatial harmonics of the closed slot models (including the original slot and the improved slot models). In addition, the amplitudes of the tooth harmonics are much larger than the others.
Fig.12
It should be noted that the stator tooth harmonics caused by stator slotting and the stator harmonics caused by stator MMF have the same frequencies and the same pole numbers. This means that results obtained by the FFT analysis of air gap flux densities, as shown in Fig.11and Fig.12, include both the stator tooth harmonics caused by stator slotting and the stator harmonics caused by stator MMF. However, at the no-load condition, the rotor speed is very close to the synchronous speed. Hence, the stator tooth harmonic field caused by stator slotting is the primary cause of the rotor surface loss at no-load condition.
The stator tooth harmonic field caused by stator slotting can be separated from the total stator tooth harmonics as shown in Fig.13. The angle between the stator tooth harmonics 
Fig.13
Fig.13
Sketch of separating stator tooth harmonic field caused by stator slotting from the total stator tooth harmonics
Fig.14
It can be seen from Fig.14 that the flux density of the stator tooth harmonic caused by stator slotting is affected by the slot opening as well as the slot type. In general, the stator tooth harmonics of the motor with large slot opening are larger than those of the motor with small slot opening. But this is not always true because the flux density of the stator tooth harmonic is affected by the harmonic air gap permeance distribution which depends on the Carter Factor and the tooth harmonic number. Moreover, the flux density of the stator tooth harmonic caused by stator slotting is not zero and the amplitude is influenced by the slot shape. According to the previous analytical method of calculating Carter Factor, the Carter Factor of the closed-stator-slot motor is equal to 1. However, the authors believe that the Carter Factor of the closed-stator-slot motor is not equal to 1 because the flux density in the slot bridge is highly saturated due to the narrow and long shape. This gives a good explanation that the stator tooth harmonic of the closed-stator-slot motor is not zero.
3.4.3 Analysis of Rotor Surface Loss
Fig.15 shows the distribution of the rotor eddy current density resulting from the 2-D finite element analysis of the motor with original stator slot. At no-load condition, the rotor eddy current losses caused by the stator slotting are mainly generated on the rotor surface. Three circles which pass through the point A, point B and point C are introduced to analyze the distribution of the eddy current at different position, as shown in Fig.16. Point A, point B and point C are located on the rotor surface, at the center of the rotor bar and at the rotor yoke, respectively.
Fig.15
Fig.16
Fig.17 shows the eddy current distribution of the models with different stator slot types at t=0.2s. It can be seen from Fig.17 that eddy current on rotor surface is much larger than that in rotor tooth and rotor yoke. In addition, the permeance caused by stator slotting affects the symmetrical characteristic of the eddy current distribution on rotor surface. Moreover, the eddy current density of the model with the full-open slot is the largest. Furthermore, the eddy current density of the model with closed stator slot is small but not equal to zero. And the eddy current of the model with improved stator slot is smaller than the others.
Fig.17
The rotor surface losses of the models with the different slot types are shown in Fig.18. In general, the results are consistent with the eddy current distributions as well as the stator tooth harmonic field. The average values are evaluated from 180ms to 200ms. The analytical results which considered the precious five harmonics and the FEM results are shown in Tab.2. The rotor surface losses of the model with the original slot and the improved slot are calculated to be zero according to the analytical method. As the authors said, the limitation is that the previous analytical method did not consider the highly saturated slot bridge. The difference between the analytical results and the FEM results can be explained because the FEM method considered the magnetic saturation and material nonlinearity. According to the finite element results, the rotor surface loss of the model with improved stator slot is much smaller than that of the open slots and original slots. This is because the slot bridge of the improved stator slot is not easily to be highly saturated.
Fig.18
Tab.2 Analytical and FEM results for the rotor surface losses of the model
| Rotor surface loss/W | ||
|---|---|---|
| Analytical | FEM | |
| Full-open slot | 155.6 | 235.8 |
| Semi-open slot | 71.6 | 89.0 |
| Original slot | 0 | 48.2 |
| Improved slot | 0 | 14.1 |
4 Conclusion
This paper studies the rotor surface losses in submersible motor with different stator slot types including full-open slot, semi-open slot, original closed slot and improved closed slot at no-load condition. Generally, the rotor surface experiences alternating flux components because of a changing permeance caused by the stator slots. And the drops in the flux density caused by slots create losses on the rotor surface.
In this paper, models with four stator slot types are introduced to analyze the rotor surface losses of the submersible motor at no-load condition. Full-open slot and semi-open slot are used for comparison and the improved slot is the optimized slot shape we proposed for reduction of rotor surface losses. According to the analytical method, the Carter Factors of the models with the full-open slot and the semi-open slot are 1.45 and 1.14, respectively. Both Carter Factors of the models with closed slot types are zero. The spatial and time-varying distributions of the tooth harmonic fields caused by stator slotting are presented by analytical method.
The difference between the analytical results and the FEM results can be explained because the FEM method considered the magnetic saturation and material nonlinearity. It is concluded that the rotor surface loss of the model with improved stator slots is much smaller than that of the open slots and original slots. This is because the slot bridge of the improved stator slot is not easily to be highly saturated. The authors believe that the Carter Factor of the closed-stator-slot motor is not equal to 1 because the flux density in the slot bridge is highly saturated due to the narrow and long shape. This gives a good explanation that the stator tooth harmonics as well as the rotor surface losses of the closed-stator-slot motor are not zero. Further work will be done in a later paper.
Appendix
APP.Tab. Specifications of the submersible motor
| Item | Specification |
|---|---|
| Rated power/kW | 30 |
| Rated voltage/V | 380 |
| Pole pairs | 3 |
| Connection type | Y |
| Outer diameter of stator/mm | 590 |
| Inner diameter of stator/mm | 445 |
| Outer diameter of rotor/mm | 443.4 |
| Core length/mm | 68 |
| Number of stator slots | 54 |
| Number of rotor slots | 58 |
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