Prediction of Eddy Current Losses in Cooling Tubes of Direct Cooled Windings in Electric Machines

: The direct coil cooling method is one of the existing cooling techniques for electric machines with concentrated windings, in which cooling tubes of conductive material are inserted between the windings. In such cases, eddy current losses are induced in those cooling tubes because of the time variant magnetic ﬁeld. To compute the cooling tubes losses, either a transient ﬁnite element simulation (mostly based on commercial software), or a full analytical method, which is more complex to be constructed, is required. Instead, this paper proposes a simple and an accurate combined semi-analytical-ﬁnite element method to calculate the losses of electric machines having cooling tubes. The 2D magnetostatic solution of the magnetic ﬁeld is obtained e.g., using the free package “FEMM”. Then, the eddy current losses in the tubes are computed using simple analytical equations. In addition, the iron core losses could be obtained. In order to validate the proposed method, two cases are investigated. In Case 1, a six-toothed stator of a switched reluctance machine (SRM), without rotor, is employed in which six cooling tubes are used while in Case 2 a complete rotating SRM is studied. The proposed method is validated by a 2D transient simulation in the commercial software “ANSYS Maxwell” and also by experimental measurements. Evidently, the proposed method is simple and fast to be constructed and it is almost free of cost.


Introduction
Electric machines of high power density and small volume have been receiving great interest in several industrial applications e.g., in electric vehicles in which there are constraints on the motor size and a necessary power density is required depending on the driving cycle [1,2]. Therefore, researchers are investigating several possibilities to improve the power per volume of the electric machines. This can be achieved by improving the electromagnetic and the thermal behaviors of the machine [1][2][3][4][5][6][7][8]. The electromagnetic behavior of the motor includes optimal geometrical selection and optimal drive system whilst the thermal behavior includes better heat extraction from the motor using a cooling system. It is necessary to combine both the electromagnetic and thermal behaviors so that an optimal motor performance can be achieved [9][10][11].
Several cooling techniques to improve the thermal behavior of electric machines can be found in the literature [12][13][14], among them the so-called "direct coil cooling technique". The direct coil cooling method is employed in several electric machines, in particular for a toothed stator with concentrated windings [15,16]. This is because in those machines, there is unused areas between the windings due to some geometric constraints; for instance, Figure 1 shows a drawing of a six-toothed stator core with concentrated windings which can be used in several electric machines e.g., switched reluctance machines, brushless DC machines, etc. [16,17]. Cooling tubes can be inserted in the space between the windings of the machine in which a liquid (water or oil) flows as sketched in Figure 1. In this way, the heat of the windings could be extracted much faster and efficiently compared to the other cooling techniques e.g., water jacket method [12,14,16]. Consequently, the current in the windings might be increased, and hence the machine power per fixed volume can be improved [16,18]. Several literature works have investigated the influence of using the direct coil (slot cooling) technique on the thermal behavior of the electric machines [15,16]. For example, in Reference [18] an interior permanent magnet synchronous machine (IPMSM) with a conventional round wire winding and water jacket cooling is compared with rectangular flat wires and slot cooling. It was found that the slot cooling technique could be considered as an effective cooling method for both stator and rotor geometries. In addition, the absence of a water jacket results in a more compact and a comparable machine cost. However, when the cooling tubes are inserted between the stator windings, induced eddy current losses occur in them because of the time variant magnetic field. To estimate the efficiency of the machine, the cooling tube losses should be obtained. Hence, a transient time solution is necessary to accurately obtain the losses of the tubes. This solution could be based on completely analytical or numerical models (finite element model) [19]. The analytical model has the advantage of the low computation time to get the solution, however it is more complex and may include some assumptions as indicated in Reference [20]; it is necessary to model not only the cooling tube but also the complete system because the leakage magnetic field in the tube is mandatory to obtain the losses. Thus, this costs time and effort to be constructed. In contrast, the finite element model (FEM) is more accurate and can be implemented faster. However, it consumes large amounts of time to obtain the solution [1][2][3]. In addition, the packages available are mostly commercial with an expensive cost. Therefore, a combined model of the analytical and free 2D FEM package, which is called a "combined semi-analytical-FEM", could be a promising method.
Mathematics 2019, 7, x FOR PEER REVIEW 2 of 12 machines, brushless DC machines, etc. [16,17]. Cooling tubes can be inserted in the space between the windings of the machine in which a liquid (water or oil) flows as sketched in Figure 1. In this way, the heat of the windings could be extracted much faster and efficiently compared to the other cooling techniques e.g., water jacket method [12,14,16]. Consequently, the current in the windings might be increased, and hence the machine power per fixed volume can be improved [16,18]. Several literature works have investigated the influence of using the direct coil (slot cooling) technique on the thermal behavior of the electric machines [15,16]. For example, in Reference [18] an interior permanent magnet synchronous machine (IPMSM) with a conventional round wire winding and water jacket cooling is compared with rectangular flat wires and slot cooling. It was found that the slot cooling technique could be considered as an effective cooling method for both stator and rotor geometries. In addition, the absence of a water jacket results in a more compact and a comparable machine cost. However, when the cooling tubes are inserted between the stator windings, induced eddy current losses occur in them because of the time variant magnetic field. To estimate the efficiency of the machine, the cooling tube losses should be obtained. Hence, a transient time solution is necessary to accurately obtain the losses of the tubes. This solution could be based on completely analytical or numerical models (finite element model) [19]. The analytical model has the advantage of the low computation time to get the solution, however it is more complex and may include some assumptions as indicated in Reference [20]; it is necessary to model not only the cooling tube but also the complete system because the leakage magnetic field in the tube is mandatory to obtain the losses. Thus, this costs time and effort to be constructed. In contrast, the finite element model (FEM) is more accurate and can be implemented faster. However, it consumes large amounts of time to obtain the solution [1][2][3]. In addition, the packages available are mostly commercial with an expensive cost. Therefore, a combined model of the analytical and free 2D FEM package, which is called a "combined semianalytical-FEM", could be a promising method.
The main contribution of this work was to propose a semi-analytical based FEM method in order to compute the eddy current losses in the cooling tubes. In addition, the proposed method could be applied to compute the iron core loses. The proposed method was based on 2D magnetostatic free package "FEMM" coupled with some simple analytical equations. This method could be constructed quickly, and generally for any electric device e.g., motor. In addition, it was almost free of cost and hence there was no need for commercial finite element packages.

Semi-Analytical Based Finite Element Loss Model
The electrical losses in electric machines include mainly the copper and the iron core losses. When a cooling tube of conductive material is inserted between the machine windings, eddy current loss will be induced, resulting in an additional loss in the machine. This loss can be computed by [21,22]:

Coil
Cooling tube Iron core The main contribution of this work was to propose a semi-analytical based FEM method in order to compute the eddy current losses in the cooling tubes. In addition, the proposed method could be applied to compute the iron core loses. The proposed method was based on 2D magnetostatic free package "FEMM" coupled with some simple analytical equations. This method could be constructed quickly, and generally for any electric device e.g., motor. In addition, it was almost free of cost and hence there was no need for commercial finite element packages.

Semi-Analytical Based Finite Element Loss Model
The electrical losses in electric machines include mainly the copper and the iron core losses. When a cooling tube of conductive material is inserted between the machine windings, eddy current loss will be induced, resulting in an additional loss in the machine. This loss can be computed by [21,22]: where J tube , σ, A, vol and P tube are the current density, material electrical conductivity, magnetic vector potential, volume and the computed eddy current loss in the tube respectively. The iron losses consist of three main components: hysteresis loss, classical eddy current loss and excess loss; each of them depends on the flux density waveform (amplitude and frequency), and also on the magnetic steel grade. Assuming a sinusoidal flux density waveform, the iron core losses in electric machines are obtained as in Reference [4] by: where k 1 , k 2 , k 3 , k 4 and ρ are material dependent parameters, B(t) and B m are the instantaneous and peak flux densities in the machine and f is the frequency of the applied field. The previous formula of the hysteresis loss i.e., P hy in Equation (3) assumes that the flux density in the machine is a pure sinusoidal waveform. This assumption is not correct in electric machines fed from an inverter. In addition, some of the electric machines, e.g., switched reluctance machine, have asymmetric flux waveforms due to their construction. This means that the formula of Equation (3) cannot predict the hysteresis loss accurately in the case of asymmetric flux waveforms or inverter fed electric machines. Consequently, in the literature improved iron loss models are presented in order to take into account the influence of the asymmetric flux waveforms and the pulse width modulation (PWM ) of the inverter as in References [23][24][25]. The improved iron loss model presented in Reference [24] is integrated in Equation (3) to consider the aforementioned effects. The instantaneous hysteresis loss is then calculated by: where H ir is the irreversible component of the magnetic field which is associated with the hysteresis loss; θ represents the position in the hysteresis loop at the current moment and k dc and d m are constants and equal 0.65 and 2.1 respectively. More details about this loss model is given in References [23][24][25].
Moreover, knowing the current and the resistance of the machine windings, the copper losses can be simply computed by: where m, I ph and R ph are the number of phases, the root mean square of the phase current and the phase resistance respectively. The resistance of the coil depends on the length, cross section area and the material resistivity of the coil, which can be obtained for the studied geometry.
From the previous equations, it was observed that the main parameters of loss model of the cooling tube and the iron core are the magnetic vector potential (A), the flux density (B) and some fixed geometrical parameters such as the volumes of the tube wall and the iron core. To obtain these parameters, the electric machine or device was simulated by 2D magnetostatic computations, e.g., using the free package "FEMM", at different time instants. The values of the magnetic vector potential, the flux density and the volume in the mesh elements of the cooling tube wall and the iron core were then recorded at each time instant. Eventually, the tube and the iron core losses were computed using Equations (1)-(9).

Case Studies
In order to examine the aforementioned proposed loss approach, two cases were studied. Case 1 considered only a stator with six concentrated coils and six cooling tubes as sketched in Figure 1. This case was referred to as "a stator setup case". A full rotating switched reluctance motor was considered as Case 2.

Stator Setup
A stator with six teeth (outer/inner diameters were 120/62.5 mm, the stack length was 80 mm and the steel type was NO20, see Table 1), was built in FEMM, in which the cooling tubes were inserted between the windings as shown in Figure 1. The maximum air gap length between the coil and the tube and the stator yoke and the tube were 0.25 and 0.7 mm respectively. Obviously, there was no rotor included; this was for an accurate characterization of the tubes' losses. The tube material was a stainless steel of an electrical conductivity of 1.35 MS/m and with a wall thickness of 1 mm. An electrical insulation material, e.g., polyimide film, covered the cooling tubes to guarantee a complete electrical insulation between the cooling tubes and the coils as seen in the experimental validation section. Each stator tooth had a concentrated coil of 25 turns. The six coils were connected in series (A-B+, B-C+, C-D+, D-E+, E-F+) as seen in Figure 2. The coils were supplied from a sinusoidal AC supply and hence a sinusoidal current passed though the coils. This resulted in flux lines distributions as seen in Figure 2. It is evident from Figure 2 that some flux lines passed through the cooling tubes i.e., called "leakage flux". The flux level varied with the time, resulting in induced eddy currents in the tubes. Thus, eddy current losses were generated in the tubes. The mesh elements of part of the stator geometry are shown in Figure 3. The flux density, magnetic vector potential and the volume of each mesh element of stator core and tubes were recorded from FEMM at each time instant. supply and hence a sinusoidal current passed though the coils. This resulted in flux lines distributions as seen in Figure 2. It is evident from Figure 2 that some flux lines passed through the cooling tubes i.e., called "leakage flux". The flux level varied with the time, resulting in induced eddy currents in the tubes. Thus, eddy current losses were generated in the tubes. The mesh elements of part of the stator geometry are shown in Figure 3. The flux density, magnetic vector potential and the volume of each mesh element of stator core and tubes were recorded from FEMM at each time instant.  For example, Figure 4 shows the flux density and magnetic vector potential as functions of time for Mesh Element 1 (see Figure 3) at 40 A and 500 Hz. Eventually, the losses could be calculated using the aforementioned equations (Section 2) as reported in Figures 5 and 6. Figures 5 and 6 show the losses of the stator iron core and the cooling tubes respectively at several supply currents (RMS) and frequencies: the values of the current (RMS) and frequency ranged up to 42 A (14.80 A/mm 2 ) and 1000 Hz respectively. It was obvious that the tubes losses ranged from about 2.40% to 14.49%, depending on the flux amplitude and frequency, compared to the iron losses for the studied case. This meant that the tube loss could be significant depending on the application. Further, Figures 5 and 6 clearly demonstrate that the computed losses using the proposed method corresponded very well with those obtained by the 2D transient FEM simulations of the "ANSYS Maxwell" package. It is important to note that to obtain an accurate loss calculation using the proposed method, the number of time steps per electric period of the field solution should be enough.   For example, Figure 4 shows the flux density and magnetic vector potential as functions of time for Mesh Element 1 (see Figure 3) at 40 A and 500 Hz. Eventually, the losses could be calculated using the aforementioned equations (Section 2) as reported in Figures 5 and 6. Figures 5 and 6 show the losses of the stator iron core and the cooling tubes respectively at several supply currents (RMS) and frequencies: the values of the current (RMS) and frequency ranged up to 42 A (14.80 A/mm 2 ) and 1000 Hz respectively. It was obvious that the tubes losses ranged from about 2.40% to 14.49%, depending on the flux amplitude and frequency, compared to the iron losses for the studied case. This meant that the tube loss could be significant depending on the application. Further, Figures 5 and 6 clearly demonstrate that the computed losses using the proposed method corresponded very well with those obtained by the 2D transient FEM simulations of the "ANSYS Maxwell" package. It is important to note that to obtain an accurate loss calculation using the proposed method, the number of time steps per electric period of the field solution should be enough.

Switched Reluctance Motor
The 6/4 switched reluctance motor (SRM) shown in Figure 7 with the geometrical parameters listed in Table 1 was simulated using both the proposed semi-analytical based FEMM and ANSYS Maxwell packages. Six cooling tubes were inserted in the triangular prism gap between the coils as seen in Figure 7. Figure 8 shows the flux lines of the SRM at a peak current of 60 A in one phase (A) using both FEMM and ANSYS Maxwell packages. In this study, an ideal current waveform was used as shown in Figure 9.

Switched Reluctance Motor
The 6/4 switched reluctance motor (SRM) shown in Figure 7 with the geometrical parameters listed in Table 1 was simulated using both the proposed semi-analytical based FEMM and ANSYS Maxwell packages. Six cooling tubes were inserted in the triangular prism gap between the coils as seen in Figure 7. Figure 8 shows the flux lines of the SRM at a peak current of 60 A in one phase (A) using both FEMM and ANSYS Maxwell packages. In this study, an ideal current waveform was used as shown in Figure 9.       Figure 10 shows the output torque of the SRM as a function of the current amplitude at 2000 rpm using both FEMM and Ansys models. The torque increased approximately in a linear way with the peak value of the current. Clearly, the output torque of both packages was similar. Figure 11 reports the core losses of the machine as a function of the current amplitude at 2000 rpm. It was noticed that there was a slight difference of about 2% in the core losses between the proposed loss method and ANSYS Maxwell. The reason was that in the proposed loss model the constant values in Equation (4) were assumed valid for different material thicknesses. However, in Reference [23] it is mentioned that the constants in Equation (4) vary with the thickness of the magnetic steel grade. Further, the used values (0.65 and 2.1) were the average values obtained by measurements for various steel grade thicknesses from Reference [23] and could be accepted. Figure 12 shows the eddy current loss of the cooling tubes as a function of the stator peak current at an SRM speed of 2000 rpm. At 60 A, the tubes

Phase current [A]
Leakage flux in the tube    Figure 10 shows the output torque of the SRM as a function of the current amplitude at 2000 rpm using both FEMM and Ansys models. The torque increased approximately in a linear way with the peak value of the current. Clearly, the output torque of both packages was similar. Figure 11 reports the core losses of the machine as a function of the current amplitude at 2000 rpm. It was noticed that there was a slight difference of about 2% in the core losses between the proposed loss method and ANSYS Maxwell. The reason was that in the proposed loss model the constant values in Equation (4) were assumed valid for different material thicknesses. However, in Reference [23] it is mentioned that the constants in Equation (4) vary with the thickness of the magnetic steel grade. Further, the used values (0.65 and 2.1) were the average values obtained by measurements for various steel grade thicknesses from Reference [23] and could be accepted. Figure 12 shows the eddy current loss of the cooling tubes as a function of the stator peak current at an SRM speed of 2000 rpm. At 60 A, the tubes

Phase current [A]
Leakage flux in the tube Figure 9. Currents in the phases of the SRM. Figure 10 shows the output torque of the SRM as a function of the current amplitude at 2000 rpm using both FEMM and Ansys models. The torque increased approximately in a linear way with the peak value of the current. Clearly, the output torque of both packages was similar. Figure 11 reports the core losses of the machine as a function of the current amplitude at 2000 rpm. It was noticed that there was a slight difference of about 2% in the core losses between the proposed loss method and ANSYS Maxwell. The reason was that in the proposed loss model the constant values in Equation (4) were assumed valid for different material thicknesses. However, in Reference [23] it is mentioned that the constants in Equation (4) vary with the thickness of the magnetic steel grade. Further, the used values (0.65 and 2.1) were the average values obtained by measurements for various steel grade thicknesses from Reference [23] and could be accepted. Figure 12 shows the eddy current loss of the cooling tubes as a function of the stator peak current at an SRM speed of 2000 rpm. At 60 A, the tubes losses of the proposed method and ANSYS Maxwell were 2.03 W and 2.13 W respectively. This showed that the proposed method was capable of predicting the eddy current losses in the cooling tube. This loss represented about 0.1% of the motor output power, which could be neglected in this case. This was because an ideal current waveform was used in this case. However, in the real current waveform there would be an interference between the currents of the phases as well as harmonics, which would result in an increase in the tube losses.

Experimental Validation
In order to validate the proposed loss computation method presented before, an experimental test bench was constructed as shown in Figure 13. Figure 13 (left) shows the complete test bench. The stator coils had 25 turns each and the coils were connected in series as shown in Figure 2. The cooling tube material was a stainless steel with a wall thickness of 1 mm. The stator winding was connected to a controlled (amplitude and frequency) pure sinusoidal voltage supply. Figure 13 (right) shows the stacked stator with inserted cooling tubes and concentrated windings.

Experimental Validation
In order to validate the proposed loss computation method presented before, an experimental test bench was constructed as shown in Figure 13. Figure 13 (left) shows the complete test bench. The stator coils had 25 turns each and the coils were connected in series as shown in Figure 2. The cooling tube material was a stainless steel with a wall thickness of 1 mm. The stator winding was connected to a controlled (amplitude and frequency) pure sinusoidal voltage supply. Figure 13 (right) shows the stacked stator with inserted cooling tubes and concentrated windings.
Firstly, the characteristics i.e., magnetization and loss curves of the steel laminations that were used were measured to obtain the parameters of the iron loss model, see Equation (3). Figure 14 shows the steel laminations of the motor. Figure 15 shows the measured and simulated specific iron losses of the laminations for different magnetic induction and frequencies up to 1.5 T and 400 Hz respectively. It was evident that the simulated loss model predicted similar iron losses value as of the measurements, with an acceptable error of maximum 10%. Firstly, the characteristics i.e., magnetization and loss curves of the steel laminations that were used were measured to obtain the parameters of the iron loss model, see Equation (3). Figure 14 shows the steel laminations of the motor. Figure 15 shows the measured and simulated specific iron losses of the laminations for different magnetic induction and frequencies up to 1.5 T and 400 Hz respectively. It was evident that the simulated loss model predicted similar iron losses value as of the measurements, with an acceptable error of maximum 10%. Figure 16 reports the measured and simulated power losses of the stator components (copper, core and tubes) at several currents and at a supply frequency of 200 Hz. Good matching between the measured and the obtained simulated losses was observed. In addition, some difference between the results was noticed at high currents. This was mainly due to the copper losses variation as a result of the variation of the coil temperature; this was because the stator coils were naturally cooled.  Figure 16 reports the measured and simulated power losses of the stator components (copper, core and tubes) at several currents and at a supply frequency of 200 Hz. Good matching between the measured and the obtained simulated losses was observed. In addition, some difference between the results was noticed at high currents. This was mainly due to the copper losses variation as a result of the variation of the coil temperature; this was because the stator coils were naturally cooled.   Figure 16 reports the measured and simulated power losses of the stator components (copper, core and tubes) at several currents and at a supply frequency of 200 Hz. Good matching between the measured and the obtained simulated losses was observed. In addition, some difference between the results was noticed at high currents. This was mainly due to the copper losses variation as a result of the variation of the coil temperature; this was because the stator coils were naturally cooled.

Conclusions
This paper proposed a simple and an accurate combined semi-analytical finite element method to calculate the losses of electric machines with cooling tubes. The 2D magnetostatic solution of the magnetic field was obtained e.g., using the free package "FEMM". Then, the eddy current loss in the

Conclusions
This paper proposed a simple and an accurate combined semi-analytical finite element method to calculate the losses of electric machines with cooling tubes. The 2D magnetostatic solution of the magnetic field was obtained e.g., using the free package "FEMM". Then, the eddy current loss in the tubes was computed based on analytical equations. In addition, the iron core loss could be obtained. The proposed method was generic and could be applied to any electromagnetic device.
To validate the proposed method, two cases were investigated i.e., a six-toothed stator of a switched reluctance electric machine (SRM), in which six cooling tubes were inserted between the coils and full rotating SRM. The results of the proposed method were validated by both the 2D transient simulations of the commercial package "ANSYS Maxwell" and also by experimental measurements. It was found that the proposed method was an effective method to calculate the losses in the cooling tubes as well as in the iron core with almost a free cost.